# Cross-Dimensional Mathematics:
A Foundation for STP/STA

Daizhan Cheng This work is supported partly by the National Natural Science
Foundation of China (NSFC) under Grants 62073315 and 62350037.Key Laboratory
of Systems and Control, Academy of Mathematics and Systems Sciences, Chinese
Academy of Sciences, Beijing 100190, (e-mail: dcheng@iss.ac.cn).

###### Abstract

A new mathematical structure, called the cross-dimensional mathematics (CDM),
is proposed. The CDM considered in this paper consists of three parts: hyper
algebra, hyper geometry, and hyper Lie group/Lie algebra. Hyper algebra
proposes some new algebraic structures such as hyper group, hyper ring, and
hyper module over matrices and vectors with mixed dimensions (MVMDs). They
have sets of classical groups, rings, and modules as their components and
cross-dimensional connections among their components. Their basic properties
are investigated. Hyper geometry starts from mixed dimensional Euclidian
space, and hyper vector space. Then the hyper topological vector space, hyper
inner product space, and hyper manifold are constructed. They have a joined
cross-dimensional geometric structure. Finally, hyper metric space,
topological hyper group and hyper Lie algebra are built gradually, and
finally, the corresponding hyper Lie group is introduced. All these concepts
are built over MVMDs, and to reach our purpose in addition to existing semi-
tensor products (STPs) and semi-tensor additions (STAs), a couple of most
general STP and STA are introduced. Some existing structures/results about
STPs/STAs have also been resumed and integrated into this CDM.

###### Index Terms:

Semi-tensor product/addition (STP/STA), MVMD, hyper group, hyper ring, hyper
manifold, hyper linear algebra and hyper linear group.

##  I Introduction

The semi-tensor product (STP) of matrices was initiated from 23 years ago[5].
It is a generalization of classical matrix product by removing the dimension
restriction of the classical matrix product. Later on, the semi-tensor
addition (STA) has also been proposed to weaken the dimension restriction of
matrix addition [11]. The past two decades have witnessed the development of
STP/STA[28, 41]. They have received various applications, including Boolean
networks [38], finite games [14], dimension-varying systems [10], engineering
problems [34], finite automata [51], coding [53], etc. In addition to
thousands of papers, there are also many STP/STA monographs [9, 6, 7, 8, 10,
15, 17, 18, 20, 21, 35, 37, 40, 52], and books with STP chapter or appendix
[2, 46].

Roughly speaking, the STPs and STAs are cross-dimensional operators, which
means they are operators acting on objects with different dimensions, i.e.
MVMDs, which can formally be described as follows:

  * (i)

Vectors with mixed dimensions:

${\mathbb{R}}^{\infty}:=\bigcup_{n=1}^{\infty}{\mathbb{R}}^{n},$

where ${\mathbb{R}}$ may be replaced by ${\mathbb{F}}$, which is a field with
characteristic number $0$. To make the discussion and notation ease,
throughout this paper ${\mathbb{F}}$ is assumed to be ${\mathbb{R}}$ unless
elsewhere stated. It is easy to see that in the following discussions
replacing ${\mathbb{R}}$ by ${\mathbb{C}}$ is mostly straightforward.

  * (ii)

Matrices with mixed dimensions:

${\mathcal{M}}=\bigcup_{m=1}^{\infty}\bigcup_{n=1}^{\infty}{\mathcal{M}}_{m\times
n}.$

In the long history of development of matrix theory, the algebraic structure
of matrices, including group, ring, module, general linear algebra, etc., and
their geometric structure, including vector space, inner product, metric,
general linear group, etc., play fundamental roles in both theoretical
investigations and practical applications. Since they were proposed two
decades ago, the researches on STP/STA are mainly concentrated on its
applications. When investigating the set of MVMDs, which are posed by cross-
dimensional operators: STPs and STAs, the existing mathematical tools seem not
very efficient. Hence, now seems to be the time to explore the mathematical
foundation of STP/STA.

When we try to pose the existing algebraic and geometric structures on
STPs/STAs with MVMDs, we found the existing mathematical framework is not
suitable for them. The most critical contradiction lies on the fact that the
existing algebraic or geometric structure, such as group or manifold, is for a
set of objects with fixed dimensions or fixed sizes, while the STPs/STAs are
cross-dimensional operators for MVMDs.

Observing carefully these objects with mixed dimensions, we found that the
main difficult point for posing an algebraic or geometric structure on them
lies on “multiply unitary” trouble. For instance, if we consider the group
structure, then in ${\mathbb{R}}^{\infty}$ there are infinite zeros, precisely
speaking, there are $0_{n}\in{\mathbb{R}}^{n}$, $n=1,2,\cdots$. Similarly,
${\mathcal{M}}$ also have infinite $0_{m\times n}$, $m,n=1,2,\cdots$. This
fact makes it almost impossible to pose a group structure to these objects
(w.r.t. addition) If we consider the set of square matrices, denoted by
${\mathcal{M}}_{1}$, then it has also infinite identities, $I_{n}$,
$n=1,2,\cdots$, which makes the group structure of matrices (w.r.t. product)
also very difficult to define.

When we consider the geometric structure, say, vector space structure, we are
also facing this “multiple zero” puzzle.

An inspiration for solving this problem comes from mechanics, where a pseudo-
metric space is used.

###### Definition I.1

[1] Consider ${\mathcal{X}}=(X,d)\\}$, where $X$ is a set, $d:X\times
X\rightarrow{\mathbb{R}}$ is a mapping.

  1. (1)

${\mathcal{X}}$ is called a metric space if $d$ satisfies

     * (i)

$d(x,y)\geq 0$, $x,y\in X$. Moreover, $d(x,y)=0$, if and only if, $x=y$;

     * (ii)

Symmetry:

$\displaystyle d(x,y)=d(y,x),\quad x,y\in X;$ (1)

     * (iii)

Triangular Inequality:

$\displaystyle d(x,y)+d(y,z)\geq d(x,z),\quad x,y,z\in X.$ (2)

  2. (2)

${\mathcal{X}}$ is a pseudo-metric space, if $d$ satisfies (ii), (iii), and

     * (i‘)

$d(x,y)\geq 0$, $x,y\in X$. Moreover, there exists a zero set, ${\bf 0}$ such
that

$\displaystyle d(x,x)=0,\quad x\in{\bf 0}.$ (3)

The concept of “pseudo-metric space” seems can provide a way to address the
distance of points with different dimensions. This stimulates us to define a
new “group” which contains multiple identities. Instead of “pseudo-group” such
a multi-identity group is called a “hyper group” later in this paper. The
reason is that a hyper group contains a set of many (might be countable)
groups as its components.

Motivated by this fact, we may modify the existing algebraic and geometric
structures to suit the MVMDs with respect to (briefly, w.r.t.) STPs/STAs. The
purpose of this paper is to propose a new mathematical framework, called the
CDM, for objects with mixed dimensions. We hope that it can provide a solid
mathematical foundation for STPs/STAs. Meanwhile, it may cause a development
of classical fixed dimensional algebra and geometry into a new cross-
dimensional algebra and geometry.

The CDM proposed in this paper consists of three parts: (1) hyper algebra; (2)
hyper geometry; (3) hyper Lie algebra and hyper Lie group. The hyper algebra
contains hyper group, hyper ring, and hyper module; the hyper geometry
includes hyper vector space, hyper inner product space, hyper manifold; and
the hyper Lie algebra and hyper Lie group start from mixed dimensional
Euclidian space, through constructing topological hyper group, and finally
lead to hyper linear group and hyper linear algebra.

The relationship of CDM with existing mathematical structures is depicted in
Figure 1.

$\begin{array}[]{c}\mbox{Elementary}\\\ \mbox{Algebra}\end{array}$Linear
AlgebraAbstract AlgebraHyper AlgebraCross-Dimensional
Mathematics$\begin{array}[]{c}\mbox{Elementary}\\\
\mbox{Geometry}\end{array}$Topology$\begin{array}[]{c}\mbox{Differential}\\\
\mbox{Geometry}\end{array}$Hyper
Manifold$\begin{array}[]{c}\mbox{Differential}\\\
\mbox{Equation}\end{array}$$\begin{array}[]{c}\mbox{Topological}\\\
\mbox{Group}\end{array}$$\begin{array}[]{c}\mbox{Lie Group}\\\ \mbox{Lie
Algebra}\end{array}$$\begin{array}[]{c}\mbox{Hyper}\\\
gl({\mathbb{R}}),GL(R)\end{array}$ Figure 1: Cross-Dimensional Mathematics vs
Existing Math.

Some key issues completed in this paper are emphasized as follows:

  * (i)

The most general matrix addition ($\hat{+}$) and matrix product
($\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$) are proposed, which make the set of
all matrices a hyper-ring
($({\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$).
Particularly, the hyper group structure of cross-dimensional permutations
${\bf S}=\bigcup_{n=1}^{\infty}{\bf S}_{n}$ is obtained.

  * (ii)

The most general Euclidian hyper vector space (${\mathbb{R}}^{\infty}$) is
constructed. The vector space structure and the differentiable structure are
proposed, which provide a coordinate frame and differentiable structure for
hyper manifold. (This function is similar to ${\mathbb{R}}^{n}$ for
$n$-dimensional manifold.)

  * (iii)

The Lie group ($GL(m\times n,{\mathbb{R}})$) and Lie algebra ($gl(m\times
n,{\mathbb{R}})$) structure for the set of $m\times n$ dimensional matrices is
constructed. Their relationship with general linear group
($GL(n,{\mathbb{R}})$) and general linear algebra ($gl(n,{\mathbb{R}})$) is
revealed. They provide a foundation for overall hyper matrix Lie group
($GL({\mathbb{R}})$ and hyper Lie algebra ($gl({\mathbb{R}})$).

The rest part of this paper is outlined as follows:

Section 2 provides some preliminaries including (i) algebraic structures of
matrices/vectors such as matrix group, ring, module, and matrix Lie algebra;
(ii) some commonly used STPs and STAs; (iii) dimension-keeping (DK-) STP and
pseudo-STP, where pseudo-STP is newly defined. Sections 3-7 are concentrated
on the hyper algebraic structure of MVMDs. Starting from lattice, Section 3
introduces hyper group, particularly, matrix/vector hyper groups. Their sub
hyper groups. The group decomposition of hyper group into component groups and
equivalence group is also presented. Section 4 considers the hyper group
structure of permutations, including left and right permutation hyper groups.
In Section 5 the hyper group homomorphisms are investigated. Three isomorphism
theorems of groups have been extended to hyper groups. Symmetric and skew
symmetric and squaring endomorphisms of matrix groups are defined and
discussed, which extend the (skew) symmetry and eigenvalue/eigenvector of
square matrices to non-square matrices. In Section 6 the hyper ring of
matrices is proposed. Its sub hyper rings, homomorphisms, and some properties
are investigated. Section 7 considers the hyper module of matrices. The
general linear (control) systems are reconsidered under matrix hyper module
framework. Sections 8-11 consider the hyper geometric structure of MVMDs. In
Section 8 the hyper vector space is introduced. The inner product of vectors
with different dimensions is introduced. Then the norm, the distance are also
obtained. In Section 9 a metric space structure is posed on dimension-free
Euclidian space ${\mathbb{R}}^{\infty}$, which turns this cross dimensional
Euclidian space a path-wise connected topological space. Furthermore, in
Section 10 a differentiable structure is also proposed to
${\mathbb{R}}^{\infty}$. The smooth functions, (co-) vector fields,
distributions, and tensor fields are all introduced for
${\mathbb{R}}^{\infty}$. Section 11 considers the hyper manifold, where the
topological and differentiable structure of ${\mathbb{R}}^{\infty}$ is
transferred to a topological manifold through homeomorphism, similarly to
$n$-dimensional manifold w.r.t. ${\mathbb{R}}^{n}$. Sections 12-15 consider
the hyper Lie group and hyper Lie algebra, which are emphasized on their
expression on MVMDs as hyper linear group and hyper linear algebra. Section 12
constructs the general linear algebra ($gl(m\times n,{\mathbb{R}})$) and
general linear group ($GL(m\times n,{\mathbb{R}})$) for non-square matrices.
Their homomorphisms to classical general linear group ($GL(t,{\mathbb{R}})$)
and general linear algebra ($gl(t,{\mathbb{R}})$) are also revealed. Section
13 considers how to construct a hyper Lie group from a topological hyper
group. Section 14 investigates the structure of hyper general linear algebras
based on hyper vector space and hyper rings. In Section 15 the hyper linear
group is proposed. Its relationship with hyper linear algebra is revealed.
Finally, Section 16 is a brief summary with some remarks.

Before ending this section we give a list of notations.

  * •

${\mathbb{R}}$: The field of real numbers.

  * •

${\mathbb{C}}$: The field of complex numbers.

  * •

${\mathbb{Q}}$: The field of rational numbers.

  * •

${\mathbb{F}}$: A field of characteristic number $0$, (in most cases
${\mathbb{F}}\in\\{{\mathbb{R}}.{\mathbb{C}}\\}$) .

  * •

${\mathbb{Z}}$ (${\mathbb{Z}}^{+}$): Set of (positive) integers.

  * •

$t|r$: $t,r\in{\mathbb{Z}}^{+}$ and $r/t\in{\mathbb{Z}}^{+}$.

  * •

${\mathbb{F}}^{n}$: $n$ dimensional Euclidian space over field ${\mathbb{F}}$.

  * •

${\mathcal{M}}^{\mathbb{F}}_{m\times n}$: set of $m\times n$ matrices over
${\mathbb{F}}$ ( the superscript ${\mathbb{F}}$ is mostly omitted).

  * •

${\mathcal{M}}_{\mu}:=\left\\{A_{m\times n}\;|\;m/n=\mu\right\\}$ where
$0<\mu\in{\mathbb{Q}}$.

  * •

$[a,b]$: the set of integers $\\{a,a+1,\cdots,b\\}$, where $a\leq b$.

  * •

$\operatorname{lcm}(n,p)$ (or $n\vee p$) : the least common multiple of $n$
and $p$.

  * •

${\mathcal{D}}=\\{1,2,\cdots,n\\}$.

  * •

$\gcd(n,p)$ (or $n\wedge p$) : the greatest common division of $n$ and $p$.

  * •

$\delta_{n}^{i}$: the $i$-th column of the identity matrix $I_{n}$.

  * •

$\Delta_{n}:=\left\\{\delta_{n}^{i}|i=1,\cdots,n\right\\}$.

  * •

${\bf 1}_{k}:=(\underbrace{1,\cdots,1}_{k})^{T}$.

  * •

${\mathcal{E}}_{n}:=\frac{1}{\sqrt{n}}{\bf 1}_{n}$.

  * •

${\mathcal{E}}_{m\times n}:=E_{m}E^{T}_{n}$.

  * •

${\mathcal{L}}_{m\times n}$: the set of logical matrices, i.e.
${\mathcal{L}}_{m\times n}=\\{A\in{\mathcal{M}}_{m\times
n}\;|\;\operatorname{Col}(A)\subset\Delta_{m}\\}$.

  * •

$\operatorname{Col}(A)$: the set of columns of matrix $A$, and
$\operatorname{Col}_{i}(A)$ is the $i$-th column of $A$.

  * •

$\operatorname{Row}(A)$: the set of rows of matrix $A$, and
$\operatorname{Row}_{i}(A)$ is the $i$-th row of $A$.

  * •

$\operatorname{Span}(\cdot)$: subspace or dual subspace generated by $\cdot$.

  * •

${\bf S}_{n}$ (${\bf S}$): order $n$ permutation group (hyper permutation
group).

  * •

${\bf A}_{n}$ (${\bf A}$): order $n$ alternative group (hyper alternative
group).

  * •

$\perp$: perpendicular relation between $x\in{\mathbb{R}}^{n}$ and
$h\in{\mathbb{R}}^{n}_{*}$.

  * •

$\bar{+}$: STA w.r.t. $I_{n\times n}$.

  * •

$\hat{+}$: STA w.r.t. ${\bf 1}_{n}$ or ${\mathcal{E}}_{n}$
(${\mathcal{E}}_{m\times n}$).

  * •

$\ltimes$: (left) matrix-matrix semi-tensor product, which is the default one.

  * •

$\rtimes$: right semi-tensor product of matrices.

  * •

$\circ$: second (left) matrix-matrix semi-tensor product.

  * •

$\vec{\ltimes}$: matrix-vector semi-tensor product.

  * •

$\vec{\circ}$: second matrix-vector semi-tensor product.

  * •

$\ltimes$ : dimension-keeping semi-tensor product of matrices.

  * •

$\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$: pseudo-STP.

  * •

$\sup(a,b)$ (or $a\vee b$): least common upper bound of $a$ and $b$.

  * •

$\inf(a,b)$ (or $a\wedge b$): greatest common lower bound of $a$ and $b$.

  * •

$\Box(A)$: squaring matrix $A$.

  * •

$GL(n,{\mathbb{R}})$ ($GL({\mathbb{R}})$): (hyper) general linear group.

  * •

$gl(n,{\mathbb{R}})$ ($gl({\mathbb{R}})$): (hyper) general linear algebra.

  * •

$H<G$ ($H\lhd G$): $H$ is a (normal) subgroup of $G$.

  * •

$H<_{h}G$ ($H\lhd_{h}G$): $H$ is a (normal) hyper-subgroup of $G$.

  * •

$Id$: identity mapping.

  * •

$Im$: image set.

  * •

$Ker$: kernel set.

##  II Preliminaries

###  II-A Algebraic Structure on Matrices/Vectors with proper dimensions

We first review some algebraic structures over matrices/vectors.

###### Definition II.1

Consider a couple ${\mathcal{G}}=(G,*)$, where $G$ is a set, and $*:G\times
G\rightarrow G$ is a binary operator.

  * (i)

${\mathcal{G}}$ is a semi-group, if

$\displaystyle(a*b)*c=a*(b*c),\quad a,b,c\in G.$ (4)

  * (ii)

A semi-group ${\mathcal{G}}$ is a monoid (i.e., a semi-group with an
identity), if there exists an identity $e\in G$, such that

$\displaystyle a*e=e*a=a,\quad\forall a\in G$ (5)

  * (iii)

A monoid ${\mathcal{G}}$ is a group, if for each $a\in G$, there exists an
$a^{-1}$, such that

$\displaystyle a*a^{-1}=a^{-1}*a=e,\quad a\in{\mathcal{G}}.$ (6)

In addition, if $a*b=b*a,\quad\forall a,b\in G$, then it is called an abelian
group.

###### Example II.2

  * (i)

$({\mathbb{R}}^{n},+)$ is an abelian group with the identity
$0_{n}\in{\mathbb{R}}^{n}$.

  * (ii)

$({\mathcal{M}}_{m\times n},+)$ is an abelian group with the identity
$0_{m\times n}\in{\mathcal{M}}_{m\times n}$.

  * (iii)

$({\mathcal{M}}_{n\times n},+)$ is an abelian group with the identity $0_{n}$.

    * –

Set

${\mathcal{S}}_{n}:=\\{A\in{\mathcal{M}}_{n\times n}\;|\;A^{T}=A\\},$

$({\mathcal{S}}_{n},+)<({\mathcal{M}}_{n\times n},+)$ is the symmetric
subgroup.

    * –

Set

${\mathcal{K}}_{n}:=\\{A\in{\mathcal{M}}_{n\times n}\;|\;A^{T}=-A\\},$

$({\mathcal{K}}_{n},+)<({\mathcal{M}}_{n\times n},+)$ is the skew-symmetric
subgroup.

    * –

Set

${\mathcal{Q}}_{n}:=\\{A\in{\mathcal{M}}_{n\times n}\;|\;Tr(A)=0\\},$

$({\mathcal{Q}}_{n},+)<({\mathcal{M}}_{n\times n},+)$ is the zero-trace
subgroup.

(There are also some other subgroups such as diagonal, upper-triangular,
lower-triangular, etc.)

  * (iv)

$({\mathcal{M}}_{n\times n},\times)$ is a monoid with the identity $I_{n}$.

  * (v)

${\mathcal{T}}_{n}:=\\{A\in{\mathcal{M}}_{n\times n}\;|\;A~{}\mbox{is
invertible.}\\}$. Then $({\mathcal{T}}_{n},\times)$ is a group with identity
$I_{n}$.

    * –

Set

${\mathcal{O}}_{n}:=\\{A\in{\mathcal{T}}_{n}\;|\;A^{T}=A^{-1}\\},$

$({\mathcal{O}}_{n},\times)<({\mathcal{T}}_{n},\times)$ is the orthogonal
subgroup.

    * –

Set

${\mathcal{H}}_{n}:=\\{A\in{\mathcal{T}}_{n}\;\det(A)=1\\},$

$({\mathcal{H}}_{n},\times)<({\mathcal{T}}_{n},\times)$ is the special linear
subgroup.

    * –

Set

${\mathcal{P}}_{n}:={\mathcal{T}}_{n}\bigcap{\mathcal{L}}_{n\times n},$

where ${\mathcal{L}}_{n\times n}$ is the set of logical matrices ($A$ is a
logical matrix if $\operatorname{Col}(A)\subset\Delta_{n}$), then
$({\mathcal{P}}_{n},\times)<({\mathcal{T}}_{n},\times)$ is the permutation
sub-group.

(there are also some other sub-groups such as non-singular diagonal, non-
singular upper-triangular, non-singular lower-triangular, etc.)

###### Definition II.3

Triple ${\mathcal{R}}=(R,+,*)$ is a ring, if

  * (i)

$(R,+)$ is an abelian group;

  * (ii)

$(R,*)$ is a semi-group;

  * (iii)

(Distributivity)

$\displaystyle\begin{array}[]{l}(a+b)*c=a*c+b*c,\quad a,b,c\in R;\\\
a*(b+c)=a*b+a*c,\quad a,b,c\in R.\end{array}$ (9)

In addition, if $(R,*)$ is a monoid, ${\mathcal{R}}$ is a unitary ring; if
$a*b=b*a,\quad\forall a,b\in R$, ${\mathcal{R}}$ is a commutative ring.

###### Definition II.4

Assume ${\mathcal{R}}=(R,+,*)$ is a ring and $H\subset R$. If $(H,+,*)$ is
also a ring, $H$ is called a sub-ring of $R$.

###### Example II.5

$({\mathcal{M}}_{n\times n},+,\times)$ is a unitary ring. It has some sub-
rings as follows.

  * (i)

$\displaystyle{\mathcal{D}}_{n}:=\\{A\in{\mathcal{M}}_{n\times
n}\;|\;A~{}\mbox{is diagonal.}\\}.$ (10)

Then $({\mathcal{D}}_{n},+,\times)$ is a sub-ring of $({\mathcal{M}}_{n\times
n},+,\times)$ .

  * (ii)

$\displaystyle{\mathcal{U}}_{n}:=\\{A\in{\mathcal{M}}_{n\times
n}\;|\;A~{}\mbox{is upper triangular.}\\}.$ (11)

Then $({\mathcal{U}}_{n},+,\times)$ is a sub-ring of $({\mathcal{M}}_{n\times
n},+,\times)$ .

  * (iii)

$\displaystyle{\mathcal{B}}_{n}:=\\{A\in{\mathcal{M}}_{n\times
n}\;|\;A~{}\mbox{is lower triangular.}\\}.$ (12)

Then $({\mathcal{B}}_{n},+,\times)$ is a sub-ring of $({\mathcal{M}}_{n\times
n},+,\times)$ .

###### Definition II.6

Let ${\mathcal{R}}=(R,+,*)$ be a ring, and ${\mathcal{G}}=(G,+)$ be an abelian
group. ${\mathcal{R}}$, ${\mathcal{G}}$ with a mapping
$\pi:{\mathcal{R}}\times{\mathcal{G}}\rightarrow{\mathcal{G}}$ is a (left)
${\mathcal{R}}-$ module, if

  * (i)

$r(a+b)=ra+rb,\quad r\in{\mathcal{R}},~{}a,b\in{\mathcal{G}};$

  * (ii)

$(r+s)a=ra+sa,\quad r,s\in{\mathcal{R}},~{}a,b\in{\mathcal{G}};$

  * (iii)

$r(sa)=(rs)a.$

In addition, if there exists $e\in{\mathcal{R}}$ such that for all $a\in G$

$ea=a,\quad\forall a\in G,$

then ${\mathcal{G}}$ is called a unitary ${\mathcal{R}}-$ module.

###### Example II.7

  * (i)

$({\mathcal{M}}_{n\times n},+,\times)$ is a ring with identity.

  * (ii)

$({\mathbb{R}}^{n},+)$ is an abelian group.

  * (iii)

Let $\pi:{\mathcal{M}}_{n\times
n}\times{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{n}$ be the classical matrix-
vector product.

Then

$\left(({\mathcal{M}}_{n\times n},+,\times),({\mathbb{R}}^{n},+),\pi\right)$

is a module.

###### Definition II.8

[3] Let $V$ be a vector space over ${\mathbb{R}}$ and there is a mapping,
called the Lie bracket and denoted by $[\cdot,\cdot]:V\times V\rightarrow V$.
$(V,[\cdot,\cdot])$ is called a Lie algebra, if the followings are satisfied.

  * (i)

Skew-Symmetry:

$\displaystyle[x,y]=-[y,x],\quad x,y\in V.$ (13)

  * (ii)

Linearity:

$\displaystyle[ax+by,z]=a[x,z]+b[y,z],\quad a,b\in{\mathbb{R}};~{}x,y.z\in V.$
(14)

  * (iii)

Jacobi Identity:

$\displaystyle[[x,y],z]+[[y,z],x]+[[z,x],y]=0.$ (15)

###### Example II.9

Consider ${\mathcal{M}}_{n\times n}$. Define

$\displaystyle[A,B]=AB-BA,\quad A,B\in{\mathcal{M}}_{n\times n}.$ (16)

Then $({\mathcal{M}}_{n\times n},[\cdot,\cdot])$ becomes a Lie algebra, called
the general linear algebra and denoted by $gl(n,{\mathbb{R}})$.

Note that all the above algebraic structures over matrices/vectors are
restricted to a particular set of matrices with certain fixed dimensions. We
refer to [31] or any other text book of abstract algebra for the related
concepts. These concepts will be extended to MVMDs in this paper.

###  II-B STP and STA of Matrices/Vectors

The main objects of linear algebra are vectors and matrices. In linear algebra
both the addition and the product of two matrices, or matrix with vector, or
two vectors, require the factors to meet certain dimension matching condition,
here the product of two vectors means the inner product.

STP/STA are proposed to deal with MVMDs. The sets of vectors with mixed
dimensions and matrices with mixed dimensions are described by
${\mathbb{R}}^{\infty}$ and ${\mathcal{M}}$ respectively. In addition we also
need

$\displaystyle{\mathcal{M}}_{\mu}=\\{A\in{\mathcal{M}}_{m\times
n}\;|\;m/n=\mu\\},\quad\mu\in{\mathbb{Q}}_{+}.$ (17)

###### Remark II.10

Consider ${\mathcal{M}}_{\mu}$, where
$\mu=\frac{\mu_{y}}{\mu_{x}}\in{\mathbb{Q}}_{+}$. Moreover, to be uniqueness,
we assume $\gcd(\mu_{y},\mu_{x})=1$. Then

  * (i)

${\mathcal{M}}_{1}$ is the set of square matrices.

  * (ii)

${\mathcal{M}}=\bigcup_{\mu\in{\mathbb{Q}}_{+}}{\mathcal{M}}_{\mu}$

is a partition.

The STP and STA have been proposed to extend the matrix (vector) product and
addition to overall (or larger then the original) sets. The followings are
some commonly used ones.

###### Definition II.11

[6, 10] Let $A,~{}B\in{\mathcal{M}}$, specified by $A\in{\mathcal{M}}_{m\times
n}$, $B\in{\mathcal{M}}_{p\times q}$, and $t=\operatorname{lcm}(n,p)$.

  * (i)

The first type of matrix-matrix (MM-) STP of $A,B$, denoted by $A\ltimes B$,
is defined by

$\displaystyle A\ltimes B:=\left(A\otimes I_{t/n}\right)\left(B\otimes
I_{t/p}\right)\in{\mathcal{M}}_{mt/n\times qt/p}.$ (18)

  * (ii)

The second type of MM-STP of $A,B$, denoted by $A\circ B$, is defined by

$\displaystyle A\circ B:=\left(A\otimes J_{t/n}\right)\left(B\otimes
J_{t/p}\right)\in{\mathcal{M}}_{mt/n\times qt/p},$ (19)

where

$J_{n}:=\frac{1}{n}{\bf 1}_{n\times n}.$

###### Definition II.12

[7] Let $A\in{\mathcal{M}}$ and $x\in{\mathbb{R}}^{\infty}$, specified by
$A\in{\mathcal{M}}_{m\times n}$, $x\in{\mathbb{R}}^{p}$, and
$t=\operatorname{lcm}(n,p)$.

  * (i)

The first type of matrix-vector (MV-) STP of $A$ and $x$, denoted by
$A\vec{\ltimes}x$, is defined by

$\displaystyle A\vec{\ltimes}x:=\left(A\otimes
I_{t/n}\right)\left(x\otimes{\bf 1}_{t/p}\right)\in{\mathbb{R}}^{mt/n}.$ (20)

  * (ii)

The second type of MV-STP of $A$ and $x$, denoted by $A\vec{\circ}x$, is
defined by

$\displaystyle A\vec{\circ}x:=\left(A\otimes J_{t/n\times
t/n}\right)\left(x\otimes{\bf 1}_{t/p}\right)\in{\mathbb{R}}^{mt/n}.$ (21)

###### Definition II.13

[7] Let $x,~{}y\in{\mathbb{R}}^{\infty}$, specified by $x\in{\mathbb{R}}^{m}$,
$y\in{\mathbb{R}}^{n}$ , and $t=\operatorname{lcm}(m,n)$. The vector-vector
(VV-) STP of $x$ and $y$, denoted by $x~{}\bar{\cdot}~{}y$, is defined by

$\displaystyle x~{}\bar{\cdot}~{}y:=\langle x\otimes{\bf 1}_{t/m},y\otimes{\bf
1}_{t/n}\rangle\in{\mathbb{R}},$ (22)

where $\langle\cdot,\cdot\rangle$ is the classical inner product on Euclidian
space.

###### Remark II.14

  * (i)

All the MM-STPs, MV-STPs, and VV-STP are the generalization of classical
matrix product. That is, when the dimension-matching conditions for classical
matrix-matrix (or matrix-vector or vector-vector) products are satisfied, they
degenerate to classical products.

  * (ii)

All the MM-STPs and MV-STPs defined in(18), (20), and (22) are called the left
STPs, because the main objects ($A$, $B$, etc.) are lying on left. They have
their corresponding right STPs. Say,

$\displaystyle(A\rtimes B):=\left(I_{t/n}\otimes A\right)\left(I_{t/p}\otimes
B\right),$ (23)

etc. Such right extensions are also available for all other STPs defined in
the sequel. And a parallel discussion shows similar properties of right STPs
corresponding to the default left STPs. (We leave such discussions for the
reader and will not mention this again.)

  * (iii)

All the MM-STPs, MV-STPs maintain most properties of the classical matrix
product available. Particularly, the associativity and the distributivity
hold. Say,

$\displaystyle(A\ltimes B)\ltimes C=A\ltimes(B\ltimes C),\quad
A,B,C\in{\mathcal{M}};$ (24)

and

$\displaystyle\begin{array}[]{l}(A+B)\ltimes C=A\ltimes C+B\ltimes C;\\\
C\ltimes(A+B)=C\ltimes A+C\ltimes B,\quad A,B,C\in{\mathcal{M}},\\\
\end{array}$ (27)

etc.

###### Definition II.15

Some STAs of matrices/vectors are defined as follows:

  * (i)

${\mathbb{R}}^{\infty}$: Assume $x\in{\mathbb{R}}^{m}$,
$y\in{\mathbb{R}}^{n}$, and $t=\operatorname{lcm}(m,n)$, then

$\displaystyle x\bar{\pm}y:=\left(x\otimes{\bf
1}_{t/m}\right)\pm\left(y\otimes{\bf 1}_{t/n}\right),$ (28)

where

${\bf 1}_{n}:=(\underbrace{1,1,\cdots,1}_{n})^{T},\quad n\in{\mathbb{Z}}_{+}.$

  * (ii)

${\mathcal{M}}_{1}$: Assume $A\in{\mathcal{M}}_{m\times m}$,
$B\in{\mathcal{M}}_{n\times n}$, and $t=\operatorname{lcm}(m,n)$, then

$\displaystyle A\bar{\pm}B:=\left(A\otimes I_{t/m}\right)\pm\left(B\otimes
I_{t/n}\right).$ (29)

An alternative STA is defined as follows.

$\displaystyle A\hat{\pm}B:=\left(A\otimes J_{t/m}\right)\pm\left(B\otimes
J_{t/n}\right).$ (30)

  * (iii)

${\mathcal{M}}_{\mu}$: $A\in{\mathcal{M}}_{m\mu_{y}\times m\mu_{x}}$,
$B\in{\mathcal{M}}_{n\mu_{y}\times n\mu_{x}}$, and
$t=\operatorname{lcm}(m,n)$: Its additivity can be define by either
$\bar{\pm}$, as in (29), or $\hat{\pm}$, as in (30).

To define “addition” for more general case, we need a new set of matrices as
“weights”. Set

$\displaystyle\begin{array}[]{l}{\mathcal{E}}_{n}:=\frac{1}{\sqrt{n}}{\bf
1}_{n};\\\ {\mathcal{E}}_{m\times
n}:={\mathcal{E}}_{m}{\mathcal{E}}^{T}_{n}=\frac{1}{\sqrt{mn}}{\bf 1}_{m\times
n},\quad m,n\in{\mathbb{Z}}_{+}.\end{array}$ (33)

Then the following properties of this set of matrices are easily verifiable,
but they are fundamental in further discussion.

###### Proposition II.16

$\displaystyle\begin{cases}{\mathcal{E}}^{T}_{n}{\mathcal{E}}_{n}=1;\\\
{\mathcal{E}}_{n}\otimes{\mathcal{E}}_{m}={\mathcal{E}}_{m}\otimes{\mathcal{E}}_{n}={\mathcal{E}}_{mn};\\\
{\mathcal{E}}_{m\times n}\otimes{\mathcal{E}}_{p\times
q}={\mathcal{E}}_{p\times q}\otimes{\mathcal{E}}_{m\times
n}={\mathcal{E}}_{mp\times nq};\\\ {\mathcal{E}}^{T}_{m\times
n}={\mathcal{E}}_{n\times m};\\\ J_{n}={\mathcal{E}}_{n\times n}.\end{cases}$
(34)

Using this set of weights, some other STAs can be defined.

###### Definition II.17

  * (i)

${\mathbb{R}}^{\infty}$: Assume $x\in{\mathbb{R}}^{m}$,
$y\in{\mathbb{R}}^{n}$, and $t=\operatorname{lcm}(m,n)$, then

$\displaystyle
x\hat{\pm}y:=\left(x\otimes{\mathcal{E}}_{t/m}\right)\pm\left(y\otimes{\mathcal{E}}_{t/n}\right),$
(35)

  * (ii)

Consider ${\mathcal{M}}$, and assume $A\in{\mathcal{M}}_{m\times n}$,
$B\in{\mathcal{M}}_{p\times q}$, $s=\operatorname{lcm}(m,p)$, and
$t=\operatorname{lcm}(n,q)$. Then we define the STA over ${\mathcal{M}}$ as
follows:

$\displaystyle A\hat{\pm}B:=\left(A\otimes{\mathcal{E}}_{s/m\times
t/n}\right)\pm\left(B\otimes{\mathcal{E}}_{s/p\times t/q}\right).$ (36)

Note that all the STPs and STAs, defined up to this time are the
generalizations of the classical matrix product/addition. That is, when the
dimension matching conditions required by the classical products are
satisfied, they degenerated to the classical matrix product/addition.

Before ending this subsection we give a formula for calculating equivalence
transformation, which will be used in the sequel.

###### Proposition II.18

Let $A\in{\mathcal{M}}_{m\times n}$, and $B\in{\mathcal{M}}_{p\times q}$. Then

$\displaystyle\begin{array}[]{l}\left(A\otimes{\mathcal{E}}_{r\times
kp}\right)\left(B\otimes{\mathcal{E}}_{kn\times
s}\right)=\left(A\otimes{\mathcal{E}}_{r\times
p}\right)\left(B\otimes{\mathcal{E}}_{n\times s}\right),\\\
\left(A\otimes{\mathcal{E}}_{kr\times
p}\right)\left(B\otimes{\mathcal{E}}_{n\times
ks}\right)=\left(A\otimes{\mathcal{E}}_{r\times
p}\right)\left(B\otimes{\mathcal{E}}_{n\times s}\right).\\\ \end{array}$ (39)

Proof. We prove the first equality only. The proof of the second one is
similar.

$\begin{array}[]{l}\mbox{LHS of
(\ref{2.2.13})}=\left(A\otimes{\mathcal{E}}_{r\times
p}\otimes{\mathcal{E}}^{T}_{k}\right)\left(B\otimes{\mathcal{E}}_{n\times
s}\otimes{\mathcal{E}}_{k}\right)\\\
~{}=\left[\left(A\otimes{\mathcal{E}}_{r\times
p}\right)\left(B\otimes{\mathcal{E}}_{n\times
s}\right)\right]\otimes({\mathcal{E}}^{T}_{k}{\mathcal{E}}_{k})\\\
~{}=\left(A\otimes{\mathcal{E}}_{r\times
p}\right)\left(B\otimes{\mathcal{E}}_{n\times s}\right)=\mbox{RHS of
(\ref{2.2.13})}.\end{array}$

$\Box$

###  II-C DK-STP and Pseudo-STP

The dimension-keeping (DK-) STP is defined as follows.

###### Definition II.19

Let $A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times q}$, and
$t=\operatorname{lcm}(n,p)$. The DK-STP of $A$ and $B$, denoted as
$A\,\rotatebox{-90.0}{$\ltimes$}\,B$, is defined by

$\displaystyle
A\,\rotatebox{-90.0}{$\ltimes$}\,B:=\left(A\otimes{\mathcal{E}}^{T}_{t/n}\right)\left(B\otimes{\mathcal{E}}_{t/p}\right).$
(40)

###### Remark II.20

DK-STP was newly proposed in [22] as follows:

$\displaystyle A\,\rotatebox{-90.0}{$\ltimes$}\,B:=\left(A\otimes{\bf
1}^{T}_{t/n}\right)\left(B\otimes{\bf 1}_{t/p}\right).$ (41)

The original one defined by (41) has very clear physical meaning. Multiplying
the columns of the first matrix and multiplying the rows of the second matrix
to least times, such that the enlarged factor matrices meet the dimension
matching condition of the classical matrix product.

In fact, the modified Definition II.19 here uses certain weights for the
multiplication, which keeps the “norms” of the enlarged matrices unchanged. In
the sequel you will see that this modification is indeed meaningful. In [23]
(40) is called a weighted DK-STP.

An obvious property of DK-STP is: if $A,B\in{\mathcal{M}}_{m\times n}$, the
$A\,\rotatebox{-90.0}{$\ltimes$}\,B\in{\mathcal{M}}_{m\times n}$. That is why
it is named by “dimension-keeping”.

The following properties about DK-STP are mimic to the one in [22].

###### Proposition II.21

Define a bridge matrix as

$\displaystyle\Psi_{n\times
p}:=\left(I_{n}\otimes{\mathcal{E}}^{T}_{t/n}\right)\left(I_{p}\otimes{\mathcal{E}}_{t/p}\right),\quad
n,p\in{\mathbb{Z}}_{+},$ (42)

where $t=\operatorname{lcm}(n,p)$, then

$\displaystyle A\,\rotatebox{-90.0}{$\ltimes$}\,B=A\Psi_{n\times p}B,\quad
A\in{\mathcal{M}}_{m\times n},B\in{\mathcal{M}}_{p\times q}.$ (43)

###### Remark II.22

Corresponding to (41), the bridge matrix becomes

$\displaystyle\Psi^{\prime}_{n\times p}:=\left(I_{n}\otimes{\bf
1}^{T}_{t/n}\right)\left(I_{p}\otimes{\bf 1}_{t/p}\right),\quad
n,p\in{\mathbb{Z}}_{+},$ (44)

which is only different from (42) by a coefficient. That is,

$\displaystyle\Psi_{n\times p}:=\frac{\sqrt{np}}{t}\Psi^{\prime}_{n\times p}.$
(45)

Using Proposition II.21, the following results are obvious.

###### Proposition II.23

  * (i)

$\ltimes$ is a generalization of classical matrix product. That is, if $n=p$
it turns out that $A\,\rotatebox{-90.0}{$\ltimes$}\,B=AB$.

  * (ii)

(Distributivity)

$\displaystyle\begin{array}[]{l}(A+B)\,\rotatebox{-90.0}{$\ltimes$}\,C=A\,\rotatebox{-90.0}{$\ltimes$}\,C+B\,\rotatebox{-90.0}{$\ltimes$}\,C,\\\
C\,\rotatebox{-90.0}{$\ltimes$}\,(A+B)=C\,\rotatebox{-90.0}{$\ltimes$}\,A+C\,\rotatebox{-90.0}{$\ltimes$}\,B,\end{array}$
(48)

(where $A$ and $B$ are of the same dimension).

  * (iii)

(Associativity)

$\displaystyle(A\,\rotatebox{-90.0}{$\ltimes$}\,B)\,\rotatebox{-90.0}{$\ltimes$}\,C=A(B\,\rotatebox{-90.0}{$\ltimes$}\,C).$
(49)

  * (iv)

$\displaystyle(A\,\rotatebox{-90.0}{$\ltimes$}\,B)^{T}=B^{T}\,\rotatebox{-90.0}{$\ltimes$}\,A^{T}.$
(50)

DK-STP brings a new algebraic structure to non-square matrices. The following
result is interesting and useful.

###### Proposition II.24

[22] $({\mathcal{M}}_{m\times n},+,\,\rotatebox{-90.0}{$\ltimes$}\,)$ is a
ring.

Using DK-STP, we can define a pseudo-STP as follows:

###### Definition II.25

Consider ${\mathcal{M}}$, and assume $A\in{\mathcal{M}}_{m\times n}$,
$B\in{\mathcal{M}}_{p\times q}$, $s=\operatorname{lcm}(m,p)$, and
$t=\operatorname{lcm}(n,q)$. Then the pseudo-STP of $A$ and $B$ is defined as
follows:

$\displaystyle
A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B:=\left(A\otimes{\mathcal{E}}_{s/m\times
t/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(B\otimes{\mathcal{E}}_{s/p\times
t/q}\right).$ (51)

###### Remark II.26

  * (i)

As a special case, we consider ${\mathcal{M}}_{\mu}$, and assume
$A\in{\mathcal{M}}_{m\mu_{y}\times m\mu_{x}}$,
$B\in{\mathcal{M}}_{n\mu_{y}\times n\mu_{x}}$, and
$t=\operatorname{lcm}(m,n)$. Then the pseudo-STP for ${\mathcal{M}}_{\mu}$
becomes:

$\displaystyle A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B:=\left(A\otimes
J_{t/m}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(B\otimes J_{t/n}\right).$
(52)

  * (ii)

Comparing (51) with (36), one sees easily that in certain sense this pseudo-
STP is “consistent” with the STA defined by (36). It will bee seen in the
sequel that this consistence is very meaningful.

  * (iii)

The reason to call (51) or (52) a pseudo-STP is: they do not always consistent
with classical matrix product even if the dimension matching condition is
satisfied. So they do not meet the fundamental requirement of STP.

Next, we explore some basic properties of pseudo-STP.

###### Proposition II.27

Let $A,~{}B\in{\mathcal{M}}$. Then

  * (i)

$\displaystyle(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)^{T}=B^{T}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{T}.$
(53)

  * (ii)

$\displaystyle(A\hat{+}B)^{T}=A^{T}\hat{+}B^{T}.$ (54)

Proof.

We prove (53) only. The proof of (54) is similar.

Assume $A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times q}$,
$\xi=\operatorname{lcm}(m,p)$, $\eta=\operatorname{lcm}(n,q)$, and
$\delta=\operatorname{lcm}(\xi,\eta)$. Then

$\begin{array}[]{l}A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B=\left(A\otimes{\mathcal{E}}_{\xi/m\times\eta/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(B\otimes{\mathcal{E}}_{\xi/p\times\eta/q}\right)\\\
~{}=\left(A\otimes{\mathcal{E}}_{\xi/m\times\eta/n}\otimes{\mathcal{E}}^{T}_{\delta/\eta}\right)\left(B\otimes{\mathcal{E}}_{\xi/p\times\eta/q}\otimes{\mathcal{E}}_{\delta/\xi}\right)\\\
~{}=\left(A\otimes{\mathcal{E}}_{\xi/m\times\delta/n}\right)\left(B\otimes{\mathcal{E}}_{\delta/p\times\eta/q}\right).\\\
\end{array}$

It follows that

$(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)^{T}=\left(B^{T}\otimes{\mathcal{E}}_{\eta/q\times\delta/p}\right).\\\
\left(A^{T}\otimes{\mathcal{E}}_{\delta/n\times\xi/m}\right)$

On the other hand,

$\begin{array}[]{l}B^{T}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{T}=\left(B^{T}\otimes{\mathcal{E}}_{\eta/q\times\xi/p}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(A^{T}\otimes{\mathcal{E}}_{\eta/n\times\xi/m}\right)\\\
~{}=\left(B^{T}\otimes{\mathcal{E}}_{\eta/q\times\delta/p}\right)\left(A^{T}\otimes{\mathcal{E}}_{\delta/n\times\xi/m}\right).\end{array}$

The conclusion follows.

$\Box$

###### Proposition II.28

$({\mathcal{M}},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$ is a semi-group. That
is, assume $A,~{}B,~{}C\in{\mathcal{M}}$, then

$\displaystyle(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C=A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}(B\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C).$
(55)

Proof. Assume

$A\in{\mathcal{M}}_{m\times n},\;B\in{\mathcal{M}}_{p\times
q},\;C\in{\mathcal{M}}_{r\times s},$

and

$\begin{array}[]{ll}m\vee p=\xi,~{}n\vee q=\eta,&\xi\vee\eta=\delta,~{}\xi\vee
r=\alpha,\\\ \eta\vee s=\beta,~{}\alpha\vee\beta=\theta,&p\vee r=a,~{}q\vee
s=b,\\\ a\vee b=e,&m\vee a=m\vee p\vee r=\alpha,\\\ n\vee b=n\vee q\vee
s=\beta,&n\vee p=\mu,\\\ q\vee r=\lambda,&{}\\\ \end{array}$

Then

$\begin{array}[]{l}A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B=\left(A\otimes{\mathcal{E}}_{\xi/m\times\eta/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(B\otimes{\mathcal{E}}_{\xi/p\times\eta/q}\right)\\\
~{}=\left(A\otimes
E_{\xi/m\times\eta/n}\otimes{\mathcal{E}}^{T}_{\delta/\eta}\right)\left(B\otimes{\mathcal{E}}_{\xi/p\times\eta/q}\otimes{\mathcal{E}}_{\delta/\xi}\right)\\\
~{}=\left(A\otimes{\mathcal{E}}_{\xi/m\times\delta/n}\right)\left(B\otimes{\mathcal{E}}_{\delta/p\times\eta/q}\right)\in{\mathcal{M}}_{\xi\times\eta},\\\
\end{array}$

it follows that

$\begin{array}[]{l}(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C=\left\\{\left[\left(A\otimes
E_{\xi/m\times\delta/n}\right)\left(B\otimes{\mathcal{E}}_{\delta/p\times\eta/q}\right)\right]\right.\\\
~{}\left.\otimes{\mathcal{E}}_{\alpha/\xi\times\beta/\eta}\right\\}\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{\alpha/r\times\beta/s}\right)\\\
=\left\\{\left[\left(A\otimes{\mathcal{E}}_{\xi/m\times\delta/n}\right)\left(B\otimes{\mathcal{E}}_{\delta/p\times\eta/q}\right)\right]\right.\\\
~{}\left.\otimes({\mathcal{E}}_{\alpha/\xi}{\mathcal{E}}^{T}_{\beta/\eta})\right\\}\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{\alpha/r\times\beta/s}\right)\\\
=\left\\{\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\delta/n}\right)\left(B\otimes{\mathcal{E}}_{\delta/p\times\beta/q}\right)\right]\right\\}\\\
~{}\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{\alpha/r\times\beta/s}\right)\\\
=\left(\left\\{\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\delta/n}\right)\left(B\otimes{\mathcal{E}}_{\delta/p\times\beta/q}\right)\right]\right\\}\otimes{\mathcal{E}}^{T}_{\theta/\beta}\right)\\\
\left(\left(C\otimes{\mathcal{E}}_{\alpha/r\times\beta/s}\right)\otimes{\mathcal{E}}_{\theta/\alpha}\right)\\\
=\left(A\otimes{\mathcal{E}}_{\alpha/m\times\delta/n}\right)\left(B\otimes{\mathcal{E}}_{\delta/p\times\theta/q}\right)\left(C\otimes{\mathcal{E}}_{\theta/r\times\beta/s}\right)\\\
=\left(A\otimes{\mathcal{E}}_{\alpha/m\times\mu/n}\right)\left(B\otimes{\mathcal{E}}_{\mu/p\times\lambda/q}\right)\left(C\otimes{\mathcal{E}}_{\lambda/r\times\beta/s}\right).\\\
\end{array}$

On the other hand,

$\begin{array}[]{l}B\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C=\left(B\otimes{\mathcal{E}}_{a/p\times
b/q}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{a/r\times
b/s}\right)\\\ ~{}=\left(B\otimes{\mathcal{E}}_{a/p\times
e/q}\right)\left(C\otimes{\mathcal{E}}_{e/r\times
b/s}\right)\in{\mathcal{M}}_{a\times b},\\\ \end{array}$

and then

$\begin{array}[]{l}A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}(B\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C)=\left(A\otimes{\mathcal{E}}_{\alpha/m\times\beta/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\\\
~{}\left\\{\left[\left(B\otimes{\mathcal{E}}_{a/p\times
e/q}\right)\left(C\otimes{\mathcal{E}}_{e/r\times
b/s}\right)\right]\otimes{\mathcal{E}}_{\alpha/a\times\beta/b}\right\\}\\\
=\left(A\otimes{\mathcal{E}}_{\alpha/m\times\beta/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left[\left(B\otimes{\mathcal{E}}_{\alpha/p\times
e/q}\right)\left(C\otimes{\mathcal{E}}_{e/r\times\beta/s}\right)\right]\\\
=\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\beta/n}\right)\otimes{\mathcal{E}}^{T}_{\theta/\beta}\right]\\\
~{}\left\\{\left[\left(B\otimes{\mathcal{E}}_{\alpha/p\times
e/q}\right)\left(C\otimes{\mathcal{E}}_{e/r\times\beta/s}\right)\right]\otimes{\mathcal{E}}_{\theta/\alpha}\right\\}\\\
=\left(A\otimes{\mathcal{E}}_{\alpha/m\times\theta/n}\right)\left(B\otimes{\mathcal{E}}_{\theta/p\times
e/q}\right)\left(C\otimes{\mathcal{E}}_{e/r\times\beta/s}\right)\\\
=\left(A\otimes{\mathcal{E}}_{\alpha/m\times\mu/n}\right)\left(B\otimes{\mathcal{E}}_{\mu/p\times\lambda/q}\right)\left(C\otimes{\mathcal{E}}_{\lambda/r\times\beta/s}\right).\\\
\end{array}$

The conclusion follows. $\Box$

Before end of this section, we would like to summarize the new algebraic
structures over MVMDs by introducing STPs and STAs.

###### Remark II.29

There are some new algebraic structures, which are caused by STPs/STAs.

  * (i)

New Algebraic Structures:

    * –

$({\mathcal{M}},\ltimes)$ is a monoid with identity $1$.

    * –

$({\mathcal{M}},\circ)$ is a semi-group.

    * –

$({\mathcal{M}}_{m\times n},+,\,\rotatebox{-90.0}{$\ltimes$}\,)$ is a ring.

    * –

$({\mathcal{M}},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$ is a semi-group.

    * –

$({\mathcal{M}}_{\mu},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})<({\mathcal{M}},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$
is a sub-semi-group.

  * (ii)

Observing the aforementioned STAs and STPs over MVMDs, in many cases it seems
difficult to introduce some algebraic structures on them, because the MVMDs
always have “multiple identities”. Recall the pseudo-metric space introduced
in the Introduction, we may introduce a new algebraic structure, which allows
multiple identities. The “hyper group”, which will be proposed in next section
is one of such objects.

##  III Hyper Group

###  III-A Lattice

As a preparation, we briefly review the basic concepts of lattice.

We refer to [4] or [27] for some basic concepts and properties of lattice.

###### Definition III.1

A partial order set $\Lambda$ is a lattice, if for any two elements
$\lambda,~{}\mu\in\Lambda$ there exist a least common upper bound, denoted by
$\sup(\lambda,\mu)$ (or $\lambda\vee\mu$), and a greatest common lower bound,
denoted by $\inf(\lambda,\mu)$ (or $\lambda\wedge\mu$).

A lattice can be described by a graph called the Hasse diagram. A Hasse
diagram is a graph with no horizontal sides. The order of nodes is decided by
oblique sides. For instance, in Figure 2 the left is a lattice, say, $a\vee
b=c$ and $a\wedge b=d$. The right is not a littice, say neither the $a\vee b$
no the $a\wedge b$ exists.

$Lattice$$NotLattice$$e$$f$$g$$h$$f$$i$$a$$c$$b$$d$$Lattice$$NotLattice$$a$$b$
Figure 2: Hasse Diagram

In the following example, we give two lattices which will be used in the
sequel.

###### Example III.2

  * (i)

Consider ${\mathbb{Z}}_{+}$. Set $a\prec b$ if $a\neq b$ and $a|b$, i.e. $a$
is a proper factor of $b$. It follows that

$a\vee b=\operatorname{lcm}(a,b);\quad a\wedge b=\gcd(a,b).$

Then $({\mathbb{Z}}_{+},\prec)$ is a lattice. We call it the multiply-division
lattice of dimension 1, briefly, MD-1 lattice.

  * (ii)

Consider ${\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}$. Set $(a,c)\prec(b,d)$ if
both $a\prec b$ and $c\prec d$ (defined as in (i)). It follows that

$\begin{array}[]{l}(a,c)\vee(b,d)=(\operatorname{lcm}(a,b),\operatorname{lcm}(c,d));\\\
(a,c)\wedge(b,d)=(\gcd(a,b),\gcd(c,d)).\end{array}$

Then $({\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+},\prec)$ is a lattice, called the
MD-2 lattice. Similarly, we can defined the MD-k lattice for $k>2$.

###### Definition III.3

Let $L$ and $H$ be two lattices.

  * (i)

A mapping $\pi:L\rightarrow H$ is said to be a lattice homomorphism if

$\displaystyle\pi(\lambda\vee\mu)=\pi(\lambda)\vee\pi(\mu),\quad\lambda,~{}\mu\in
L;$ (56)

and

$\displaystyle\pi(\lambda\wedge\mu)=\pi(\lambda)\wedge\pi(\mu).$ (57)

  * (ii)

A homomorphism $\pi:L\rightarrow H$ is said to be a lattice isomorphism if it
is bijective and its inverse is also a homomorphism.

  * (iii)

A mapping $\pi:L\rightarrow H$ is said to be an order-preserving mapping, if

$\displaystyle a\prec_{L}b\Rightarrow\pi(a)\prec_{H}\pi(b),\quad a,b\in L.$
(58)

The following result comes from definition immediately.

###### Proposition III.4

Let $L$ and $H$ be two lattices, and $\pi:L\rightarrow H$ be a bijective
mapping. If both $\pi$ and $\pi^{-1}$ are order-preserving mappings, then
$\pi$ is a lattice isomorphism.

###### Definition III.5

Let $(L,\sup,\inf)$ be a lattice.

  * (i)

$H\subset L$, $H$ is called a sublattice of $L$, if $(H,\sup,\inf)$ is also a
lattice.

  * (ii)

$H$ is a sublattice of $L$. If for any $a\in L$, there exists an $h\in H$ such
that $a\prec h$ implies $a\in H$, then $H$ is called an ideal of $L$.

  * (iii)

$H$ is a sublattice of $L$. If for any $a\in L$, there exists an $h\in H$ such
that $a\succ h$ implies $a\in H$, then $H$ is called a filter of $L$.

###  III-B Hyper Group – A“Group” with Multiple Identities

This subsection proposes a “group” with a set of identities, called the
identity set. To begin with, we define an identity set.

###### Definition III.6

Consider a semi-group ${\mathcal{G}}=(G,*)$ . ${\bf e}\subset G$ is called an
identity set of ${\mathcal{G}}$, if

  * (i)

$\displaystyle g*e=e*g,\quad\forall g\in G,~{}\forall e\in{\bf e}.$ (59)

  * (ii)

There is a lattice $\Lambda$ as the index set of ${\bf e}$, that is,

${\bf e}=\\{e_{\lambda}\;|\;\lambda\in\Lambda\\}.$

Moreover,

$\displaystyle
e_{\lambda}*e_{\mu}=e_{\lambda\vee\mu},\quad\lambda,\mu\in\Lambda.$ (60)

  * (iii)

For each $g\in G$ there exists a unique $e_{g}=e_{\lambda_{g}}\in{\bf e}$ such
that

$\displaystyle e_{\lambda}*g=g,$ (61)

if and only if, $\lambda\prec\lambda_{g}$, (including $\lambda=\lambda_{g}$).

###### Definition III.7

Consider a semi-group ${\mathcal{G}}=(G,*)$ .

  * (i)

${\mathcal{G}}=(G,*)$ is a hyper-monoid, if there exists an identity set ${\bf
e}\subset G$.

  * (ii)

A hyper-Monoid ${\mathcal{G}}=(G,*)$ is a hyper group, if for each $g\in G$,
there exists a $g^{-1}\in G$ such that

$\displaystyle g*g^{-1}=g^{-1}*g=e_{g}.$ (62)

In addition, if $a*b=b*a,\quad\forall a,b\in G$, then it is called an abelian
hyper-monoid/hyper group.

###  III-C Hyper Groups Over MVMDs

Recall the additions over MVMDs, defined in Definitions II.15 and II.17, we
have the following result.

###### Proposition III.8

  * (i)

$({\mathbb{R}}^{\infty},\bar{+})$ with $\bar{+}$ defined by (28) and
$({\mathbb{R}}^{\infty},\hat{+})$ with $\hat{+}$ defined by (35) are hyper
groups, where the identity set is

$\displaystyle{\bf
e}=\\{0_{n}\in{\mathbb{R}}^{n}\;|\;n\in{\mathbb{Z}}_{+}\\},$ (63)

w.r.t. MD-1 lattice.

  * (ii)

$({\mathcal{M}}_{1},\bar{+})$ is a hyper group with $\bar{+}$ defined by (29),
where the identity set is

$\displaystyle{\bf e}=\\{0_{n\times n}\in{\mathcal{M}}_{n\times
n}\;|\;n\in{\mathbb{Z}}_{+}\\},$ (64)

w.r.t. MD-1 lattice.

Also, $({\mathcal{M}}_{1},\hat{+})$ is a hyper group with $\hat{+}$ defined by
(30), where the identity set is the same as (64).

  * (iii)

$({\mathcal{M}}_{\mu},\bar{+})$ (or $({\mathcal{M}}_{\mu},\hat{+})$) is a
hyper group with $\bar{+}$ defined by (29) (or $\hat{+}$ defined by (30)),
where the identity set is

$\displaystyle{\bf e}=\\{0_{n\mu_{y}\times
n\mu_{x}}\in{\mathcal{M}}_{n\mu_{y}\times
n\mu_{x}}\;|\;n\in{\mathbb{Z}}_{+}\\},$ (65)

w.r.t. MD-1 lattice.

  * (iv)

$({\mathcal{M}},\hat{+})$ is a hyper group with $\hat{+}$ defined by (35),
where the identity set is

$\displaystyle{\bf e}=\\{0_{m\times n}\in{\mathcal{M}}_{m\times
n}\;|\;(m,n)\in{\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}\\},$ (66)

w.r.t. MD-2 lattice.

Proof. We prove (iv) only. The proofs for others are similar.

First, we show that $({\mathcal{M}},\hat{+})$ is a semi-group. Assume
$A\in{\mathcal{M}}_{m\times n}$, $A\in{\mathcal{M}}_{m\times n}$,
$B\in{\mathcal{M}}_{p\times q}$, $C\in{\mathcal{M}}_{r\times s}$, and

$\begin{array}[]{l}m\vee p=a,~{}p\vee r=c,~{}m\vee p\vee r=\xi,\\\ n\vee
q=b,~{}q\vee s=d,~{}n\vee q\vee s=\eta.\end{array}$

Then

$\begin{array}[]{l}\left(A\hat{+}B\right)\hat{+}C=\left[(A\otimes{\mathcal{E}}_{a/m\times
b/n})+(B\otimes{\mathcal{E}}_{a/p\times b/q})\right]\hat{+}C\\\
~{}=\left[(A\otimes{\mathcal{E}}_{a/m\times
b/n})+(B\otimes{\mathcal{E}}_{a/p\times
b/q})\right]\otimes{\mathcal{E}}_{\xi/a\times\eta/b}\\\
~{}+C\otimes{\mathcal{E}}_{\xi/r\times\eta/s}\\\
~{}=(A\otimes{\mathcal{E}}_{\xi/m\times\eta/n})+(B\otimes{\mathcal{E}}_{\xi/p\times\eta/q})+(C\otimes{\mathcal{E}}_{\xi/r\times\eta/s})\\\
\end{array}$

A similar calculation shows that

$\begin{array}[]{l}A\hat{+}\left(B\hat{+}C\right)=(A\otimes{\mathcal{E}}_{\xi/m\times\eta/n})\\\
~{}+(B\otimes{\mathcal{E}}_{\xi/p\times\eta/q})+(C\otimes{\mathcal{E}}_{\xi/r\times\eta/s})\\\
\end{array}$

Hence $({\mathcal{M}},\hat{+})$ is a semi-group, which is obviously abelian.

Then we consider the identity set ${\bf e}$.

Let $A\in{\mathcal{M}}_{m\times n}$, $0_{p\times q}\in{\bf e}$, $m\vee p=a$,
and $n\vee q=b$. Then a straightforward computation shows that

$A\hat{+}0_{p\times q}=0_{p\times q}\hat{+}A=A\otimes{\mathcal{E}}_{a/m\times
b/n}.$

(59) holds.

Next, it is ready to verify that the lattice structure of (66) satisfies (60).

Finally, assume $A$ as before, then it is obvious that $e_{A}=0_{m\times
n}\in{\bf e}$, which leads to

$A+(-A)=e_{A}.$

(61) is satisfied.

$\Box$

###### Remark III.9

All the additive hyper groups are abelian.

Next, we consider the product hyper groups.

Set

${\mathcal{T}}:=\bigcup_{i=1}^{\infty}{\mathcal{T}}_{n}.$

Then we have the product hyper group of matrices as follows.

###### Proposition III.10

$({\mathcal{T}},\ltimes)$ is a hyper group with $\ltimes$ defined by (18),
where the identity set is

$\displaystyle{\bf e}=\\{I_{n}\in{\mathcal{M}}_{n\times
n}\;|\;n\in{\mathbb{Z}}_{+}\\},$ (67)

w.r.t. MD-1 lattice.

Proof.

It is well known that $({\mathcal{M}}.\ltimes)$ is a semi-group. Let
$A,B\in{\mathcal{T}}$. It is easy to verify that if $A\ltimes
B\in{\mathcal{T}}$. Hence, $({\mathcal{T}},\ltimes)$ is a sub-semi-group.

Let $A\in{\mathcal{T}}_{m}$, $I_{n}\in{\bf e}$, and $t=m\vee n$. Then

$A\ltimes I_{n}=I_{n}\ltimes A=A\otimes I_{t/m}.$

(59) holds.

It is obvious that $e_{A}=I_{m}$ and

$A\ltimes A^{-1}=AA^{-1}=I_{m}.$

(60) holds.

Finally, the lattice structure of the identity set is obvious. Moreover,

$I_{m}\ltimes I_{n}=I_{t}.$

(61) is also true.

$\Box$

To emphasize the identity set of a hyper group, we may also denote a hyper
group by a triple $(G,*,{\bf e})$.

###  III-D Component Groups and Equivalence Group

This subsection considers the component groups and equivalence group of a
hyper group.

###### Proposition III.11

Assume ${\mathcal{G}}=(G,*,{\bf e})$ is a hyper group, where ${\bf e}$ is
determined by a lattice $\Lambda$. Assign for each $e_{\lambda}\in{\bf e}$,
$\lambda\in\Lambda$ a set $G_{\lambda}\subset G$ as

$\displaystyle G_{\lambda}:=\\{x\in G\;|\;e_{x}=e_{\lambda}\\}.$ (68)

Then

  * (i)

For each $\lambda\in\Lambda$, ${\mathcal{G}}_{\lambda}=(G_{\lambda},*)$ is a
group, called the component group of ${\mathcal{G}}$, and its identity is
$e_{\lambda}$.

  * (ii)

$\displaystyle G=\bigcup_{\lambda\in\Lambda}G_{\lambda}$ (69)

is a partition.

  * (iii)

If $x\in G_{\lambda}$ and $y\in G_{\mu}$, then $x*y\in G_{\lambda\vee\mu}$.

Proof.

  * (i)

To see $G_{\lambda}$ is a group, we have to show that if $x,y\in G_{\lambda}$,
then $x*y\in G_{\lambda}$ . It obvious that

$x*y*e_{\lambda}=e_{\lambda}*x*y.$

We also have

$x*y*y^{-1}x^{-1}=x*e_{\lambda}x^{-1}=e_{\lambda},$

and similarly, we have

$y^{-1}x^{-1}*x*y=e_{\lambda}.$

Hence, we have that both $(x*y)^{-1}=y^{-1}*x^{-1}$ and $x*y$ are $\in
G_{\lambda}$.

It is clear from definition that if $x\in G_{\lambda}$ then $x^{-1}\in
G_{\lambda}$.

Then the associativity and the uniqueness of the inverse are inherited from
the properties of hyper group.

We conclude that $G_{\lambda}$ is a group.

  * (ii)

The partition (69) comes from definition immediately.

  * (iii)

Since

$\begin{array}[]{l}x*y*y^{-1}*x^{-1}=x*e_{\mu}*x^{-1}=e_{\mu}*x*x^{-1}\\\
=e_{mu}*e_{\lambda}=e_{\lambda\vee\mu}.\end{array}$

By definition $x*y\in G_{\lambda\vee\mu}$.

$\Box$

###### Definition III.12

Consider a hyper group ${\mathcal{G}}=(G,*,{\bf e})$ . Let $x,~{}y\in G$. $x$
and $y$ are said to be equivalent, denoted by $x\sim y$, if there exist
$e_{\alpha}$ and $e_{\beta}$ such that

$\displaystyle x*e_{\alpha}=y*e_{\beta}.$ (70)

The equivalence class of $x$, denoted by $\bar{x}$, is

$\bar{x}:=\\{y\in G\;|\;y\sim x\\}.$

Note that we have to show that (70) is an equivalence relation. A
straightforward verification ensures this.

###### Definition III.13

Consider a hyper group ${\mathcal{G}}=(G,*)$. The equivalence $\sim$ is said
to be consistent with the group operator $*$, if the followings are satisfied.

  * (i)

If $x_{1}\sim x_{2}$ and $y_{1}\sim y_{2}$, then

$\displaystyle x_{1}*y_{1}\sim x_{2}*y_{2}.$ (71)

  * (ii)

If there exists $e\in{\bf e}$ such that $x\sim e$, then $x\in{\bf e}$.

Then we have the identity set as one element.

###### Proposition III.14

Consider a hyper group ${\mathcal{G}}=(G,*,{\bf e})$. If the equivalence is
consistent w.r.t. the operator, then

$\displaystyle\bar{e}={\bf e},\quad e\in{\bf e}.$ (72)

Proof. By definition we have

$e_{\lambda}\sim e_{\mu},\quad\lambda,\mu\in\Lambda,$

According to (ii) of Definition III.13, we also know that if $a\not\in{\bf e}$
then for any $e\in{\bf e}$, $a\not\sim e$. (72) follows. $\Box$

Using Proposition III.14, we have the following result immediately.

###### Proposition III.15

Consider a hyper group ${\mathcal{G}}=(G,*,{\bf e})$. Assume the equivalence
$\sim$ defined by (70) is consistent with $*$. Then

$\displaystyle\bar{x}*\bar{y}:=\overline{x*y},$ (73)

is properly defined. Moreover, $\bar{\mathcal{G}}=(\bar{G},*)$ is a group,
called the equivalence group of ${\mathcal{G}}$, denoted by

$\bar{\mathcal{G}}:={\mathcal{G}}/\sim=(\bar{G},*),$

where $\bar{G}=\\{\bar{x}\;|\;x\in G\\}$.

Observing Propositions III.11 and III.15, a geometric picture for the
structure of a hyper group, called the group decomposition of hyper group, is
depicted by Figure 3.

$G_{e}$$\bar{{\mathcal{G}}}$$G^{1}_{e}$$G^{2}_{e}$$G^{n}_{e}$$\vdots$$\vdots$${\mathcal{G}}$$0$
Figure 3: Group Decomposition of a Hyper-Group ${\mathcal{G}}$

Since the existence of the equivalence hyper group depends on the consistence
of the equivalence w.r.t. the group operator, the consistence is the key issue
for the structure analysis of hyper groups. It is worth showing the
consistence of the hyper groups over MVMDs. To this end we need a preparation.

###### Lemma III.16

  * (i)

Consider $x,y\in{\mathbb{R}}^{\infty}$, which are specified by
$x\in{\mathbb{R}}^{m}$, $y\in{\mathbb{R}}^{n}$, $m\vee n=t$, and $m\wedge
n=s$. Assume there are ${\bf 1}_{p}$ and ${\bf 1}_{q}$ (or ${\mathcal{E}}_{p}$
and ${\mathcal{E}}_{q}$) such that

$\displaystyle x\otimes{\bf 1}_{p}=y\otimes{\bf
1}_{q},~{}(\mbox{or}~{}x\otimes{\mathcal{E}}_{p}=y\otimes{\mathcal{E}}_{q}),$
(74)

then there exists a $z\in{\mathbb{R}}^{s}$ such that

$\displaystyle x=z\otimes{\bf 1}_{m/s};\quad y=z\otimes{\bf 1}_{n/s}.$ (75)

  * (ii)

Consider $A,B\in{\mathcal{M}}_{\mu}$, which are specified by
$A\in{\mathcal{M}}_{m\mu_{y}\times m\mu_{x}}$,
$B\in{\mathcal{M}}_{n\mu_{y}\times n\mu_{x}}$, $m\vee n=a$, and $m\wedge n=b$.
Assume there are $H_{r}=I_{r}$ and $H_{s}=I_{s}$ (or $H_{r}=J_{r}$ and
$H_{s}=J_{s}$ ) such that

$\displaystyle A\otimes H_{r}=B\otimes H_{s},$ (76)

then there exists a $C\in{\mathcal{M}}_{b\mu_{y}\times b\mu_{x}}$ such that

$\displaystyle A=C\otimes H_{m/b};\quad B=C\otimes H_{n/b}.$ (77)

  * (iii)

Consider $A,B\in{\mathcal{M}}$, which are specified by
$A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times q}$, $m\wedge
p=r$, and $n\wedge q=s$. Assume there are ${\mathcal{E}}_{a\times b}$ and
${\mathcal{E}}_{c\times d}$ such that

$\displaystyle A\otimes{\mathcal{E}}_{a\times
b}=B\otimes{\mathcal{E}}_{c\times d},$ (78)

then there exists a $C\in{\mathcal{M}}_{r\times s}$ such that

$\displaystyle A=C\otimes{\mathcal{E}}_{m/r\times n/s};\quad
B=C\otimes{\mathcal{E}}_{p/r\times q/s}.$ (79)

Proof. In fact, (i) and (ii) have been proved in [11]. The proof of (iii) is
similar. We sketch it. First, note that $ma=pc$ and $nb=qd$. By deleting
common factors we can assume, without loss of generality, that

$a=\frac{m\vee p}{m},\quad b=\frac{n\vee q}{n},\quad c=\frac{m\vee p}{p},\quad
d=\frac{n\vee q}{q}.$

Using the above $a,b,c,d$ and comparing both sides of (78) carefully, the
conclusion follows. The detailed arguments are similar to those for (i) and
(ii) in [11]. $\Box$

###### Example III.17

Consider the hyper group ${\mathcal{G}}=({\mathcal{T}},\ltimes,{\bf e})$,
where the identity set is defined by (67). Then we have

  * (i)

The component groups are

$\displaystyle{\mathcal{T}}_{n}=GL(n,{\mathbb{R}}),\quad
n\in{\mathbb{Z}}_{+},$ (80)

where $GL(n,{\mathbb{R}})$ are the standard general linear groups, and

${\mathcal{T}}:=\bigcup_{i=1}^{\infty}GL(n,{\mathbb{R}}).$

  * (ii)

Using Lemma III.16, it is easy to verify that the equivalence is consistent
with the STP, hence the equivalence quotient group exists.

$\overline{\mathcal{T}}:=\bigcup_{n=1}^{\infty}\\{\bar{A}_{n}\;|\;A_{n}\in{\mathcal{M}}_{n\times
n}~{}\mbox{is invertible.}\\}.$

If we define $\pi:{\mathcal{T}}\rightarrow\overline{\mathcal{T}}$ by

$\pi(A)=\bar{A},\quad A\in{\mathcal{T}},$

then it is easy to verify that $\pi$ is a hyper group homomorphism.

In the following we consider the additive hyper groups. Recall the additive
hyper groups discussed in Proposition III.8.

###### Example III.18

In the following we assume the consistence of the equivalence w.r.t. addition
in each case holds.

  * (i)

Consider $({\mathbb{R}}^{\infty},\bar{+})$ with $\bar{+}$ defined by (28). It
is easy to verify that the component groups are $({\mathbb{R}}^{n},+)$,
$n\in{\mathbb{Z}}_{+}$.

Next, consider the equivalence hyper group. Let $x\in{\mathbb{R}}^{m}$,
$y\in{\mathbb{R}}^{n}$, and $t=\operatorname{lcm}(m.n)$. Assume $x$ and $y$
are equivalent, denoted by $x\leftrightarrow y$. By definition, there exist
$0_{p}$ and $0_{q}$, such that

$\displaystyle x\bar{+}0_{p}=y\bar{+}0_{q}.$ (81)

Comparing the dimensions and deleting common factor ${\bf 1}_{s}$, where
$s=p\wedge q$, we have

$\displaystyle x\otimes{\bf 1}_{t/m}=y\otimes{\bf 1}_{t/n}.$ (82)

In fact, (82) is the definition of equivalence of two vectors in [10]. Note
that the consistence of the equivalence w.r.t. $\bar{+}$ is assumed, (In fact,
it has already proved in [10]), then the equivalence group exists, which is
denoted by

$\displaystyle\Omega=\left({\mathbb{R}}^{\infty}/\leftrightarrow\right)=(\\{\bar{x}\;|\;x\in{\mathbb{R}}^{\infty}\\},\bar{+}).$
(83)

Similarly, we also have $x\leftrightharpoons y$, if

$\displaystyle x\hat{+}0_{p}=y\hat{+}0_{q},$ (84)

which leads to

$\displaystyle x\otimes{\mathcal{E}}_{t/m}=y\otimes{\mathcal{E}}_{t/n}.$ (85)

And finally we have the equivalence group as

$\displaystyle\Omega^{\prime}=\left({\mathbb{R}}^{\infty}/\leftrightharpoons\right)=(\\{\bar{x}\;|\;x\in{\mathbb{R}}^{\infty}\\},\hat{+}).$
(86)

  * (ii)

Consider $({\mathcal{M}}_{1},\bar{+})$. Let $A\in{\mathcal{M}}_{m\times m}$,
$B\in{\mathcal{M}}_{n\times n}$, and $t=\operatorname{lcm}(m,n)$. Then $A\sim
B$ if there exist $0_{p\times p}$ and $0_{q\times q}$ such that

$\displaystyle A\bar{+}0_{p\times p}=B\bar{+}0_{q\times q},$ (87)

which leads to

$\displaystyle A\otimes I_{t/m}=B\otimes I_{t/n}.$ (88)

And finally we have the equivalence group as

$\displaystyle\Sigma=({\mathcal{M}}_{1}/\sim)=(\\{\bar{A}\;|\;A\in{\mathcal{M}}_{1}\\},\bar{+}).$
(89)

Similarly, consider $({\mathcal{M}}_{1},\hat{+})$. We have $A\approx B$ if

$\displaystyle A\hat{+}0_{p\times p}=B\hat{+}0_{q\times q},$ (90)

which leads to

$\displaystyle A\otimes J_{t/m}=B\otimes J_{t/n}.$ (91)

And finally we have the equivalence group as

$\displaystyle\Sigma^{\prime}=({\mathcal{M}}_{1}/\approx)=(\\{\bar{A}\;|\;A\in{\mathcal{M}}_{1}\\},\hat{+}).$
(92)

  * (iii)

Consider $({\mathcal{M}}_{\mu}.\hat{+})$. Let
$A\in{\mathcal{M}}_{m\mu_{y}\times m\mu_{x}}$,
$B\in{\mathcal{M}}_{n\mu_{y}\times n\mu_{x}}$, and
$t=\operatorname{lcm}(m,n)$. $A\approx B$ if

$\displaystyle A\hat{+}0_{p\mu_{y}\times p\mu_{x}}=B\hat{+}0_{q\nu_{y}\times
q\mu_{x}},$ (93)

which leads to

$\displaystyle A\otimes J_{(t/m)\mu_{y}\times(t/m)\mu_{x}}=B\otimes
J_{(t/n)\mu_{y}\times(t/n)\mu_{x}}.$ (94)

Finally we have

$\displaystyle\Sigma_{\mu}=({\mathcal{M}}_{\mu}/\approx)=(\\{\bar{A}\;|\;A\in{\mathcal{M}}_{\mu}\\},\hat{+}).$
(95)

  * (iv)

Consider $({\mathcal{M}},\hat{+})$. Let $A\in{\mathcal{M}}_{m\times n}$,
$B\in{\mathcal{M}}_{p\times q}$, $A\sim B$ if there exist $0_{r\times s}$ and
$0_{a\times b}$ such that

$\displaystyle A\hat{+}0_{r\times s}=B\hat{+}0_{a\times b}.$ (96)

Assume $\alpha=m\vee p$ and $\beta=n\vee q$, then (90) leads to

$\displaystyle
A\otimes{\mathcal{E}}_{(\alpha/m)\times(\beta/n)}=B\otimes{\mathcal{E}}_{(\alpha/p)\times(\beta/q)n)}.$
(97)

Finally we have

$\displaystyle\Sigma=({\mathcal{M}}/\approx)=(\\{\bar{A}\;|\;A\in{\mathcal{M}}\\},\hat{+}).$
(98)

In the above example we assume the consistence always exist. Using Lemma
III.16, this can be proved one by one:

###### Proposition III.19

  * (i)

Consider $({\mathbb{R}}^{\infty},\bar{+})$ (or
$({\mathbb{R}}^{\infty},\hat{+})$), its equivalence is consistent with its
operator $\bar{+}$ ( or $\hat{+}$).

  * (ii)

Consider $({\mathcal{M}}_{1},\bar{+})$ (or $({\mathcal{M}}_{1},\hat{+})$ ),
its equivalence is consistent with its operator $\bar{+}$ (or $\hat{+}$).

  * (iii)

Consider $({\mathcal{M}}_{\mu},\hat{+})$, its equivalence is consistent with
its operator $\hat{+}$.

  * (iv)

Consider $({\mathcal{M}},\hat{+})$, its equivalence is consistent with its
operator $\hat{+}$.

Proof. We prove (i) and (iv). The proofs for (ii) and (iii) are similar.

  * (i)

First, to use Lemma III.16, we show that if $x,y\in{\mathbb{R}}^{\infty}$ and
$x\leftrightarrow y$, then there exist ${\bf 1}_{p}$ and ${\bf 1}_{q}$ such
that

$x\otimes{\bf 1}_{p}=y\otimes{\bf 1}_{q}.$

Assume $x\in{\mathbb{R}}^{m}$ and $y\in{\mathbb{R}}^{n}$. By definition, there
exist $0_{p}$ and $0_{q}$, such that

$\begin{array}[]{l}x\bar{+}0_{p}=\left(x\otimes{\bf 1}_{(m\vee
p)/m}\right)+\left(0_{p}\otimes{\bf 1}_{(m\vee p)/p}\right)\\\
\left(x\otimes{\bf 1}_{(m\vee p)/m}\right)\\\
=y\bar{+}0_{q}=\left(y\otimes{\bf 1}_{(n\vee q)/n}\right).\\\ \end{array}$

Set $t=m\vee n$. Then the above equality can be simplified to

$\begin{array}[]{l}x\otimes{\bf 1}_{t/m}=y\otimes{\bf 1}_{t/n}.\end{array}$

Next, assume $x_{1}\sim x_{2}$ and $y_{1}\sim y_{2}$, we have to show that

$\displaystyle x_{1}\bar{+}y_{1}\sim x_{2}\bar{+}y_{2}.$ (99)

Suppose $x_{1}\in{\mathbb{R}}^{m}$, $x_{2}\in{\mathbb{R}}^{n}$,
$y_{1}\in{\mathbb{R}}^{p}$, $y_{2}\in{\mathbb{R}}^{q}$, and

$\begin{array}[]{ccc}m\vee p=\alpha,&n\vee q=\beta,&m\wedge n=r,\\\ p\wedge
q=s,&r\vee s=\xi.&\alpha=\xi a,\\\ \beta=\xi b,&{}\hfil&\\\ \end{array}$

According to Lemma III.16 there exist $x_{0}\in{\mathbb{R}}^{r}$ and
$y_{0}\in{\mathbb{R}}^{s}$ such that

$\begin{array}[]{cc}x_{1}=x_{0}\otimes{\bf 1}_{m/r},&x_{2}=x_{0}\otimes{\bf
1}_{n/r},\\\ y_{1}=y_{0}\otimes{\bf 1}_{p/s},&y_{2}=y_{0}\otimes{\bf
1}_{q/s}.\\\ \end{array}$

Then

$\begin{array}[]{l}x_{1}\bar{+}y_{1}=x_{0}\otimes{\bf 1}_{m/r}\otimes{\bf
1}_{\alpha/m}+y_{0}\otimes{\bf 1}_{p/s}\otimes{\bf 1}_{\alpha/p}\\\
~{}=x_{0}\otimes{\bf 1}_{\alpha/r}+y_{0}\otimes{\bf 1}_{\alpha/s}\\\
~{}=\left(x_{0}\otimes{\bf 1}_{\xi/r}+y_{0}\otimes{\bf
1}_{\xi/s}\right)\otimes{\bf 1}_{a}.\\\ \end{array}$

Similarly,

$\begin{array}[]{l}x_{2}\bar{+}y_{2}=x_{0}\otimes{\bf
1}_{\beta/r}+y_{0}\otimes{\bf 1}_{\beta/s}\\\ ~{}=\left(x_{0}\otimes{\bf
1}_{\xi/r}+y_{0}\otimes{\bf 1}_{\xi/s}\right)\otimes{\bf 1}_{b}.\\\
\end{array}$

We conclude that

$x_{1}\bar{+}y_{1}\sim\left(x_{0}\otimes{\bf 1}_{\xi/r}+y_{0}\otimes{\bf
1}_{\xi/s}\right)\sim x_{2}\hat{+}y_{2}.$

Next, we check condition (ii) of Definition III.13. Assume
$x\in{\mathbb{R}}^{m}$ and $x\sim 0_{a}\in{\bf e}$, then there exist ${\bf
1}_{s}$ and ${\bf 1}_{t}$, such that

$x\otimes{\bf 1}_{s}=0_{a}\otimes{\bf 1}_{t}=0_{at}.$

It is easy to see that $x=0_{m}\in{\bf e}$.

  * (iv)

To prove the consistence of the addition and the product with the equivalence
we need to show that for $A_{1}\sim A_{2}$ and $B_{1}\sim B_{2}$ we have

$\displaystyle A_{1}\hat{+}B_{1}\sim A_{2}\hat{+}B_{2}.$ (100)

Assume $A_{1}\in{\mathcal{M}}_{m\times n}$, $A_{2}\in{\mathcal{M}}_{p\times
q}$, $B_{1}\in{\mathcal{M}}_{\alpha\times\beta}$, and
$B_{2}\in{\mathcal{M}}_{\gamma\times\delta}$, and

$\begin{array}[]{ccc}m\vee\alpha=a,&n\vee\beta=b,&p\vee\gamma=c,\\\
q\vee\delta=d,&m\wedge p=s,&n\wedge q=t,\\\
\alpha\wedge\gamma=\xi,&\beta\wedge\delta=\eta,&{}\hfil\\\
s\vee\xi=u,&t\vee\eta=v.&{}\hfil\\\ \end{array}$

Using Lemma III.16, we know that there exist $A_{0}\in{\mathcal{M}}_{s\times
t}$ and $B_{0}\in{\mathcal{M}}_{\xi\times\eta}$ such that

$\begin{array}[]{l}A_{1}=A_{0}\otimes{\mathcal{E}}_{m/s\times n/t},\quad
A_{2}=A_{0}\otimes{\mathcal{E}}_{p/s\times q/t};\\\
B_{1}=B_{0}\otimes{\mathcal{E}}_{\alpha/\xi\times\beta/\eta},\quad
B_{2}=B_{0}\otimes{\mathcal{E}}_{\gamma/\xi\times\delta/\eta}.\\\ \end{array}$

Now

$\begin{array}[]{l}A_{1}\hat{+}B_{1}=A_{1}\otimes{\mathcal{E}}_{a/m\times
b/n}+B_{1}\otimes{\mathcal{E}}_{a/\alpha\times b/\beta}\\\
~{}=A_{0}\otimes{\mathcal{E}}_{a/s\times
b/t}+B_{0}\otimes{\mathcal{E}}_{a/\xi\times b/\eta}\\\
~{}=\left(A_{0}\otimes{\mathcal{E}}_{u/s\times
v/t}+B_{0}\otimes{\mathcal{E}}_{u/\xi\times
v/\eta}\right)\otimes{\mathcal{E}}_{a/u\times b/v}.\end{array}$
$\begin{array}[]{l}A_{2}\hat{+}B_{2}=A_{2}\otimes{\mathcal{E}}_{c/p\times
d/q}+B_{2}\otimes{\mathcal{E}}_{c/\gamma\times d/\delta}\\\
~{}=A_{0}\otimes{\mathcal{E}}_{c/s\times
d/t}+B_{0}\otimes{\mathcal{E}}_{c/\xi\times d/\eta}\\\
~{}=\left(A_{0}\otimes{\mathcal{E}}_{u/s\times
v/t}+B_{0}\otimes{\mathcal{E}}_{u/\xi\times
v/\eta}\right)\otimes{\mathcal{E}}_{c/u\times d/v}.\end{array}$

It follows immediately that

$A_{1}\hat{+}B_{1}\sim A_{2}\hat{+}B_{2}.$

The condition (ii) of Definition III.13 is straightforward verifiable. $\Box$

We conclude that for these additive hyper-vector spaces the equivalence vector
spaces always exist.

###  III-E Sub Hyper Group

###### Definition III.20

Let ${\mathcal{G}}=(G,*,{\bf e})$ be a hyper group. ${\mathcal{H}}=(H,*,{\bf
e}_{0})$ is called a sub hyper group, if $H\subset G$ is a subset, ${\bf
e}_{0}\subset{\bf e}$ is a sub-lattice, and $(H,*,{\bf e}_{0})$ is a hyper
group. Denoted the sub hyper group by ${\mathcal{H}}<_{h}{\mathcal{G}}$. If
${\mathcal{H}}<_{h}{\mathcal{G}}$ and $gH_{e_{g}}=H_{e_{g}}g$, $\forall g\in
G$ and $e_{g}\in{\bf e}_{0}$, then $(H,*,{\bf e}_{0})$ is a normal sub hyper
group, and denoted by ${\mathcal{H}}\lhd_{h}{\mathcal{G}}$.

The following result comes from the definition immediately.

###### Proposition III.21

Let ${\mathcal{G}}=(G,*,{\bf e})$ be a hyper group.

  * (i)

Assume ${\bf e}_{0}\prec{\bf e}$ is a sub-lattice. Set

$\displaystyle G_{0}:=\bigcup_{e\in{\bf e}_{0}}G_{e},$ (101)

then

$\displaystyle{\mathcal{G}}_{0}:=(G_{0},*,{\bf e}_{0})<_{h}(G,*,{\bf e}).$
(102)

  * (ii)

Consider ${\mathcal{G}}_{0}$ as in (i). Assume $H_{e}<G_{e}$, $\forall
e\in{\bf e}$, and

$\displaystyle H^{0}:=\bigcup_{e\in{\bf e}_{0}}H_{e}.$ (103)

Then

$\displaystyle{\mathcal{H}}_{0}:=(H^{0},*,{\bf e}_{0})<_{h}{\mathcal{G}}_{0}.$
(104)

  * (iii)

Conversely, if ${\bf e}_{0}\subset{\bf e}$ is a sublattice, $\emptyset\neq
H_{e}\subset G_{e}$, $e\in{\bf e}_{0}$. $H^{0}$ is defined by (103), and
${\mathcal{H}}_{0}:=(H^{0},*,{\bf e}_{0})<_{h}{\mathcal{G}}$, then

$\displaystyle{\mathcal{H}}_{e}:=(H_{e},*)<{\mathcal{G}}_{e},\quad\forall
e\in{\bf e}_{0}.$ (105)

  * (iv)

If ${\mathcal{H}}_{0}:=(H^{0},*,{\bf e}_{0})\lhd_{h}{\mathcal{G}}$, then

$\displaystyle{\mathcal{H}}_{e}:=(H_{e},*)\lhd{\mathcal{G}}_{e},\quad\forall
e\in{\bf e}_{0}.$ (106)

Fig. 4 depicts these relations.

${\mathcal{H}}_{0}=\\{H_{e}\;|\;e\in{\bf
e}_{0}\\}$${\mathcal{H}}=\\{H_{e}\;|\;e\in{\bf
e}\\}$${\mathcal{G}}_{0}=\\{G_{e}\;|\;e\in{\bf
e}_{0}\\}$${\mathcal{G}}=\\{G_{e}\;|\;e\in{\bf e}\\}$${\bf e}_{0}\prec{\bf
e}$$<$$<$$\vee$$\vee$$H_{e}<G_{e}$ Figure 4: Sub Hyper Groups

###### Example III.22

  * (i)

A lattice ${\mathcal{H}}$ is constructed by

$H=\\{1,2,3,4,6,9,12,18,36\\}\subset Z_{+}$

with sublattice structure of $(Z_{+},\sup=\vee,\inf=\wedge)$. Then its Hasse
diagram is depicted by the left graph of Figure 2, where

$\begin{array}[]{ccc}e=36,&c=12,&f=18,\\\ a=4,&h=6,&g=3,\\\ i=2,&b=3,&d=1.\\\
\end{array}$

Set

${\mathcal{T}}_{H}:=\\{{\mathcal{T}}_{n}\;|\;n\in H\\},$

and

${\bf e}_{h}=(\\{I_{n}\;|\;n\in H\\},\vee,\wedge\\}).$

Then ${\mathcal{H}}:=({\mathcal{T}}_{H},\ltimes,{\bf
e}_{h})<_{h}{\mathcal{G}}$ is a sub-hyper group, w.r.t. MD-1 lattice.
${\mathcal{G}}=({\mathcal{T}},\ltimes,{\bf e})$ is defined in Example III.17.

  * (ii)

Consider $M_{1}=\bigcup_{n=1}^{\infty}{\mathcal{M}}_{n\times n}$, $\hat{+}$ is
defined by (30), and

${\bf e}=\\{0_{n\times n}\in{\mathcal{M}}_{n\times
n}\;|\;n\in{\mathbb{Z}}_{+}\\},$

with lattice structure $({\mathbb{Z}}_{+},\prec)$ defined in (i) of Example
III.2. Then ${\mathcal{M}}_{1}=(M_{1},\hat{+},{\bf e})$ is a hyper group.

    * –

Set

$S_{n}:=\\{A\in{\mathcal{M}}_{n\times n}\;|\;A^{T}=A\\},$

and

$S=\bigcup_{n=1}^{\infty}S_{n}.$

Then ${\mathcal{S}}:=(S,\hat{+},{\bf e})$ is a hyper group, and

${\mathcal{S}}\lhd_{h}{\mathcal{M}}_{1}.$

    * –

Set

$K_{n}:=\\{A\in{\mathcal{M}}_{n\times n}\;|\;A^{T}=-A\\},$

and

$K=\bigcup_{n=1}^{\infty}K_{n}.$

Then ${\mathcal{K}}:=(K,\hat{+},{\bf e})$ is a hyper group, and

${\mathcal{K}}\lhd_{h}{\mathcal{M}}_{1}.$

    * –

Set

$D_{n}:=\\{A\in{\mathcal{M}}_{n\times n}\;|\;A~{}\mbox{is diagonal.}\\},$

and

$D=\bigcup_{n=1}^{\infty}D_{n}.$

Then ${\mathcal{D}}:=(D,\hat{+},{\bf e})$ is a hyper group, and

${\mathcal{D}}\lhd_{h}{\mathcal{M}}_{1}.$

  * (iii)

Consider $T_{n}$, $n\in{\mathbb{Z}}_{+}$ as the set of $n\times n$ non-
singular matrices, and set

$T:=\bigcup_{i=1}^{\infty}T_{n},$

with $\ltimes$ defined by (18), and ${\bf e}$ as in (67). Then
${\mathcal{T}}:=(T,\ltimes,{\bf e})$ is a hyper group.

    * –

Set

$O_{n}:=\\{A\in{\mathcal{T}}_{n}\;|\;A^{T}=A^{-1}\\},$
$O=\bigcup_{n=1}^{\infty}O_{n}.$

Then ${\mathcal{O}}:=({O}_{n},\ltimes,{\bf e})$ is a hyper group, and

${\mathcal{O}}<_{h}{\mathcal{T}}.$

    * –

Set

$E_{n}:=\\{A\in{\mathcal{M}}_{n\times n}\;|\;\det(A)=1\\},$

and

$E=\bigcup_{n=1}^{\infty}E_{n}.$

Then ${\bf E}=(E,\ltimes,{\bf e})$ is a hyper group, and

${\bf E}<_{h}{\mathcal{T}}.$

###### Proposition III.23

Let ${\mathcal{G}}=(G.*,{\bf e})$ be a hyper group, and $H\subset G$.
${\mathcal{H}}=(H.*,{\bf e}_{H})$ is a sub-hyper group, if and only if,

  * (i)

$\displaystyle{\bf e}_{H}=\\{e_{x}\;|\;x\in H\\},$ (107)

and which is a sub-lattice;

  * (ii)

$\displaystyle x*y^{-1}\in H,\quad\forall x,y\in H.$ (108)

Proof. The necessity is obvious. To see the sufficiency, assume $x,y\in
H_{e}$, then $x*y^{-1}\in H_{e}$. That is, $H_{e}<G_{e}$ is a sub-group.
Hence, $H$ is a sub-hyper group of $G$. $\Box$

##  IV Permutation Hyper Group

###  IV-A Left Permutation Hyper Group

Permutation group is one of the most important finite groups, because any
finite group can be isomorphic to a subgroup of certain permutation group
[26]. In this section we construct a hyper group structure for permutations
with different orders.

Let ${\bf S}_{n}$ be the permutation group of order $n$ with the group
operator $\circ$ as the compound of two permutations. Set

${\bf S}:=\bigcup_{n=1}^{\infty}{\bf S}_{n}.$

To build a cross-order product of permutations with different orders, we need
to construct a cross-dimensional mapping.

###### Definition IV.1

Assume $m|n$, and $km=n$, then we define a mapping, called the left embedding,
as $\varphi^{m}_{n}:{\bf S}_{m}\rightarrow{\bf S}_{n}$ as follows: Assume
$\sigma\in{\bf S}_{m}$, then

$\displaystyle\begin{array}[]{l}\varphi^{m}_{n}(\sigma)((i-1)k+s):=(\sigma(i)-1)k+s,\\\
\quad\sigma\in{\bf S}_{m},\;s\in[1,k],\;i\in[1,m].\\\ \end{array}$ (111)

Note that when $m=n$ $\varphi^{m}_{m}$ is the identity mapping. When $m<n$, to
understand this mapping, we recall permutation matrices. Recall the definition
of permutation matrices in Example II.2. A permutation matrix
$P\in{\mathcal{M}}_{n\times n}$ is a non-singular logical matrix. A logical
matrix $A\in{\mathcal{L}}_{n\times n}$ satisfies
$\operatorname{Col}(A)\subset{\mathcal{D}}_{n}$, hence it has the form that
$A=[\delta_{n}^{i_{1}},\delta_{n}^{i_{2}},\cdots,\delta_{n}^{i_{n}}]$ and can
be briefly denoted by

$A=\delta_{n}[i_{1},i_{2},\cdots,i_{n}].$

Recall that the set of $n\times n$ permutation matrices form a subgroup

${\mathcal{P}}_{n}<{\mathcal{T}}_{n}.$

For each $\sigma\in{\bf S}_{n}$, its permutation matrix is expressed as

$\displaystyle P_{\sigma}:=\delta_{n}[\sigma(1),\sigma(2),\cdots,\sigma(n)].$
(112)

According to the construction of a permutation matrix, we have the following
result.

###### Proposition IV.2

  * (i)

Assume a permutation $\sigma\in{\bf S}_{n}$ and its permutation matrix is
$P_{\sigma}$. Then

$\displaystyle P_{\sigma}\delta_{n}^{i}=\delta_{n}^{\sigma(i)},\quad
i\in[1,n].$ (113)

  * (ii)

Let $\sigma,~{}\mu\in{\bf S}_{n}$. Then

$\displaystyle M_{\sigma\circ\mu}=M_{\sigma}M_{\mu}.$ (114)

  * (iii)

Assume $m|n$, $km=n$, and $\sigma\in{\bf S}_{m}$. Then

$\displaystyle P_{\varphi(\sigma)}=P_{\sigma}\otimes I_{k}.$ (115)

###### Remark IV.3

  * (i)

Define $\pi:{\bf S}_{n}\rightarrow{\mathcal{P}}_{n}$ by $\pi:\sigma\mapsto
P_{\sigma}$ as shown in (112). Then Proposition IV.2 shows that $\pi$ is a
group isomorphism.

  * (ii)

Assume $m|n$, the $\varphi^{m}_{n}$ is an one-to-one homomorphism from ${\bf
S}_{m}$ to ${\bf S}_{n}$. In other words, $\varphi^{m}_{n}$ is an isomorphism
from ${\bf S}_{m}$ to its image $\varphi({\bf S}_{m})<{\bf S}_{n}$. It can
also be considered as an “embedding”.

Now we are ready to define a cross-order product of permutations as follows.

###### Definition IV.4

Let $\sigma,~{}\mu\in{\bf S}$, say, $\sigma\in{\bf S}_{m}$, $\mu\in{\bf
S}_{n}$, and $t=\operatorname{lcm}(m,n)$. Then the product, called the left
product, of $\sigma$ and $\mu$, denoted by $\sigma\odot\mu$, is defined by

$\displaystyle\sigma\odot\mu=\varphi^{m}_{t}(\sigma)\circ\varphi^{n}_{t}(\mu)\in{\bf
S}_{t}.$ (116)

The following result can be verified by the construction easily.

###### Proposition IV.5

Let $\sigma,~{}\mu\in{\bf S}$ with their permutation matrices $M_{\sigma}$ and
$M_{\mu}$ respectively. Then

$\displaystyle M_{\sigma\odot\mu}=M_{\sigma}\ltimes M_{\mu}.$ (117)

$\varphi^{m}_{n}$ is called the left embedding because it is determined by
this left STP of their permutation matrices.

Define the set of cross-order permutation matrices as

${\mathcal{P}}:=\bigcup_{n=1}^{\infty}{\mathcal{P}}_{n}.$

Then it is easy to verify the following.

###### Proposition IV.6

  * (i)

$({\mathcal{P}},\ltimes,{\bf e})$ is a hyper group, where ${\bf e}$ is defined
by (67).

  * (ii)

$\displaystyle({\mathcal{P}},\ltimes,{\bf e})<_{h}{\mathcal{T}},$ (118)

where $({\mathcal{T}}=(T,\ltimes,{\bf e})$ is defined in Proposition III.10.

Using Propositions IV.5 and IV.6, the following result is obvious.

###### Theorem IV.7

${\bf S}^{L}:=({\bf S},\odot,{\bf e})$ is a hyper group, where the identity
set is

$\displaystyle{\bf e}=\\{e_{n}=Id_{n}\in{\bf
S}_{n}\;|\;n\in{\mathbb{Z}}_{+}\\},$ (119)

w.r.t. MD-1 lattice $({\mathbb{Z}}_{+},\prec)$. w.

Next, we consider some basic properties of hyper group ${\bf S}^{L}$.

###### Proposition IV.8

Set the alternative set of permutations by

$\displaystyle{\bf A}:=\bigcup_{n=1}^{\infty}{\bf A}_{n}.$ (120)

  * (i)

${\bf A}^{L}$ is a sub-hyper group of ${\bf S}^{L}$. Precisely speaking,

$\displaystyle\left({\bf A},\odot,{\bf e}\right)<_{h}\left({\bf S},\odot,{\bf
e}\right).$ (121)

  * (ii)

${\bf A}^{L}$ is a normal sub-hyper group of ${\bf S}^{L}$. That is,

$\displaystyle\left({\bf A},\odot,{\bf e}\right)\lhd_{h}\left({\bf
S},\odot,{\bf e}\right).$ (122)

###### Remark IV.9

The definition of normal hyper subgroup can not be defined in the same way as
for normal subgroup. It is defined in each component groups. For instance,
note that ${\bf A}\lhd_{h}{\bf S}$ does not mean

$\sigma\circ{\bf A}={\bf A}\circ\sigma,\forall\sigma\in{\bf S}.$

It means

$\sigma\circ{\bf A}_{n}={\bf A}_{n}\circ\sigma,\forall\sigma\in{\bf
S}_{n},\forall n\in{\mathbb{Z}}_{+}.$

In fact, ${\bf A}$ and $g\circ{\bf A}$, where $g\in{\bf S}\backslash{\bf A}$,
is not a partition of ${\bf S}$.

Finally, we consider the consistence of the equivalence w.r.t. $\odot$.

  * (i)

Assume $\sigma_{1}\sim\sigma_{2}$ and $\mu_{1}\sim\mu_{2}$, we verify if

$\displaystyle\sigma_{1}\odot\mu_{1}\sim\sigma_{2}\odot\mu_{2}.$ (123)

Using the isomorphism between permutations and their structure matrices, (123)
is equivalent to

$\displaystyle M_{\sigma_{1}}M_{\mu_{1}}\sim M_{\sigma_{1}}M_{\mu_{1}}.$ (124)

Assume $\sigma_{1}\in{\bf S}_{m}$, $\sigma_{2}\in{\bf S}_{n}$, $\mu_{1}\in{\bf
S}_{p}$, $\mu_{2}\in{\bf S}_{q}$, $m\wedge n=\alpha$, $p\wedge q=\beta$,
$m\vee p=\xi$, $n\vee q=\eta$, and $\alpha\vee\beta=\lambda$. According to
Lemma III.16, there exist $M_{\alpha}\in{\mathcal{P}}_{\alpha}$ and
$M_{\beta}\in{\mathcal{P}}_{\beta}$ such that

$\begin{array}[]{cc}M_{\sigma_{1}}=M_{\alpha}\otimes
I_{m/\alpha},&M_{\sigma_{2}}=M_{\alpha}\otimes I_{n/\alpha},\\\
M_{\mu_{1}}=M_{\beta}\otimes I_{p/\beta},&M_{\mu_{2}}=M_{\beta}\otimes
I_{q/\beta}.\\\ \end{array}$

Then

$\begin{array}[]{l}M_{\sigma_{1}}M_{\mu_{1}}=\left(M_{\alpha}\otimes
I_{m/\alpha}\otimes I_{\xi/m}\right)\left(M_{\beta}\otimes I_{p/\beta}\otimes
I_{\xi/p}\right)\\\ ~{}=\left(M_{\alpha}\otimes
I_{xi/\alpha}\right)\left(M_{\beta}\otimes I_{\xi/\beta}\right)\\\
~{}=\left[\left(M_{\alpha}\otimes
I_{\lambda/\alpha}\right)\left(M_{\beta}\otimes
I_{\lambda/\beta}\right)\right]\otimes I_{\xi/\lambda}.\\\ \end{array}$

Similarly, we have

$\begin{array}[]{l}M_{\sigma_{1}}M_{\mu_{1}}\\\
~{}=\left[\left(M_{\alpha}\otimes
I_{\lambda/\alpha}\right)\left(M_{\beta}\otimes
I_{\lambda/\beta}\right)\right]\otimes I_{\eta/\lambda}.\\\ \end{array}$

(124) follows.

  * (ii)

Verify the (ii) of Definition III.13. Assume $x\sim I_{r}$. Then there exist
$I_{\alpha}$ and $I_{\beta}$ such that

$e\otimes I_{\alpha}=I_{r}\otimes I_{\beta}.$

Then it is clear that $e=I_{s}\in{\bf e}$

$\Box$

Hence we have the following result.

###### Proposition IV.10

There exists the equivalence hyper group

$\displaystyle\overline{\bf S}^{L}:={\bf S}^{L}/\sim.$ (125)

###### Remark IV.11

We are surely interested in what are the elements in $\overline{\bf S}^{L}$.
Let $\sigma\in{\bf S}$. $\sigma$ is said to be reducible, if $\sigma\in{\bf
S}_{n}$ and there are $m<n$, $m|n$ and a $\sigma_{0}\in{\bf S}_{m}$ such that

$\sigma=\varphi^{m}_{n}(\sigma_{0}).$

Let $\sigma\in{\bf S}_{m}$ be irreducible and define

$\overline{\sigma}=\\{\varphi^{m}_{km}(\sigma)\;|\;k\in{\mathbb{Z}}_{+}\\}.$

Then it is easy to verify that

$\displaystyle\overline{\bf S}^{L}:=\\{\overline{\sigma}\;|\;\sigma\in{\bf
S}~{}\mbox{is irreducible}\\}.$ (126)

###  IV-B Right Permutation Hyper Group

Alternatively, we can also define a right permutation hyper group. We need to
construct another cross-dimensional mapping.

###### Definition IV.12

Assume $m|n$, and $km=n$, then we define a mapping, called the right
embedding, as $\psi^{m}_{n}:{\bf S}_{m}\rightarrow{\bf S}_{n}$ as follows:
Assume $\sigma\in{\bf S}_{m}$, then

$\displaystyle\begin{array}[]{l}\psi^{m}_{n}(\sigma)((s-1)m+i):=(k-s)m+\sigma(i),\\\
\quad\sigma\in{\bf S}_{m},\;s\in[1,k],\;i\in[1,m].\\\ \end{array}$ (129)

Similarly to left embedding case, we have the following result.

###### Proposition IV.13

Assume $m|n$, $km=n$, and $\sigma\in{\bf S}_{m}$. Then

$\displaystyle P_{\psi(\sigma)}=I_{k}\otimes P_{\sigma}.$ (130)

###### Remark IV.14

Assume $m|n$, similarly to $\varphi^{m}_{n}$, the $\psi^{m}_{n}$ is an one-to-
one homomorphism from ${\bf S}_{m}$ to ${\bf S}_{n}$. In other words,
$\psi^{m}_{n}$ is an isomorphism from ${\bf S}_{m}$ to its image $\varphi({\bf
S}_{m})<{\bf S}_{n}$. It can also be considered as an “embedding”.

Next, we define another cross-order product of permutations as follows.

###### Definition IV.15

Let $\sigma,~{}\mu\in{\bf S}$, say, $\sigma\in{\bf S}_{m}$, $\mu\in{\bf
S}_{n}$, and $t=\operatorname{lcm}(m,n)$. Then the right product of $\sigma$
and $\mu$, denoted by $\sigma\odot_{r}\mu$, is defined by

$\displaystyle\sigma\odot_{r}\mu=\psi^{m}_{t}(\sigma)\circ\psi^{n}_{t}(\mu)\in{\bf
S}_{t}.$ (131)

The following result is obvious.

###### Proposition IV.16

Let $\sigma,~{}\mu\in{\bf S}$ with their permutation matrices $M_{\sigma}$ and
$M_{\mu}$ respectively. Then

$\displaystyle M_{\sigma\odot_{r}\mu}=M_{\sigma}\rtimes M_{\mu}.$ (132)

$\psi^{m}_{n}$ is called the right embedding because it is determined by the
right STP of their permutation matrices.

Using Propositions IV.16 and IV.6, the following result is obvious.

###### Theorem IV.17

${\bf S}^{R}=({\bf S},\odot_{r},{\bf e})$ is a hyper group, where the identity
set is

$\displaystyle{\bf e}=\\{e_{n}=Id_{n}\in{\bf
S}_{n}\;|\;n\in{\mathbb{Z}}_{+}\\},$ (133)

w.r.t. MD-1 lattice $({\mathbb{Z}}_{+},\prec)$.

Correspondingly, the right hyper group ${\bf S}^{R}$ has the following
properties.

###### Proposition IV.18

  * (i)

${\bf A}^{R}$ is a sub-hyper group of ${\bf S}^{R}$. Precisely speaking,

$\displaystyle\left({\bf A},\odot_{r},{\bf e}\right)<_{h}\left({\bf
S},\odot_{r},{\bf e}\right).$ (134)

  * (ii)

${\bf A}^{R}$ is a normal sub-hyper group of ${\bf S}^{R}$. That is,

$\displaystyle\left({\bf A},\odot_{r},{\bf e}\right)\lhd_{h}\left({\bf
S},\odot_{r},{\bf e}\right).$ (135)

$\sigma$ and $\mu$ are said to be right-equivalent, denoted by
$\sigma\sim_{r}\mu$, if there exsit $I_{r}$ and $I_{s}$ such that

$\displaystyle I_{r}\otimes M_{\sigma}=I_{s}\otimes M_{\mu}.$ (136)

Then we can also prove that the equivalence is consistent with respect to the
$\odot_{r}$, hence we have

###### Proposition IV.19

There exists the equivalence hyper group

$\displaystyle\overline{\bf S}^{R}:={\bf S}^{R}/\sim_{r}.$ (137)

Denote by

$\tilde{\sigma}=\\{\mu\;|\;\mu\sim_{r}\sigma\\},$

then we have

###### Remark IV.20

$\displaystyle\overline{\bf S}^{R}:=\\{\tilde{\sigma}\;|\;\sigma\in{\bf
S}~{}\mbox{is irreducible}\\}.$ (138)

##  V Homomorphism

###  V-A Hyper Group Homomorphism/Isomorphism

###### Definition V.1

Let ${\mathcal{G}}_{i}=(G^{i},\times^{i},{\bf e}^{i})$, $i=1,2$ be two hyper
groups. $\pi:G^{1}\rightarrow G^{2}$ is called a hyper group homomorphism, and
${\mathcal{G}}_{1}$ and ${\mathcal{G}}_{2}$ are said to be homomorphic,
denoted by ${\mathcal{G}}_{1}\simeq_{h}{\mathcal{G}}_{2}$, if the followings
are satisfied:

  * (i)

$\displaystyle\pi(x\times^{1}y)=\pi(x)\times^{2}\pi(y),\quad x,y\in G_{1}.$
(139)

  * (ii)

$\displaystyle\pi(e^{1})\in{\bf e}^{2},\quad e^{1}\in{\bf e}^{1},$ (140)

and $\pi:{\bf e}^{1}\rightarrow{\bf e}^{2}$ is a lattice order-preserving
mapping.

  * (iii)

If a hyper group homomorphism $\pi:G_{1}\rightarrow G_{2}$ is bijective, and
$\pi^{-1}:G_{2}\rightarrow G_{1}$ is also a homomorphism, then $\pi$ is called
a hyper group isomorphism, and ${\mathcal{G}}_{1}$ and ${\mathcal{G}}_{2}$ are
said to be isomorphic, denoted by
${\mathcal{G}}_{1}\cong_{h}{\mathcal{G}}_{2}$.

###### Example V.2

Recall ${\mathcal{T}}_{H}$ defined in Example III.22.

  * (i)

Let $Id:{\mathcal{T}}_{H}\hookrightarrow{\mathcal{T}}$ be the including
mapping. Then $Id$ is a hyper group homomorphism, and
${\mathcal{T}}_{H}\simeq_{h}{\mathcal{T}}$.

  * (ii)

Let $\pi_{k}:{\mathcal{T}}_{H}\rightarrow{\mathcal{T}}$ be defined by
$A\mapsto A\otimes I_{k}$. Then
$(\pi_{k}({\mathcal{T}}_{H}),\ltimes)\simeq_{h}{\mathcal{T}}$.

  * (iii)

Since $Id$ is one-to-one. Then it is easy to verify that

${\mathcal{T}}_{H}\cong_{h}(\pi_{k}({\mathcal{T}}_{H}),\ltimes).$

###### Example V.3

Consider $({\mathcal{M}}_{\mu},\bar{+})$ (or $({\mathcal{M}}_{\mu},\hat{+})$).
Let $\pi:A\mapsto A^{T}$. Then $\pi$ is a hyper group isomorphism. That is,

$\displaystyle\begin{array}[]{l}({\mathcal{M}}_{\mu},\bar{+})\cong_{h}({\mathcal{M}}_{1/\mu},\bar{+});\\\
(\mbox{or}~{}({\mathcal{M}}_{\mu},\hat{+})\cong_{h}({\mathcal{M}}_{1/\mu},\hat{+})).\\\
\end{array}$ (143)

###### Example V.4

Consider $({\bf S}^{L},\odot)$ and $({\mathcal{P}},\ltimes)$ (or $({\bf
S}^{R},\odot_{r})$ and $({\mathcal{P}},\rtimes)$). Then $\pi:\sigma\mapsto
M_{\sigma}$ is a hyper group isomorphism. That is,

$\displaystyle\begin{array}[]{l}({\bf
S}^{L},\odot)\cong_{h}({\mathcal{P}},\ltimes);\\\ (\mbox{or}~{}({\bf
S}^{R},\odot_{r})\cong_{h}({\mathcal{P}},\rtimes)).\\\ \end{array}$ (146)

###  V-B Quotient Hyper Group Homomorphism/Isomorphism

Assume $\varphi:H\rightarrow G$ is a hyper group homomorphism. Set

$\displaystyle\begin{array}[]{l}Im(\varphi):=\\{g\in G\;|\;\exists h\in
H~{}\mbox{such that}~{}g=\varphi(h)\\},\\\ Ker(\varphi):=\\{h\in
H\;|\;\varphi(h)\in{\bf e}_{G}\\}.\end{array}$ (149)

Then we have the following result.

###### Proposition V.5

  * (i)

$Im(\varphi)<_{h}G$.

  * (ii)

$Ker(\varphi)\lhd_{h}H$.

Proof.

  * (i)

Since $\varphi:{\bf e}_{H}\rightarrow{\bf e}_{G}$ is a lattice homomorphism,
it is easy to see that $\varphi({\bf e}_{H})$ is a sub-lattice of ${\bf
e}_{G}$. Let $x,y\in Im(\varphi)$. Then there exist $a,b\in H$ such that
$\varphi(a)=x$, $\varphi(b)=y$. Since $H$ is a hyper group, $y^{-1}\in H$.
Then $\varphi(ab^{-1})=xy^{-1}\in Im(\varphi)$. We conclude that
$Im(\varphi)<_{h}G$.

  * (ii)

Since ${\bf e}_{H}\subset K:=\ker(\varphi)$, ${\bf e}_{K}={\bf e}_{H}$. It is
obvious that $K<_{h}H$. To see $K$ is a normal hyper-sub-group, we need to
show that for any $a\in H$ $aKa^{-1}=K$. This is true because

$\varphi(aka^{-1})=\varphi(a)e(\varphi(a))^{-1}=e\varphi(a)(\varphi(a))^{-1}\in{\bf
e}_{G}.$

$\Box$

###### Proposition V.6

Let $\varphi:H\rightarrow G$ be a hyper group homomorphism, and for each
$e\in{\bf e}_{H}$, $\varphi_{e}:H_{e}\rightarrow G$ be the restriction of
$\varphi$ on $H_{e}$. Then $\varphi_{e}:H_{e}\rightarrow G_{\varphi{e}}$ is a
group homomorphism. Moreover, for each $e\in{\bf e}_{H}$

  * (i)

$Im(\varphi_{e})<G_{e}$;

  * (ii)

$Ker(\varphi_{e})\lhd H$.

Proof. We claim that $\varphi_{e}(H_{e})\subset G_{\varphi(e)}$. If this claim
is true, all the remaining statements are obvious. Let $h\in H_{e}$, then

$\varphi(hh^{-1})=\varphi(h)\varphi(h^{-1})=\varphi(e)\in{\bf e}_{G}.$

Hence $\varphi(h)\in G_{\varphi(e)}$, which proves the claim. $\Box$

The three Isomorphism Theorems for groups can be modified to suit the hyper
groups.

###### Theorem V.7

(First Isomorphism Theorem)

  * (i)

Assume $\varphi:H\rightarrow G$ is a hyper group homomorphism. For each
$e\in{\bf e}_{H}$, set $K_{e}=Ker(\varphi_{e})$ and $Im_{e}=Im(\varphi_{e})$.
Define a mapping $\theta_{e}:H_{e}/K_{e}\rightarrow Im_{e}$ as
$h_{e}K_{e}\mapsto\varphi_{e}(h_{e})$, $h_{e}\in H_{e}$. Then $\theta_{e}$ is
an isomorphism. That is.

$\displaystyle H_{e}/K_{e}\cong Im_{e},\quad e\in{\bf e}_{H}.$ (150)

  * (ii)

Let $H\lhd_{h}G$ be a normal hyper-sub-group. For each $e\in{\bf e}_{H}$
define $\pi_{e}:G_{\varphi(e)}\rightarrow G_{\varphi(e)}/H_{e}$ by
$g_{e}\mapsto g_{e}H_{e}$. Then $\pi_{e}$ is a surjective homomorphism with
$Ker(\pi_{e})=H_{e}$.

###### Theorem V.8

(Second Isomorphism Theorem) Assume $H,~{}N,~{}G$ are hyper groups.

  * (i)

If $H<_{h}G$ and $N\lhd_{h}G$, then for each $e\in{\bf e}_{H}\bigcap{\bf
e}_{N}$

$\displaystyle N_{e}\bigcap H_{e}\lhd_{h}H_{e}.$ (151)

  * (ii)

Set $NH=\\{nh\;|\;n\in N,~{}h\in H\\}$, where $N,H$ are as in (i). Then for
each $e\in{\bf e}_{H}\bigcap{\bf e}_{N}$

$\displaystyle N_{e}H_{e}<_{h}G_{e}.$ (152)

  * (iii)

Define $\pi_{e}:H_{e}/(N_{e}\bigcap H_{e})\rightarrow N_{e}H_{e}/N_{e}$ by

$h_{e}(N_{e}\bigcap H_{e})\mapsto h_{e}N_{e},\quad h_{e}\in H_{e},$

where $e=e_{h}e_{h}$. Then $\pi_{e}$ is an isomorphism. That is

$\displaystyle H_{e}/(H_{e}\bigcap N_{e})\cong N_{e}H_{e}/N_{e}.$ (153)

###### Theorem V.9

(Third Isomorphism Theorem) Let $M,N,G$ be hyper groups, $M\lhd_{h}G$,
$N\lhd_{h}G$, and $N<_{h}M$. Then for eacg $e\in{\bf e}_{N}$

  * (i)

$\displaystyle M_{e}/N_{e}\lhd_{h}G_{e}/N_{e}.$ (154)

  * (ii)

$\displaystyle(G_{e}/N_{e})/(M_{e}/N_{e})\cong G_{e}/M_{e}.$ (155)

###### Example V.10

Consider $\Lambda=\\{(2,2),(2,4),(4,2),(4,4)\\}$ as a sub-lattice of
${\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}$,
$H=\bigcup_{\lambda\in\Lambda}{\mathcal{M}}_{\lambda}$, and

${\mathcal{H}}:=(H,\hat{+})<_{h}{\mathcal{G}}=({\mathcal{M}},\hat{+}).$

Define a hyper group homomorphism:
$\varphi:{\mathcal{H}}\rightarrow{\mathcal{G}}$ as

$\varphi(A)=A\hat{-}A^{T}.$

A straightforward computation shows that

$K:=\ker(\varphi)=K_{4\times 4}\bigcup K_{4\times 2}\bigcup K_{2\times
4}\bigcup K_{2\times 2},$

where

$\begin{array}[]{l}K_{4\times 4}=\\{A\in{\mathcal{M}}_{4\times
4}\;|\;A^{T}=A\\};\\\ K_{4\times 2}=\\{A\otimes{\bf
1}_{2}\;|\;A\in{\mathcal{M}}_{2\times 2},~{}\mbox{and}~{}A^{T}=A\\};\\\
K_{2\times 4}=\\{A\otimes{\bf 1}^{T}_{2}\;|\;A\in{\mathcal{M}}_{2\times
2},~{}\mbox{and}~{}A^{T}=A\\};\\\ K_{2\times 2}=\\{A\in{\mathcal{M}}_{2\times
2}\;|\;A^{T}=A\\}.\\\ \end{array}$ $Im:=Im(\varphi)=Im_{4\times 4}\bigcup
Im(2\times 2),$

where

$\begin{array}[]{l}Im_{4\times 4}=\\{A\in{\mathcal{M}}_{4\times
4}\;|\;A^{T}=-A\\};\\\ Im_{2\times 2}=\\{A\in{\mathcal{M}}_{2\times
2}\;|\;A^{T}=-A\\}.\\\ \end{array}$

Since

$K_{\alpha\times\beta}\hat{+}K_{\gamma\times\delta}=\begin{cases}K_{4\times
4},\quad\max(\alpha,\beta,\gamma,\delta)=4,\\\ K_{2\times
2},\quad\max(\alpha,\beta,\gamma,\delta)=2,\\\ \end{cases}$

$K\lhd_{h}{\mathcal{H}}$ is a hyper-sub-group. It is normal because
${\mathcal{H}}$ is abelian. Since

$Im_{\alpha\times\alpha}\hat{+}Im_{\beta\times\beta}=\begin{cases}Im_{4\times
4},\quad\max(\alpha,\beta)=4,\\\ Im_{2\times 2},\quad\max(\alpha,\beta)=2,\\\
\end{cases}$

$Im\lhd_{h}{\mathcal{G}}$ is a hyper-sub-group. Note that the lattice order-
preserving mapping is:

$\pi(e_{\lambda})=\begin{cases}(4,4),\quad\lambda\in\\{(2,4),(4,2),(4,4)\\},\\\
(2,2),\quad\lambda=(2,2).\end{cases}$

Next, we verify First Isomorphism Theorem. Consider each case when
$\lambda\in{\bf e}_{H}$.

  * (i)

$\lambda=(4,4)$.

$K_{\lambda}=\\{A\in{\mathcal{M}}_{4\times 4}\;|\;A^{T}=A\\}.$
$Im_{\lambda}=\\{A\in{\mathcal{M}}_{4\times 4}\;|\;A^{T}=-A\\}.$

Let $A\in H_{\lambda}={\mathcal{M}}_{4\times 4}$. Since

$A=\frac{1}{2}(A+A^{T})+\frac{1}{2}(A-A^{T}),$
$AK_{\lambda}=\frac{1}{2}(A+A^{T})+\frac{1}{2}(A-A^{T})+K_{\lambda}=\frac{1}{2}(A-A^{T})+K_{\lambda}.$

Define $\pi:H_{\lambda}/K_{\lambda}\rightarrow Im_{\lambda}$ as

$\pi(AK_{\lambda}):=\frac{1}{2}(A-A^{T}).$

Then $\pi$ is an isomorphism. That is,

$H_{\lambda}/K_{\lambda}\cong Im_{\lambda}.$

  * (ii)

$\lambda=(4,2)$. A straightforward calculation shows that

$K_{\lambda}=\left\\{\left.\begin{bmatrix}a&a&b&b\\\
b&b&c&c\end{bmatrix}\;\right|\;a,b,c\in{\mathbb{R}}\right\\}.$

Set

$\begin{array}[]{l}A=\begin{bmatrix}a_{1,1}&a_{1,2}&a_{1,3}&a_{1,4}\\\
a_{2,1}&a_{2,2}&a_{2,3}&a_{2,4}\\\ \end{bmatrix}\\\
~{}=\begin{bmatrix}a_{1,1}&a_{1,1}&a_{1,3}&a_{1,3}\\\
a_{1,3}&a_{1,3}&a_{2,3}&a_{2,3}\\\ \end{bmatrix}\\\
~{}+\begin{bmatrix}0&a_{1,2}-a_{1,1}&0&a_{1,4}-a_{1,3}\\\
a_{2,1}-a_{1,3}&a_{2,2}-a_{1,3}&0&a_{2,4}-a_{1,3}\\\ \end{bmatrix}\\\
~{}:=\begin{bmatrix}a_{1,1}&a_{1,1}&a_{1,3}&a_{1,3}\\\
a_{1,3}&a_{1,3}&a_{2,3}&a_{2,3}\\\ \end{bmatrix}+\begin{bmatrix}0&d&0&e\\\
f&g&0&h\\\ \end{bmatrix}\\\ \end{array}$

Hence

$AK_{\lambda}=\begin{bmatrix}0&d&0&e\\\ f&g&0&h\\\ \end{bmatrix}+K_{\lambda}.$

The $Im_{\lambda}$ is calculated as

$\begin{array}[]{ccl}Im_{\lambda}&=&\left\\{\left.\begin{bmatrix}0&d&-f&e-f\\\
-d&0&-g&e-g\\\ f&g&0&h\\\ f-e&g-e&-h&0\end{bmatrix}\;\right|\right.\\\
{}\hfil&{}\hfil&\left.\;d,e,f,g,h\in{\mathbb{R}}\right\\}\lhd{\mathcal{K}}_{4}.\end{array}$

That is, $Im_{\lambda}$ is a (normal) sub-group of ${\mathcal{K}}_{4}$.

Define $\pi:H_{\lambda}/K_{\lambda}\rightarrow Im_{\lambda}$ as

$\begin{bmatrix}0&d&0&e\\\ f&g&0&h\\\
\end{bmatrix}+K_{\lambda}\mapsto\begin{bmatrix}0&d&-f&e-f\\\ -d&0&-g&e-g\\\
f&g&0&h\\\ f-e&g-e&-h&0\end{bmatrix}$

Then it is easy to verify that $\pi$ is an isomorphism.That is,

$H_{\lambda}/K_{\lambda}\cong Im_{\lambda}.$

  * (iii)

$\lambda=(2,4)$. Similarly to (ii), we have

$K_{\lambda}=\left\\{\left.\begin{bmatrix}a&b\\\ a&b\\\ b&c\\\
b&c\end{bmatrix}\;\right|\;a,b,c\in{\mathbb{R}}\right\\}.$
$AK_{\lambda}=\begin{bmatrix}0&f\\\ d&g\\\ 0&0\\\ e&h\\\
\end{bmatrix}+K_{\lambda}.$
$\begin{array}[]{ccl}Im_{\lambda}&=&\left\\{\left.\begin{bmatrix}0&-d&f&f-e\\\
d&0&g&g-e\\\ -f&-g&0&-h\\\ e-f&e-g&h&0\end{bmatrix}\;\right|\right.\\\
{}\hfil&{}\hfil&\left.\;d,e,f,g,h\in{\mathbb{R}}\right\\}<{\mathcal{K}}_{4}.\end{array}$

That is, $Im_{\lambda}$ is a sub-group of ${\mathcal{K}}_{4}$.

Define $\pi:H_{\lambda}/K_{\lambda}\rightarrow Im_{\lambda}$ as

$\begin{bmatrix}0&f\\\ d&g\\\ 0&0\\\ e&h\\\
\end{bmatrix}+K_{\lambda}\mapsto\begin{bmatrix}0&-d&f&f-e\\\ d&0&g&g-e\\\
-f&-g&0&-h\\\ e-f&e-g&h&0\end{bmatrix}$

Then $\pi$ is an isomorphism. That is,

$H_{\lambda}/K_{\lambda}\cong Im_{\lambda}.$

  * (iv)

$\lambda=(2,2)$.

$K_{\lambda}=\\{A\in{\mathcal{M}}_{2\times 2}\;|\;A^{T}=A\\}.$
$Im_{\lambda}=\\{A\in{\mathcal{M}}_{2\times 2}\;|\;A^{T}=-A\\}.$

Let $A\in H_{\lambda}={\mathcal{M}}_{2\times 2}$. Since

$A=\frac{1}{2}(A+A^{T})+\frac{1}{2}(A-A^{T}),$
$AK_{\lambda}=\frac{1}{2}(A+A^{T})+\frac{1}{2}(A-A^{T})+K_{\lambda}=\frac{1}{2}(A-A^{T})+K_{\lambda}.$

Define $\pi:H_{\lambda}/K_{\lambda}\rightarrow Im_{\lambda}$ as

$\pi(AK_{\lambda}):=\frac{1}{2}(A-A^{T}).$

Then $\pi$ is an isomorphism. That is,

$H_{\lambda}/K_{\lambda}\cong Im_{\lambda}.$

###  V-C Symmetric and Skew symmetric Matrices

###### Definition V.11

Consider the hyper group ${\mathcal{G}}_{M}:=({\mathcal{M}},\hat{+})$.

  * (i)

A mapping $\pi_{s}:{\mathcal{G}}_{M}\rightarrow{\mathcal{G}}_{M}$, called the
symmetrizing mapping, is defined by

$\displaystyle\pi_{s}:A\mapsto\frac{1}{2}\left(A\hat{+}A^{T}\right),\quad
A\in{\mathcal{M}}.$ (156)

  * (ii)

A mapping $\pi_{a}:{\mathcal{G}}_{M}\rightarrow{\mathcal{G}}_{M}$, called the
alternating mapping, is defined by

$\displaystyle\pi_{a}:A\mapsto\frac{1}{2}\left(A\hat{-}A^{T}\right),\quad
A\in{\mathcal{M}}.$ (157)

The following result makes the symmetry and skew-symmetry of matrices
meaningful for non-square matrices.

###### Theorem V.12

Consider the hyper group ${\mathcal{G}}_{M}:=({\mathcal{M}},\hat{+})$.

  * (i)

Both $\pi_{s}$ and $\pi_{a}$ are hyper group homomorphisms (precisely
speaking, endomorphisms). According to Proposition V.5, the kernels of
$\pi_{s}$ and $\pi_{a}$ are hyper groups.

  * (ii)

The kernel of $\pi_{s}$, called the set of skew-symmetric matrices, denoted by
${\mathcal{K}}$, is $\pi_{a}$ invariant in the sense

$\displaystyle\pi_{a}(A)=A\otimes{\mathcal{E}}_{\alpha/m\times\alpha/n},\quad
A\in{\mathcal{K}}.$ (158)

  * (iii)

${\mathcal{K}}_{n\times n}$=${\mathcal{K}}_{n}$. That is, when $m=n$ the
component group ${\mathcal{K}}_{m\times n}$ coincides the classical set of
skew-symmetric matrices.

  * (iv)

The kernel of $\pi_{a}$, called the set of symmetric matrices, denoted by
${\mathcal{S}}$, is $\pi_{s}$ invariant in the sense

$\displaystyle\pi_{s}(A)=A\otimes{\mathcal{E}}_{\alpha/m\times\alpha/n},\quad
A\in{\mathcal{S}}.$ (159)

  * (v)

${\mathcal{S}}_{n\times n}$=${\mathcal{S}}_{n}$. That is, when $m=n$ the
component group ${\mathcal{S}}_{m\times n}$ coincides the classical set of
symmetric matrices.

Proof.

  * (i)

We show that $\pi_{s}$ is a hyper group homomorphism. Let
$A\in{\mathcal{M}}_{m\times n}$, and $t=\operatorname{lcm}(m,n)$. Note that

$A\hat{+}A^{T}=A\otimes{\mathcal{E}}_{t/m\times
t/n}+A^{T}\otimes{\mathcal{E}}_{t/n\times t/m}\in{\mathcal{M}}_{t\times t}.$

It is clear that $\pi_{s}:0_{m\times n}\rightarrow 0_{m\vee n}$, which is
order-keeping.

Next, we assume $A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times
q}$, $s=m\vee p$, $t=n\vee q$, $\xi=s\vee t$, $\alpha=m\vee n$, and
$\beta=p\vee q$. Then

$\begin{array}[]{l}\pi_{s}(A\hat{+}B)=\pi_{s}\left[\left(A\otimes{\mathcal{E}}_{s/m\times
t/n}\right)+\left(B\otimes{\mathcal{E}}_{s/p\times t/q}\right)\right]\\\
~{}=\frac{1}{2}\left[\left(A\otimes{\mathcal{E}}_{s/m\times
t/n}\right)+\left(B\otimes{\mathcal{E}}_{s/p\times t/q}\right)\right]\\\
~{}\hat{+}\frac{1}{2}\left[\left(A^{T}\otimes{\mathcal{E}}_{t/n\times
s/m}\right)+\left(B^{T}\otimes{\mathcal{E}}_{t/q\times s/p}\right)\right]\\\
~{}=\frac{1}{2}\left[\left(A\otimes{\mathcal{E}}_{s/m\times
t/n}\right)+\left(B\otimes{\mathcal{E}}_{s/p\times
t/q}\right)\right]\otimes{\mathcal{E}}_{\xi/s\times\xi/t}\\\
~{}+\frac{1}{2}\left[\left(A^{T}\otimes{\mathcal{E}}_{t/n\times
s/m}\right)+\left(B^{T}\otimes{\mathcal{E}}_{t/q\times
s/p}\right)\right]\otimes{\mathcal{E}}_{\xi/t\times\xi/s}\\\
~{}=\frac{1}{2}\left[\left(A\otimes{\mathcal{E}}_{\xi/m\times\xi/n}\right)+\left(A^{T}\otimes{\mathcal{E}}_{\xi/n\times\xi/m}\right)\right]\\\
~{}+\frac{1}{2}\left[\left(B\otimes{\mathcal{E}}_{\xi/p\times\xi/q}\right)+\left(B^{T}\otimes{\mathcal{E}}_{\xi/q\times\xi/p}\right)\right]\\\
~{}=\frac{1}{2}\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\alpha/n}\right)+\left(A^{T}\otimes{\mathcal{E}}_{\alpha/n\times\alpha/m}\right)\right]\\\
~{}~{}\otimes{\mathcal{E}}_{\xi/\alpha\times\xi/\alpha}+\frac{1}{2}\left[\left(B\otimes{\mathcal{E}}_{\beta/p\times\beta/q}\right)\right.\\\
~{}\left.+\left(B^{T}\otimes{\mathcal{E}}_{\beta/q\times\beta/p}\right)\right]\otimes{\mathcal{E}}_{\xi/\beta\times\xi/\beta}\\\
~{}=\frac{1}{2}\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\alpha/n}\right)+\left(A^{T}\otimes{\mathcal{E}}_{\alpha/n\times\alpha/m}\right)\right]\\\
~{}\hat{+}\frac{1}{2}\left[\left(B\otimes{\mathcal{E}}_{\beta/p\times\beta/q}\right)+\left(B^{T}\otimes{\mathcal{E}}_{\beta/q\times\beta/p}\right)\right]\\\
~{}=\pi_{s}(A)\hat{+}\pi_{s}(B).\end{array}$

We conclude that $\pi_{s}$ is a hyper group homomorphism. Similarly, we can
prove $\pi_{a}$ is also a hyper group homomorphism.

  * (ii)

Assume $A\in{\mathcal{K}}=Ker(\pi_{s})$. Then

$A\otimes{\mathcal{E}}_{\alpha/m\times\alpha/n}=-A^{T}\otimes{\mathcal{E}}_{\alpha/n\times\alpha/m}.$

(158) follows immediately.

All the rest parts can be proved similarly. $\Box$

###### Corollary V.13

Let $A\in{\mathcal{M}}_{m\times n}$ and $t=\operatorname{lcm}(m,n)$. Then
there exist unique symmetric matrix $A_{s}\in{\mathcal{S}}_{t}$ and a skew-
symmetric matrix $A_{a}\in{\mathcal{K}}_{t}$ such that

$\displaystyle A\otimes E_{t/m\times t/n}=A_{s}+A_{a},$ (160)

where

$\displaystyle A_{s}=\frac{1}{2}\left(A\hat{+}A^{T}\right),\quad
A_{a}=\frac{1}{2}\left(A\hat{-}A^{T}\right).$ (161)

###  V-D Squaring Matrix

###### Definition V.14

Let $A\in{\mathcal{M}}_{m\times n}\subset{\mathcal{M}}$ and
$t=\operatorname{lcm}(m.n)$. A mapping
$\Box:{\mathcal{M}}\rightarrow{\mathcal{M}}_{1}$, called a squaring mapping,
is defined by

$\displaystyle\Box(A):=A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}I_{t}\in{\mathcal{M}}_{t\times
t}.$ (162)

###### Remark V.15

  * (i)

It is obvious that if $A$ is a square matrix, then

$\displaystyle\Box(A)=A,\quad A\in{\mathcal{M}}_{1}.$ (163)

  * (ii)

Let $A\in{\mathcal{M}}_{m\times n}$ and $t=\operatorname{lcm}(m.n)$. Then

$\displaystyle\Box(A)=A\otimes{\mathcal{E}}_{t/m\times t/n}.$ (164)

Using squaring mapping, (158)-(160) can be rewritten as (165)-(167)
respectively as

$\displaystyle\pi_{a}(A)=\Box(A),\quad A\in{\mathcal{K}}.$ (165)
$\displaystyle\pi_{s}(A)=\Box(A),\quad A\in{\mathcal{S}}.$ (166)
$\displaystyle\Box(A)=A_{s}+A_{a}.$ (167)

Hence, we have the following result.

###### Proposition V.16

A non-square matrix $A$ is symmetric (skew-symmetric), if and only if, its
squaring matrix $\Box(A)$ is symmetric (skew-symmetric).

###### Proposition V.17

$\Box:({\mathcal{M}},\hat{+})\rightarrow({\mathcal{M}}_{1},\hat{+})$ is a
hyper group homomorphism. Hence,

$\displaystyle\Box(A\hat{+}B)=\Box(A)\hat{+}\Box(B),\quad
A,B\in{\mathcal{M}}.$ (168)

The proof is similar to the proof of Theorem V.12.

###### Definition V.18

Let $A\in{\mathcal{M}}_{m\times n}$ and $t=\operatorname{lcm}(m.n)$.

  * (i)

$\displaystyle p(\lambda):=\det(\lambda I_{t}\hat{-}A)$ (169)

is called the s-characteristic function of $A$.

  * (ii)

$\sigma\in{\mathbb{C}}$ is called a s-eigenvalue of $A$, if $p(\sigma)=0$.

  * (iii)

$X\in{\mathbb{C}}^{t}$ is called an s-eigenvector of $A$ w.r.t. $\sigma$, if

$\displaystyle\Box(A)X=\sigma X.$ (170)

We have the s-Cayley-Hamilton theorem as follows.

###### Theorem V.19

Let $p(\lambda)$ be the s-characteristic function of $A$, then

$\displaystyle p(\Box(A))=0.$ (171)

###### Remark V.20

The squaring mapping poses a non-square matrix all the properties of square
matrix. For instance, we can define

$\displaystyle Tr^{\Box}(A)=Tr(\Box(A)),\quad A\in{\mathcal{M}}.$ (172)

Then it is clear that if $Tr^{\Box}(A)=0$, then
$\det\left(\exp(\Box(A))\right)=1$.

###  V-E Homomorphisms Among Component Groups

###### Proposition V.21

Let $(G,*)$ be a hyper group with identity set ${\bf
e}=\\{e_{\lambda}\;|\;\lambda\in\Lambda\\}$, where $\Lambda$ is the lattice of
identity set. Assume $H<G_{\lambda}$ (or $H\lhd G_{\lambda}$) and
$e_{\mu}\in{\bf e}$, then

$\displaystyle H*e_{\mu}<G_{\lambda\vee\mu},(H*e_{\mu}\lhd
G_{\lambda\vee\mu}).$ (173)

###### Example V.22

  * (i)

Since ${\bf A}_{n}\lhd{\bf S}_{n}$,

${\bf A}\odot Id_{s}\lhd{\bf S}_{n\vee s}.$

  * (ii)

Since ${\mathcal{P}}_{n}<{\mathcal{T}}_{n}$.

$\begin{array}[]{l}{\mathcal{P}}_{n}\ltimes I_{s}<{\mathcal{T}}_{n\vee s};\\\
{\mathcal{P}}_{n}\rtimes I_{t}<{\mathcal{T}}_{n\vee t}.\\\ \end{array}$

###### Proposition V.23

Let $(G,*)$ be a hyper group with identity set ${\bf
e}-\\{e_{\lambda}\;|\;\lambda\in\Lambda\\}$, where $\Lambda$ is the lattice of
identity set. Assume $H<G_{\lambda}$, $N\lhd G_{\mu}$, then

$\displaystyle HN<G_{\lambda\vee\mu}.$ (174)

##  VI Ring and Hyper Ring of Matrices

###  VI-A Matrix Ring

It has been demonstrated in Example II.5 that $({\mathcal{M}}_{n\times
n},+,\times)$ is a unitary ring, and it has some sub-rings as (i) diagonal
sub-ring ${\mathcal{D}}_{n}$; (ii) upper triangular sub-ring
${\mathcal{U}}_{n}$; (iii) lower triangular sub-ring ${\mathcal{B}}_{n}$; etc.

In addition to these rings with square matrices, are there any rings with non-
square matrices?

Recall the DK-STP defined by Definition II.19. It leads to a perfect ring
structure.

###### Theorem VI.1

[22] Consider ${\mathcal{R}}:=({\mathcal{M}}_{m\times
n},+,\,\rotatebox{-90.0}{$\ltimes$}\,)$, where $+$ is the classical matrix
addition/subtraction. Then ${\mathcal{R}}$ is a ring.

###  VI-B Hyper Ring

###### Definition VI.2

A triple ${\mathcal{R}}:=(R,+,\times)$ is called a hyper-ring, if the
following conditions are satisfied.

  * (i)

$(R,+)$ is an Abelien hyper group.

  * (ii)

$(R,\times)$ is a semi-group.

  * (iii)

$\displaystyle\begin{array}[]{l}(a+b)\times c=a\times c+b\times c;\\\
c\times(a+b)=c\times a+c\times b,\quad a,b,c\in R.\end{array}$ (177)

If $(R,\times)$ is a hyper-monoid, ${\mathcal{R}}$ is called a unitary hyper-
ring. If $(R,\times)$ is commutative, ${\mathcal{R}}$ is called a commutative
hyper-ring.

In the following we construct the most general matrix ring, which covers all
matrices.

###### Theorem VI.3

The triple $({\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$ is
a hyper-ring, where the addition $\hat{+}$ is defined by (35) and the product
$\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$ is defined by (51).

Proof. It was shown in Proposition II.28 that
$({\mathcal{M}},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$ is a semi-group, and
in Proposition III.8 (iv) that $({\mathcal{M}},\hat{+})$ is an abelian hyper
group. Hence what remaining to be proved is the distributivity, that is,

$\displaystyle\begin{array}[]{l}(A\hat{+}B)\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C=A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C\hat{+}B\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C;\\\
C\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}(A\hat{+}B)=C\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A\hat{+}C\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B.\\\
\end{array}$ (180)

We prove the first equality only. Using the first equality and Proposition
II.27, the second equality is obvious.

Assume $A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times q}$,
$C\in{\mathcal{M}}_{r\times s}$, and

$\begin{array}[]{c}\begin{array}[]{cc}\xi=m\vee p,&\eta=n\vee q,\\\
\alpha=\xi\vee r=m\vee p\vee r,&\beta=\eta\vee s=n\vee q\vee s,\\\
\end{array}\\\ \theta=\alpha\vee\beta=m\vee p\vee r\vee n\vee q\vee s,\\\
\begin{array}[]{cc}u=m\vee r,&v=n\vee s,\\\ \end{array}\\\ w=u\vee v=m\vee
n\vee r\vee s,\\\ \begin{array}[]{cc}a=p\vee r,&b=q\vee s,\\\ \end{array}\\\
e=a\vee b=p\vee r\vee q\vee s,\\\ u\vee a=m\vee p\vee r=\alpha,\\\ v\vee
b=n\vee q\vee s=\beta,\\\ \begin{array}[]{cc}\mu=r\vee q,&\lambda=r\vee n.\\\
\end{array}\\\ \end{array}$

First we have

$A\hat{+}B=\left(A\otimes{\mathcal{E}}_{\xi/m\times\eta/n}\right)+\left(B\otimes{\mathcal{E}}_{\xi/p\times\eta/q}\right).$

Using formula (39), we can calculate that

$\begin{array}[]{l}(A\hat{+}B)\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C=\left\\{\left[\left(A\otimes{\mathcal{E}}_{\xi/m\times\eta/n}\right)+\left(B\otimes{\mathcal{E}}_{\xi/p\times\eta/q}\right)\right]\right.\\\
~{}\left.\otimes{\mathcal{E}}_{\alpha/\xi\times\beta/\eta}\right\\}\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{\alpha/r\times\beta/s}\right)\\\
=\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\beta/n}\right)+\left(B\otimes{\mathcal{E}}_{\alpha/p\times\beta/q}\right)\right]\\\
~{}\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{\alpha/r\times\beta/s}\right)\\\
=\left\\{\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\beta/n}\right)+\left(B\otimes{\mathcal{E}}_{\alpha/p\times\beta/q}\right)\right]\otimes
E^{T}_{\theta/\beta}\right\\}\\\
\left[\left(C\otimes{\mathcal{E}}_{\alpha/r\times\beta/s}\right)\otimes{\mathcal{E}}_{\theta/\alpha}\right]\\\
=\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\theta/n}\right)+\left(B\otimes{\mathcal{E}}_{\alpha/p\times\theta/q}\right)\right]~{}\left(C\otimes{\mathcal{E}}_{\theta/r\times\beta/s}\right)\\\
=\left(A\otimes{\mathcal{E}}_{\alpha/m\times\theta/n}\right)\left(C\otimes{\mathcal{E}}_{\theta/r\times\beta/s}\right)\\\
~{}+\left(B\otimes{\mathcal{E}}_{\alpha/p\times\theta/q}\right)\left(C\otimes{\mathcal{E}}_{\theta/r\times\beta/s}\right)\\\
=\left(A\otimes{\mathcal{E}}_{\alpha/m\times\lambda/n}\right)\left(C\otimes{\mathcal{E}}_{\lambda/r\times\beta/s}\right)\\\
~{}+\left(B\otimes{\mathcal{E}}_{\alpha/p\times\mu/q}\right)\left(C\otimes{\mathcal{E}}_{\mu/r\times\beta/s}\right).\\\
\end{array}$

On the other hand,

$\begin{array}[]{l}A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C=\left(A\otimes{\mathcal{E}}_{u/m\times
v/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{u/r\times
v/s}\right).\\\ =\left[\left(A\otimes{\mathcal{E}}_{u/m\times
v/n}\right)\otimes{\mathcal{E}}^{T}_{w/v}\right]~{}\left[\left(C\otimes{\mathcal{E}}_{u/r\times
v/s}\right)\otimes{\mathcal{E}}_{w/u}\right]\\\
=\left(A\otimes{\mathcal{E}}_{u/m\times
w/n}\right)\left(C\otimes{\mathcal{E}}_{w/r\times v/s}\right)\\\
=\left(A\otimes{\mathcal{E}}_{u/m\times\lambda/n}\right)\left(C\otimes{\mathcal{E}}_{\lambda/r\times
v/s}\right).\\\ \end{array}$
$\begin{array}[]{l}B\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C=\left(B\otimes{\mathcal{E}}_{a/p\times
b/q}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{a/r\times
b/s}\right).\\\ =\left[\left(B\otimes{\mathcal{E}}_{a/p\times
b/q}\right)\otimes{\mathcal{E}}^{T}_{e/b}\right]~{}\left[\left(C\otimes{\mathcal{E}}_{a/r\times
b/s}\right)\otimes{\mathcal{E}}_{e/a}\right]\\\
=\left(B\otimes{\mathcal{E}}_{a/p\times
e/q}\right)\left(C\otimes{\mathcal{E}}_{e/r\times b/s}\right)\\\
=\left(B\otimes{\mathcal{E}}_{a/p\times\mu/q}\right)\left(C\otimes{\mathcal{E}}_{\mu/r\times
b/s}\right).\\\ \end{array}$

Then

$\begin{array}[]{l}\left(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C\right)\hat{+}\left(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C\right)\\\
=\left[\left(A\otimes{\mathcal{E}}_{u/m\times\lambda/n}\right)\left(C\otimes{\mathcal{E}}_{\lambda/r\times
v/s}\right)\right]~{}\otimes E_{\alpha/u\times\beta/v}+\\\
\left[\left(B\otimes{\mathcal{E}}_{a/p\times\mu/q}\right)\left(C\otimes{\mathcal{E}}_{\mu/r\times
b/s}\right)\right]~{}\otimes{\mathcal{E}}_{\alpha/a\times\beta/b}\\\
=\left(A\otimes{\mathcal{E}}_{\alpha/m\times\lambda/n}\right)\left(C\otimes{\mathcal{E}}_{\lambda/r\times\beta/s}\right)\\\
~{}+\left(B\otimes{\mathcal{E}}_{\alpha/p\times\mu/q}\right)\left(C\otimes{\mathcal{E}}_{\mu/r\times\beta/s}\right).\end{array}$

The first equality of (216) follows.

$\Box$

###### Definition VI.4

Consider a hyper-ring $(R,+,\times)$. Assume the identity set of its addition
hyper group is ${\bf e}=\\{e_{\lambda}\;|\;\lambda\in\Lambda\\}$.

  * (i)

If for each $\lambda\in\Lambda$ the corresponding component group
$(R_{\lambda},+)$ satisfies that

$\displaystyle x\times y\in R_{\lambda},\quad x,y\in R_{\lambda},$ (181)

it is said that the addition $+$ and the product $\times$ of $R$ are
consistent.

  * (ii)

$x,~{}y\in R$ are said to be equivalent, if there exist
$e_{\alpha},e_{\beta}\in{\bf e}$ such that

$\displaystyle x+e_{\alpha}=y+e_{\beta}.$ (182)

The equivalence is said to be consistent with operators, if $x_{1}\sim x_{2}$
and $y_{1}\sim y_{2}$ imply

$\displaystyle x_{1}+y_{1}\sim x_{2}+y_{2};\quad x_{1}\times y_{1}\sim
x_{2}\times y_{2}.$ (183)

###### Proposition VI.5

Consider $({\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$. (i)
$\hat{+}$ and $\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$ are consistent; and
(ii) the equivalence is consistent with operators.

Proof.

  * (i)

Assume $A_{1}\sim A_{2}$ and $B_{1}\sim B_{2}$. Using all the
notations/parameters in the proof of Proposition III.19 (iv), we have

$A_{1}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{1}=(A_{0}\otimes{\mathcal{E}}_{a/s\times
b/t})\,\rotatebox{-90.0}{$\ltimes$}\,(B_{0}\otimes{\mathcal{E}}_{a/\xi\times
b/\eta})$

Assume

$x:=a\vee b=m\vee\alpha\vee n\vee\beta:=b\mu.$

Then

$\begin{array}[]{l}A_{1}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{1}=(A_{0}\otimes{\mathcal{E}}_{a/s\times
b/t}\otimes{\mathcal{E}}^{T}_{x/b})\\\ (B_{0}\otimes{\mathcal{E}}_{a/\xi\times
b/\eta}\otimes{\mathcal{E}}_{x/a})\\\
~{}=(A_{0}\otimes{\mathcal{E}}_{a/s\times
x/t})(B_{0}\otimes{\mathcal{E}}_{x/\xi\times b/\eta})\\\
~{}=(A_{0}\otimes{\mathcal{E}}_{a/s\times
b/t}\otimes{\mathcal{E}}^{T}_{\mu})\\\
\frac{}{}(B_{0}\otimes{\mathcal{E}}_{b/\xi\times
b/\eta}\otimes{\mathcal{E}}_{\mu})\\\
~{}=(A_{0}\otimes{\mathcal{E}}_{u/s\times v/t}\otimes{\mathcal{E}}_{a/u\times
b/v})\\\ ~{}(B_{0}\otimes{\mathcal{E}}_{u/\xi\times
v/\eta}\otimes{\mathcal{E}}_{a/u\times b/v})\\\
~{}=\left[(A_{0}\otimes{\mathcal{E}}_{u/s\times
v/t})(B_{0}\otimes{\mathcal{E}}_{u/\xi\times
v/\eta})\right]\otimes{\mathcal{E}}_{a/u\times b/v}).\\\ \end{array}$

Similarly,

$\begin{array}[]{l}A_{2}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{2}=\left[(A_{0}\otimes{\mathcal{E}}_{u/s\times
v/t})(B_{0}\otimes{\mathcal{E}}_{u/\xi\times v/\eta})\right]\\\
~{}~{}\otimes{\mathcal{E}}_{c/u\times d/v}).\end{array}$

Hence

$A_{1}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{1}\sim
A_{2}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{2}.$

  * (ii)

The fact that the equivalence is consistent w.r.t. $\hat{+}$ was proved in
Proposition III.19, and in (i) it has just been proved that the equivalence is
consistent w.r.t. $\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$.

$\Box$

Then similar to the decomposition result of hyper group, we have the following
result.

###### Proposition VI.6

Consider a hyper-ring $(R,+,\times)$ with the identity set of its addition
hyper group as ${\bf e}=\\{e_{\lambda}\;|\;\lambda\in\Lambda\\}$.

  * (i)

If the addition $+$ and the product $\times$ of $R$ are consistent, then each
$(R_{\lambda},+,\times)$ is a ring, called the component ring of the hyper-
ring $(R,+,\times)$ .

  * (ii)

In addition to (i), if the equivalence is consistent with the operators, then
the operators over quotient space can be defined properly by

$\displaystyle\begin{array}[]{l}\bar{x}+\bar{y}:=\overline{x+y},\quad x,y\in
R,\\\ \bar{x}\times\bar{y}:=\overline{x\times y}.\end{array}$ (186)

Therefore, the quotient space becomes a ring, called the equivalence ring.

Similarly to hyper group, a geometric picture for the structure of a hyper-
ring is shown by Figure 5.

$R_{e}$$\bar{{\mathcal{R}}}$$R^{1}_{e}$$R^{2}_{e}$$R^{n}_{e}$$\vdots$$\vdots$${\mathcal{R}}$$0$
Figure 5: Ring Decomposition of a Hyper-Ring ${\mathcal{R}}$

###### Example VI.7

Consider
${\mathcal{R}}:=\left({\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}\right)$.
To see the consistence of two operators, let $A,B\in M_{e}$, where
$e=0_{m\times n}$. Then

$A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}=A\,\rotatebox{-90.0}{$\ltimes$}\,B\in{\mathcal{M}}_{m\times
n}+{\mathcal{M}}_{e}.$

The consistence of two operators is proved. Secondly, it was proved in
Proposition VI.5 that the equivalence is consistent with operators. Hence, its
decomposition exists. Then we have the following decomposed rings:

  * (i)

Component Rings:

${\bf e}=\left\\{e_{m\times n}=0_{m\times
n}\;|\;(m,n)\in{\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}\right\\}.$

Each component ring is:

$\displaystyle{\mathcal{R}}_{e_{m\times n}}=\left({\mathcal{M}}_{m\times
n},+,\,\rotatebox{-90.0}{$\ltimes$}\,\right),\quad m,n\in{\mathbb{Z}}_{+}.$
(187)

This set of special rings have been described in the beginning of this
section.

  * (ii)

Equivalence Ring.

By definition, two matrices $A$ and $B$ are equivalent, if there are two
identity elements $0_{m\times n}$ and $0_{p\times q}$, such that

$A\hat{+}0_{m\times n}=B\hat{+}0_{p\times q},$

which leads to the following equivalent statement: Let
$A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times q}$,
$s=\operatorname{lcm}(m,p)$ and $t=\operatorname{lcm}(n,q)$. Then $A\sim B$,
if and only if,

$\displaystyle A\otimes{\mathcal{E}}_{s/m\times
t/n}=B\otimes{\mathcal{E}}_{s/p\times t/q}.$ (188)

$A\in{\mathcal{M}}_{m\times n}$ is reducible, if there exist $(a,b)\prec(m,n)$
and $(a,b)\neq(m,n)$ and $A_{0}\in{\mathcal{M}}_{a\times b}$, such that

$A=A_{0}\otimes{\mathcal{E}}_{m/a\times n/b}.$

The equivalence class of $A$ is denoted by

$\bar{A}=\\{B\;|\;B\sim A\\}.$

Denote the equivalence ring by

$\displaystyle\overline{\mathcal{R}}:=\left(\overline{\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}\right),$
(189)

then

$\overline{\mathcal{M}}=\left\\{\bar{A}\;|\;A~{}\mbox{is
irreducible}\right\\}.$

###  VI-C Square Hyper Ring

It is well known that $({\mathcal{M}}_{n\times n},+,\times)$ is a ring, where
$+$ and $\times$ are classical matrix addition and product.

This subsection try to extend this ring structure to a larger set of matrices,
which are of mixed dimensions.

###### Proposition VI.8

Assume ${\mathcal{R}}=({\mathcal{M}}_{1},\bar{+},\ltimes)$, where
${\mathcal{M}}_{1}$ is the set of square matrices; $\bar{+}$ is defined by
(29). Then ${\mathcal{R}}$ is a hyper-ring.

Proof.

It was shown in Proposition III.8 (ii) that $({\mathcal{M}}_{1},\bar{+})$ is a
commutative hyper group. It is obvious that $({\mathcal{M}}_{1},\ltimes)$ is a
semi-group.

To see that ${\mathcal{R}}$ is a hyper-ring, we have only to show the
distributivity.

Let $A\in{\mathcal{M}}_{m\times m}$, $B\in{\mathcal{M}}_{n\times n}$,
$C\in{\mathcal{M}}_{r\times r}$, $\operatorname{lcm}(m,n)=t$,
$\operatorname{lcm}(t,r)=\operatorname{lcm}(m,n,r)=s$,
$\operatorname{lcm}(m,r)=\alpha$, $\operatorname{lcm}(n,r)=\beta$, and
$\operatorname{lcm}(\alpha,\beta)=\operatorname{lcm}(m,n,r)=s$. Then we have

$\begin{array}[]{l}(A\bar{+}B)\ltimes C=(A\otimes I_{t/m}+B\otimes
I_{t/n})\ltimes C\\\ ~{}=[(A\otimes I_{t/m}+B\otimes I_{t/n})\otimes
I_{s/t}](C\otimes I_{s/r})\\\ ~{}=(A\otimes I_{s/m})(C\otimes
I_{s/r})+(B\otimes I_{s/n})(C\otimes I_{s/r}).\\\ \end{array}$
$\begin{array}[]{l}(A\ltimes C)\bar{+}(B\ltimes C)=(A\otimes
I_{\alpha/m})(C\otimes I_{\alpha/r})\\\ ~{}\bar{+}(B\otimes
I_{\beta/n})(C\otimes I_{\beta/r})\\\ ~{}=[(A\otimes I_{\alpha/m})(C\otimes
I_{\alpha/r})]\otimes(I_{s/\alpha})\\\ ~{}+[(B\otimes I_{\beta/n})(C\otimes
I_{\beta/r})]\otimes I_{s/\beta}\\\ ~{}=(A\otimes I_{s/m})(C\otimes
I_{s/r})+(B\otimes I_{s/n})(C\otimes I_{s/r}).\\\ \end{array}$

That is,

$(A\bar{+}B)\ltimes C=(A\ltimes C)\bar{+}(B\ltimes C).$

Similarly, we can show

$C\ltimes(A\bar{+}B)=(C\ltimes A)\bar{+}(C\ltimes B).$

We conclude that ${\mathcal{R}}=({\mathcal{M}}_{1},\bar{+},\ltimes)$ is a
hyper-ring.

Finally, we consider the decomposition of this hyper-ring. The identity set is

${\bf e}={\bf 0}=\\{0_{n\times n}\;|\;n=1,2,\cdots\\}.$

Then

  * (i)

For each $e_{n}=0_{n\times n}\in{\bf 0}$,
${\mathcal{R}}_{e_{n}}=({\mathcal{M}}_{n\times
n},\bar{+},\ltimes)=({\mathcal{M}}_{n\times n},+,\times)$ is a component ring.

  * (ii)

It is easy to see that $A\sim B$, if and only if, there exist $I_{\alpha}$ and
$I_{\beta}$ such that

$A\otimes I_{\alpha}=B\otimes I_{\beta}.$

This is the one defined in [10]. And the consistence of the addition with the
product and the their consistence w.r.t. the equivalence have already been
proved in [10]. We conclude that
$\overline{{\mathcal{R}}}=(\overline{{\mathcal{M}}}_{1},\bar{+},\ltimes)$ is
the equivalence ring.

$\Box$

###### Proposition VI.9

Assume ${\mathcal{R}}=({\mathcal{M}}_{1},\hat{+},\circ)$, where the product
$\circ$ is defined by (19) and the addition is defined as follows: Assume
$A\in{\mathcal{M}}_{m\times m}$, $B\in{\mathcal{M}}_{n\times n}$, and
$t=\operatorname{lcm}(m,n)$, then

$\displaystyle A\hat{+}B:=(A\otimes J_{t/m})+(B\otimes J_{t/m}).$ (190)

Then ${\mathcal{R}}$ is a hyper-ring.

Proof.

A straightforward verification shows that

  * (i)

$({\mathcal{M}}_{1},\hat{+})$ is a hyper group, where the identity set is

${\bf e}=\\{0_{n\times n}\in{\mathcal{M}}_{n\times n}\;|\;n=1,2,\cdots\\}.$

  * (ii)

$({\mathcal{M}}_{1},\circ)$ is a semi-group.

To see that ${\mathcal{R}}$ is a hyper-ring, it is enough to show the
distributivity.

Assume $C\in{\mathcal{M}}_{p\times p}$, and $\alpha=\operatorname{lcm}(m,p)$,
$\beta=\operatorname{lcm}(n,p)$, and $s=\operatorname{lcm}(m,n,p)$. Then

$\begin{array}[]{l}(A\hat{+}B)\circ C=(A\otimes J_{t/m}+B\otimes J_{t/n})\circ
C\\\ ~{}=[(A\otimes J_{t/m}+B\otimes J_{t/n})\otimes J_{s/t}](C\otimes
J_{s/p})\\\ ~{}=[(A\otimes J_{s/m}+B\otimes J_{s/n})(C\otimes J_{s/p})\\\
~{}=(A\otimes J_{s/m})(C+B\otimes J_{t/n})\otimes J_{s/t}](C\otimes
J_{s/p})\\\ ~{}=(A\otimes J_{s/m})(C\otimes J_{s/p})+(B\otimes
J_{s/n})(C\otimes J_{s/p}).\\\ \end{array}$

Meanwhile,

$\begin{array}[]{l}(A\circ C)\hat{+}(B\circ C)=(A\otimes
J_{\alpha/m})(C\otimes J_{\alpha/p}\\\ \hat{+}(B\otimes J_{\beta/n})(C\otimes
J_{\beta/p}\\\ ~{}=(A\otimes J_{\alpha/m})(C\otimes J_{\alpha/p})\otimes
J_{s/\alpha}\\\ ~{}+(B\otimes J_{\beta/n})(C\otimes J_{b/p})\otimes
J_{s/\beta}\\\ ~{}=(A\otimes J_{\alpha/m})(C\otimes J_{\alpha/p})\otimes
J^{2}_{s/\alpha}\\\ ~{}+(B\otimes J_{\beta/n})(C\otimes J_{b/p})\otimes
J^{2}_{s/\beta}\\\ ~{}=(A\otimes J_{s/m})(C\otimes J_{s/p}+(B\otimes
J_{s/n})(C\otimes J_{s/p}).\\\ \end{array}$

We conclude that

$(A\hat{+}B)\circ C=(A\circ C)\hat{+}(B\circ C).$

Similarly, we can also show that

$C\circ(A\hat{+}B)=(C\circ A)\hat{+}(C\circ B).$

We conclude that ${\mathcal{R}}=({\mathcal{M}}_{1},\hat{+},\circ)$ is a hyper-
ring.

Finally, we consider the decomposition of this hyper-ring. The identity set is

${\bf e}={\bf 0}=\\{0_{n\times n}\;|\;n=1,2,\cdots\\}.$

Then

  * (i)

For each $e_{n}=0_{n\times n}\in{\bf 0}$,
${\mathcal{R}}_{e_{n}}=({\mathcal{M}}_{n\times
n},\hat{+},\circ)=({\mathcal{M}}_{n\times n},+,\times)$ is a component ring.

  * (ii)

Since $A\sim B$ means there exist $J_{\alpha}$ and $J_{\beta}$ such that

$A\otimes J_{\alpha}=B\otimes J_{\beta}.$

To prove the consistence of the addition and the product with the equivalence
we need to show that for $A_{1}\sim A_{2}$ and $B_{1}\sim B_{2}$ we have

$\displaystyle A_{1}\hat{+}B_{1}\sim A_{2}\hat{+}B_{2};$ (191)

and

$\displaystyle A_{1}\circ B_{1}\sim A_{2}\circ B_{2}.$ (192)

We prove (192) only. The proof of (191) is similar.

Assume $A_{1}\in{\mathcal{M}}_{m\times m}$, $A_{2}\in{\mathcal{M}}_{n\times
n}$, $B_{1}\in{\mathcal{M}}_{r\times r}$, and $B_{2}\in{\mathcal{M}}_{s\times
s}$ with

$A_{1}\sim A_{2};\quad B_{1}\sim B_{2}.$

Denote $\gcd(m,n)=\alpha$, and $\gcd(r,s)=\beta$. Using Lemma III.16, we know
that there exist $A_{0}\in{\mathcal{M}}_{\alpha\times\alpha}$ and
$B_{0}\in{\mathcal{M}}_{\beta\times\beta}$ such that

$\begin{array}[]{l}A_{1}=A_{0}\otimes J_{m/\alpha},\quad A_{2}=A_{0}\otimes
J_{n/\alpha};\\\ B_{1}=B_{0}\otimes J_{r/\beta},\quad B_{2}=B_{0}\otimes
J_{s/\beta}.\\\ \end{array}$

Set $\operatorname{lcm}(m,r)=p$, $\operatorname{lcm}(n,s)=q$,
$\operatorname{lcm}(p,q)=\mu$, then we have

$\begin{array}[]{l}A_{1}\circ B_{1}=(A_{0}\otimes J_{m/\alpha}\otimes
J_{p/m})(B_{0}\otimes J_{r/\beta}\otimes J_{p/r})\\\ ~{}=(A_{0}\otimes
J_{p/\alpha})(B_{0}\otimes J_{p/\beta}).\\\ \end{array}$

Similarly, we have

$\begin{array}[]{l}A_{2}\circ B_{2}=(A_{0}\otimes J_{n/\alpha}\otimes
J_{q/n})(B_{0}\otimes J_{s/\beta}\otimes J_{q/s})\\\ ~{}=(A_{0}\otimes
J_{q/\alpha})(B_{0}\otimes J_{q/\beta}).\\\ \end{array}$

Note that

$\begin{array}[]{l}(A_{1}\circ B_{1})\otimes J_{\mu/p}=(A_{0}\otimes
J_{p/\alpha})(B_{0}\otimes J_{p/\beta})\otimes J^{2}_{\mu/p}\\\
~{}=(A_{0}\otimes J_{\mu/\alpha})(B_{0}\otimes J_{\mu/\beta});\end{array}$

and

$\begin{array}[]{l}(A_{2}\circ B_{2})\otimes J_{\mu/q}=(A_{0}\otimes
J_{q/\alpha})(B_{0}\otimes J_{q/\beta})\otimes J^{2}_{\mu/q}\\\
~{}=(A_{0}\otimes J_{\mu/\alpha})(B_{0}\otimes J_{\mu/\beta}).\end{array}$

(192) follows.

We conclude that
$\overline{{\mathcal{R}}}=(\overline{{\mathcal{M}}}_{1},\hat{+},\circ)$ is the
equivalence ring.

$\Box$

###  VI-D Sub Hyper Ring

###### Definition VI.10

Let $(R,+,\times)$ be a hyper-Ring and $H\subset R$. $(H,+,\times)$ is said to
be a sub-hyper-ring of $(R,+.\times)$, if (i) $(H,+)<_{h}(R,+)$ is a sub-hyper
group, and $(H,\times)$ is a sub-semi-group of $(R,\times)$.

###### Proposition VI.11

  * (i)

${\mathcal{R}}_{\mu}=({\mathcal{M}}_{\mu},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$
is a hyper-ring, where $\hat{+}$ is defined by (30) and
$\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$ is defined by (52).

  * (ii)

${\mathcal{R}}_{\mu}$ is a sub-hyper-ring of
${\mathcal{R}}=({\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$.

Proof.

  * (i)

It has been proved in Proposition III.8 that $({\mathcal{M}}_{\mu},\hat{+})$
is a hyper group, and in Remark II.29 that
$({\mathcal{M}}_{\mu},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$ is a hyper-
semigroup. The proof of distributivity is exactly the same as the one for
$({\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$.

  * (ii)

A careful observation shows that the $\hat{+}$ defined by (30) and
$\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$ defined by (52) respectively are
exactly the same as the restrictions of the corresponding $\hat{+}$ and
$\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$ for ${\mathcal{M}}$ on
${\mathcal{M}}_{\mu}$. Then the conclusion follows immediately.

$\Box$

###### Example VI.12

Define

$\begin{array}[]{l}U^{1}=\\{A\in{\mathcal{M}}_{1}\;|\;A~{}\mbox{is upper
triangular.}\\};\\\ B^{1}=\\{A\in{\mathcal{M}}_{1}\;|\;A~{}\mbox{is lower
triangular.}\\};\\\ D^{1}=\\{A\in{\mathcal{M}}_{1}\;|\;A~{}\mbox{is
diagonal.}\\};\\\ \end{array}$

Then (i) $(U^{1},\bar{+},\ltimes)$ is a sub-hyper ring of
$({\mathcal{M}}_{1},\bar{+},\ltimes)$. (ii) $(B^{1},\bar{+},\ltimes)$ is a
sub-hyper ring of $({\mathcal{M}}_{1},\bar{+},\ltimes)$. (iii)
$(D^{1},\bar{+},\ltimes)$ is a sub-hyper ring of
$({\mathcal{M}}_{1},\bar{+},\ltimes)$.

###  VI-E Hyper Ring Homomorphism

###### Definition VI.13

Let ${\mathcal{N}}=(N,+,\times)$, and , ${\mathcal{R}}=(R,+,\times)$ be two
hyper rings. A mapping $\varphi:N\rightarrow R$ is called a hyper-ring
homomorphism, if $\varphi:(N,+)\rightarrow(R,+)$ is a hyper group
homomorphism, and

$\displaystyle\varphi(x\times y)=\varphi(x)\times\varphi(y),\quad x,y\in N.$
(193)

###### Proposition VI.14

Consider
${\mathcal{R}}=({\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$
and ${\mathcal{N}}:=({\mathcal{M}}_{1},\hat{+},\circ)$. Let
$\Box:{\mathcal{R}}\rightarrow{\mathcal{N}}$. Then $\Box$ is a hyper-ring
homomorphism.

Proof. It was proved in Proposition V.17 that
$\Box:({\mathcal{M}},\hat{+})\rightarrow({\mathcal{M}}_{1},\hat{+})$ is a
hyper group homomorphism. So we have only to prove (193).

Assume $A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times q}$,
$s=m\vee p$, $t=n\vee q$, $\alpha=m\vee n$, $\beta=p\vee q$, and $\xi=s\vee
t=\alpha\vee\beta$. Then

$\begin{array}[]{l}\Box(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)=\Box\left[\left(A\otimes{\mathcal{E}}_{s/m\times
t/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(B\otimes{\mathcal{E}}_{s/p\times
t/q}\right)\right]\\\
~{}=\Box\left[\left(A\otimes{\mathcal{E}}_{s/m\times\xi/n}\right)\left(B\otimes{\mathcal{E}}_{\xi/p\times
t/q}\right)\right]\\\
~{}=\left(A\otimes{\mathcal{E}}_{\xi/m\times\xi/n}\right)\left(B\otimes{\mathcal{E}}_{\xi/p\times\xi/q}\right).\end{array}$
$\begin{array}[]{l}\Box(A)=A\otimes{\mathcal{E}}_{\alpha/m\times\alpha/n};\\\
\Box(B)=B\otimes{\mathcal{E}}_{\beta/p\times\beta/q}.\\\ \end{array}$
$\begin{array}[]{l}\Box(A)\circ\Box(B)=\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\alpha/n}\right)\otimes
J_{\xi/\alpha}\right]\\\
~{}\left[\left(B\otimes{\mathcal{E}}_{\beta/p\times\beta/q}\right)\otimes
J_{\xi/\beta}\right]\\\
~{}=\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\alpha/n}\right)\otimes{\mathcal{E}}_{\xi/\alpha\times\xi/\alpha}\right]\\\
~{}\left[\left(B\otimes{\mathcal{E}}_{\beta/p\times\beta/q}\right)\otimes{\mathcal{E}}_{\xi/\beta\times\xi/\beta}\right]\\\
~{}=\left(A\otimes{\mathcal{E}}_{\xi/m\times\xi/n}\right)\left(B\otimes{\mathcal{E}}_{\xi/p\times\xi/q}\right).\end{array}$

Hence,

$\Box(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)=\Box(A)\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}\Box(B).$

$\Box$

##  VII Hyper Module and General Linear (Control) System

###  VII-A Hyper Module