11institutetext: Department of Pure Mathematics
University of Calcutta
9/1B, Jatin Bagchi Road
Kolkata(Calcutta)-700029, India
Email: 11email<EMAIL_ADDRESS>
Homepage: http://www.logicamani.in

# On Deductive Systems of AC Semantics for Rough Sets

A. Mani

###### Abstract

Antichain based semantics for general rough sets were introduced recently by
the present author. In her paper two different semantics, one for general
rough sets and another for general approximation spaces over quasi-equivalence
relations, were developed. These semantics are improved and studied further
from a lateral algebraic logic perspective in this research. The main results
concern the structure of the algebras and deductive systems in the context.

ough Objects, Granular operator Spaces, Maximal Antichains, Algebraic Logic,
Granular Rough Semantics, Deductive Systems

###### Keywords:

R

##  1 Introduction

It is well known that sets of rough objects (in various senses) are quasi or
partially orderable. Specifically in classical or Pawlak rough sets [1], the
set of roughly equivalent sets has a Quasi Boolean order on it while the set
of rough and crisp objects is Boolean ordered. In the classical semantic
domain or classical meta level, associated with general rough sets, the set of
crisp and rough objects is quasi or partially orderable. Under minimal
assumptions on the nature of these objects, many orders with rough ontology
can be associated - these necessarily have to do with concepts of
discernibility. Concepts of rough objects, in these contexts, depend
additionally on approximation operators and granulations used. These were part
of the motivations of the development of the concept of granular operator
spaces by the present author in [2].

In quasi or partially ordered sets, sets of mutually incomparable elements are
called _antichains_ (for basics see[3, 4]). The possibility of using
antichains of rough objects for a possible semantics was mentioned in [5, 6,
7] by the present author and developed in [2]. The semantics is applicable for
a large class of operator based rough sets including specific cases of RYS [8]
and other general approaches like [9].

The semantics of [2], as improved in the present paper, is very general, open
ended, extendable and optimal for lateral studies. In the same framework, the
machinery for isolation of deductive systems is developed and studied from a
purely algebraic logic point of view. New results on representation are also
proved.

### Background

Let $S$ be any set and $l,u$ be lower and upper approximation operators on
$\mathcal{S}\subseteq\wp(S)$ that satisfy monotonicity and $(\forall
A\subseteq S)\,A\subseteq A^{u}$. An element $A\in\mathcal{S}$ will be said to
be _lower definite_ (resp. _upper definite_) if and only if $A^{l}=A$ (resp.
$A^{u}=A$) and _definite_ , when it is both lower and upper definite. For
possible concepts of rough objects [2, 8] may be consulted. Finiteness of $S$
and granular operator spaces, defined below, will be assumed (though not
always essential) unless indicated otherwise.

Set framework with operators will be used as all considerations will require
quasi orders in an essential way. The evolution of the operators need not be
induced by a cover or a relation (corresponding to cover or relation based
systems respectively), but these would be special cases. The generalization to
some rough Y-systems RYS (see [8] for definitions), will of course be possible
as a result.

######  Definition 1 ([2])

A _Granular Operator Space_ $S$ will be a structure of the form
$S\,=\,\left\langle\underline{S},\mathcal{G},l,u\right\rangle$ with
$\underline{S}$ being a set, $\mathcal{G}$ an _admissible granulation_(defined
below) over $S$ and $l,u$ being operators
$:\wp(\underline{S})\longmapsto\wp(\underline{S})$ satisfying the following
($\underline{S}$ will be replaced with $S$ if clear from the context. Lower
case alphabets will often be used for subsets ):

$\displaystyle A^{l}\subseteq A\,\&\,A^{ll}=A^{l}\,\&\,A^{u}\subset A^{uu}$
$\displaystyle(A\subseteq B\longrightarrow A^{l}\subseteq
B^{l}\,\&\,A^{u}\subseteq B^{u})$
$\displaystyle\emptyset^{l}=\emptyset\,\&\,\emptyset^{u}=\emptyset\,\&\,\underline{S}^{l}\subseteq
S\,\&\,\underline{S}^{u}\subseteq S.$

Here, Admissible granulations are granulations $\mathcal{G}$ that satisfy the
following three conditions (Relative RYS [8], $\mathbf{P}=\subseteq$,
$\mathbb{P}=\subset$) and $t$ is a term operation formed from set operations):

$\displaystyle(\forall x\exists y_{1},\ldots
y_{r}\in\mathcal{G})\,t(y_{1},\,y_{2},\ldots\,y_{r})=x^{l}$
$\displaystyle\mathrm{and}\>(\forall x)\,(\exists
y_{1},\,\ldots\,y_{r}\in\mathcal{G})\,t(y_{1},\,y_{2},\ldots\,y_{r})=x^{u},$
(Weak RA, WRA) $\displaystyle{(\forall
y\in\mathcal{G})(\forall{x\in\underline{S}})\,(y\subseteq
x\,\longrightarrow\,y\subseteq(x^{l})),}$ (Lower Stability, LS)
$\displaystyle{(\forall x,\,y\in\mathcal{G})(\exists
z\in\underline{S})\,x\subset z,\,y\subset z\,\&\,z^{l}=z^{u}=z,}$ (Full
Underlap, FU)

In the present context, these conditions mean that every approximation is
somehow representable by granules, that granules are lower definite, and that
all pairs of distinct granules are contained in definite objects.

On $\wp(\underline{S})$, the relation $\sqsubset$ is defined by

$A\sqsubset B\text{ if and only if }A^{l}\subseteq B^{l}\,\&\,A^{u}\subseteq
B^{u}.$

The rough equality relation on $\wp(\underline{S})$ is defined via $A\approx
B\text{ if and only if }A\sqsubset B\,\&\,B\sqsubset A$.

Regarding the quotient $\underline{S}|\approx$ as a subset of
$\wp(\underline{S})$, the order $\Subset$ will be defined as per

$\alpha\Subset\beta\text{ if and only if
}\alpha^{l}\subseteq\beta^{l}\,\&\,\alpha^{u}\subseteq\beta^{u}.$

Here $\alpha^{l}$ is being interpreted as the lower approximation of $\alpha$
and so on. $\Subset$ will be referred to as the _basic rough order_.

###### Definition 2

By a _roughly consistent object_ will be meant a set of subsets of
$\underline{S}$ of the form $H=\\{A;(\forall B\in
H)\,A^{l}=B^{l},A^{u}=B^{u}\\}$. The set of all roughly consistent objects is
partially ordered by the inclusion relation. Relative this maximal roughly
consistent objects will be referred to as _rough objects_. By _definite rough
objects_ , will be meant rough objects of the form $H$ that satisfy

$(\forall A\in H)\,A^{ll}=A^{l}\,\&\,A^{uu}=A^{u}.$

###### Proposition 1

$\Subset$ is a bounded partial order on $\underline{S}|\approx$.

###### Theorem 1.1

  * $\bullet$

The poset $AC_{m}(X)$ of all maximum sized antichains of a poset $X$ is a
distributive lattice.

  * $\bullet$

For every finite distributive lattice $L$ and every chain decomposition ${C}$
of $J_{L}$ (the set of join irreducible elements of $L$), there is a poset
$X_{C}$ such that $L\cong AC_{m}(X_{C})$.

Links to proofs can be found in [2].

##  2 Deductive Systems

In this section, key aspects of the approach to ternary deductive systems in
[10, 11] are presented. These are intended as natural generalizations of the
concepts of ideals and filters and classes of congruences that can serve as
subsets or subalgebras closed under consequence operations or relations (also
see [12]).

###### Definition 3

Let $\mathbb{S}=\left\langle S,\Sigma\right\rangle$ be an algebra, then the
set of term functions over it will be denoted by
$\mathbf{T}^{\Sigma}(\mathbb{S})$ and the set of $r$-ary term functions by
$\mathbf{T}^{\Sigma}_{r}(\mathbb{S})$. Further let

$\displaystyle g\in\mathbf{T}^{\Sigma}_{1}(\mathbb{S}),\;z\in
S,\;\tau\subset\mathbf{T}^{\Sigma}_{3}(\mathbb{S}),$ (0) $\displaystyle
g(z)\in\Delta\subset S,$ (1) $\displaystyle(\forall
t\in\tau)(a\in\Delta\,\&\,t(a,b,z)\in\Delta\longrightarrow b\in\Delta),$ (2)
$\displaystyle(\forall t\in\tau)(b\in\Delta\longrightarrow
t(g(z),b,z)\in\Delta),$ (3)

then $\Delta$ is a $(g,z)-$_$\tau$ -deductive system_ of $\mathbb{S}$. If
further for each $k$-ary operation $f\in\Sigma$

$(\forall a_{i},b_{i}\in
S)(\&_{i=1}^{k}p(a_{i},b_{i},z)\in\Delta\longrightarrow
p(f(a_{1},\ldots,a_{k}),f(f(a_{1},\ldots,b_{k}),x)\in\Delta),$

then $\Delta$ is said to be compatible.

$\tau$ is said to be a _g-difference system_ for $\mathbb{S}$ if $\tau$ is
finite and the condition

$(\forall t\in\tau)t(a,b,c)=g(c)\text{ if and only if }a=b\text{ holds.}$

A variety $\mathcal{V}$ of algebras is regular with respect to a unary term
$g$ if and only if for each $S\in\mathcal{V}$,

$(\forall b\in S)(\forall\sigma,\rho\in
con(S))([g(b)]_{\sigma}=[g(b)]_{\rho}\longrightarrow\sigma=\rho).$

It should be noted that in the above $\tau$ is usually taken to be a finite
subset and a variety has a g-difference system if and only if it is regular
with respect to $g$.

###### Proposition 2

In the above definition, it is provable that

$(\forall t\in\tau)(t(g(z),b,z)\in\Delta\longrightarrow b\in\Delta).$

###### Definition 4

In the context of Def.3, $\Theta_{Delta,z}$ shall be a relation induced on $S$
by $\tau$ as per the following

$(a,b)\in\Theta_{\Delta,z}\text{ if and only if }(\forall
t\in\tau)\,t(a,b,z)\in\Delta.$

###### Proposition 3

In the context of Def.4, $\Delta=[g(z)]_{\Theta_{\Delta,z}}$.

###### Proposition 4

Let $\tau\subset T_{3}^{\Sigma}(\mathbb{S})$ with the algebra
$\mathbb{S}=\left\langle S,\Sigma\right\rangle$, $v\in
T_{1}^{\Sigma}(\mathbb{S})$, $e\in S$, $K\subseteq S$ and let $\Theta_{K,e}$
be induced by $\tau$. If $\Theta_{K,e}$ is a reflexive and transitive relation
such that $K=[v(e)]_{Theta_{K,e}}$ , then $K$ is a $(v,e)$\- $\tau$-deductive
system of $\mathbb{S}$.

###### Theorem 2.1

Let $h$ is a unary term of a variety $\mathcal{V}$ and $\tau$ a h-difference
system for $\mathcal{V}$. If $\mathbb{S}\in\mathcal{V}$m $\Theta\in
Con(\mathbb{S})$, $z\in S$ and $\Delta=[h(z)]_{\Theta}$, then
$\Theta_{\Delta,z}=\Theta$ and $\Delta$ is a compatible
$(h,z)$-$\tau$-deductive system of $\mathbb{S}$.

The converse holds in the following sense:

###### Theorem 2.2

If $h$ is a unary term of a variety $\mathcal{V}$, $\tau$ is a $h$-difference
system in it, $\mathbb{S}\in\mathcal{V}$, $z\in S$ and if $\Delta$ is a
compatible $(h,z)$-$\tau$-deductive system of $\mathbb{S}$, then
$\Theta_{\Delta,z}\in Con(\mathbb{S})$ and
$\Delta=[g(z)]_{\Theta_{\Delta,z}}$.

When $\mathcal{V}$ is regular relative $h$, then $\mathcal{V}$ has a
$h$-difference system relative $\tau$ and for each $\mathbb{S}\in\mathcal{V}$,
$z\in S$ and $\Delta\subset S$, $\Delta=[h(z)]$ if and only if $\Delta$ is a
$(h,z)$\- $\tau$-deductive system of $\mathbb{S}$.

In each case below, $\\{t\\}$ is a $h$-difference system ($x\oplus y=((x\wedge
y^{*})^{*}\wedge(x^{*}\wedge y)^{*})^{*}$):

$\displaystyle h(z)=z\;\&\;t(a,b,c)=a-b+c$ (Variety of Groups) $\displaystyle
h(z)=z\;\&\;t(a,b,c)=a\oplus b\oplus c$ (Variety of Boolean Algebras)
$\displaystyle h(z)=z^{**}\;\&\;t(a,b,c)=(a+b)+c$ (Variety of p-Semilattices)

##  3 Anti Chains for Representation

In this section, the main algebraic semantics of [2] is summarized, extended
to _AC_ -algebras and relative properties are studied. It is also proved that
the number of maximal antichains required to generate the AC-algebra is rather
small.

###### Definition 5

$\mathbb{A,B}\in\underline{S}|\approx$, will be said to be _simply
independent_ (in symbols $\Xi(\mathbb{A,B})$)if and only if

$\neg(\mathbb{A}\Subset\mathbb{B})\text{ and
}\neg(\mathbb{B}\Subset\mathbb{A}).$

A subset $\alpha\subseteq\underline{S}|\approx$ will be said to be _simply
independent_ if and only if

$(\forall\mathbb{A,B}\in\alpha)\,\Xi(\mathbb{A,B})\,\vee\,(\mathbb{A=B}).$

The set of all simply independent subsets shall be denoted by
$\mathcal{SY}(S)$.

A _maximal simply independent subset_ , shall be a simply independent subset
that is not properly contained in any other simply independent subset. The set
of maximal simply independent subsets will be denoted by
$\mathcal{SY}_{m}(S)$. On the set $\mathcal{SY}_{m}(S)$, $\ll$ will be the
relation defined by

$\alpha\ll\beta\text{ if and only if
}(\forall\mathbb{A}\in\alpha)(\exists\mathbb{B}\in\beta)\,\mathbb{A\Subset
B}.$

###### Theorem 3.1

$\left\langle\mathcal{SY}_{m}(S),\ll\right\rangle$ is a distributive lattice.

Analogous to the above, it is possible to define essentially the same order on
the set of maximal antichains of $\underline{S}|\approx$ denoted by
$\mathfrak{S}$ with the $\Subset$ order. This order will be denoted by
$\lessdot$ \- this may also be seen to be induced by maximal ideals.

###### Theorem 3.2

If
$\alpha=\\{\mathbb{A}_{1},\mathbb{A}_{2},\ldots,\mathbb{A}_{n},\dots\\}\in\mathfrak{S}$,
and if $L$ is defined by

$L(\alpha)=\\{\mathbb{B}_{1},\mathbb{B}_{2},\ldots,\mathbb{B}_{n},\ldots\\}$

with $X\in\mathbb{B}_{i}$ if and only if
$X^{l}=\mathbb{A}_{i}^{ll}=\mathbb{B}_{i}^{l}$ and
$X^{u}=\mathbb{A}_{i}^{lu}=\mathbb{B}_{i}^{u}$, then $L$ is a partial
operation in general.

###### Definition 6

Let $\chi(\alpha\cap\beta)=\\{\xi;\;\xi\text{ is a maximal antichain
}\&\,\alpha\cap\beta\subseteq\xi\\}$ be the set of all possible extensions of
$\alpha\cap\beta$. The function
$\delta:\mathfrak{S}^{2}\longmapsto\mathfrak{S}$ corresponding to _extension
under cognitive dissonance_ will be defined as per
$\delta(\alpha,\beta)\,\in\,\chi(\alpha\cap\beta)$ and (LST means _maximal
subject to_)

$\delta(\alpha,\beta)\,=\,\left\\{\begin{array}[]{ll}\xi,&\text{ if
}\xi\cap\beta\text{ is a maximum subject to }\xi\neq\beta\text{ and }\xi\text{
is unique },\\\ \xi,&\text{ if }\xi\cap\beta\,\&\,\xi\cap\alpha\,\text{ are
LST }\xi\neq\beta,\alpha\text{ and }\xi\text{ is unique },\\\ \beta,&\text{ if
}\xi\cap\beta\,\&\,\xi\cap\alpha\text{ are LST }\&\,\xi\neq\beta,\alpha\text{
but }\xi\text{ is not unique },\\\ \beta,&\text{ if
}\chi(\alpha\cap\beta)=\\{\alpha,\beta\\}.\end{array}\right.$

###### Definition 7

In the context of the above definition, the function
$\varrho:\mathfrak{S}^{2}\longmapsto\mathfrak{S}$ corresponding to _radical
extension_ will be defined as per
$\varrho(\alpha,\beta)\,\in\,\chi(\alpha\cap\beta)$ and (MST means _minimal
subject to_)

$\varrho(\alpha,\beta)\,=\,\left\\{\begin{array}[]{ll}\xi,&\text{ if
}\xi\cap\beta\text{ is a minimum under }\xi\neq\beta\text{ and }\xi\text{ is
unique },\\\ \xi,&\text{ if }\xi\cap\beta\,\&\,\xi\cap\alpha\text{ are MST
}\xi\neq\beta,\alpha\text{ and }\xi\text{ is unique },\\\ \alpha,&\text{ if
}(\exists\xi)\,\xi\cap\beta\,\&\,\xi\cap\alpha\text{ are MST
}\xi\neq\beta,\alpha\text{ but }\xi\text{ is not unique },\\\ \alpha,&\text{
if }\chi(\alpha\cap\beta)=\\{\alpha,\beta\\}.\end{array}\right.$

###### Theorem 3.3

The operations $\varrho,\,\delta$ satisfy all of the following:

  1. 1.

$\varrho,\,\delta$ are groupoidal operations,

  2. 2.

$\varrho(\alpha,\alpha)=\alpha$,

  3. 3.

$\delta(\alpha,\alpha)=\alpha$,

  4. 4.

$\delta(\alpha,\beta)\cap\beta\subseteq\delta(\delta(\alpha,\beta),\beta)\cap\beta$,

  5. 5.

$\delta(\delta(\alpha,\beta),\beta)=\delta(\alpha,\beta)$

  6. 6.

$\varrho(\varrho(\alpha,\beta),\beta)\cap\beta\subseteq\varrho(\alpha,\beta)\cap\beta$.

In general, a number of possibilities (potential non-implications) like the
following are satisfied by the algebra:
$\alpha\lessdot\beta\,\&\,\alpha\lessdot\gamma\,\nrightarrow\,\alpha\lessdot\delta(\beta,\gamma)$.
Given better properties of $l$ and $u$, interesting operators can be induced
on maximal antichains towards improving the properties of $\varrho$ and
$\delta$. The key constraint hindering the definition of total $l,u$ induced
operations can be avoided in the following way:

###### Definition 8

In the context of Thm 3.2, operations $\square,\Diamond$ can be defined as
follows:

  * $\bullet$

Given
$\alpha=\\{\mathbb{A}_{1},\mathbb{A}_{2},\ldots,\mathbb{A}_{n},\dots\\}\in\mathfrak{S}$,
form the set
$\gamma(\alpha)=\\{\mathbb{A}_{1}^{l},\mathbb{A}_{2}^{l},\ldots,\mathbb{A}_{n}^{l},\dots\\}$.
If this is an antichain, then $\alpha$ would be said to be _lower pure_.

  * $\bullet$

Form the set of all relatively maximal antichains $\gamma_{+}(\alpha)$ from
$\gamma(\alpha)$.

  * $\bullet$

Form all maximal antichains $\gamma_{*}(\alpha)$ containing elements of
$\gamma_{+}(\alpha)$ and set $\square(\alpha)=\wedge\gamma_{*}(\alpha)$

  * $\bullet$

For $\Diamond$, set
$\pi(\alpha)=\\{\mathbb{A}_{1}^{u},\mathbb{A}_{2}^{u},\ldots,\mathbb{A}_{n}^{u},\dots\\}$.
If this is an antichain, then $\alpha$ would be said to be _upper pure_.

  * $\bullet$

Form the set of all relatively maximal antichains $\pi_{+}(\alpha)$ from
$\pi(\alpha)$

  * $\bullet$

Form all maximal antichains $\pi_{*}(\alpha)$ containing elements of
$\pi_{+}(\alpha)$ and set $\Diamond(\alpha)=\vee\pi_{*}(\alpha)$

###### Theorem 3.4

In the context of the above definition, the following hold:

$\displaystyle\alpha\lessdot\beta\longrightarrow\square(\alpha)\lessdot\square(\beta)\,\&\,\Diamond(\alpha)\lessdot\Diamond(\beta)$
$\displaystyle\square(\alpha)\lessdot\alpha\lessdot\Diamond(\alpha),\;\square(0)=0\,\&\,\Diamond(1)=1$

Based on the above properties, the following algebra can be defined.

###### Definition 9

By a _Concrete AC_ algebra (AC -algebra) will be meant an algebra of the form
$\left\langle\mathfrak{S},\varrho,\delta,\vee,\wedge,\square,\Diamond,0,1\right\rangle$
associated with a granular operator space $S$ satisfying all of the following:

  * $\bullet$

$\left\langle\mathfrak{S},\vee,\wedge\right\rangle$ is a bounded distributive
lattice derived from a granular operator space as in the above.

  * $\bullet$

$\varrho,\delta,\square,\Diamond$ are as defined above

The following concepts of ideals and filters are of interest as deductive
systems in a natural sense and relate to ideas of rough consequence (detailed
investigation will appear separately).

###### Definition 10

By a _LD-ideal_ (resp. LD-filter)) $K$ of an AC-algebra $\mathfrak{S}$ will be
meant a lattice ideal (resp. filter) that satisfies:

$(\forall\alpha\in K)\,\square(\alpha),\Diamond(\alpha)\in K$

By a _VE-ideal_ (resp. VE-filter)) $K$ of an AC-algebra $\mathfrak{S}$ will be
meant a lattice ideal (resp. filter) that satisfies:

$(\forall\xi\in\mathfrak{S})(\forall\alpha\in
K)\,\varrho(\xi,\alpha),\delta(\xi,\alpha)\in K$

###### Proposition 5

Every VE filter is closed under $\Diamond$

###  3.1 Generating AC-Algebras

Now it will be shown below that specific subsets of AC-algebras suffice to
generate the algebra itself and that the axioms satisfied by the granulation
affect the generation process and properties of AC-algebras and forgetful
variants thereof.

An element $x\in\mathfrak{S}$ will be said to be _meet irreducible_ (resp.
join irreducible) if and only if $\wedge\\{x_{i}\\}=x\longrightarrow(\exists
i)\,x_{i}=x$ (resp. $\vee\\{x_{i}\\}=x\longrightarrow(\exists i)\,x_{i}=x$).
Let $W(S),\,J(S)$ be the set of meet and join irreducible elements of
$\mathfrak{S}$ and let $l(\mathfrak{S})$ be the length of the distributive
lattice.

###### Theorem 3.5

All of the following hold:

  * $\bullet$

$\left(\mathfrak{S},\vee,\wedge,0,1\right)$ is a isomorphic to the lattice of
principal ideals of the poset of join irreducibles.

  * $\bullet$

$l(\mathfrak{S})=\\#(J(S))=\\#(W(S))$

  * $\bullet$

$J(S)$ is not necessarily the set of sets of maximal antichains of granules in
general.

  * $\bullet$

When $\mathcal{G}$ satisfies mereological atomicity that is $(\forall
a\in\mathcal{G})(\forall b\in S)(\mathbf{P}ba,\,a^{l}=a^{u}=a\longrightarrow
a=b)$, and all approximations are unions of granules, then elements of $J(S)$
are proper subsets of $\mathcal{G}$.

  * $\bullet$

In the previous context, $W(S)$ must necessarily consist of two subsets of $S$
that are definite and are not parts of each other.

###### Proof

  * $\bullet$

The first assertion is a well known.

  * $\bullet$

Since the lattice is distributive and finite, its length must be equal to the
number of elements in $J(S)$ and $W(S)$. For a proof see [13].

  * $\bullet$

Under the minimal assumptions on $\mathcal{G}$, it is possible for definite
elements to be properly included in granules as in esoteric or prototransitive
rough sets [14, 7]. These provide the required counterexamples.

  * $\bullet$

The rest of the assertions follows from the nature of maximal antichains and
the constructive nature of approximations.

∎

###### Theorem 3.6

In the context of the previous theorem if $R(\Diamond),\,R(\square)$ are the
ranges of the operations $\Diamond,\square$ respectively, then these have a
induced lattice order on them. Denoting the associated lattice operations by
$\curlyvee,\curlywedge$ on $R(\Diamond)$, it can be shown that

  * $\bullet$

$R(\Diamond)$ can be reconstructed from $J(R(\Diamond))\cup W(R(\Diamond))$.

  * $\bullet$

$R(\square)$ can be reconstructed from $J(R(\square))\cup W(R(\square))$.

  * $\bullet$

When $\mathcal{G}$ satisfies mereological atomicity and absolute crispness
(i.e. $(\forall x\in\mathcal{G})\,x^{l}=x^{u}=x$), then $R(\Diamond)$ are
lattices which are constructible from two sets $A$, $C$ (with
$A=\\{\mathcal{G}\cup\\{g_{1}\cup
g_{2}\\}^{u}\setminus\\{g_{1},g_{2}\\};\,g_{1},g_{2}\in\mathcal{G}\\}$ and $C$
being the set of two element maximal antichains formed by sets that are upper
approximations of other sets).

###### Proof

It is clear that $R(\Diamond)$ is a lattice in the induced order with
$J(R(\Diamond))$ and $W(R(\Diamond))$ being the partially ordered sets of join
and meet irreducible elements respectively. This holds because the lattice is
finite.

The reconstruction of the lattice can be done through the following steps:

  * $\bullet$

Let $Z=J(R(\Diamond))\cup W(R(\Diamond))$. This is a partially ordered set in
the order induced from $R(\Diamond)$.

  * $\bullet$

For $b\in J(R(\Diamond))$ and $a\in W(R(\Diamond))$, let $b\prec a$ if and
only if $a\neq b$ in $R(\Diamond)$.

  * $\bullet$

On the new poset $Z$ with $\prec$, form sets including elements of
$W(R(\Diamond))$ connected to it.

  * $\bullet$

The set of union of all such sets including empty set ordered by inclusion
would be isomorphic to the original lattice. [13]

  * $\bullet$

Under additional assumptions on $\mathcal{G}$, the structure of $Z$ can be
described further.

When the granulation satisfies the properties of crispness and mereological
atomicity, then $A=J(R(\Diamond))$ and $C=W(R(\Diamond))$. So the third part
holds as well. ∎

The results motivate this concept of purity: A maximal antichain will be said
to _pure_ if and only if it is both lower and upper pure.

###  3.2 Enhancing the Anti Chain Based Representation

An integration of the orders on sets of maximal antichains or antichains and
the representation of rough objects and possible orders among them leads to
interesting multiple orders on the resulting structure. A major problem is
that of defining the orders or partials thereof in the first place among the
various possibilities.

###### Definition 11

By the _rough interpretation of an antichain_ will be meant the sequence of
pairs obtained by substituting objects in the rough domain in place of objects
in the classical perspective. Thus if $\alpha=\\{a_{1},a_{2}.\ldots,a_{p}\\}$
is a antichain, then its rough interpretation would be
($\pi(a_{i})=(a_{i}^{l},a_{i}^{u})$ for each $i$)

$\underline{\alpha}=\\{\pi(a_{1}),\pi(a_{2}),\ldots,\pi(a_{p})\\}.$

###### Proposition 6

It is possible that some rough objects are not representable by maximal
antichains.

###### Proof

Suppose the objects represented by the pairs $(a,b)$ and $(e,f)$ are such that
$a=e$ and $b\subset f$, then it is clear that any maximal antichain containing
$(e,f)$ cannot contain any element from $\\{x\,:x^{l}=a\,\&\,x^{u}=b\\}$. This
situation can happen, for example, in the models of proto transitive rough
sets [6, 5]. Concrete counterexamples can be found in the same paper. ∎

###### Definition 12

A set of maximal antichains $V$ will be said to be _fluent_ if and only if
$(\forall x)(\exists\alpha\in
V)(\exists(a,b)\in\alpha)\,x^{l}=a\,\&\,x^{u}=b$.

It will be said to be _well fluent_ if and only if it is fluent and no proper
subset of it is fluent.

A related problem is of finding conditions on $\mathcal{G}$, that ensure that
$V$ is fluent.

##  4 Ternary Deduction Terms

Since AC-algebras are distributive lattices with additional operations, a
natural strategy should be to consider terms similar to Boolean algebras and
p-Semilattices. For isolating deductive systems in the sense of Sec.2, a
strategy can be through complementation-like operations. This motivates the
following definition:

###### Definition 13

In a AC-algebra $\mathfrak{S}$, if an antichain
$\alpha=(X_{1},X_{2},\ldots,X_{n})$, then some possible general complements on
the schema $\alpha^{c}=\mathfrak{H}(X_{1}^{c},X_{2}^{c},\ldots,X_{n}^{c})$ are
as follows:

$\displaystyle X_{i}^{*}=\\{w;\,(\forall a\in
X_{i})\,\neg\mathbf{P}aw\,\&\,\neg\mathbf{P}wa\\}$ (Class A) $\displaystyle
X_{i}^{\\#}=\\{w;\,(\forall a\in X_{i})\,\neg a^{l}=w^{l}\text{ or }\neg
a^{u}=w^{u}\\}$ (Light) $\displaystyle X_{i}^{\flat}=\\{w;\,(\forall a\in
X_{i})\,\neg a^{l}=w^{l}\text{ or }\neg a^{uu}=w^{uu}\\}$ (UU)

$\mathfrak{H}$ is intended to mean the maximal antichain containing the set if
that is definable.

Note that under additional assumptions (similarity spaces), the light
complementation is similar to the preclusivity operation in [15] for Quasi BZ-
lattice or Heyting-Wajsburg semantics and variants.

The above operations on $\alpha$ are partial in general, but can be made total
with the help of an additional order on $\alpha$ and the following procedure:

  1. 1.

Let $\alpha=\\{X_{1},X_{2},\ldots,X_{n}\\}$ be a finite sequence,

  2. 2.

Form $\alpha^{c}$ and split into longest ACs in sequence,

  3. 3.

Form maximal ACs containing each AC in sequence

  4. 4.

Join resulting maximal ACs.

###### Proposition 7

Every general complement defined by the above procedure is well defined.

###### Proof

  * $\bullet$

Suppose $\\{X_{1}^{c},X_{2}^{c}\\},\\{X_{3}^{c},\ldots X_{n}^{c}\\}$ form
antichains, but $\\{X_{1}^{c},X_{2}^{c},X_{3}^{c}\\}$ is not an antichain.

  * $\bullet$

Then form the maximal antichains $\eta_{1},\ldots,\eta_{p}$ containing either
of the two antichains.

  * $\bullet$

The join of this finite set of maximal antichains is uniquely defined. By
induction, it follows that the operations are well defined.

∎

###  4.1 Translations

As per the approach of Sec 2, possible definitions of translations are as
follows:

###### Definition 14

A _translation_ in a AC-algebra $\mathfrak{S}$ is a
$\sigma:\mathfrak{S}\longmapsto\mathfrak{S}$ that is defined in one of the
following ways (for a fixed $a\in\mathfrak{S}$):

$\displaystyle\sigma_{\theta}(x)=\theta(a,x)\;;\theta\in\\{\vee,\wedge,\varrho,\delta\\}$
$\displaystyle\sigma_{\mu}(x)=\mu(x)\;;\mu\in\\{\square,\Diamond\\}$
$\displaystyle\sigma_{t}(x)=(x\oplus a)\oplus b\text{ for fixed a, b}$
$\displaystyle\sigma_{t+}(x)=(a\oplus b)\oplus x\text{ for fixed a, b}$

###### Theorem 4.1

$\displaystyle\sigma_{\vee}(0)=a=\sigma_{\vee}(a\,;\sigma_{\vee}(1)=1$
$\displaystyle Ran(\sigma_{\vee})\text{ is the principal filter generated by
}a$ $\displaystyle Ran(\sigma_{\wedge})\text{ is the principal ideal generated
by }a$ $\displaystyle x\lessdot
w\longrightarrow\sigma_{\vee}(x)\lessdot\sigma_{\vee}(w)\,\&\,\sigma_{\wedge}(x)\lessdot\sigma_{\wedge}(w)$

###### Proof

  * $\bullet$

Let $\mathbb{F}(a)$ be the principal lattice filter generated by $a$.

  * $\bullet$

If $a\lessdot w$, then $a\vee w=w=\sigma_{\vee}(w)$. So $w\in
Ran(\sigma_{\vee})$.

  * $\bullet$

$\sigma_{\vee}(x)\wedge\sigma_{\vee}(w)=(a\vee x)\wedge(a\vee w)=a\vee(x\wedge
w)=\sigma_{\vee}(x\wedge w)$.

  * $\bullet$

So if $x,w\in Ran(\sigma_{\vee})$, then $x\wedge w,x\vee w\in
Ran(\sigma_{\vee})$

  * $\bullet$

Simly it is provable that $Ran(\sigma_{\wedge})$ is the principal ideal
generated by $a$.

∎

###  4.2 Ternary Terms and Deductive Systems

Possible ternary terms that can cohere with the assumptions of the semantics
include the following $t(a,b,z)=a\wedge b\wedge z$, $t(a,b,z)=a\oplus b\oplus
z$ ($\oplus$ being as indicated earlier) and $t(a,b,z)=\square(a\wedge
b)\wedge z$. These have admissible deductive systems associated. Further under
some conditions on granularity, the distributive lattice structure associated
with $\mathfrak{S}$ becomes pseudo complemented.

###### Theorem 4.2

If $t(a,b,z)=a\wedge b\wedge z$, $\tau=\\{t\\}$, $z\in H$, $h(x)=x$
$\sigma(x)=x\wedge z$ and if $H$ is a ternary $\tau$-deduction system at $z$,
then it suffices that $H$ be an filter.

###### Proof

All of the following must hold:

  * $\bullet$

If $a\in H$, $t(z,a,z)=a\wedge z\in H$

  * $\bullet$

If $t(a,b,z)\in H$, then $t(\sigma(a),\sigma(b),z)=t(a,b,z)\in H$

  * $\bullet$

If $a,t(a,b,z)\in H$ then $t(a,b,z)=(a\wedge z)\wedge b\in H$. But $H$ is a
filter, so $b\in H$.

∎

###### Theorem 4.3

If $t(a,b,z)=(a\vee(\square b))\wedge z$, $\tau=\\{t\\}$, $z\in H$, $h(x)=x$
$\sigma(x)=x\wedge z$ and if $H$ is a ternary $\tau$-deduction system at $z$,
then it suffices that $H$ be a principal LD-filter generated by $z$.

###### Proof

All of the following must hold:

  * $\bullet$

If $a\in H$, $t(z,a,z)=(z\vee(\square a))\wedge z\in H$ because
$(z\vee(\square a))\in H$.

  * $\bullet$

If $t(a,b,z)\in H$, then $t(\sigma(a),\sigma(b),z)=t((a\wedge z),(b\wedge
z),z)=((a\wedge z)\vee\square(b\wedge z))\wedge z\in H$

  * $\bullet$

If $a,t(a,b,z)\in H$ then $t(a,b,z)=(a\vee\square(b))\wedge z=(a\wedge
z)\vee(\square(b)\wedge z)\in H$. But $H$ is a LD-filter, so
$a\vee\square(b)\in H$. This implies $\square(b)\in H$, which in turn yields
$b\in H$.

∎

In the above two theorems, the conditions on $H$ can be weakened considerably.
The converse questions are also of interest.

The existence of pseudo complements can also help in defining ternary terms
that determine deductive systems (or subsets closed under consequence). In
general, pseudo complementation $\circledast$ is a partial unary operation on
$\mathfrak{S}$ that is defined by $x^{\circledast}=\max\\{a\,;\,a\wedge
x=0\\}$ (if the greatest element exists).

There is no one answer to the question of existence as it depends on the
granularity assumptions of representation and stability of granules. The
following result guarantees pseudo complementation (in the literature, there
is no universal approach - it has always been the case that in some case they
exist):

###### Theorem 4.4

In the context of AC-algebras, if the granulation satisfies mereological
atomicity and absolute crispness, then a pseudo complementation is definable.

###### Proof

Under the conditions on the granulation, it is possible to form the rough
interpretation of each antichain. Moreover the granules can be moved in every
case to construct the pseudo complement. The inductive steps in this proof
have been omitted.

## Concluding Remarks

In this research, the problem of finding deductive systems in the context of
antichain based semantics for general rough sets has been explored and key
results have been proved by the present author. The lateral approach used by
her is justified by the wide variety of possible concepts of rough consequence
in the general setting. In a forthcoming paper, the framework of granular
operator spaces has been expanded with definable parthood relations and
semantics has been considered through counting strategies. All this will be
explored in greater detail in future work.

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