# A Logic for Dually Hemimorphic Semi-Heyting Algebras
and
Axiomatic Extensions

Juan M. Cornejo and Hanamantagouda P. Sankappanavar

###### Abstract

Semi-Heyting algebras were introduced by the second-named author during
1983-85 as an abstraction of Heyting algebras. The first results on these
algebras, however, were published only in 2008 (see [San08]). Three years
later, in [San11], he initiated the investigations into the variety
$\mathbb{DHMSH}$ of dually hemimorphic semi-Heyting algebras obtained by
expanding semi-Heyting algebras with a dually hemimorphic operation. His
investigations were continued in a series of papers thereafter. He also had
raised the problem of finding logics corresponding to subvarieties of
$\mathbb{DHMSH}$, such as the variety $\mathbb{DMSH}$ of De Morgan semi-
Heyting algebras, and $\mathbb{DPCSH}$ of dually pseudocomplemented semi-
Heyting algebras, as well as logics to 2, 3, and 4-valued
$\mathbb{DHMSH}$-matrices.

In this paper, we first present a Hilbert-style axiomatization of a new
implicative logic called “Dually hemimorphic semi-Heyting logic”
($\mathcal{DHMSH}$, for short)” as an expansion of semi-intuitionistic logic
by a dual hemimorphism as negation and prove that it is complete with respect
to the variety $\mathbb{DHMSH}$ of dually hemimorphic semi-Heyting algebras as
its equivalent algebraic semantics (in the sense of Abstract Algebraic Logic).
Secondly, we characterize the (axiomatic) extensions of $\mathcal{DHMSH}$ in
which the Deduction Theorem holds. Thirdly, we present several logics,
extending the logic $\mathcal{DHMSH}$, corresponding to several important
subvarieties of the variety $\mathbb{DHMSH}$, thus solving the problem
mentioned earlier. We also provide new axiomatizations for Moisil’s logic and
the 3-valued Lukasiewicz logic.

Keywords: dually hemimorphic semi-Heyting logic, De Morgan semi-Heyting logic,
De Morgan semi-Heyting algebra, Implicative logic, Semi-intuitionistic logic,
logics having finitely many finite characteristic matrices.

2010 AMS subject class: Primary: 03G25,06D20,06D15; Secondary: 08B26, 08B15.

##  1 Introduction

Semi-Heyting algebras were introduced by the second-named author during
1983-85 as an abstraction of Heyting algebras, and, later in 1987, some of the
early results were announced in [San87a]. However, the results with their
proofs on these algebras were published much later in 2008 (see [San08]).

An algebra $L=\langle L;\land,\lor,\to,0,1\rangle$ is a semi-Heyting algebra
if the following conditions hold:

  1. (SH1)

$\langle L;\land,\lor,0,1\rangle$ is a lattice with $0$ and $1$,

  2. (SH2)

$x\land(x\to y)\approx x\land y$,

  3. (SH3)

$x\land(y\to z)\approx x\land[(x\land y)\to(x\land z)]$,

  4. (SH4)

$x\to x\approx 1$.

A semi-Heyting algebra is a Heyting algebra if it satisfies the identity:
$(x\land y)\to x\approx 1$. We will denote the variety of semi-Heyting
algebras by $\mathbb{SH}$ and that of Heyting algebras by $\mathbb{H}.$ Semi-
Heyting algebras share some important properties with Heyting algebras; for
instance, semi-Heyting algebras are distributive and pseudocomplemented, with
the pseudocomplement $x^{*}:x\to 0$; the congruences on them are determined by
filters and the variety of semi-Heyting algebras is arithmetical (for more
properties of $\mathbb{SH}$, see [San08])).

It is well known that Heyting algebras are the algebraic semantics of the
intuitionistic propositional logic. In 2011, the first-named author of this
paper defined, in [Cor11], a new logic called “semi-intuitionistic logic”
($\mathcal{SI}$, for short) and showed that the variety of semi-Heyting
algebras is an algebraic semantics for this logic and that the intuitionistic
logic is a (axiomatic) extension of it. The axioms of this logic, however,
were expressed in the language that was not the same as that of semi-Heyting
algebras. In [CV15], a much simpler, but equivalent, set of axioms for
$\mathcal{SI}$, was presented (in the same language as that of semi-Heyting
algebras).

In 1942, Moisil [Moi42] defined a logic called “Logique modale”
($\mathcal{LM}$). He also introduced Heyting algebras endowed with an
involution in [Moi42a], as the algebraic models of an expansion of
intuitionistic propositional calculus by a De Morgan negation. These algebras
were investigated further by Monteiro [Mon80] under the name of symmetric
Heyting algebras. In particular, he presents a proof of an algebraic
completeness theorem for Moisil’s calculus by showing that the variety of
symmetric Heyting algebras as its equivalent algebraic semantics.

Independently, motivated purely by algebraic considerations, in [San87], the
second-named author defined and studied De Morgan Heyting algebras, in 1987,
by expanding Heyting algebras by a De Morgan operation. Already in 1985, the
second-named author had also introduced and investigated, in [San85], the
variety of Heyting algebras with a dual pseudocomplementation. In 2011,
motivated by the similarities of the results and proofs in [San85] and
[San87], the second-named author introduced in [San11] a more general variety
of algebras called “dually hemimorphic semi-Heyting algebras” as expansions of
semi-Heyting algebras by a dual hemimorphism– a common generalization of De
Morgan operation and the dual psedocomplementation.

###### Definition 1.1

[San11] An algebra $\mathbf{A}=\langle A;\wedge,\lor,\to,^{\prime},0,1\rangle$
is a semi-Heyting algebra with a dual hemimorphism (or dually hemimorphic
semi-Heyting algebra) if $\mathbf{A}$ satisfies the following conditions:

(E1): $\langle A,\wedge,\lor,\to,0,1\rangle$ is a semi-Heyting algebra,

(E2): $0^{\prime}\approx 1$,

(E3): $1^{\prime}\approx 0$,

(E4): $(x\land y)^{\prime}\approx x^{\prime}\lor y^{\prime}$ ($\land$-De
Morgan law).

A dually hemimorphic semi-Heyting algebra $\mathbf{A}$ is a De Morgan semi-
Heyting algebra if $\mathbf{A}$ satisfies

(E5): $(x\lor y)^{\prime}\approx x^{\prime}\land y^{\prime}$ ($\lor$-De Morgan
law) , and

(E6): $x^{\prime\prime}\approx x$.
A De Morgan semi-Heyting algebra $\mathbf{A}$ is a De Morgan Heyting algebra
if $\mathbf{A}$ satisfies:
(H) $(x\land y)\to x\approx 1$.
A dually hemimorphic semi-Heyting algebra $\mathbf{A}$ is a dually
pseudocomplemented if $\mathbf{A}$ satisfies
$x\lor x^{\prime}\approx 1$.

We will denote the varieties of dually hemimorphic semi-Heyting algebras, De
Morgan semi-Heyting algebras, and dually pseudocomplemented semi-Heyting
algebras by $\mathbb{DHMSH}$, $\mathbb{DMSH}$, and $\mathbb{DPCSH}$,
respectively. Similarly, $\mathbb{DHMH}$, $\mathbb{DMH}$ and $\mathbb{DPCH}$
denote, respectively, the varieties of dually hemimorphic Heyting algebras, De
Morgan Heyting algebras and dually pseudocomplemented Heyting algebras.

Inspired by Moisil’s logic and the completeness theorem for that logic
mentioned earlier, we were naturally led to the following problem (already in
2011):

PROBLEM 1: Find a propositional logic in the language
$\\{\lor,\land,\to,\neg,\bot,\top\\}$ with the following properties:
(1) It has the variety $\mathbb{DHMSH}$ of dually hemimorphic semi-Heyting
algebras as its algebraic semantics and
(2) It has Moisil’s logic as one of its (axiomatic) extensions.

Also, the following problem was raised in [San11, Section 8] and [San14b] by
the second author.

PROBLEM 2: Find Hilbert-type axiomatization for logics corresponding to two-
valued, three-valued and four-valued dually hemimorphic semi-Heyting algebras,
viewed as logical matrices with $\\{1\\}$ as the distinguished subset.

The purpose of this paper is four-fold: Firstly, we give a solution to PROBLEM
1. More specifically, we present a Hilbert-style presentation of a new logic
called “$\mathcal{DHMSH}$” as an expansion of semi-intuitionistic logic and
prove that it is implicative, in the sense of Rasiowa, and that it is complete
with respect to the variety $\mathbb{DHMSH}$ of dually hemimorphic semi-
Heyting algebras as its equivalent algebraic semantics (in the abstract
algebraic sense). Secondly, we characterize the (axiomatic) extensions of
$\mathcal{DHMSH}$ in which the Deduction Theorem holds. Thirdly, we present
logics, extending the logic $\mathcal{DHMSH}$, corresponding to several
interesting subvarieties of the variety of hemimorphic semi-Heyting and
Heyting algebras. Fourthly, as further applications, we provide
axiomatizations for logics corresponding to two-valued, three-valued and four-
valued dually hemimorphic semi-Heyting algebras, thus solving PROBLEM 2.

The paper is organized as follows: Section 2 contains definitions, notation
and some preliminary results that are needed later in the paper. In Section 3,
we present a Hilbert-style axiomatization for the new logic $\mathcal{DHMSH}$.
We expand the language of semi-intuitionistic logic $\mathcal{SI}$ by a (weak)
negation called _d_ ually hemimorphic negation” and expand the axioms of the
logic $\mathcal{SI}$ by adding new axioms and a new inference rule. We then
prove that the logic $\mathcal{DHMSH}$ is an implicative logic with respect to
the connective $\to_{H}$ (see Section 3 for definition of $\to_{H}$.) In
Section 4, we prove the completeness theorem for the logic $\mathcal{DHMSH}$:
The logic $\mathcal{DHMSH}$ is complete with respect to the variety
$\mathbb{DHMSH}$ of dually hemimorphic semi-Heyting algebras. In Section 5, we
characterize the (axiomatic) extensions of $\mathcal{DHMSH}$ in which the
Deduction Theorem holds. In the rest of the paper (Sections 6-12) we give
Hilbert-style axiomatizations for several important extensions of the logic
$\mathcal{DHMSH}$, as applications of the results proved in earlier sections
and the algebraic results proved in [San11], as well as some significant
improvements on Deduction Theorem (Theorem 5.7) when restricted to certain
extensions of $\mathcal{DHMSH}$. More specifically, In Sections 6, we present
axiomatizations for several important extensions of the logic
$\mathcal{DHMSH}$, including the dually hemimorphic Heyting logic
$\mathcal{DHMH}$, the De Morgan semi-Heyting logic $\mathcal{DMSH}$, the
dually pseudocomplemented semi-Heyting logic $\mathcal{DPCSH}$, the dually
pseudocomplemented Heyting logic $\mathcal{DPCH}$, the logic $\mathcal{DQDSH}$
corresponding to the variety generated by all dually quasi-De Morgan semi-
Heyting algebras, and the logic $\mathcal{DPCSH}$ corresponding to the variety
of dually pseudocomplemented semi-Heyting algebras. We also show that Moisil’s
“logique modale” ${\mathcal{LM}}$ is equivalent to the logic $\mathcal{DMH}$,
thus giving a different axiomatization of ${\mathcal{LM}}$ and a new proof of
Monteiro’s completeness theorem for ${\mathcal{LM}}$ mentioned earlier. In
Section 7, we present a significant improvement on the description of the
Deduction Theorem when restricted to the logic $\mathcal{DQDSH}$.

Sections 8-15 present further applications. In Sections 8, we present
axiomatizations of some extensions of the logic $\mathcal{DMSH}$ having
finitely many finite characteristic matrices, including the ones corresponding
to the ${\mathbb{DMSH}}$-matrices of order 3. In Section 9 we revisit the
3-valued Lukasiewicz logic and give an alternate definition for it. In fact,
we show that the logic corresponding to the 3-valued De Morgan Heyting algebra
is equivalent to the 3-valued Łukasiewicz logic. Section 10 presents
axiomatizations of some extensions of the logic $\mathcal{DPCSH}$
corresponding to the variety generated by all 3-element
$\mathcal{DPCSH}$-chains and also for the 3-valued logicscorresponding to each
of the 3-element $\mathcal{DPCSH}$-matrices. Section 11 presents extensions of
$\mathcal{DMSH}$ cooresponding to the variety generated by De Morgan Boolean
semi-Heyting algebras and to the 4-valued individual
$\mathcal{DMBSH}$-matrices. Section 12 axiomatizes De Morgan Goedel logic and
dually pseudocomplemented Goedel logic corresponding to the varietiy generated
by the De Morgan Heyting chains and the varietiy generated by the dually
pseudocomplemented Heyting chains, as well as the logics corresponding to
their subvarieties.

In Section 13, we present axiomatizations for logics corresponding to several
subvarieties of the variety $\mathbb{RDQDS}\rm t\mathbb{SH}_{1}$ of regular
dually quasi-De Morgan Stone semi-Heyting algebras. Section 14 presents logics
corresponding to several subvarieties of $\mathbb{RDMSH}_{1}.$ In Section 15,
we consider axiomatizations for the logic $\mathcal{JIDSH}$ and its extensions
that correspond to some of the subvarieties of the variety $\mathbb{JIDSH}$.
We conclude the paper with a direct and self-sufficient proof, in Section 16,
of Theorem 4.6 as it applies to the lattice of extensions of the logic
$\mathcal{DHMSH}$ and that of the subvarieties of the variety
$\mathbb{DHMSH}$.

##  2 Preliminaries

A language $\mathbf{L}$ is a set of finitary operations (or connectives), each
with a fixed arity $n\geq 0$. In this paper, we identify $\bot$ and $\top$
with $0$ and $1$ respectively and thus consider the languages
$\\{\land,\lor,\to,^{\prime},\bot,\top\\}$ and
$\\{\land,\lor,\to,^{\prime},0,1\\}$ as the same; however, we use the former
in the context of logics and the latter in the context of algebras. For a
countably infinite set Var of propositional variables, the formulas of the
logical language $\mathbf{L}$ are inductively defined as usual. A logic in the
language $\mathbf{L}$ is a pair $\mathcal{L}=\langle\mbox{\it Fm}_{\,\bf
L},\vdash_{\mathcal{L}}\rangle$, where $\mbox{\it Fm}_{\,\bf L}$ is the set of
formulas and $\vdash_{\mathcal{L}}$ is a substitution-invariant consequence
relation on $\mbox{\it Fm}_{\,\bf L}$. We will present logics by means of
their “Hilbert style” axioms and inference rules.

The set of formulas $\mbox{\it Fm}_{\,\bf L}$ can be turned into an algebra in
the usual way. Throughout the paper, $\Gamma$ denotes a set of formulas. The
homomorphisms from the formula algebra ${\bf Fm}_{\bf L}$ into an
$\mathbf{L}$-algebra $\mathbf{A}$ are called interpretations (or valuations).
The set of all such interpretations is denoted by $Hom({\bf Fm}_{\bf
L},\mathbf{A})$. If $h\in Hom({\bf Fm}_{\bf L},\mathbf{A})$ then the
interpretation of a formula $\alpha$ under $h$ is its image $h\alpha\in A$.
$h\Gamma$ denotes the set $\\{h\phi\ |\ \phi\in\Gamma\\}$.

Identities are ordered pairs of formulas that will be written in the form
$\alpha\approx\beta$. An identity $\alpha\approx\beta$ is satisfied by the
interpretation $h$ in $\mathbf{A}$ if $h\alpha=h\beta$. We denote this
satisfaction relation by the notation:
$\mathbf{A}\models_{h}\alpha\approx\beta$. An algebra $\mathbf{A}$ satisfies
the equation $\alpha\approx\beta$ if all the interpretations in $\mathbf{A}$
satisfy it; in symbols,

$\mathbf{A}\models\alpha\approx\beta\mbox{ if and only if
}\mathbf{A}\models_{h}\alpha\approx\beta,\mbox{ for all }h\in Hom({\bf
Fm}_{\bf L},\mathbf{A}).$

A class $\mathbb{K}$ of algebras satisfies the equation $\alpha\approx\beta$
when all the algebras in $\mathbb{K}$ satisfy it; i.e.

$\mathbb{K}\models\alpha\approx\beta\mbox{ if and only if
}\mathbf{A}\models\alpha\approx\beta,\mbox{ for all }\mathbf{A}\in\mathbb{K}.$

We abbreviate by $\Gamma\models_{\mathbb{K}}\phi$ the statement:

$\mbox{for all }\mathbf{A}\in\mathbb{K}\mbox{ and for all h}\in Hom(\mbox{\it
Fm}_{\bf L},\mathbf{A}),\mbox{ if }h\Gamma\subseteq\\{\top\\}\mbox{ then
}h\phi=\top.$

We present below a list of formulas that will be useful in later sections of
this paper. To state them, it will be convenient to use $\alpha\to_{H}\beta$
as an abbreviation for $\alpha\to(\alpha\land\beta)$ so that the axioms given
later are easier to read.

Let $\Sigma_{1}$ denote the following list of formulas.

AXIOMS:

(A1): $\alpha\to_{H}(\alpha\vee\beta)$

(A2): $\beta\to_{H}(\alpha\vee\beta)$

(A3):
$(\alpha\to_{H}\gamma)\to_{H}[(\beta\to_{H}\gamma)\to_{H}((\alpha\vee\beta)\to_{H}\gamma)]$

(A4): $(\alpha\wedge\beta)\to_{H}\alpha$

(A5):
$(\gamma\to_{H}\alpha)\to_{H}[(\gamma\to_{H}\beta)\to_{H}(\gamma\to_{H}(\alpha\wedge\beta))]$

(A6): $\top$

(A7): $\bot\to_{H}\alpha$

(A8):
$((\alpha\wedge\beta)\to_{H}\gamma)\to_{H}(\alpha\to_{H}(\beta\to_{H}\gamma))$

(A9):
$(\alpha\to_{H}(\beta\to_{H}\gamma))\to_{H}((\alpha\wedge\beta)\to_{H}\gamma)$

(A10):
$(\alpha\to_{H}\beta)\to_{H}((\beta\to_{H}\alpha)\to_{H}((\alpha\to\gamma)\to_{H}(\beta\to\gamma)))$

(A11):
$(\alpha\to_{H}\beta)\to_{H}((\beta\to_{H}\alpha)\to_{H}((\gamma\to\beta)\to_{H}(\gamma\to\alpha)))$

(A12): $\top\to_{H}\bot^{\prime}$

(A13): $\top^{\prime}\to_{H}\bot$

(A14):
$(\alpha\wedge\beta)^{\prime}\to_{H}(\alpha^{\prime}\vee\beta^{\prime})$

The following list of subsets of $\Sigma_{1}$ will be useful later to define
the main logic of this paper.

SETS OF FORMULAS:

  * $T=\\{{\rm(A\ref{axioma_supremo_izq})},\cdots,{\rm(A\ref{axioma_ImplicAIzquierda})}\\}$

  * $T_{0}=T\cup\\{{\rm(A\ref{axiom_bot_neg}),(A\ref{axiom_top_neg}),(A\ref{axiom_DM_law1})\\}}$

The following rules of inference are also needed to define some
(propositional) logics, both hitherto known and new ones, that appear later in
the paper.

RULES OF INFERENCE:

Let $\\{\phi,\gamma,\phi^{\prime},\gamma^{\prime}\\}\subseteq\mbox{\it Fm}$.

  * (SMP)

From $\phi$ and $\phi\to_{H}{\gamma}$, deduce $\gamma$ (semi-Modus Ponens).

  * (SCP)

From $\phi\to_{H}\gamma$, deduce $\gamma^{\prime}\to_{H}\phi^{\prime}$ (semi
contraposition rule).

As mentioned in the introduction, generalizing the intuitionistic calculus
$\mathcal{I}$, semi-intuitionistic logic $\mathcal{SI}$ was introduced in
[Cor11] in the language $\\{\wedge,\vee,\to,\neg\\}$ using a Hilbert-style
calculus such that the intuitionistic calculus $\mathcal{I}$ is essentially an
extension of it. A new, more streamlined set of axioms for semi-intuitionistic
logic in the language $\\{\wedge,\vee,\to,\bot,\top\\}$ was presented in
[CV15].

###### Definition 2.1

[CV15] The semi-intuitionistic logic $\mathcal{SI}$ is defined in the language
$\\{\wedge,\vee,\to,\bot,\top\\}$ and has $T$ as its set of axioms, and (SMP)
as its (only) inference rule.

The following theorem, proved in [CV15], is crucial in what follows.

###### Theorem 2.2

[CV15]

$\mbox{For all }\,\Gamma\cup\\{\alpha\\}\subseteq\mbox{\it
Fm},\,\Gamma\vdash_{\mathcal{SI}}\alpha\mbox{ if and only if
}\,\Gamma\models_{\mathbb{SH}}\alpha.$

###### Lemma 2.3

[CV15] The following statements hold in the logic $\mathcal{SI}$:

  1. (a)

If $\Gamma\vdash_{\mathcal{SI}}\psi$ then
$\Gamma\vdash_{\mathcal{SI}}\alpha\to_{H}\psi$.

  2. (b)

$\vdash_{\mathcal{SI}}\alpha\to_{H}\alpha$

  3. (c)

$\vdash_{\mathcal{SI}}(\alpha\wedge\beta)\to_{H}\beta$

  4. (d)

If $\vdash_{\mathcal{SI}}\alpha\to_{H}\beta$ then
$\vdash_{\mathcal{SI}}(\alpha\wedge\gamma)\to_{H}(\beta\wedge\gamma)$ and
$\vdash_{\mathcal{SI}}(\gamma\wedge\alpha)\to_{H}(\gamma\wedge\beta)$.

  5. (e)

$\vdash_{\mathcal{SI}}(\alpha\to_{H}\beta)\to_{H}[(\beta\to_{H}\gamma)\to_{H}(\alpha\to_{H}\gamma)]$

  6. (f)

If $\Gamma\vdash_{\mathcal{SI}}\alpha$ and $\Gamma\vdash_{\mathcal{SI}}\beta$
then $\Gamma\vdash_{\mathcal{SI}}\alpha\wedge\beta$.

In 1942, Moisil presented in [Moi42] (see also [Mon80]), a propositional logic
called, “Logique modale”, as an expansion of the positive intuitionistic logic
of Hilbert and Bernays [HB39] by a De Morgan negation. We will refer to it as
“$\mathcal{LM}$”. The axioms and the rules of inference for $\mathcal{LM}$ are
given below.

###### Definition 2.4

[Moi42] (see also [Mon80]) The logic, $\mathcal{LM}$, is defined in the
language
$\\{\wedge,\vee,\to,^{\prime}\\}$, as follows:

AXIOMS:

  1. (B1)

$\alpha\to(\beta\to\alpha)$,

  2. (B2)

$(\alpha\to(\beta\to\gamma))\to((\alpha\to\beta)\to(\alpha\to\gamma))$,

  3. (B3)

$(\alpha\wedge\beta)\to\alpha$,

  4. (B4)

$(\alpha\wedge\beta)\to\beta$,

  5. (B5)

$((\alpha\to\beta)\to(\alpha\to\gamma))\to(\alpha\to(\beta\wedge\gamma))$,

  6. (B6)

$\alpha\to(\alpha\vee\beta)$,

  7. (B7)

$\beta\to(\alpha\vee\beta)$,

  8. (B8)

$((\alpha\to\gamma)\to(\beta\to\gamma))\to((\alpha\vee\beta)\to\gamma)$,

  9. (B9)

$\alpha\to\alpha^{\prime\prime}$,

  10. (B10)

$\alpha^{\prime\prime}\to\alpha$.

RULES OF INFERENCE:

Let $\Gamma\cup\\{\phi,\gamma\\}\subseteq\mbox{\it Fm}$.

  * (MP)

From $\phi$ and ${\phi}\to\gamma$, conclude $\gamma$ (modus ponens),

  * (CP)

From $\alpha\to\beta$, conclude $\beta^{\prime}\to\alpha^{\prime}$
(contraposition rule).

Note that the axioms (B1-B8) and the rule (MP) form the Positive
intuitionistic propositional logic of Hilbert-Bernays ([HB39]). Monteiro
[Mon80] proved that the variety $\mathbb{DMH}$ is an algebraic semantics for
the logic $\mathcal{LM}$.

We conclude this section with some useful information regarding semi-Heyting
algebras.

A key feature of semi-Heyting algebras is the following [ACDV13]: Every semi-
Heyting algebra $\langle A,\lor,\land,\to,\top,\bot\rangle$ gives rise
naturally to a Heyting algebra $\langle A,\lor,\land,\to_{H},\top,\bot\rangle$
by defining the implication $x\to_{H}y$ as $x\to(x\land y)$.

The following lemmas will be useful in the sequel.

###### Lemma 2.5

Let $\mathbf{L}=\langle L;\land,\lor,\to,0,1\rangle$ be a semi-Heyting
algebra. For $a,b,c\in L,a\land b\leq c$ if and only if $a\leq b\to(b\land
c)$.

###### Lemma 2.6

[Cor11, Corollary 3.9] Let $\mathbf{L}=\langle L;\land,\lor,\to,0,1\rangle$ be
a semi-Heyting algebra. For $a,b\in L$, $a\to_{H}b=\top$ if and only if $a\leq
b$.

##  3 The logic $\mathcal{DHMSH}$

In 2011, at the end of Section 8 of [San11], the second-named author had
expressed his hope that the algebras discussed in that paper (i.e.
$\mathbb{DHMSH}$) will inspire new logics. Indeed, this section will turn his
hope into a reality.

In this section we present a new propositional logic which we call “dually
hemimorphic semi-Heyting logic” (or, “semi-Heyting logic with a dually
hemimorphic negation”) denoted by $\mathcal{DHMSH}$. We will show, as one of
our main results, that the variety $\mathbb{DHMSH}$ is the algebraic semantics
for the logic $\mathcal{DHMSH}$.

###### Definition 3.1

The dually hemimorphic semi-Heyting logic, $\mathcal{DHMSH}$, is defined in
the language $\\{\wedge,\vee,\to,^{\prime},\bot,\top\\}$, with $T_{0}$ (see
page 2) as its set of axioms and with (SMP) and (SCP) as its inference rules.

Since the axioms and the inference rule of the logic $\mathcal{SI}$ are
included in the logic $\mathcal{DHMSH}$, the following result is immediate.

###### Theorem 3.2

Let $\Gamma\cup\\{\alpha\\}\subseteq\mbox{\it Fm}$. If
$\Gamma\vdash_{\mathcal{SI}}\alpha$ then
$\Gamma\vdash_{\mathcal{DHMSH}}\alpha$.

###### Lemma 3.3

Let $\Gamma\cup\\{\alpha,\beta,\gamma,\psi\\}\subseteq\mbox{\it Fm}$. The
following statements hold in the logic $\mathcal{DHMSH}$:

  1. (a)

If $\Gamma\vdash_{\mathcal{DHMSH}}\psi$, then
$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}\psi$,

  2. (b)

$\vdash_{\mathcal{DHMSH}}\alpha\to_{H}\alpha$,

  3. (c)

$\vdash_{\mathcal{DHMSH}}(\alpha\wedge\beta)\to_{H}\beta$,

  4. (d)

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\to_{H}\beta)\to_{H}[(\beta\to_{H}\gamma)\to_{H}(\alpha\to_{H}\gamma)]$,

  5. (e)

$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}\top$,

  6. (f)

$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\wedge\beta$ if and only if
$\Gamma\vdash_{\mathcal{DHMSH}}\alpha$ and
$\Gamma\vdash_{\mathcal{DHMSH}}\beta$,

  7. (g)

If $\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}\beta$, then
$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\wedge\gamma)\to_{H}(\beta\wedge\gamma)$
and
$\Gamma\vdash_{\mathcal{DHMSH}}(\gamma\wedge\alpha)\to_{H}(\gamma\wedge\beta)$.

Proof  Items (a), (b), (c) and (d) follow from Theorem 3.2 and items (a), (b),
(c) and (e) of Lemma 2.3, respectively. The proof of (e) is immediate using
(a). By Theorem 3.2 and Lemma 2.3 (f), we have that if
$\Gamma\vdash_{\mathcal{DMSH}}\alpha$ and
$\Gamma\vdash_{\mathcal{DHMSH}}\beta$ then
$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\wedge\beta$. The converse of the item
(f) is easy to verify by using axiom (A2), item (c) and (SMP). Item (g)
follows from Lemma 2.3 (d). $\hfill\square$

###### Lemma 3.4

Let $\Gamma\cup\\{\alpha,\beta\\}\subseteq\mbox{\it Fm}$. Then

  1. (a)

If $\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}\beta$ and
$\Gamma\vdash_{\mathcal{DHMSH}}\beta\to_{H}\gamma$ then
$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}\gamma$.

  2. (b)

$\Gamma,\beta\to_{H}\alpha\vdash_{\mathcal{DHMSH}}\alpha^{\prime}\to_{H}\beta^{\prime}$,

  3. (c)

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\vee\beta)^{\prime}\to_{H}(\alpha^{\prime}\wedge\beta^{\prime})$.

  4. (d)

If $\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}\beta$ then
$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\lor\gamma)\to_{H}(\beta\lor\gamma)$ and
$\Gamma\vdash_{\mathcal{DHMSH}}(\gamma\lor\alpha)\to_{H}(\gamma\lor\beta)$.

  5. (e)

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha^{\prime}\vee\beta^{\prime})\to_{H}(\alpha\wedge\beta)^{\prime}$.

Proof

  1. (a)

    1. 1.

$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}\beta$ by hypothesis.

    2. 2.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\to_{H}\beta)\to_{H}[(\beta\to_{H}\gamma)\to_{H}(\alpha\to_{H}\gamma)]$
by 3.3 (d)

    3. 3.

$\Gamma\vdash_{\mathcal{DHMSH}}(\beta\to_{H}\gamma)\to_{H}(\alpha\to_{H}\gamma)$
by (SMP) in a1 and a2

    4. 4.

$\Gamma\vdash_{\mathcal{DHMSH}}\beta\to_{H}\gamma$ by hypothesis

    5. 5.

$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}\gamma$ by (SMP) in a3 and a4.

  2. (b)

This is immediate from (SCP).

  3. (c)

    1. 1.

$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}(\alpha\vee\beta)$ by (A2)

    2. 2.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\vee\beta)^{\prime}\to_{H}\alpha^{\prime}$
by (SCP) with c1

    3. 3.

$\Gamma\vdash_{\mathcal{DHMSH}}\beta\to_{H}(\alpha\vee\beta)$ by (A2)

    4. 4.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\vee\beta)^{\prime}\to_{H}\beta^{\prime}$
by (SCP) with c3

    5. 5.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\vee\beta)^{\prime}\to_{H}(\alpha^{\prime}\wedge\beta^{\prime})$
by (A2) and (SMP) with c2 and c4.

  4. (d)

    1. 1.

$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}\beta$ by hypothesis.

    2. 2.

$\Gamma\vdash_{\mathcal{DHMSH}}\beta\to_{H}(\beta\lor\gamma)$ by (A2)

    3. 3.

$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}(\beta\lor\gamma)$ by (a) and
(SMP) in d1 and d2.

    4. 4.

$\Gamma\vdash_{\mathcal{DHMSH}}\gamma\to_{H}(\beta\lor\gamma)$ by (A2)

    5. 5.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\to_{H}(\beta\lor\gamma))\to_{H}[(\gamma\to_{H}(\beta\lor\gamma))\to_{H}((\alpha\vee\gamma)\to_{H}(\beta\lor\gamma))]$
by (A2)

    6. 6.

$\Gamma\vdash_{\mathcal{DHMSH}}(\gamma\to_{H}(\beta\lor\gamma))\to_{H}((\alpha\vee\gamma)\to_{H}(\beta\lor\gamma))$
by (SMP) in d3 and d5.

    7. 7.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\vee\gamma)\to_{H}(\beta\lor\gamma)$ by
(SMP) in d4 and d6.

    8. 8.

$\Gamma\vdash_{\mathcal{DHMSH}}\gamma\to_{H}(\gamma\lor\beta)$ by (A2)

    9. 9.

$\Gamma\vdash_{\mathcal{DHMSH}}\beta\to_{H}(\gamma\lor\beta)$ by (A2)

    10. 10.

$\Gamma\vdash_{\mathcal{DHMSH}}\alpha\to_{H}(\gamma\lor\beta)$ by (a) and
(SMP) in d1 and d9.

    11. 11.

$\Gamma\vdash_{\mathcal{DHMSH}}(\gamma\to_{H}(\gamma\lor\beta))\to_{H}[(\alpha\to_{H}(\gamma\lor\beta))\to_{H}((\gamma\vee\alpha)\to_{H}(\gamma\lor\beta))]$
by (A2)

    12. 12.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\to_{H}(\gamma\lor\beta))\to_{H}((\gamma\vee\alpha)\to_{H}(\gamma\lor\beta))$
by (SMP) in d8 and d11.

    13. 13.

$\Gamma\vdash_{\mathcal{DHMSH}}(\gamma\vee\alpha)\to_{H}(\gamma\lor\beta)$ by
(SMP) in d10 and d12.

  5. (e)

    1. 1.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\wedge\beta)\to_{H}\alpha$ by axiom
(A2).

    2. 2.

$\Gamma\vdash_{\mathcal{DHMSH}}\alpha^{\prime}\to_{H}(\alpha\wedge\beta)^{\prime}$
by (SCP).

    3. 3.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha\wedge\beta)\to_{H}\beta$ by 3.3 (c).

    4. 4.

$\Gamma\vdash_{\mathcal{DHMSH}}\beta^{\prime}\to_{H}(\alpha\wedge\beta)^{\prime}$
by (SCP).

    5. 5.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha^{\prime}\to_{H}(\alpha\wedge\beta)^{\prime})\to_{H}[(\beta^{\prime}\to_{H}(\alpha\wedge\beta)^{\prime})\to_{H}((\alpha^{\prime}\vee\beta^{\prime})\to_{H}(\alpha\wedge\beta)^{\prime})]$
by axiom (A2).

    6. 6.

$\Gamma\vdash_{\mathcal{DHMSH}}(\beta^{\prime}\to_{H}(\alpha\wedge\beta)^{\prime})\to_{H}((\alpha^{\prime}\vee\beta^{\prime})\to_{H}(\alpha\wedge\beta)^{\prime})$
by (SMP) in e2 and e5.

    7. 7.

$\Gamma\vdash_{\mathcal{DHMSH}}(\alpha^{\prime}\vee\beta^{\prime})\to_{H}(\alpha\wedge\beta)^{\prime}$
by (SMP) in e4 and e6.

$\hfill\square$

The following definition is well known (see [Ras74, page 179]).

###### Definition 3.5

[Ras74] Let $\mathcal{L}$ be a logic in a language that includes a binary
connective $\to$, either primitive or defined by a term in exactly two
variables. Then $\mathcal{L}$ is called an implicative logic with respect to
the binary connective $\to$ if the following conditions are satisfied:

  * (IL1)

$\vdash_{\mathcal{L}}\alpha\to\alpha$.

  * (IL2)

$\alpha\to\beta,\beta\to\gamma\vdash_{\mathcal{L}}\alpha\to\gamma$.

  * (IL3)

For each connective $f$ in the language of arity $n>0$,
$\left\\{\begin{array}[]{c}\alpha_{1}\to\beta_{1},\ldots,\alpha_{n}\to\beta_{n},\\\
\beta_{1}\to\alpha_{1},\ldots,\beta_{n}\to\alpha_{n}\end{array}\right\\}\vdash_{\mathcal{L}}f(\alpha_{1},\ldots,\alpha_{n})\to
f(\beta_{1},\ldots,\beta_{n})$.

  * (IL4)

$\alpha,\alpha\to\beta\vdash_{\mathcal{L}}\beta$.

  * (IL5)

$\alpha\vdash_{\mathcal{L}}\beta\to\alpha$.

The following theorem was proved in [CV15, Lemma 4.6].

###### Theorem 3.6

[CV15, Lemma 4.6] The logic $\mathcal{SI}$ is implicative with respect to the
connective $\to_{H}$.

The following theorem is immediate from Theorem 3.2, Lemma 3.4 (b) and Theorem
3.6.

###### Theorem 3.7

The logic $\mathcal{DHMSH}$ is implicative with respect to the connective
$\to_{H}$.

##  4 Completeness of $\mathcal{DHMSH}$

The goal of this section is to prove that the logic $\mathcal{DHMSH}$ is
complete with respect to the variety $\mathbb{DHMSH}$.

###### Definition 4.1

[Ras74, Definition 6, page 181] Let $\mathcal{L}$ be an implicative logic in
the language $\mathbf{L}$ with an implication connective $\to$. An
$\mathcal{L}$-algebra is an algebra $\mathbf{A}$ in the language $\mathbf{L}$
that has an element $\top$ with the following properties:

>   * (LALG1)
>
> For all $\Gamma\cup\\{\phi\\}\subseteq\mbox{\it Fm}$ and all $h\in Hom({\bf
> Fm}_{\bf L},\mathbf{A})$, if $\Gamma\vdash_{\mathcal{L}}\phi$ and
> $h\Gamma\subseteq\\{\top\\}$ then $h\phi=\top$
>
>   * (LALG2)
>
> For all $a,b\in A$, if $a\to b=\top$ and $b\to a=\top$ then $a=b$.
>
>

The class of $\mathcal{L}$-algebras is denoted by
$\mathbf{Alg^{*}\mathcal{L}}$.

###### Theorem 4.2

[CV15, Corollary 4.8] $\mathbf{Alg^{*}{\mathcal{SH}}}=\mathbb{SH}$.

Since $\mathcal{DHMSH}$ is an implicative logic with respect to the binary
connective $\to_{H}$ by Theorem 3.7, we have the next result in view of
[Ras74, Theorem 7.1, pag 222].

###### Theorem 4.3

The logic $\mathcal{DHMSH}$ is complete with respect to the class
$\sf{A}lg^{*}{\mathcal{DHMSH}}$. In other words,

$\mbox{for all }\Gamma\cup\\{\phi\\}\subseteq\mbox{\it Fm,
}\Gamma\vdash_{\mathcal{DHMSH}}\phi\mbox{ if and only if
}\Gamma\models_{\mathbb{DHMSH}}\phi.$

We will now aim to prove that the logic $\mathcal{DHMSH}$ is complete with
respect to the variety $\mathbb{DHMSH}$.

Recall that every semi-Heyting algebra $\langle A,\lor,\land,\to,0,1\rangle$
gives rise naturally to a Heyting algebra $\langle
A,\lor,\land,\to_{H},0,1\rangle$ by defining the implication $x\to_{H}y$ as
$x\to(x\land y)$ (see[ACDV13]).

###### Lemma 4.4

$\mathbb{DHMSH}\subseteq\sf{A}lg^{*}{\mathcal{DHMSH}}$.

Proof  Let $\mathbf{A}\in\mathbb{DHMSH}$,
$\Gamma\cup\\{\phi\\}\subseteq\mbox{\it Fm}$ and $h\in Hom(\mbox{\it Fm}_{\bf
L},\mathbf{A})$, such that $\Gamma\vdash_{\mathcal{DMSH}}\phi$ and
$h\Gamma\subseteq\\{\top\\}$. We need to check that $h\phi=\top$. We will
proceed by induction on the length of the proof of
$\Gamma\vdash_{\mathcal{DMSH}}\phi$.

  * •

Assume that $\phi$ is an axiom.

If $\phi$ is one of the axioms (A2) to (A2) then $\vdash_{\mathcal{SI}}\phi$.
Then, by theorem 2.2, $\models_{\mathbb{DHMSH}}\phi$ and, hence,
$h(\phi)=\top$.

If $\phi$ is the axiom (A2) then, using (E1.1), we have that
$h(\phi)=h(\top\to_{H}\bot^{\prime})=\top\to_{H}\bot^{\prime}=\top\to_{H}\top=\top$.

If $\phi$ is the axiom (A2) then, using (E1.1), we have that
$h(\phi)=h(\top^{\prime}\to_{H}\bot)=\top^{\prime}\to_{H}\bot=\bot\to_{H}\bot=\top$.

If $\phi$ is the axiom (A2) then, using (E1.1), we have that
$h(\phi)=h((\alpha\wedge\beta)^{\prime}\to_{H}(\alpha^{\prime}\vee\beta^{\prime}))=(h(\alpha)\wedge
h(\beta))^{\prime}\to_{H}(h(\alpha)^{\prime}\vee
h(\beta)^{\prime})=(h(\alpha)\wedge h(\beta))^{\prime}\to_{H}(h(\alpha)\wedge
h(\beta))^{\prime}=\top$.

  * •

If $\phi\in\Gamma$ then $h(\phi)=\top$ by hypothesis.

  * •

Assume now that $\Gamma\vdash_{\mathcal{L}}\phi$ comes from an application of
(SMP). Then there exist a formula $\psi$ such that
$\Gamma\vdash_{\mathcal{L}}\psi$ and
$\Gamma\vdash_{\mathcal{L}}\psi\to_{H}\phi$. By induction, $h(\psi)=\top$ and
$h(\psi\to_{H}\phi)=\top$. Then
$h(\phi)=\top\to_{H}h(\phi)=h(\psi)\to_{H}h(\phi)=h(\psi\to_{H}\phi)=\top$.

  * •

Assume that $\Gamma\vdash_{\mathcal{L}}\phi$ comes from an application of the
rule (SCP). Then for $\alpha,\beta\in\mbox{\it Fm}$,
$\phi=\beta^{\prime}\to_{H}\alpha^{\prime}$ and
$\Gamma\vdash_{\mathcal{L}}\alpha\to_{H}\beta$. By induction
$\top=h(\alpha\to_{H}\beta)=h(\alpha)\to_{H}h(\beta)$ and, consequently
$h(\alpha)\leq h(\beta)$. Then, using condition (E1.1), $h(\beta)^{\prime}\leq
h(\alpha)^{\prime}$. Hence $h(\beta)^{\prime}\to_{H}h(\alpha)^{\prime}=\top$.
Therefore
$h(\phi)=h(\beta^{\prime}\to_{H}\alpha^{\prime})=h(\beta)^{\prime}\to_{H}h(\alpha)^{\prime}=\top$.

From the previous cases we concludes that $\mathbf{A}$ satisfies (LALG1). It
is easy to see that the condition (LALG2) also holds. $\hfill\square$

Recall that the following statements hold in a semi-Heyting algebra
$\mathbf{A}$:
For all $a,b\in A$:

  * •

$a=b$ if and only if $(a\to_{H}b)\wedge(b\to_{H}a)=\top$ and

  * •

$a\to b\leq a\to_{H}b$ [ACDV13].

###### Lemma 4.5

$\sf{A}lg^{*}{\mathcal{DHMSH}}\subseteq\mathbb{DHMSH}$.

Proof  Let $\mathbf{A}=\langle
A;\vee,\wedge,\to,^{\prime},0,1\rangle\in\sf{A}lg^{*}{\mathcal{DHMSH}}$.
Notice that $\langle
A;\vee,\wedge,\to,0,1\rangle\in\sf{A}lg^{*}{\mathcal{\mathcal{SI}}}$. By
Theorem 4.2, $\langle A;\vee,\wedge,\to,0,1\rangle\in\mathbb{SH}$. Now, it
only remains to show that $\mathbf{A}$ satisfies the conditions (E1.1) to
(E1.1).

In view of axiom $(A\ref{axiom_bot_neg})$, by condition (LALG1), we have that
$\mathbf{A}\models\top\to_{H}\bot^{\prime}\approx\top$. Using (LALG1) and
Lemma 3.3 (e), $\mathbf{A}\models\bot^{\prime}\to_{H}\top\approx\top$. Then by
(LALG2), $\mathbf{A}\models\top\approx\bot^{\prime}$.

In view of axioms $(A\ref{axioma_bot})$ and $(A\ref{axiom_top_neg})$, by
condition (LALG1), we have that
$\mathbf{A}\models\bot\to_{H}\top^{\prime}\approx\top$ and
$\mathbf{A}\models\top^{\prime}\to_{H}\bot\approx\top$. Then by (LALG2),
$\mathbf{A}\models\top^{\prime}\approx\bot$.

By Lemma 3.4 (e) and the condition (LALG1), $\mathbf{A}$ satisfies the
identity $(x^{\prime}\vee y^{\prime})\to_{H}(x\wedge y)^{\prime}\approx\top$.
In view of axiom $(A\ref{axiom_DM_law1})$, by condition (LALG1), $\mathbf{A}$
satisfies the identity $(x\vee y)^{\prime}\to_{H}(x^{\prime}\wedge
y^{\prime})\approx\top$. Applying (LALG2), the algebra holds (E1.1).
Consequently $\mathbf{A}\in\mathbb{DHMSH}$. $\hfill\square$

We are now ready to give our first main result.

###### Theorem 4.6

The logic $\mathcal{DHMSH}$ is complete with respect to the variety
$\mathbb{DHMSH}$.

Proof  From Lemmas 4.4 and 4.5 we have
$\sf{A}lg^{*}{\mathcal{DHMSH}}=\mathbb{DHMSH}$. The conclusion follows from
Theorem 4.3. $\hfill\square$

###  4.1 Axiomatic Extensions of the logic $\mathcal{DHMSH}$

In the sequel, we use the term “extension” for “axiomatic extension”. It is
clear that the extensions of the logic $\mathcal{DHMSH}$ form a (complete)
lattice. Let $\mathbf{L_{V}(\mathbb{DHMSH})}$ denote the lattice of
subvarieties of $\mathbb{DHMSH}$ and $Ext(\mathcal{DHMSH})$ the lattice of
extensions of the logic $\mathcal{DHMSH}$.

The following crucial theorem is a consequence of a well known theorem (see
[Fo17]) in Abstract Algebraic Logic and Theorem 4.6.

###### Theorem 4.7

$Ext(\mathcal{DHMSH})$ is dually isomorphic to
$\mathbf{L_{V}(\mathbb{DHMSH})}$.

The above theorem justifies the use of the phrase “the logic corresponding to
a subvariety $\mathbb{V}$ of $\mathbb{DHMSH}$.

The following theorem is an immediate consequence of Theorem 4.7 and plays an
important role in the rest of the paper.

###### Theorem 4.8

Every extension $\mathcal{E}$ of the logic $\mathcal{DHMSH}$ is complete with
respect to the (correspnding) variety $\mathbf{Mod}(\mathcal{E})$, where
$\mathbf{Mod}(\mathcal{E}):=\\{\mathbf{A}\in\mathbb{DHMSH}:\mathbf{A}\models\delta\approx\top,\mbox{
for every }\delta\in\mathcal{E}\\}$.

##  5  The Deduction Theorem in some extensions of $\mathcal{DHMSH}$

In this section we first show that the “usual” form of the Deduction Theorem”
fails in the logic $\mathcal{DHMSH}$ and then characterize those extensions of
$\mathcal{DHMSH}$ where it does hold.

A logic $\mathcal{L}$ is said to have the Deduction Property for the
connective $\to$ if the following statement holds:

$\Gamma,\alpha\vdash_{\mathcal{L}}\beta\text{ if and only if
}\Gamma\vdash_{\mathcal{L}}\alpha\to\beta,$

for all $\Gamma\cup\\{\alpha,\beta\\}\subseteq\mbox{\it Fm}$.

In the logic $\mathcal{SI}$ the Deduction Property for the conective $\to_{H}$
is known to hold [Cor11, Theorem 3.18]. But, this property fails in the logic
$\mathcal{DHMSH}$, as shown in the following remark.

###### Remark 5.1

By Lemma 3.4 (b) we have that

(5.1) $x\to_{H}y\ \vdash_{\mathcal{DHMSH}}\ y^{\prime}\to_{H}x^{\prime}.$

Consider the algebra $\mathbf{L}_{1}^{dm}$ introduced on Page 8.2. It is clear
that $\mathbf{L}_{1}^{dm}\in\mathbb{DHMSH}$. Observe that
$\mathbf{L}_{1}^{dm}\ \not\models_{\mathbb{DHMSH}}\
(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime})\approx 1$ by taking $x=1$ and
$y=a$. Thus
$\mathbb{DHMSH}\not\models(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime})\approx
1$ and therefore, by Theorem 4.6,

$\not\vdash_{\mathcal{DHMSH}}(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime}).$

Thus, Deduction Property fails in $\mathcal{DHMSH}$, in view of (5.1).

We now wish to characterize the extensions of $\mathcal{DHMSH}$ in which the
Deduction Property holds. For this, we need a preliminary lemma.

###### Lemma 5.2

Let $\mathcal{E}$ be an extension of the logic $\mathcal{DHMSH}$ such that
$\vdash_{\mathcal{E}}(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime})$, then
$\mathcal{E}$ satisfies the Deduction Property for the connective $\to_{H}$.

Proof  Assume that $\Gamma\cup\\{\phi\\}\vdash_{\mathcal{E}}\psi$. We shall
prove $\Gamma\vdash_{\mathcal{E}}\phi\to_{H}\psi$ by induction on the proof
for $\psi$. By hypothesis,

(5.2) $\vdash_{\mathcal{E}}(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime}).$

If $\psi$ is an axiom of $\mathcal{E}$ or a formula in $\Gamma$, then
$\Gamma\vdash_{\mathcal{E}}\psi$. By Lemma 3.3, part (a) we have
$\Gamma\vdash_{\mathcal{E}}\phi\to_{H}\psi.$

Let us assume that $\Gamma\cup\\{\phi\\}\vdash_{\mathcal{E}}\psi$ is the
result of applying the rule (SMP). Then we may assume that there is some
formula $\alpha$ such that $\Gamma\cup\\{\phi\\}\vdash_{\mathcal{E}}\alpha$
and $\Gamma\cup\\{\phi\\}\vdash_{\mathcal{E}}\alpha\to_{H}\psi$. So, by
inductive hypothesis, we have,

  1. 1.

$\Gamma\vdash_{\mathcal{E}}\phi\to_{H}\alpha$, and

  2. 2.

$\Gamma\vdash_{\mathcal{E}}\phi\to_{H}(\alpha\to_{H}\psi),$

  3. 3.

$\Gamma\vdash_{\mathcal{E}}\phi\to_{H}\phi$ by Lemma 3.3, part (b),

  4. 4.

$\Gamma\vdash_{\mathcal{E}}\phi\to_{H}(\phi\wedge\alpha)$ by (A2) and SMP
applied to 1 and 3,

  5. 5.

$\Gamma\vdash_{\mathcal{E}}(\phi\wedge\alpha)\to_{H}\psi$ by (A2) and SMP
applied to 2,

  6. 6.

$\Gamma\vdash_{\mathcal{E}}\phi\to_{H}\psi$ by 3.3 (d) and SMP applied to 4
and 5.

Assume that the last rule used in the proof of
$\Gamma\cup\\{\phi\\}\vdash_{\mathcal{E}}\psi$ was (SCP). Hence
$\psi=\beta^{\prime}\to_{H}\alpha^{\prime}$ and
$\Gamma\cup\\{\phi\\}\vdash_{\mathcal{E}}\alpha\to_{H}\beta$. By induction we
have that

  1. 1.

$\Gamma\vdash_{\mathcal{E}}\phi\to_{H}(\alpha\to_{H}\beta)$,

  2. 2.

$\Gamma\vdash_{\mathcal{E}}(\alpha\to_{H}\beta)\to_{H}(\beta^{\prime}\to_{H}\alpha^{\prime})$
by (5.2),

  3. 3.

$\Gamma\vdash_{\mathcal{E}}\phi\to_{H}(\beta^{\prime}\to_{H}\alpha^{\prime})$
by 3.3 (d) and SMP applied to 1 and 2.

For the other implication, we assume that
$\Gamma\vdash_{\mathcal{E}}\phi\to_{H}\psi$. Then
$\Gamma\cup\\{\phi\\}\vdash_{\mathcal{E}}\phi\to_{H}\psi$. Since
$\Gamma\cup\\{\phi\\}\vdash_{\mathcal{E}}\phi$, we have
$\Gamma\cup\\{\phi\\}\vdash_{\mathcal{E}}\psi$ by (SMP). $\hfill\square$

###### Theorem 5.3

The Deduction Property holds in an extension $\mathcal{E}$ of the logic
$\mathcal{DHMSH}$ for the connective $\to_{H}$ if and only if
$\mathcal{E}\vdash(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime})$.

Proof  Let us assume that the Deduction Property holds in $\mathcal{E}$ for
the conective $\to_{H}$. Note that $x\to_{H}y\vdash_{\mathcal{E}}x\to_{H}y$
and $x\to_{H}y\vdash_{\mathcal{E}}y^{\prime}\to_{H}x^{\prime}$ by (SCP). Hence
$\vdash_{\mathcal{E}}(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime})$ by
Deduction Property, or equivalently,
$\mathcal{E}\vdash_{\mathcal{DHMSH}}(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime})$.

For the converse, let us assume that
$\mathcal{E}\vdash(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime})$. By Lemma
5.2, the Deduction Property holds in $\mathcal{E}$ for the conective
$\to_{H}$. $\hfill\square$

Recall that semi-Heyting algebras are pseudocomplemented with $x^{*}:=x\to 0$
as the pseudocomplement of $x$, and a semi-Heyting algebra
$\mathbf{L}\in\mathbb{S}t\mathbb{SH}$ if $\mathbf{L}$ satisfies the Stone
identity: $x^{*}\lor x^{**}\approx 1$. Recall also that if $\mathbf{A}$ is a
semi-Heyting algebra, then $\langle A,\lor,\land,\to_{H}0,1\rangle$ is a
Heyting algebra.

###### Lemma 5.4

Let $\mathbf{A}\in\mathbb{DHMSH}$. If $\mathbf{A}$ satisfies the identity
$(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime})\approx 1$ then

  1. (a)

$\mathbf{A}\models x\wedge x^{\prime}\approx 0.$

  2. (b)

$\mathbf{A}\models x^{*}\approx x^{\prime}$

  3. (c)

$\mathbf{A}\models x^{\ast}\lor x^{\ast\ast}\approx 1.$

Proof

Let $a\in A$.

  1. (a)

Since
$a\to_{H}(a^{\prime}\to_{H}0)=(1\to_{H}a)\to_{H}(a^{\prime}\to_{H}0)=(1\to_{H}a)\to_{H}(a^{\prime}\to_{H}1^{\prime})=1$
in view of hypothesis, we have that
$a\wedge(a^{\prime}\to_{H}0)=a\wedge(a\to_{H}(a^{\prime}\to_{H}0))=a\wedge
1=a$. Hence

(5.3) $a\wedge(a^{\prime}\to_{H}0)=a.$

Then

$\begin{array}[]{lcll}a\wedge a^{\prime}&=&a\wedge(a^{\prime}\to_{H}0)\wedge
a^{\prime}&\mbox{by (\ref{equation_260219_02})}\\\ &=&a\wedge(a^{\prime}\to
0)\wedge a^{\prime}&\mbox{}\\\ &=&a\wedge a^{\prime}\wedge 0&\mbox{}\\\
&=&0.&\mbox{}\end{array}$

  2. (b)

Observe that
$a^{*}\to_{H}a^{\prime}=a^{*}\to_{H}(1\to_{H}a^{\prime})=(a\to_{H}0)\to_{H}(1\to_{H}a^{\prime})=(a\to_{H}0)\to_{H}(0^{\prime}\to_{H}a^{\prime})=1$
by hypothesis. Hence

(5.4) $\mathbf{A}\models x^{*}\leq x^{\prime}.$

Next,

$\begin{array}[]{lcll}a^{\prime}\wedge a^{*}&=&a^{\prime}\wedge(a\to
0)&\mbox{}\\\ &=&a^{\prime}\wedge((a^{\prime}\wedge a)\to(a^{\prime}\wedge
0))&\mbox{}\\\ &=&a^{\prime}\wedge(0\to 0)&\mbox{by (\ref{270219_01})}\\\
&=&a^{\prime}.&\mbox{}\end{array}$

Hence. $\mathbf{A}\models x^{\prime}\leq x^{*}.$ Now, using (5.4) we conclude
that $a^{\prime}=a^{*}$.

  3. (c)

$\begin{array}[]{lcll}a^{\ast}\lor a^{\ast\ast}&=&a^{\prime}\lor
a^{\prime\prime}&\mbox{by (\ref{260219_03})}\\\ &=&(a\wedge
a^{\prime})^{\prime}&\mbox{by (E\ref{infDMlaw})}\\\ &=&0^{\prime}&\mbox{by
(\ref{270219_01})}\\\ &=&1.&\mbox{}\end{array}$

$\hfill\square$

###### Lemma 5.5

Let $\mathbf{A}\in\mathbb{DHMSH}$. Then the following conditions are
equivalent in the algebra $\mathbf{A}$.

  1. (1)

$(x\to_{H}y)\to_{H}(y^{\prime}\to_{H}x^{\prime})\approx 1$

  2. (2)

$x^{*}\approx x^{\prime}$

Proof  Let $a,b\in A$. Observe that (1) implies (2) from Lemma 5.4.

On other hand, if $\mathbf{A}$ satisfies the identity (2), using the fact that
$\to_{H}$ is a Heyting implication, and (SH3), we have that
$(a\to_{H}b)\to_{H}(b^{\prime}\to_{H}a^{\prime})=(a\to_{H}b)\to_{H}(b^{\ast}\to_{H}a^{\ast})=(b^{\ast}\wedge(a\to_{H}b))\to_{H}a^{\ast}=(b^{\ast}\wedge(((b^{\ast}\wedge
a)\to_{H}(b^{\ast}\wedge b))\to_{H}a^{\ast}=(b^{\ast}\wedge((b^{\ast}\wedge
a)\to_{H}0))\to_{H}a^{\ast}=(b^{\ast}\wedge(a\to_{H}0))\to_{H}a^{\ast}=(b^{\ast}\wedge
a^{\ast})\to_{H}a^{\ast}=1$. $\hfill\square$

###### Lemma 5.6

Let $\mathbf{L}$ be a Stone semi-Heyting algebras. Let $\mathbf{L}^{e}$ be the
expansion of $\mathbf{L}$ to the language
$\langle\lor,\land,\to,^{\prime},0,1\rangle$, where we define ′ by:
$x^{\prime}:=x^{*}$. Then

  1. (1)

$\mathbf{L}^{e}\in\mathbb{DHMSH}$ and satisfies the identity:
$x^{\prime}\approx x^{*}$,

  2. (2)

$\mathbf{L}^{e}\models(x\lor y)^{\prime\prime}\approx x^{\prime\prime}\lor
y^{\prime\prime}$.

Proof  The lemma clearly follows from the well-known facts that
$\mathbf{L}\models(x\lor y)^{*}\approx x^{*}\land y^{*}$ and
$\mathbf{L}\models(x\land y)^{*}\approx x^{*}\lor y^{*}$. $\hfill\square$

We will refer to the algebra $\mathbf{L}^{e}$ as an “essentially Stone semi-
Heyting algebra”.

Let $\mathbb{V}$ be a subvariety of the variety of Stone semi-Heyting algebras
and let

$\mathbb{V}^{e}:=\\{\mathbf{L}^{e}:\ \mathbf{L}\in\mathbb{V}\\}.$

It is clear that $\mathbb{V}^{e}$ is a subvariety of $\mathbb{DHMSH}$.

We are now ready to present our second main result that describes precisely
those extensions of the logic $\mathcal{DHMSH}$ that have the Deduction
Property.

###### Theorem 5.7

The Deduction Property holds in an extension $\mathcal{E}$ of the logic
$\mathcal{DHMSH}$ for the connective $\to_{H}$ if and only if the
corresponding variety $\mathbb{E}$ is of the form $\mathbb{V}^{e}$, where
$\mathbb{V}\subseteq\mathbb{S}t\mathbb{SH}$.

##  6 Applications: Important Extensions of $\mathcal{DHMSH}$

In the rest of this paper, we give Hilbert-style axiomatizations for several
important extensions of the logic $\mathcal{DHMSH}$, as applications of the
results of Section 4 and Section 5, together with the algebraic results proved
in [San11], as well as some significant improvements on Deduction Theorem
(Theorem 5.7) when restricted to certain extensions of $\mathcal{DHMSH}$.

To facilitate the presentation of the extensions of the logic
$\mathcal{DHMSH}$, we first list several subvarieties of the variety
$\mathbb{DHMSH}$ of dually hemimorphic semi-Heyting algebras that were
introduced (or implicit) in [San11].

LIST 1: SUBVARIETIES OF THE VARIETY $\mathbb{DHMSH}$

  1. 1.

$\mathbb{DHMH}:$ Dually hemimorphic Heyting algebras are
$\mathbb{DHMSH}$-algebras satisfying the identity:

(H): $(x\wedge y)\to x\approx 1$,

  2. 2.

$\mathbb{OCKSH}:$ Ockham semi-Heyting algebras are $\mathbb{DHMSH}$-algebras
satisfying the identity:

(E7): $(x\lor y)^{\prime}\approx x^{\prime}\land y^{\prime}$,

  3. 3.

$\mathbb{D}ms\mathbb{SH}:$ Dually ms semi-Heyting algebras are
$\mathbb{OCKSH}$-algebras satisfying the identity:

(E8): $x^{\prime\prime}\leq x$,

  4. 4.

$\mathbb{DMSH}:$ De Morgan semi-Heyting algebras are $\mathbb{OCKSH}$-algebras
(or $\mathbb{DHMSH}$-algebras) satisfying the identity (E1.1).

  5. 5.

$\mathbb{DMH}:$ De Morgan Heyting algebras are $\mathbb{DMSH}$-algebras
satisfying the identity (H).

  6. 6.

$\mathbb{DSDSH}:$ Dually semi-De Morgan semi-Heyting algebras are
$\mathbb{DHMSH}$-algebras satisfying the identities:

(E9): $(x\vee y)^{\prime\prime}\approx x^{\prime\prime}\vee y^{\prime\prime}$,

(E10): $x^{\prime\prime\prime}\approx x^{\prime}$,

  7. 7.

$\mathbb{DQDSH}:$ Dually quasi-De Morgan semi-Heyting algebras are
$\mathbb{DSDSH}$-algebras satisfying the identity (E3),

  8. 8.

$\mathbb{DPCSH}:$ Dually pseudocomplemented semi-Heyting algebras are
$\mathbb{DQDSH}$-algebras satisfying the identity:

(E11): $x\vee x^{\prime}\approx 1$,

  9. 9.

$\mathbb{DPCH}:$ Dually pseudocomplemented Heyting algebras are
$\mathbb{DPCSH}$-algebras satisfying the identity (H).

  10. 10.

$\mathbb{BDQDSH}:$ Blended dually quasi-De Morgan semi-Heyting algebras  are
$\mathbb{DQDSH}$-algebras satisfying the identity:

(E12): $(x\vee x^{*})^{\prime}\approx x^{\prime}\wedge x{{}^{*}}^{\prime}$
(Blended $\vee$-De Morgan law),

  11. 11.

$\mathbb{SBDQDSH}:$ Strongly blended dually quasi-De Morgan semi-Heyting
algebras are $\mathbb{DQDSH}$-algebras satisfying the identity:

(E13): $(x\vee y^{*})^{\prime}\approx x^{\prime}\wedge(y^{*})^{\prime}$
(Strongly blended $\vee$-De Morgan law),

  12. 12.

$\mathbb{DQDBSH}:$ Dually quasi-De Morgan Boolean semi-Heyting algebras are
$\mathbb{DQDSH}$-algebras satisfying the identity:

(E14): $x\vee x^{*}\approx 1$,

  13. 13.

$\mathbb{DQSSH}:$ Dually quasi-Stone Semi-Heyting algebras are
$\mathbb{DHMSH}$-algebras satisfying the identities (E3),

(E15): $(x\vee y^{\prime})^{\prime}\approx x^{\prime}\wedge y^{\prime\prime}$
(weak $\vee$-De Morgan law),

(E16): $x^{\prime}\wedge x^{\prime\prime}\approx 0$ (Dual Stone identity).

  14. 14.

$\mathbb{DSSH}:$ Dually Stone semi-Heyting algebras are
$\mathbb{DQSSH}$-algebras satisfying the identity (E2),

  15. 15.

$\mathbb{BDQSSH}:$ Blended dually quasi-Stone Semi-Heyting algebras are
$\mathbb{DQSSH}$-algebras satisfying the identity (E10),

  16. 16.

$\mathbb{SBDQSSH}:$ Strongly blended dual quasi-Stone Semi-Heyting algebras
are $\mathbb{DQSSH}$-algebras satisfying the identitiy (E11),

  17. 17.

$\mathbb{DSCSH}:$ Semi-Heyting algebras with a dual semicomplementation are
$\mathbb{DHMSH}$-algebras satisfying the identity (E8),

  18. 18.

$\mathbb{DDPCSH}:$ Dually demi-pseudocomplemented Semi-Heyting algebras are
$\mathbb{DHMSH}$-algebras satisfying the identity:

(E17): $x^{\prime}\vee x^{\prime\prime}\approx 1$,

  19. 19.

$\mathbb{DAPCSH}:$ Dually almost-pseudocomplemented Semi-Heyting algebras are
$\mathbb{DDPCSH}$-algebras satisfying the identity (E3).

Next, we present Hilbert-type axiomatization of some new logics that are
extensions of $\mathcal{DHMSH}$.

LIST 2: EXTENSIONS OF $\mathcal{DHMSH}$

  1. 1.

$\mathcal{DHMH}:$ The dually hemimorphic Heyting logic is the extension of
$\mathcal{DHMSH}$ given by

(A15): $(\alpha\wedge\beta)\to\alpha$ ,

  2. 2.

$\mathcal{OCKSH}:$ The Ockham semi-Heyting logic is the extension of
$\mathcal{DHMSH}$ given by

(A16):
$(\alpha\vee\beta)^{\prime}\to_{H}(\alpha^{\prime}\wedge\beta^{\prime})$ ,

(A17):
$(\alpha^{\prime}\wedge\beta^{\prime})\to_{H}(\alpha\vee\beta)^{\prime}$ ,

  3. 3.

$\mathcal{D}ms\mathcal{SH}:$ The dually ms semi-Heyting logic is the extension
of $\mathcal{OCKSH}$ given by

(A18): $\alpha^{\prime\prime}\to_{H}\alpha$ ,

  4. 4.

$\mathcal{DMSH}:$ The De Morgan semi-Heyting logic is the extension of
$\mathcal{OCKSH}$ given by (A3) and

(A19): $\alpha\to_{H}\alpha^{\prime\prime}$ ,

  5. 5.

$\mathcal{DMH}:$ The De Morgan Heyting logic is the extension of
$\mathcal{DMSH}$ given by (A1),

  6. 6.

$\mathcal{DSDSH}:$ The dually semi-De Morgan semi-Heyting logic is the
extension of $\mathcal{DHMSH}$ given by

(A20):
$(\alpha\vee\beta)^{\prime\prime}\to_{H}(\alpha^{\prime\prime}\vee\beta^{\prime\prime})$
,

(A21):
$(\alpha^{\prime\prime}\vee\beta^{\prime\prime})\to_{H}(\alpha\vee\beta)^{\prime\prime}$
,

(A22): $\alpha^{\prime\prime\prime}\to_{H}\alpha^{\prime}$ ,

(A23): $\alpha^{\prime}\to_{H}\alpha^{\prime\prime\prime}$ ,

  7. 7.

$\mathcal{DQDSH}:$ The dually quasi-De Morgan semi-Heyting logic is the
extension of $\mathcal{DSDSH}$ given by (A3),

  8. 8.

$\mathcal{DPCSH}:$ The dually pseudocomplemented semi-Heyting logic is the
extension of $\mathcal{DQDSH}$ given by

(A24): $\alpha\vee\alpha^{\prime}$ ,

  9. 9.

$\mathcal{DPCH}:$ The dually pseudocomplemented Heyting logic is the extension
of $\mathcal{DPCSH}$ given by (A1),

  10. 10.

$\mathcal{BDQDSH}:$ The blended dually quasi-De Morgan semi-Heyting logic is
the extension of $\mathcal{DQDSH}$ given by

(A25):
$(\alpha\vee(\alpha\to\bot))^{\prime}\to_{H}(\alpha^{\prime}\wedge(\alpha\to\bot)^{\prime})$
,

(A26):
$(\alpha^{\prime}\wedge(\alpha\to\bot)^{\prime})\to_{H}(\alpha\vee(\alpha\to\bot))^{\prime}$
,

  11. 11.

$\mathcal{SBDQDSH}:$ The strongly blended quasi-De Morgan semi-Heyting logic
is the extension of $\mathcal{DQDSH}$ given by

(A27):
$(\alpha\vee(\beta\to\bot))^{\prime}\to_{H}(\alpha^{\prime}\wedge(\beta\to\bot)^{\prime})$
,

(A28):
$(\alpha^{\prime}\wedge(\beta\to\bot)^{\prime})\to_{H}(\alpha\vee(\beta\to\bot))^{\prime}$
,

  12. 12.

$\mathcal{DQDBSH}:$ The quasi-De Morgan Boolean semi-Heyting logic is the
extension of $\mathcal{DQDSH}$ given by

(A29): $\alpha\vee(\alpha\to\bot)$ ,

  13. 13.

$\mathcal{DQSSH}:$ The dually quasi-Stone semi-Heyting logic is the extension
of $\mathcal{DHMSH}$ given by (A3) and the following axioms:

(A30):
$(\alpha\vee\beta^{\prime})^{\prime}\to_{H}(\alpha^{\prime}\wedge\beta^{\prime\prime})$
,

(A31):
$(\alpha^{\prime}\wedge\beta^{\prime\prime})\to_{H}(\alpha\vee\beta^{\prime})^{\prime}$
,

(A32): $(\alpha^{\prime}\wedge\alpha^{\prime\prime})\to_{H}\bot$ ,

  14. 14.

$\mathcal{DSSH}:$ The dually Stone semi-Heyting logic is the extension of
$\mathcal{DQSSH}$ given by (A2) and (A2),

  15. 15.

$\mathcal{BDQSSH}:$ The blended dually quasi-Stone semi-Heyting logic is the
extension of $\mathcal{DHMSH}$ given by (A3), (A10), (A10), (A13), (A13) and
(A13).

  16. 16.

$\mathcal{SBDQSSH}:$ The strongly blended dually quasi-Stone semi-Heyting
logic is the extension of $\mathcal{DHMSH}$ given by (A3), (A11), (A11),
(A13), (A13) and (A13),

  17. 17.

$\mathcal{DSCSH}:$ The dually semicomplemented semi-Heyting logic is the
extension of $\mathcal{DHMSH}$ given by (A8),

  18. 18.

$\mathcal{DDPCSH}:$ The dually demipseudocomplemented semi-Heyting logic is
the extension of $\mathcal{DHMSH}$ given by

(A33): $\alpha^{\prime}\vee\alpha^{\prime\prime}$ ,

  19. 19.

$\mathcal{DAPCSH}:$ The dually almost-pseudocomplemented Semi-Heyting logic is
the extension of $\mathcal{DDPCSH}$ given by
$\alpha^{\prime\prime}\to_{H}\alpha$.

Recall that in a semi Heyting algebra $\mathbf{A}$, we have the following
conditions:

  * •

$a\to_{H}b=1$ if and only if $a\leq b$

  * •

$(a\to b)\wedge(b\to a)=1$ if and only if $(a\to_{H}b)\wedge(b\to_{H}a)=1$.

Hence, the following theorem is immediate from Theorem 4.6.

###### Theorem 6.1

Let $\mathbb{V}_{i}$ be the $i$-th variety of algebras in LIST 1 and
$\mathcal{V}_{i}$ the logic in LIST 2. Then, the logic $\mathcal{V}_{i}$
corresponds to the variety $\mathbb{V}_{i}$ in the sense that $\mathbb{V}_{i}$
is its algebraic semantics for $\mathcal{V}_{i}$.

###### Corollary 6.2

The (Moisil’s) logic $\mathcal{LM}$ is equivalent to the logic
$\mathcal{DMH}$.

Proof  It is known from Moisil’s result (or Monteiro’s result) that
$\mathcal{LM}$ correspond to $\mathbb{DMH}$. Also, observe from Theorem 6.1
that the logic $\mathcal{DMH}$ correspond to $\mathbb{DMH}$ as well.
$\hfill\square$

We conclude this section by noting that the lattice of extensions of
$\mathcal{DMH}$ is an interval of the lattice of extensions of
$\mathcal{DMSH}$, which, in turn, is an interval in the lattice of extensions
of $\mathcal{DHMSH}$.

##  7 Deduction Theorem in the Extensions of the logic $\mathcal{DQDSH}$

In this section we show that Theorem 5.7 can be significantly improved for the
logic $\mathcal{DQDSH}$. We shall give an explicit description of the
extensions of the logic $\mathcal{DQDSH}$ in which the Deduction Property
holds.

For this purpose the following algebras, $\mathbf{2^{e}}$ and
$\mathbf{\bar{2}^{e}}$, which are the only two 2-element algebras in
$\mathbb{DQDSH}$ (in fact, in $\mathbb{DMSH}$), will be useful.

$\mathbf{2}^{e}$:  $\to$: 0 1 0 1 1 1 0 1 ′: 0 1 1 0 $\mathbf{\bar{2}^{e}}$:
$\to$: 0 1 0 1 0 1 0 1 ′: 0 1 1 0

Figure 1

###### Lemma 7.1

Let $\mathbb{V}$ be a subvariety of $\mathbb{DQDSH}$. If
$V\models(x\to_{H}y)\to(y^{\prime}\to_{H}x^{\prime})\approx 1$, then

$\mathbb{V}\subseteq\mathbb{V}(\mathbf{2^{e}},\mathbf{\bar{2^{e}}})$, where
$\mathbb{V}(\mathbf{2^{e}},\mathbf{\bar{2^{e}}})$ denotes the variety
generated by $\\{\mathbf{2^{e}},\mathbf{\bar{2^{e}}}\\}$.

Proof  The hypothesis and Lemma 5.5 (b) imply that $\mathbb{V}\models
x^{\prime}\approx x^{*}$. Hence
$\mathbb{V}=\mathbb{V}(\mathbf{2^{e}},\mathbf{\bar{2^{e}}})$ by [San11,
Theorem 5.11]. $\hfill\square$

The following theorem describes precisely those extensions of
$\mathcal{DQDSH}$ in which the Deduction Property holds. Let $\mathbf{T}$
denote the trivial subvariety of $\mathbb{DHMSH}$.

###### Theorem 7.2

The Deduction Property holds in a logic $\mathcal{E}\in Ext(\mathcal{DQDSH})$
for $\to_{H}$ if and only if the corresponding variety is either either
$\mathbf{T}$ or $\mathbb{V}(\mathbf{2^{e}})$ or
$\mathbb{V}(\mathbf{\bar{2^{e}}})$ or
$\mathbb{V}(\mathbf{2^{e}},\mathbf{\bar{2^{e}}})$.

Proof  The theorem is immediate in view of Theorem 4.8, Theorem 5.3, Lemma
5.5, and Lemma 7.1. $\hfill\square$

Since $\mathbb{DMSH}\subseteq\mathbb{DQDSH}$ and
$\mathbb{DPCSH}\subseteq\mathbb{DQDSH}$, the following corollaries are
immediate.

###### Corollary 7.3

The Deduction Property holds in a logic $\mathcal{E}\in Ext(\mathcal{DMSH})$
for $\to_{H}$ if and only if the corresponding variety is either $\mathbf{T}$
or $\mathbb{V}(\mathbf{2^{e}})$ or $\mathbb{V}(\mathbf{\bar{2^{e}}})$ or
$\mathbb{V}(\mathbf{2^{e}},\mathbf{\bar{2^{e}}})$.

Since $\mathbb{DPCSH}\subset\mathbb{DQDSH}$, the following corollary is
immediate.

###### Corollary 7.4

The Deduction Property holds in a logic $\mathcal{E}\in Ext(\mathcal{DPCSH})$
for $\to_{H}$ if and only if the corresponding variety is either $\mathbf{T}$
or $\mathbb{V}(\mathbf{2^{e}})$ or $\mathbb{V}(\mathbf{\bar{2^{e}}})$ or
$\mathbb{V}(\mathbf{2^{e}},\mathbf{\bar{2^{e}}})$.

##  8  Logics having finitely many finite characteristic matrices

In this section we will present several new non-classical logics as extensions
of the logic $\mathcal{DHMSH}$ with finitely many finite characteristic
matrices or with a single finite characteristic matrix, as applications of
Theorem 4.8 and results from [San11]. Thus we solve PROBLEM 2 which includes
the questions raised in [San11, Section 8] and [San14b].

###  8.1 2-valued Logics: Extensions corresponding to the non-trivial
subvarieties of $\mathbb{V}(\mathbf{2^{e}},\mathbf{\bar{2^{e}}})$

The algebras $\mathbf{2^{e}}$ and $\mathbf{\bar{2}^{e}}$ were defined earlier
in Figure 2 of Section 7. Earlier we showed that the Deduction Property holds
in an extension of the logic $\mathcal{DHMSH}$ iff the corresponding variety
is a subvariety of $\mathbb{V}(\mathbf{2^{e}},\mathbf{\bar{2}^{e}})$. So, it
is but natural to ask for the axiomatization of these extensions.

The variety $\mathbb{V}(\mathbf{2^{e}},\mathbf{\bar{2}^{e}})$ was axiomatized
in [San11, Theorem 5.11] by the identity: $x^{\prime*}\approx x$, relative to
the variety $\mathbb{DHMSH}$. From this it is easy to see that the varieties
$\mathbb{V}(\mathbf{2^{e}})$ and $\mathbb{V}(\mathbf{\bar{2}^{e}})$ are
defined, respectively, by the identities: $0\to 1\approx 1$ and $0\to 1\approx
0$, relative to $\mathbb{V}(\mathbf{2}^{e},\mathbf{\bar{2}^{e}})$. In view of
these observations, we obtain, using Theorem 4.8, the following corollaries
defining their corresponding logics.

Let $\mathcal{L}(2^{e},\bar{2}^{e})$ be the extension of the logic
$\mathcal{DHMSH}$ corresponding to the variety
$\mathbb{V}(\mathbf{2}^{e},\mathbf{\bar{2}^{e}})$. Let
$\alpha\leftrightarrow_{H}\gamma$ denote the formula:
$(\alpha\rightarrow_{H}\beta)\land(\beta\rightarrow_{H}\alpha)$.

###### Corollary 8.1

The logic $\mathcal{L}(2^{e},\bar{2}^{e})$ is defined, relative to the logic
$\mathcal{DHMSH}$, by the following two axioms:

  1. (i)

$(\phi^{\prime}\to\bot)\to_{H}\phi$,

  2. (ii)

$\phi\to_{H}(\phi^{\prime}\to\bot)$,

or equivalently, by

$(\phi^{\prime}\to\bot)\leftrightarrow_{H}\phi$.

Let $\mathcal{L}({2^{e}})$ and $\mathcal{L}({\bar{2}^{e}})$ denote,
respectively, the extensions of the logic $\mathcal{L}({2^{e},\bar{2}^{e}})$
corresponding to the varieties $\mathbb{V}(\mathbf{2}^{e})$ and
$\mathbb{V}(\mathbf{\bar{2}^{e}})$.

###### Corollary 8.2

The logic $\mathcal{L}({2^{e}})$ is defined, relative to the logic
$\mathcal{L}({2^{e},\bar{2}^{e}})$, by the axiom:

  1. (iii)

$\bot\to\top$.

Thus, $\mathcal{L}({2^{e}})$ is yet another axiomatization of the classical
logic.

###### Corollary 8.3

The logic $\mathcal{L}({\bar{2}^{e}})$ is defined, relative to the logic
$\mathcal{L}({2^{e},\bar{2}^{e}})$, by the axiom:

  1. (iv)

$(\bot\to\top)\to_{H}\bot$.

It should perhaps be remarked here that, in view of (A7), it is not necessary
to include the formula

$\bot\to_{H}(\bot\to_{H}\top)$

as an axiom in the above corollary. It should also be pointed out here that
the logic $\mathcal{L}({\bar{2}^{e}})$ is “anti-classical logic” since $F\to
T=F$ and the implication commutes in this logic.

###### Remark 8.4

The Deduction Theorem holds only in the preceding three non-trivial logics in
the lattice of extensions of the logic $\mathcal{DQDSH}$, in view of Theorem
7.2.

###  8.2 Some 3-valued Logics: Extensions of the logic $\mathcal{DMSH}$

It was shown in [San08] that there are, up to isomorphism, ten $3$-element
semi-Heyting algebras, as defined in figure $3$.

$0$

$a$

$1$

${\bf L}_{1}$:

$\to$ |  0 |  $a$ |  1
---|---|---|---
0 | 1 | 1 | 1
$a$ | 0 | 1 | 1
1 | 0 | $a$ | 1

$0$

$a$

$1$

${\bf L}_{2}$:

$\to$ |  0 |  $a$ |  1
---|---|---|---
0 | 1 | $a$ | 1
$a$ | 0 | 1 | 1
1 | 0 | $a$ | 1

$0$

$a$

$1$

${\bf L}_{3}$:

$\to$ |  0 |  $a$ |  1
---|---|---|---
0 | 1 | 1 | 1
$a$ | 0 | 1 | $a$
1 | 0 | $a$ | 1

$0$

$a$

$1$

${\bf L}_{4}$:

$\to$ |  0 |  $a$ |  1
---|---|---|---
0 | 1 | $a$ | 1
$a$ | 0 | 1 | $a$
1 | 0 | $a$ | 1

$0$

$a$

$1$

${\bf L}_{5}$:

$\to$ |  0 |  $a$ |  1
---|---|---|---
0 | 1 | $a$ | $a$
$a$ | 0 | 1 | 1
1 | 0 | $a$ | 1

$0$

$a$

$1$

${\bf L}_{6}$:

$\to$ |  0 |  $a$ |  1
---|---|---|---
0 | 1 | 1 | $a$
$a$ | 0 | 1 | 1
1 | 0 | $a$ | 1

$0$

$a$

$1$

${\bf L}_{7}$:

$\to$ |  0 |  $a$ |  1
---|---|---|---
0 | 1 | $a$ | $a$
$a$ | 0 | 1 | $a$
1 | 0 | $a$ | 1

$0$

$a$

$1$

${\bf L}_{8}$:

$\to$ |  0 |  $a$ |  1
---|---|---|---
0 | 1 | 1 | $a$
$a$ | 0 | 1 | $a$
1 | 0 | $a$ | 1

$0$

$a$

$1$

${\bf L}_{9}$:

$\to$ |  0 |  $a$ |  1
---|---|---|---
0 | 1 | 0 | 0
$a$ | 0 | 1 | 1
1 | 0 | $a$ | 1

$0$

$a$

$1$

${\bf L}_{10}$:

$\to$ |  0 |  $a$ |  1
---|---|---|---
0 | 1 | 0 | 0
$a$ | 0 | 1 | $a$
1 | 0 | $a$ | 1

Figure 2

It is clear that there are exactly two expansions on each of the above 10
semi-Heyting algebras by a unary operation ′ that give rise to twenty
$\mathbb{DHMSH}$-algebras. Ten of these, that correspond to $a^{\prime}=a$ for
$a\neq 0,1$ are in $\mathbb{DMSH}$, will be considered now, and the other ten,
that correspond to $a^{\prime}=a$ for $a\neq 0,1$ are in $\mathbb{DPCSH}$,
will be considered later.

More precisely, let ${\bf L}_{i}^{dm}$, $i=1,2,\dots,10$, denote the expansion
of ${\bf L}_{i}$ by adding a unary operation ′ such that $0^{\prime}=1$,
$1^{\prime}=0$, and $a^{\prime}=a$, for $a\neq 0,1$. Similarly, let ${\bf
L}_{i}^{dp}$, $i=1,2,\dots,10$, denote the expansion of ${\bf L}_{i}$ by
adding a unary operation ′ such that $0^{\prime}=1$, $1^{\prime}=0$, and
$a^{\prime}=1$. Then, clearly, ${\bf L}_{i}^{dm}\in\mathbf{DMSH}$ and ${\bf
L}_{i}^{dp}\in\mathbf{DPCSH}$. Let $\mathbf{C}^{dm}=\\{{\bf
L}_{i}^{dm}:i=1,2,\dots,10\\}$, $\mathbf{C}^{dp}=\\{{\bf
L}_{i}^{dm}:i=1,2,\dots,10\\}$ and
$\mathbf{C}_{20}=\mathbf{C}^{dm}\cup\mathbf{C}^{dp}$. Finally, let
$\mathbb{DQDSHC}_{3}(=\mathbb{V}(\mathbf{C}_{20})$) and
$\mathbb{DMSHC}_{3}(=\mathbb{V}(\mathbf{C}^{dm})$). The algebras
$\mathbf{L}_{i}^{dm}$, $i=1,2,\dots,10$, can be viewed as logical matrices
with 1 as the distinguished subset. We now present axiomatizations of the
logics corresponding to the 3-valued $\mathbb{DMSH}$-matrices.

The following theorem is immediate from [San11, Lemma 10.2, Theorem 10.3,
Corollary 10.4 and Theorem 11.1].

###### Theorem 8.5

A base for $\mathbb{DQDSHC}_{3}$, relative to $\mathbb{DQDSH}$, is given by:

  1. (i)

$x^{**}\approx x{{}^{*}}^{\prime}$,

  2. (ii)

$x\land x^{\prime}{{}^{*}}^{\prime}\leq y\lor y^{*}$.

The following theorem, which follows from Theorem 4.8, presents an
axiomatization of the logic corresponding to $\mathbb{DMSHC}_{3}$. .

###### Theorem 8.6

The logic $\mathcal{DQDSHC}_{3}$ corresponding to the variety
$\mathbb{DQDSHC}_{3}$ is defined, as an extension of $\mathcal{DQDSH}$, by the
following axioms:

  * •

$((\phi\to\bot)\to\bot)\to_{H}(\phi\to\bot)^{\prime}$,

  * •

$(\phi\to\bot)^{\prime}\to_{H}((\phi\to\bot)\to\bot)$,

  * •

$(\phi\land(\phi^{\prime}\to\bot)^{\prime})\to_{H}(\psi\lor(\psi\to\bot))$.

Since the variety $\mathbb{DQDSHC}_{3}$ is generated by finitely many finite
algebras, the following corollary is now immediate.

###### Corollary 8.7

The logic $\mathcal{DQDSHC}_{3}$ is decidable.

Using the results of Section 4 and the equational axiomatizations obtained in
[San11], we will now present axiomatizations for

  1. (1)

the extension $\mathcal{DMSHC}_{3}$ of the logic $\mathcal{DMSH}$
corresponding to the variety $\mathbb{DMSHC}_{3}$ and

  2. (2)

the extensions corresponding to the varieties ${\bf L}_{i}^{dm}$ generated by
3-element $\mathbb{DMSH}$-chains.

The following theorem is immediate from Theorem 8.5.

###### Theorem 8.8

A base for $\mathbb{DMSHC}_{3}$ relative to $\mathbb{DQDSHC}_{3}$, is given
by:

  1. $x^{\prime\prime}\approx x$.

The following corollary is immediate from Theorem 4.8 and Theorem 8.8 and
presents an axiomatization of the logic corresponding to $\mathbb{DMSHC}_{3}$.

###### Theorem 8.9

The logic $\mathcal{DMSHC}_{3}$ corresponding to the variety
$\mathbb{DMSHC}_{3}$ is defined, as an extension of $\mathcal{DQDSHC}_{3}$ by
the following axioms:

  * •

$((\phi\to\bot)\to\bot)\to_{H}\phi$,

  * •

$\phi\to_{H}((\phi\to\bot)\to\bot)$.

We need to recall another result from [San11] that gives a base for each of
ten 3-chains in $\mathbf{C}^{dm}$. To this end, we need the following
identities:

  1. (C1)

$x\vee(x\to y)\approx(x\to y)^{*}\to x$

  2. (C2)

$x\vee[y\to(x\vee y)]\approx(0\to x)\vee(x\to y)$

  3. (C3)

$x\vee(y\to x)\approx[(x\to y)\to y]\to x$

  4. (C4)

$x\vee(x\to y)\approx x\to[x\vee(y\to 1)]$

  5. (C5)

$(x\to y)\to(0\to y)\approx x\vee[(x\wedge y)\to 1]$

  6. (C6)

$x^{*}\vee(x\to y)\approx(x\vee y)\to y$

  7. (C7)

$x\vee(0\to x)\vee(y\to 1)\approx x\vee[(x\to 1)\to(x\to y)]$

  8. (C8)

$x\vee y\vee(x\to y)\approx x\vee[(x\to y)\to 1]$

  9. (C9)

$x\vee[(0\to y)\to y]\approx x\vee[(x\to 1)\to y]$

  10. (C10)

$x\vee[x\to(y\wedge(0\to y))]\approx x\to[(x\to y)\to y]$

  11. (C11)

$(0\to 1)^{*}=0$

  12. (C12)

$x\vee y\vee[y\to(y\to x)]\approx x\to[x\vee(0\to y)]$

  13. (C13)

$x\vee[y\to(0\to(y\to x))]\approx x\vee y\vee(y\to x)$

  14. (C14)

$x\vee(x\to y)\approx x\vee[(x\to y)\to 1]$

  15. (C15)

$0\to 1\approx 0$ (FTF identity)

  16. (C16)

$x\to y\approx y\to x$ (commutative identity).

In the next theorem, we abbreviate “is a base, relative to
$\mathbb{DMSHC}_{3}$” to just “is a base”. The following theorem is a special
case of [San11, Theorem 11.1].

###### Theorem 8.10

We have

  1. (i)

{(C1)} is a base for the variety $\mathbf{V(L_{1}^{dm})}$,

  2. (ii)

{(C2), (C3)} is a base for the variety $\mathbf{V(L_{2}^{dm})}$,

  3. (iii)

{(C2), (C4)} is a base for the variety $\mathbf{V(L_{3}^{dm})}$,

  4. (iv)

{(C4), (C5)} is a base for the variety $\mathbf{V(L_{4}^{dm})}$,

  5. (v)

{(C7)} is a base for the variety $\mathbf{V(L_{5}^{dm})}$,

  6. (vi)

{(C8)} is a base for the variety $\mathbf{V(L_{6}^{dm})}$,

  7. (vii)

{(C9), (C10)} is a base for the variety $\mathbf{V(L_{7}^{dm})}$,

  8. (viii)

{(C11), (C12)} or {(C11), (C13)} is a base for the variety
$\mathbf{V(L_{8}^{dm})}$,

  9. (ix)

{(C6), (C14), (C15)} is a base for the variety $\mathbf{V(L_{9}^{dm})}$,

  10. (x)

{(C16)} is a base for the variety $\mathbf{V(L_{10}^{dm})}$.

We are now ready to present the axiomatizations of the logics corresponding to
the 3-valued $\mathbb{DMSH}$-matrices. The following theorems will follow as a
consequence of Theorem 4.8, Theorem 8.8 and [San11, Theorem 11.2].

Let $\mathcal{L}(\mathbf{L}_{i}^{dm})$ be the extension of the logic
$\mathcal{L}(\mathbb{C}^{dm})$ corresponding to the variety
$\mathbb{V}(\mathbf{L}_{i}^{dm})$, for $i=1,2,\cdots,10$.

###### Theorem 8.11

$\mathcal{L}(\mathbb{L}_{1}^{dm})$ is defined by the following axioms:

  * •

$[\phi\lor(\phi\to\psi)]\to_{H}[((\phi\to\psi)\to\bot)\to_{H}\phi]$,

  * •

$[((\phi\to\psi)\to\bot)\to_{H}\phi]\to_{H}[\phi\lor(\phi\to\psi)]$

###### Theorem 8.12

$\mathcal{L}(\mathbf{L}_{2}^{dm})$ is defined by the following axioms:

  * •

$[\phi\lor\\{\psi\to(\phi\vee\psi)\\}]\to_{H}[(\bot\to\phi)\vee(\phi\to\psi)]$,

  * •

$[(\bot\to\phi)\vee(\phi\to\psi)]\to_{H}[\phi\lor\\{\psi\to(\phi\vee\psi)\\}]$,

  * •

$[\phi\vee(\psi\to\phi)]\to_{H}[\\{(\phi\to\psi)\to\psi\\}\to\phi]$,

  * •

$[\\{(\phi\to\psi)\to\psi\\}\to\phi]\to_{H}[\phi\vee(\psi\to\phi)]$.

###### Theorem 8.13

$\mathcal{L}(\mathbf{L}_{3}^{dm})$ is defined by the following axioms:

  * •

$[\phi\lor\\{\psi\to(\phi\vee\psi)\\}]\to_{H}[(\bot\to\phi)\vee(\phi\to\psi)]$,

  * •

$[(\bot\to\phi)\vee(\phi\to\psi)]\to_{H}[\phi\lor\\{\psi\to(\phi\vee\psi)\\}]$,

  * •

$[\phi\vee(\phi\to\psi)]\to_{H}[\phi\to\\{\phi\vee(\psi\ \to\top)\\}]$,

  * •

$[\phi\to\\{\phi\vee(\psi\ \to\top)\\}]\to_{H}[\phi\vee(\phi\to\psi)]$.

###### Theorem 8.14

$\mathcal{L}(\mathbf{L}_{4}^{dm})$ is defined by the following axioms:

  * •

$[\phi\vee(\phi\to\psi)]\to_{H}[\phi\to\\{\phi\vee(\psi\ \to\top)\\}]$,

  * •

$[\phi\to\\{\phi\vee(\psi\ \to\top)\\}]\to_{H}[\phi\vee(\phi\to\psi)]$,

  * •

$[(\phi\to\psi)\to(\bot\to\psi)]\to_{H}[\phi\vee\\{(\phi\land\psi)\to\top\\}]$,

  * •

$[\phi\vee\\{(\phi\land\psi)\to\top\\}]\to_{H}[(\phi\to\psi)\to(\bot\to\psi)]$.

###### Theorem 8.15

$\mathcal{L}(\mathbf{L}_{5}^{dm})$ is defined by the following axioms:

  * •

$[\phi\vee(\bot\to\phi)\vee(\psi\to\top)]\to_{H}[\phi\vee\\{(\phi\to\top)\to(\phi\
\to\psi)\\}]$,

  * •

$[\phi\vee\\{(\phi\to\top)\to(\phi\
\to\psi)\\}]\to_{H}[\phi\vee(\bot\to\phi)\vee(\psi\to\top)]$.

###### Theorem 8.16

$\mathcal{L}(\mathbf{L}_{6}^{dm})$ is defined by the following axioms:

  * •

$[\phi\vee\psi\vee(\phi\to\psi)]\to_{H}[\phi\vee\\{(\phi\to\psi)\to\top\\}]$,

  * •

$[\phi\vee\\{(\phi\to\psi)\to\top\\}]\to_{H}[\phi\vee\psi\vee(\phi\to\psi)]$.

###### Theorem 8.17

$\mathcal{L}(\mathbf{L}_{7}^{dm})$ is defined by the following axioms:

  * •

$[\phi\vee\\{(\bot\to\psi)\to\psi\\}]\to_{H}[\phi\vee\\{(\phi\to\top)\to\psi\\}]$,

  * •

$[\phi\vee\\{(\phi\to\top)\to\psi\\}]\to_{H}[\phi\vee\\{(\bot\to\psi)\to\psi\\}]$,

  * •

$[(\phi\vee\\{\phi\to(\psi\land(\bot\to\psi))\\}]\to_{H}[\phi\to\\{(\phi\to\psi)\to\psi\\}]$,

  * •

$[\phi\to\\{(\phi\to\psi)\to\psi\\}]\to_{H}[(\phi\vee\\{\phi\to(\psi\land(\bot\to\psi))\\}]$.

###### Theorem 8.18

$\mathcal{L}(\mathbf{L}_{8}^{dm})$ is defined by the following axioms:

  * •

$((\bot\to\top)\to\bot)\to_{H}\bot$,

  * •

$[(\phi\vee\psi\vee\\{\psi\to(\psi\to\phi)\\}]\to_{H}[\phi\to\\{\phi\vee(\bot\to\psi)\\}]$,

  * •

$[\phi\to\\{\phi\vee(\bot\to\psi)\\}]\to_{H}[(\phi\vee\psi\vee\\{\psi\to(\psi\to\phi)\\}]$.

###### Theorem 8.19

$\mathcal{L}(\mathbf{L}_{9}^{dm})$ is defined by the following axioms:

  * •

$[\phi^{*}\vee(\phi\to\psi)]\to_{H}[(\phi\vee\psi)\to\psi]$,

  * •

$[(\phi\vee\psi)\to\psi]\to_{H}[\phi^{*}\vee(\phi\to\psi)]$,

  * •

$[\phi\vee(\phi\to\psi)]\to_{H}[\phi\vee\\{(\phi\to\psi)\to\top\\}]$,

  * •

$[\phi\vee\\{(\phi\to\psi)\to\top\\}]\to_{H}[\phi\vee(\phi\to\psi)]$,

  * •

$(\bot\to\top)\to_{H}\bot$.

###### Theorem 8.20

$\mathcal{L}(\mathbf{L}_{10}^{dm})$ defined by the following axioms:

  * •

$(\phi\to\psi)\to_{H}(\psi\to\phi)$,

  * •

$(\psi\to\phi)\to_{H}(\phi\to\psi)$.

We conclude this section by mentioning that all the logics mentioned in this
section are decidable as their corresponding varieties are finitely generated.

##  9 $3$-valued Łukasiewicz logic Revisited

It is worthwhile to point out that the logic
$\mathcal{L}\mathbf{(L_{1}^{dm})}$, defined earlier, has an interesting
relationship with the well-known 3-valued Łukasiewicz logic. Let us recall the
definition of 3-valued Łukasiewicz algebras.

An algebra $\mathbf{A}=\langle A,\land,\lor,^{\prime},d_{1},d_{2},0,1\rangle$
is a 3-valued Łukasiewicz algebras if

  1. (1)

$\langle A,\land,\lor,^{\prime},0,1\rangle$ is a De Morgan algebra,

  2. (2)

$d_{i}(x\lor y)=d_{i}(x)\lor d_{i}(y)$, for $i=1,2$,

  3. (3)

$d_{i}(x)\lor(d_{i}(x))^{\prime}=1$, for $i=1,2$,

  4. (4)

$d_{i}(d_{j}(x))=d_{j}(x)$, for $i=1,2$

  5. (5)

$d_{i}(x^{\prime})=(d_{3-i}(x))^{\prime}$, for $i=1,2$,

  6. (6)

$d_{1}(x)\leq d_{2}(x)$,

  7. (7)

If $d_{1}(x)=d_{1}(y)$ and $d_{2}(x)=d_{2}(y)$ then $x=y$.

Let
$\mathbf{{L}}=\langle\\{0,a,1\\},\land,\lor,^{\prime}.d_{1},d_{2},0,1\rangle$
be the algebra such that $\langle\\{0,a,1\\},\land,\lor,^{\prime},0,1\rangle$
is a 3-element Kleene algebra with $0<a<1$, and $d_{1}$ and $d_{2}$ are unary
operations defined as follows: $d_{1}(0)=d_{1}(a)=0,d_{1}(1)=1$, and
$d_{2}(0)=0$, and $d_{2}(1)=d_{2}(a)=1$. Then it is routine to verify that
$\mathbf{{L}}$ is a 3-valued Łukasiewicz algebra. It is well-known that
$\mathbb{V}\mathbf{({L})}$ is precisely the variety of all 3-valued
Łukasiewicz algebras.

###### Theorem 9.1

The logic $\mathcal{L}\mathbf{(L_{1}^{dm})}$ is equivalent to the 3-valued
Łukasiewicz logic.

Proof  It suffices to prove that the variety $\mathbb{V}\mathbf{(L_{1}^{dm})}$
is term-equivalent to the variety $\mathbb{V}\mathbf{({\L})}$. Without loss of
generality, we can assume that $\mathbf{L_{1}^{dm}}$ and $\mathbf{{\L}}$ have
the same universe, say $L=\\{0,a,1\\}$ with $0<a<1$. Given
$\mathbf{L_{1}^{dm}}$, define the unary operations $d_{1}$ and $d_{2}$ on L
by: $d_{1}(x)=x^{\prime*}$ and $d_{2}=x{{}^{*}}^{\prime}$. Then it is
straightforward to verify that $\langle
L;\land,\lor,^{\prime},d_{1},d_{2},0,1\rangle=\mathbf{{\L}}$. To prove the
converse, let us first define the unary function ∗ on $\mathbf{(L}$ by:
$x^{*}=d_{1}((d_{2}(x))^{\prime})$. Using ∗ we can now define the
Katriňák${}^{\prime}s\to$ by:

$x\to y:=(x^{*}\lor y^{**})\land[(x\lor x^{*})^{\prime}{{}^{*}}^{\prime}\lor
x^{*}\lor y\lor y^{*}].$

Then, $\to$ is the Heyting implication (see [Ka73]). Hence, $\langle
L;\land,\lor,\to,^{\prime},0,1\rangle=\mathbf{L_{1}^{dm}}$. The theorem now
follows. $\hfill\square$

##  10 More 3-valued Logics: Extensions of the logic $\mathcal{DPCSH}$

Recall from Section 8.2 that ${\bf L}_{i}^{dp}$, $i=1,2,\dots,10$, are the
expansions of ${\bf L}_{i}$ by adding a unary operation ′ such that
$0^{\prime}=1$, $a^{\prime}=1$ and $1^{\prime}=0$ and $\mathbf{C^{dp}=\\{\bf
L}_{i}^{dp}:i=1,2,\dots,10\\}$. Then, clearly, ${\bf
L}_{i}^{dp}\in\mathbf{DPCSH}$. The algebras $\mathbf{L}_{i}^{dp}$,
$i=1,2,\dots,10$, can be viewed as logical matrices with 1 as the
distinguished subset. Let $\mathbf{C}^{dp}=\\{\mathbf{L}_{i}^{dp}\
:i=1,2,\dots,10\\}$ and let $\mathbb{DPCSHC}_{3}$ denote the variey generaed
by $\mathbf{C}^{dp}$.

We now present axiomatizations of the logics corresponding to the 3-valued
$\mathbb{DPCSH}$-matrices.

The following theorem is immediate from Theorem 8.5.

###### Theorem 10.1

A base for $\mathbb{DPCSHC}_{3}$ relative to $\mathbb{DQDSHC}_{3}$, is given
by:

  1. $x\lor x^{\prime}\approx 1$.

The following theorems will follow from Theorem 4.8 and Theorem 10.1. Let
$\mathcal{DPCSHC}_{3}$ be the extension of the logic $\mathcal{DPCSH}$
corresponding to the variety $\mathbb{DPCSHC}_{3}$.

###### Theorem 10.2

$\mathcal{DPCSHC}_{3}$ is defined, relaive to the logic $\mathcal{DQDSHC}_{3}$
by the following axiom:

  * •

$\phi\lor\phi^{\prime}$.

In the next theorem, we abbreviate “is a base, relative to
$\mathbb{DPCSHC}_{3}$” to just “is a base”. The following theorem is a special
case of [San11, Theorem 11.1]. It refers to the identities given in section 8.

###### Theorem 10.3

We have

  1. (i)

{(C1) is a base for the variety $\mathbf{V(L_{1}^{dp})}$,

  2. (ii)

{(C2), (C3)} is a base for the variety $\mathbf{V(L_{2}^{dp})}$,

  3. (iii)

{(C2), (C4)} is a base for the variety $\mathbf{V(L_{3}^{dp})}$,

  4. (iv)

{(C4), (C5)} is a base for the variety $\mathbf{V(L_{4}^{dp})}$,

  5. (v)

{(C7)} is a base for the variety $\mathbf{V(L_{5}^{dp})}$,

  6. (vi)

{(C8)} is a base for the variety $\mathbf{V(L_{6}^{dp})}$,

  7. (vii)

{(C9), (C10)} is a base for the variety $\mathbf{V(L_{7}^{dp})}$,

  8. (viii)

{(C11), (C12)} or {(C11), (C13)} is a base for the variety
$\mathbf{V(L_{8}^{dp})}$,

  9. (ix)

{(C6), (C14), (C15)} is a base for the variety $\mathbf{V(L_{9}^{dp})}$,

  10. (x)

{(C16)} is a base for the variety $\mathbf{V(L_{10}^{dp})}$.

Let $\mathcal{L}(\mathbf{L_{i}^{dp}})$ be the extension of the logic
$\mathcal{L}(\mathbf{C}^{dp})$ corresponding to the variety
$\mathbb{V}(\mathbf{L}_{i}^{dp})$, for $i=1,2,\cdots,10$.

The following theorems will follow from Theorem 4.8, Theorem LABEL:3dpbases
and Theorem 10.3.

###### Theorem 10.4

$\mathcal{L}(\mathbf{L}_{1}^{dp})$ is defined by the following axioms:

  * •

$[\phi\lor(\phi\to\psi)]\to_{H}[((\phi\to\psi)\to\bot)\to\phi]$,

  * •

$[((\phi\to\psi)\to\bot)\to\phi]\to_{H}[\phi\lor(\phi\to\psi)].$

###### Theorem 10.5

$\mathcal{L}(\mathbf{L}_{2}^{dp})$ is defined by the following axioms:

  * •

$[\phi\lor\\{\psi\to(\phi\vee\psi)\\}]\to_{H}[(\bot\to\phi)\vee(\phi\to\psi)]$,

  * •

$[(\bot\to\phi)\vee(\phi\to\psi)]\to_{H}[\phi\lor\\{\psi\to(\phi\vee\psi)\\}]$,

  * •

$[\phi\vee(\psi\to\phi)]\to_{H}[\\{(\phi\to\psi)\to\psi\\}\to\phi]$,

  * •

$[\\{(\phi\to\psi)\to\psi\\}\to\phi]\to_{H}[\phi\vee(\psi\to\phi)]$.

###### Theorem 10.6

$\mathcal{L}(\mathbf{L}_{3}^{dp})$ is defined by the following two axioms:

  * •

$[\phi\lor\\{\psi\to(\phi\vee\psi)\\}]\to_{H}[(\bot\to\phi)\vee(\phi\to\psi)]$,

  * •

$[(\bot\to\phi)\vee(\phi\to\psi)]\to_{H}[\phi\lor\\{\psi\to(\phi\vee\psi)\\}]$,

  * •

$[\phi\vee(\phi\to\psi)]\to_{H}[\phi\to\\{\phi\vee(\psi\ \to\top)\\}]$,

  * •

$[\phi\to\\{\phi\vee(\psi\ \to\top)\\}]\to_{H}[\phi\vee(\phi\to\psi)]$.

###### Theorem 10.7

$\mathcal{L}(\mathbf{L}_{4}^{dp})$ is defined by the following axioms:

  * •

$[\phi\vee(\phi\to\psi)]\to_{H}[\phi\to\\{\phi\vee(\psi\ \to\top)\\}]$,

  * •

$[\phi\to\\{\phi\vee(\psi\ \to\top)\\}]\to_{H}[\phi\vee(\phi\to\psi)]$,

  * •

$[(\phi\to\psi)\to(\bot\to\psi)]\to_{H}[\phi\vee\\{(\phi\land\psi)\to\top\\}]$,

  * •

$[\phi\vee\\{(\phi\land\psi)\to\top\\}]\to_{H}[(\phi\to\psi)\to(\bot\to\psi)]$.

###### Theorem 10.8

$\mathcal{L}(\mathbf{L}_{5}^{dp})$ is defined by the following axioms:

  * •

$[\phi\vee(\bot\to\phi)\vee(\psi\to\top)]\to_{H}[\phi\vee\\{(\phi\to\top)\to(\phi\
\to\psi)\\}]$,

  * •

$[\phi\vee\\{(\phi\to\top)\to(\phi\
\to\psi)\\}]\to_{H}[\phi\vee(\bot\to\phi)\vee(\psi\to\top)]$.

###### Theorem 10.9

$\mathcal{L}(\mathbf{L}_{6}^{dp})$ is defined by the following two axioms:

  * •

$[\phi\vee\psi\vee(\phi\to\psi)]\to_{H}[\phi\vee\\{(\phi\to\psi)\to\top\\}]$,

  * •

$[\phi\vee\\{(\phi\to\psi)\to\top\\}]\to_{H}[\phi\vee\psi\vee(\phi\to\psi)]$.

###### Theorem 10.10

$\mathcal{L}(\mathbf{L}_{7}^{dp})$ is defined by the following axioms:

  * •

$[\phi\vee\\{(\bot\to\psi)\to\psi\\}]\to_{H}[\phi\vee\\{(\phi\to\top)\to\psi\\}]$,

  * •

$[\phi\vee\\{(\phi\to\top)\to\psi\\}]\to_{H}[\phi\vee\\{(\bot\to\psi)\to\psi\\}]$.

  * •

$[(\phi\vee\\{\phi\to(\psi\land(\bot\to\psi))\\}]\to_{H}[\phi\to\\{(\phi\to\psi)\to\psi\\}]$,

  * •

$[\phi\to\\{(\phi\to\psi)\to\psi\\}]\to_{H}[(\phi\vee\\{\phi\to(\psi\land(\bot\to\psi))\\}]$.

###### Theorem 10.11

$\mathcal{L}(\mathbf{L}_{8}^{dp})$ is defined by the following axioms:

  * •

$((\bot\to\top)\to\bot)\to_{H}\bot$,

  * •

$[(\phi\vee\psi\vee\\{\psi\to(\psi\to\phi)\\}]\to_{H}[\phi\to\\{\phi\vee(\bot\to\psi)\\}]$,

  * •

$[\phi\to\\{\phi\vee(\bot\to\psi)\\}]\to_{H}[(\phi\vee\psi\vee\\{\psi\to(\psi\to\phi)\\}]$.

###### Theorem 10.12

$\mathcal{L}(\mathbf{L}_{9}^{dp})$ is defined by the following axioms:

  * •

$[(\phi\to\bot)\vee(\phi\to\psi)]\to_{H}[(\phi\vee\psi)\to\psi]$,

  * •

$[(\phi\vee\psi)\to\psi]\to_{H}[(\phi\to\bot)\vee(\phi\to\psi)]$,

  * •

$[\phi\vee(\phi\to\psi)]\to_{H}[\phi\vee\\{(\phi\to\psi)\to\top\\}]$,

  * •

$[\phi\vee\\{(\phi\to\psi)\to\top\\}]\to_{H}[\phi\vee(\phi\to\psi)]$.

  * •

$(\bot\to\top)\to_{H}\bot$,

###### Theorem 10.13

$\mathcal{L}(\mathbf{L}_{10}^{dp})$ is defined by the following axioms:

  * •

$(\phi\to\psi)\to_{H}(\psi\to\phi)$,

  * •

$(\psi\to\phi)\to_{H}(\phi\to\psi)$.

We conclude this section by mentioning that all the logics mentioned in this
section are decidable as their corresponding varieties are finitely generated.

##  11 4-valued Logics: Extensions of Dually quasi-De Morgan Boolean semi-
Heyting logic

An algebra $\mathbf{L}=\langle
L,\lor,\land,\to,^{\prime},0,1\rangle\in\mathbb{DQDSH}$ is a dually quasi-De
Morgan Boolean semi-Heyting algebra ($\mathbb{DQDBSH}$, for short) if $\langle
L,\lor,\land,\to,0,1\rangle$ is a Boolean semi-Heyting algebra, that is,
$\mathbf{L}\models x\lor x^{*}\approx 1$. The variety of such algebras is
denoted by $\mathbb{DQDBSH}$.

The following corollary is now immediate, in view of Theorem 4.8.

###### Theorem 11.1

The logic $\mathcal{DQDBSH}$ is defined, relative to $\mathcal{DQDSH}$ by the
following axiom:

  * (B)

$\phi\lor(\phi\to\bot).$

To understand the structure of the variety $\mathbb{DQDBSH}$ better, the
following three algebras will be helpful.

Figure 5 defines the $\to$ operation on the three $\rm{4}$-element algebras
$\mathbf{D_{1}}$, $\mathbf{D_{2}}$ and $\mathbf{D_{3}}$, each of whose lattice
reduct is the 4-element Boolean lattice having the universe $\\{0,a,b,1\\}$,
with $b$ as the complement of $a$, and ′ is defined as follows:
$a^{\prime}=a$, $b^{\prime}=b$, $0^{\prime}=1$ and $1^{\prime}=0$.

\begin{picture}(15.0,4.0)\par\par\put(1.0,2.5){$\bf D_{1}$ \ :}
\par\put(8.5,2.5){$\bf D_{2}$ \ :}
\par\par\put(2.2,2.5){\begin{tabular}[]{c|cccc}$\to$&\ 0&\ $1$&\ $a$&\ $b$\\\
\hline\cr&&&&\\\ 0&1&0&$b$&$a$\\\ &&&&\\\ $1$&0&1&$a$&$b$\\\ &&&&\\\
$a$&$b$&$a$&1&0\\\ &&&&\\\ $b$&$a$&$b$&0&1\\\ \end{tabular} }
\put(9.5,2.5){\begin{tabular}[]{c|cccc}$\to$&\ 0&\ $1$&\ $a$&\ $b$\\\
\hline\cr&&&&\\\ 0&1&1&1&1\\\ &&&&\\\ $1$&0&1&$a$&$b$\\\ &&&&\\\
a&$b$&$1$&1&$b$\\\ &&&&\\\ $b$&$a$&1&$a$&1\\\ \end{tabular} } \end{picture}

$\bf D_{1}$:

$\bf D_{2}$:

$\to$ |  0 |  $1$ |  $a$ |  $b$
---|---|---|---|---
0 | 1 | 0 | $b$ | $a$
$1$ | 0 | 1 | $a$ | $b$
$a$ | $b$ | $a$ | 1 | 0
$b$ | $a$ | $b$ | 0 | 1
$\to$ |  0 |  $1$ |  $a$ |  $b$
---|---|---|---|---
0 | 1 | 1 | 1 | 1
$1$ | 0 | 1 | $a$ | $b$
a | $b$ | $1$ | 1 | $b$
$b$ | $a$ | 1 | $a$ | 1

$\bf D_{3}$:

$\to$ |  0 |  $1$ |  $a$ |  $b$
---|---|---|---|---
0 | 1 | $a$ | 1 | $a$
$1$ | 0 | 1 | $a$ | $b$
$a$ | $b$ | $a$ | 1 | 0
$b$ | $a$ | 1 | $a$ | 1
Figure 3

The algebras $\mathbf{D_{1}}$, $\mathbf{D_{2}}$, and $\mathbf{D_{3}}$ can be
viewed as logical matrices with 1 as the distinguished element. So, we will
now turn our attention to the axiomatization of logics corresponding to the
varieties generated by these matrices. For this purpose it is convenient to
recall the following algebraic results proved in [San11].

The following theorem, which follows immediately from [San11, Corollary 9.4],
reveals the structure of $\mathbb{DQDBSH}$.

###### Theorem 11.2

$\mathbb{DQDBSH}=\mathbb{V}(\mathbf{D_{1},D_{2},D_{3}})=\mathbb{DMBSH}$.

The above theorem leads us to the following decidability result, in view of
Theorem 4.8.

###### Corollary 11.3

The logic $\mathcal{DQDBSH}$ is decidable.

The following theorem is taken from [San11, Theorem 9.5].

###### Theorem 11.4

We have:

  1. (1)

A base for the variety $\mathbb{V}(\mathbf{D_{1}})$, modulo $\mathbb{DQDBSH}$,
is given by
$0\to 1\approx 0$,

  2. (2)

A base for $\mathbb{V}(\mathbf{D_{2}})$, modulo $\mathbb{DQDBSH}$, is given by
$0\to 1\approx 1$,

  3. (3)

A base for the variety $\mathbb{V}(\mathbf{D_{3}})$, modulo $\mathbb{DQDBSH}$,
is given by
$(0\to 1)^{\prime}\approx 0\to 1$.

The following theorems will follow as applications of Theorem 4.8, Theorem
11.2 and Theorem 11.4.

Let $\mathcal{L}(\mathbf{D_{i}})$ denote the extension of the logic
$\mathcal{DMSH}$ corresponding to the variety $\mathbb{V}(\mathbf{D_{i}})$ for
$i=1,2,3$. In what follows we use
$\alpha\leftrightarrow_{H}\beta:=(\alpha\to\beta)\wedge(\beta\to\alpha)$.

###### Theorem 11.5

We have

  1. (1)

The logic $\mathcal{L}(\mathbf{D_{1}})$ is an extension of $\mathcal{DQDBSH}$
defined by the axiom:

     * $(\bot\to\top)\to_{H}\bot$,

  2. (2)

The logic $\mathcal{L}(\mathbf{D_{2}})$ is an extension of $\mathcal{DQDBSH}$
defined by the axiom:

     * $\bot\to\top$.

  3. (3)

The logic $\mathcal{L}(\mathbf{D_{3}})$ is an extension of $\mathcal{DQDBSH}$
defined by the axioms:

     * $(\bot\to\top)^{\prime}\leftrightarrow_{H}(\bot\to\top)$.

It is clear that each of the logics
$\mathcal{L}(\mathbf{D_{i}}),i\in\\{1,2,3\\}$ is decidable.

##  12 Infinite Chains of Logics in the lattice of extensions of the logic
$\mathcal{DHMH}$

Recall that the logic $\mathcal{DHMH}$ corresponds to the variety of dually
hemimorphic Heyting algebras. In this section, we present two infinite chains
of logics that are extensions of the logic $\mathcal{DHMH}$.

###  12.1 Logics corresponding to the subvarities of the variety generated by
all De Morgan Heyting Chains

Let $\mathbb{DMHC}$ denote the subvariety of $\mathbb{DMH}$ generated by the
De Morgan Heyting chains. It is easy to see, using the results of ([San11,
Section 12]) that the subvarieties of the variety $\mathbb{DMHC}$ form an
$\omega+1$-chain. Let $\mathcal{DMHC}$ denote the extension corresponding to
$\mathbb{DMHC}$. Then it follows that the corresponding extensions of
$\mathcal{DMHC}$ form a dual chain.

In this subsection, we present axiomatizations for the logics corresponding to
the subvarieties of $\mathbb{DMHC}$. The following theorem was proved in
[San11, Theorem 12.3].

###### Theorem 12.1

A base for $\mathbf{DMHC}$, relative to $\mathbf{DMSH}$, is given by:

  * (1)

$x{{}^{*}}^{\prime}\approx x^{**}$,

  * (2)

$(x\to y)\lor(y\to x)\approx 1$.

We now give an aximatization for $\mathcal{DMHC}$, relative to the logic
$\mathcal{DMSH}$.

###### Theorem 12.2

The logic $\mathcal{DMHC}$, relative to the logic $\mathcal{DMSH}$ is given by

  * •

$(\alpha\to\bot)^{\prime}\to_{H}((\alpha\to\bot)\to\bot)$,

  * •

$((\alpha\to\bot)\to\bot)\to_{H}(\alpha\to\bot)^{\prime}$,

  * •

$[(\alpha\to\beta)\lor\beta\to\alpha]\to_{H}\top$,

  * •

$\top\to_{H}[(\alpha\to\beta)\lor\beta\to\alpha]$.

Next we will present an axiomatization for the logic corresponding to each
subvariety $\mathbb{DMHC}_{n}$ of $\mathbb{DMHC}$ generated by the $n$-element
$\mathbb{DMH}$-chain, where $n\in\mathbb{N}$.

###### Theorem 12.3

[San11, Theorem 12.4] Let $n\in\omega$ such that $n\geq 2$. Then
$\\{{\rm(DMHC3}_{n}\rm{)}\\}$ is an equational base, mod $\mathbf{DMHC}$, for
$\mathbb{DMHC}_{n}$, where (DMHC3)n is the axiom:

$(\bigvee_{i=1}^{i=n}x_{i})\lor(\bigvee_{i=1}^{i=n-1}(x_{i}\to
x_{i+1}))\approx 1.$

Hence we have the following axiomatization of the logic $\mathcal{DMHC}_{n}$.

###### Theorem 12.4

The logic $\mathcal{DMHC}_{n}$, relative to the logic $\mathcal{DMHC}$ is
given by

$(\bigvee_{i=1}^{i=n}\alpha_{i})\lor(\bigvee_{i=1}^{i=n-1}(\alpha_{i}\to_{H}\alpha_{i+1})).$

###  12.2 Logics corresponding to the variety generated by Dually
Pseudocomplemented Heyting Chains

An algebra $\mathbf{L}=\langle L,\lor,\land,\to,^{\prime},0,1\rangle$, in
which $\langle L,\lor,\land,\to,0,1\rangle$ is a Heyting chain, and the unary
operation ′ is the dual pseudocomplement, is called a $\mathbb{DPCH}$-chain.
It was proved in [San17, Theorem 4.6] (and was implicit in [San11, Section
13]) that the subvarieties of the variety $\mathbb{DPCHC}$ (of dually
pseudocomplemented Heyting algebras) generated by all dually
pseudocomplemented Heyting Chains form an $\omega+1$-chain. We let
$\mathcal{DPCHC}$ denote the extension of $\mathbb{DPCH}$ corresponding to
$\mathbb{DPCHC}$. It then follows that the corresponding extensions of
$\mathcal{DPCHC}$ form a dual chain. In this subsection, we present
axiomatizations for the logics corresponding to the subvarieties of
$\mathbb{DPCHC}$. The following theorem was proved in [San11, Theorem 13.2].
Let $x^{+}:=x^{\prime}{{}^{*}}^{\prime}$.

###### Theorem 12.5

The following form a base, mod $\mathbf{DQDSH}$, for $\mathbf{DPCHC}$.

  1. (i)

$x^{+}\approx x^{\prime}$,

  2. (ii)

$(x\to y)\lor(y\to x)\approx 1$.

###### Corollary 12.6

The logic $\mathcal{DPCHC}$ corresponding to the variety $\mathbb{DPCHC}$ is
defined, relative to the logic $\mathcal{DQDSH}$ by

  1. (i1)

$\alpha^{+}\to_{H}\alpha^{\prime}$,

  2. (i2)

$\alpha^{\prime}\to_{H}\alpha^{+}$,

  3. (ii)

$(\alpha\to\beta)\lor(\beta\to\alpha)$,

where $\alpha^{+}:=(\alpha^{\prime}\to\bot)^{\prime}$.

Let $\mathbb{DPCHC}_{n}$ denote the variety generated by the $n$-element
$\mathbb{DPCSH}$-chain. The following theorem, which follows from [San11,
Theorem 13.3], gives an equational base for each subvariety
$\mathbb{DPCHC}_{n}$ of $\mathbb{DPCHC}$, where $n\in\omega$. Since
$\mathbb{DPCHC}_{2}=\mathbb{V}(\mathbf{2^{e}})$, a base for it is already
given.

###### Theorem 12.7

Let $n\in\omega$ such that $n\geq 3$. Then, $\\{{\rm(An)}\\}$ is an equational
base, mod $\mathbb{DPCHC}$, for $\mathbb{DPCHC}_{n}$, where (An) is the
following axiom:

$\bigvee_{j=1}^{j=n}x_{j}\lor\bigvee_{j=1}^{j=n-1}(x_{j}\to x_{j+1})\approx
1$.

Let $\mathbb{DPCHC_{n}}$ denote the variety generated by the $n$-element
$\mathbb{DPCSH}$-chain.

###### Corollary 12.8

Let $n\in\omega$ such that $n\geq 3$. Then the logic $\mathcal{DPCHC}_{n}$
corresponding to the variety $\mathbb{DPCHC}_{n}$ is defined, relative to the
logic $\mathcal{DPCHC}$ by

  1. ($\Lambda_{n}$)a

$[\bigvee_{j=1}^{j=n}\alpha_{j}\lor\bigvee_{j=1}^{j=n-1}(\alpha_{j}\to\alpha_{j+1})]\to_{H}\top,$

  2. ($\Lambda_{n}$)b

$\top\to_{H}[\bigvee_{j=1}^{j=n}\alpha_{j}\lor\bigvee_{j=1}^{j=n-1}(\alpha_{j}\to\alpha_{j+1})]$.

Note that since $\mathbb{DPCHC}_{2}=\mathbb{V}(\mathbf{2^{e}})$, the
correspnding logic is the classical logic.

##  13 Logics corresponding to subvarieties of regular dually quasi-De Morgan
Stone semi-Heyting algebras

An algebra $\mathbf{A}\in\mathbb{DQDSH}$ is of level 1 if $\mathbf{A}$
satisfies:

$x\land x^{\prime*}\approx(x\land x^{\prime*})^{\prime*},$

where $x^{+}:=x^{\prime}{{}^{*}}^{\prime}$. Let $\mathbb{DQDSH}_{1}$ denote
the variety of $\mathbb{DQDSH}$-algebras of level 1. An algebra
$\mathbf{A}\in\mathbb{DQDSH}$ is called regular if $\mathbf{A}$ satisfies:

$x\land x^{+}\leq y\vee y^{*}.$

Let $\mathbb{DQDS}\rm t\mathbb{SH}$ denote the subvariety of
$\mathbb{DQDSH}$-algebras that satisfy the Stone identity

$(St)\qquad x^{\ast}\vee x^{\ast\ast}\approx 1.$

$\mathbb{DQDS}\rm t\mathbb{SH}_{1}$ denotes the subvariety of
$\mathbb{DQDS}\rm t\mathbb{SH}$ of level 1, while $\mathbb{RDQDS}\rm
t\mathbb{SH}_{1}$ denotes the subvariety consisting of regular members of
$\mathbb{DQDS}\rm t\mathbb{SH}_{1}$.

In this section we present axiomatizations for logics corresponding to several
subvarieties of the variety $\mathbb{RDQDS}\rm t\mathbb{SH}_{1}$ of regular
dually quasi-De Morgan Stone semi-Heyting algebras.

Recall $\mathbf{C}^{dm}$ and $\mathbf{C}^{dp}$ were defined, respectively, in
Section 8 and Section 10. Let
$\mathbf{C}_{20}=\mathbf{C}^{dm}\cup\mathbf{C}^{dp}$. Let
$\mathbb{V}(\mathbf{C}_{20})$ denote the variety generated by
$\mathbf{C}_{20}$. Recall also that the three $4$-element algebras
$\mathbf{D}_{1}$, $\mathbf{D}_{2}$ and $\mathbf{D}_{3}$ were defined in Figure
3. The variety $\mathbb{V}(\mathbf{C}_{20})$ was axiomatized in [San11].

In what follows, $\mathcal{V}$ denotes the logic corresponding to the variety
$\mathbf{V}$ of $\mathbf{DQDSH}$-algebras.

Thus, for example, the logic $\mathcal{DQDS}\mathbf{t}\mathcal{SH}_{1}$
corresponds to the variety $\mathbb{DQDS}\mathbf{t}\mathbb{SH}_{1}$.

###### Theorem 13.1

We have
(a) The logic $\mathcal{DQDS}\mathbf{t}\mathcal{SH}_{1}$ is defined, relative
to the logic $\mathcal{DQDSH}$ by the following axioms:

  * (1)

$(\alpha\wedge\alpha^{\prime*})^{\prime*}\ \to_{H}\
(\alpha\wedge\alpha^{\prime*})$,

  * (2)

$(\alpha\wedge\alpha^{\prime*})\ \to_{H}\
(\alpha\wedge\alpha^{\prime*})^{\prime*},$

  * (3)

$\alpha^{*}\lor\alpha^{**}.$

(b) The logic $\mathcal{RDQDS}\mathbf{t}\mathcal{SH}_{1}$ is defined, relative
to the logic $\mathbb{DQDS}\mathbf{t}\mathbb{SH}_{1}$ by the following axiom:

  * $(\alpha\land\alpha^{+})\ \to_{H}\ (\alpha\lor\alpha^{*})$.

The following result is taken from [San14b, Theorem 3.1].

###### Theorem 13.2

We have

$\displaystyle\mathbb{RDQDS}\rm t\mathbb{SH}_{1}$ $\displaystyle=$
$\displaystyle\mathbb{V}(\mathbf{C}_{20}\cup\\{\mathbf{D}_{1},\mathbf{D}_{2},\mathbf{D}_{3}\\}).$

It follows from Theorem that the variety $\mathbb{RDQDS}\rm t\mathbb{SH}_{1}$
is generated by a finite set of finite algebras. Hence the following corollary
is immediate.

###### Corollary 13.3

The logic $\mathcal{RDQDS}\mathbf{t}\mathcal{SH}_{1}$ is decidable.

In view of the above corollary, the following problem is of interest.

PROBLEM Is the logic $\mathcal{DQDS}\mathbf{t}\mathcal{SH}_{1}$ decidable? In
particular, is the logic $\mathcal{DQDS}\mathbf{t}\mathcal{H}_{1}$ decidable?

Let $\mathbb{RDQDS}t\mathbb{H}_{1}$ denote the subvariety of
$\mathbb{RDQDS}t\mathbb{SH}_{1}$ defined by:

(H) $(x\land y)\to x\approx 1$.

###### Theorem 13.4

The logic $\mathcal{RDQDS}{\rm t}\mathcal{H}_{1}$ is defined, relative to
$\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * (i)

$(\alpha\land\beta)\to\alpha.$

If we restrict the underlying semi-Heyting algebras to Heyting algebras,
Theorem 13.2 simplifies to the following theorem.

###### Theorem 13.5

$\displaystyle\mathbb{RDQDS}{\rm t}\mathbb{H}_{1}$ $\displaystyle=$
$\displaystyle\mathbb{V}(\\{\mathbf{L}^{dm}_{1},\mathbf{L}^{dp}_{1},\mathbf{D}_{2}\\}).$

###### Corollary 13.6

The logic $\mathcal{RDQDS}\rm t\mathcal{H}_{1}$ is dcidable.

In the rest of this section, all the (algebraic) theorems, which give bases to
subvarieties of $\mathbb{RDQDS}\rm t\mathbb{H}_{1}$, are recalled from
[San14b] for the convenience of the reader. Each of the corollaries given
below follow from the immediately preceding theorem and Theorem 4.8.

In the theorems below the reader should interpret “defined by” as “defined,
modulo $\mathbb{RDQDS}\rm t\mathbb{SH}_{1}$, by”.

###### Theorem 13.7

$\mathbb{RDQDS}\rm t\mathbb{H}_{1}$ is also defined, modulo $\mathbb{RDQDS}\rm
t\mathbb{SH}_{1}$, by the “semi-linearity” identity:

  * $(x\to y)\lor(y\to x)\approx 1$.

It is also defined by:

  * $x\lor(x\to y)=(x\to y)^{*}\to x$.

###### Corollary 13.8

The logic $\mathcal{RDQDS}\rm t\mathcal{H}_{1}$ is defined, relative to
$\mathcal{RDQDS}\rm t\mathcal{SH}_{1}$, by

  * $(\alpha\to\beta)\lor(\beta\to\alpha)$,

It is also defined, modulo $\mathcal{RDQDS}\rm t\mathcal{SH}_{1}$, by

  * (1a)

$(\alpha\lor(\alpha\to\beta))\to_{H}((\alpha\to\beta)^{*}\to\alpha)$,

  * (1b)

$((\alpha\to\beta)^{*}\to\alpha)\to_{H}(\alpha\lor(\alpha\to\beta))$.

Let $\mathbb{RDMS}\rm t\mathbb{SH}_{1}$ and $\mathbb{RDMS}\rm t\mathbb{H}_{1}$
denote, respectively, the varieties of regular De Morgan Stone semi-Heyting
algebras and regular De Morgan Stone semi-Heyting algebras. Similarly, The
varieties $\mathbb{RDPCS}\rm t\mathbb{SH}_{1}$ and $\mathbb{RDPCS}\rm
t\mathbb{H}_{1}$ denote, respectively, the varieties of regular dually
pseudocomplemented Stone semi-Heyting algebras and regular De dually
pseudocomplemented Stone Heyting algebras.

###### Corollary 13.9

We have
(a)The logic $\mathcal{RDMS}\rm t\mathcal{SH}_{1}$ is defined, relative to
$\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * (i)

$\alpha^{\prime\prime}\ \to_{H}\alpha$,

  * (ii)

$\alpha\ \to_{H}\alpha^{\prime\prime}$,

(b) The logic $\mathcal{RDPCS}\rm t\mathcal{SH}_{1}$ is defined, relative to
$\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * (i)

$\alpha\ \lor\ \alpha^{\prime}$,

(c) The logic $\mathcal{RDMS}\rm t\mathcal{H}_{1}$ is defined, relative to
$\mathcal{RDMS}\rm t\mathbb{SH}_{1}$, by

  * (1a)

$(\alpha\land\beta)\to_{H}\alpha$,

(d) The logic $\mathcal{RDPCS}\rm t\mathcal{H}_{1}$ is defined, relative to
$\mathcal{RDPCS}{\rm t}\mathcal{SH}_{1}$, by

  * (1a)

$(\alpha\land\beta)\to_{H}\alpha$,

###### Theorem 13.10

We have

  * (a)

$\mathbb{RDMS}{\rm
t}\mathbb{SH}_{1}=\mathbb{V}(\mathbf{C}^{dm})\lor\mathbb{V}(\\{\mathbf{D_{1}},\mathbf{D_{2}},\mathbf{D_{3}}\\})$,

  * (b)

$\mathbb{RDPCS}\rm t\mathbb{SH}_{1}=\mathbb{V}(\mathbf{C}^{dp})$,

  * (c)

$\mathbb{RDMS}\rm
t\mathbb{H}_{1}=\mathbb{V}(\\{\mathbf{L}^{dm}_{1},\mathbf{D}_{2}\\})=\mathbb{V}(\mathbf{L}^{dm}_{1})\lor\mathbb{V}(\mathbf{D}_{2})$,

  * (d)

$\mathbb{RDPCS}\rm t\mathbb{H}_{1}=\mathbb{V}(\mathbf{L}^{dp}_{1})$.

###### Corollary 13.11

The logics $\mathcal{RDMS}{\rm t}\mathbb{SH}_{1}$, $\mathcal{RDPCS}\rm
t\mathcal{SH}_{1}$, $\mathcal{RDMS}\rm t\mathcal{H}_{1}$, and
$\mathcal{RDPCS}\rm t\mathbb{H}_{1}$ are decidable.

Let $\mathbb{RDQD}{\rm cm}\mathbb{S}{\rm t}\mathbb{SH}_{1}$ be the subvariety
of $\mathbb{RDQD}\mathbb{S}{\rm t}\mathbb{SH}_{1}$ defined by the commutative
law: $x\to y\approx y\to x$.

###### Theorem 13.12

We have

  * (a)

$\mathbb{RDQD}{\rm cm}\mathbb{S}{\rm t}\mathbb{SH}_{1}$
=$\mathbb{V}(\mathbf{L}^{dm}_{10})\lor\mathbb{V}(\mathbf{L}^{dp}_{10})\lor\mathbb{V}(\mathbf{D}_{1})$

  * (b)

$\mathbb{RDM}{\rm cm}\mathbb{S}{\rm
t}\mathbb{SH}_{1}=\mathbb{V}(\\{\mathbf{L}^{dm}_{10},\mathbf{D}_{1}\\})$

  * (c)

$\mathbb{RDPC}{\rm cm}\mathbb{S}{\rm
t}\mathbb{SH}_{1}=\mathbb{V}((\mathbf{L}^{dp}_{10})$

  * (d)

$\mathbb{RDM}{\rm cm}\mathbb{S}{\rm t}\mathbb{SH}_{1}\cap\mathbb{RDPC}{\rm
cm}\mathbb{S}{\rm t}\mathbb{SH}_{1}=\mathbb{V}(\bar{2}^{e})$.

###### Corollary 13.13

(a) The logic $\mathcal{RDQD}\rm cm\mathcal{S}{\rm t}\mathcal{H}_{1}$ is
defined, relative to $\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * •

$(\alpha\to\beta)\to_{H}\ (\beta\to\alpha)$.

###### Theorem 13.14

The variety
$\mathbf{V(\\{L^{dm}_{1},L^{dp}_{1},L^{dm}_{3},L^{dp}_{3},D_{2}\\})}$ is
defined by the identity:

  * $(x\to y)\to(0\to y)\approx(x\to y)\to 1$.

###### Corollary 13.15

The logic
$\mathcal{L}(\mathbb{V}\mathbf{(L_{1}^{dm},L_{1}^{dp},L_{3}^{dm},L_{3}^{dp},D_{2}})$
is defined, relative to $\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * (1a)

$[(\alpha\to\beta)\to(\bot\to\beta)]\to_{H}((\alpha\to\beta)\to\top)$,

  * (1b)

$[(\alpha\to\beta)\to\top]\to_{H}[(\alpha\to\beta)\to(\bot\to\beta)]$.

The variety generated by $\mathbf{D}_{1}$ was axiomatized earlier in Section
11. Here are two more bases for it.

###### Theorem 13.16

$\mathbb{V}(\mathbf{D}_{1})$ is defined by

  * $x\to(y\to z)\approx z\to(x\to y)$.

It is also defined by

  * $(x\to y)\to(u\to w)\approx(x\to u)\to(y\to w)$ (Medial Law).

###### Corollary 13.17

The logic $\mathcal{L}(\mathbb{V}\mathbf{(D_{1}}))$ is defined, relative to
$\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * (a)

$\alpha\to(\beta\to\gamma)\to_{H}(\gamma\to(\alpha\to\beta)$,

  * (b)

$(\gamma\to(\alpha\to\beta))\to_{H}(\alpha\to(\beta\to\gamma))$.

It is also defined by

  * (c)

$(\alpha\to\beta)\to(\gamma\to\delta)\to_{H}((\alpha\to\gamma)\to(\beta\to\delta))$,

  * (d)

$((\alpha\to\gamma)\to(\beta\to\delta))\to_{H}(\alpha\to\beta)\to(\gamma\to\delta)$.

###### Theorem 13.18

The variety
$\mathbf{V(\\{L^{dm}_{1},L^{dp}_{1},L^{dm}_{2},L^{dp}_{2},D_{2}\\})}$ is
defined by:

  * $y\leq x\to y$.

It is also defined by:

  * $[(x\to y)\to y]\to(x\to y)\approx x\to y$.

It is also defined by

  * $x\to(y\to z)\approx(x\to y)\to(x\to z)$ (Left distributive law).

###### Corollary 13.19

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{1},L^{dp}_{1},L^{dm}_{2},L^{dp}_{2},D_{2}\\})$
is defined, relative to $\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * $\beta\land(\alpha\to\beta)$.

It is also defined by:

  * (a)

$[(\alpha\to\beta)\to\beta]\to(\alpha\to\beta)\to_{H}(\alpha\to\beta)$,

  * (b)

$(\alpha\to\beta)\to_{H}[(\alpha\to\beta)\to\beta]\to(\alpha\to\beta)]$.

It is also defined by

  * (a)

$\alpha\to(\beta\to\gamma)\to_{H}[(\alpha\to\beta)\to(\alpha\to\gamma)$,

  * (b)

$[(\alpha\to\beta)\to(\alpha\to\gamma)\to_{H}(\alpha\to(\beta\to\gamma))$.

###### Theorem 13.20

The variety
$\mathbf{V(\\{L^{dm}_{1},L^{dp}_{1},L^{dm}_{2},L^{dp}_{2},L^{dm}_{5},L^{dp}_{5},L^{dm}_{6},L^{dp}_{6},D_{2})\\}}$
is defined by:

  * $[x\to(y\to x)]\to x\approx x$,

###### Corollary 13.21

The logic
$\mathcal{L}(\mathbf{V(\\{L^{dm}_{1},L^{dp}_{1},L^{dm}_{2},L^{dp}_{2},L^{dm}_{5},L^{dp}_{5},L^{dm}_{6},L^{dp}_{6},D_{2})\\}})$
is defined, relative to $\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * (a)

$[[\alpha\to(\beta\to\alpha)]\to\alpha]\to_{H}\alpha$,

  * (b)

$\alpha\to_{H}[\alpha\to(\beta\to\alpha)]\to\alpha]$.

###### Theorem 13.22

$\mathbf{V(\\{L^{dp}_{1},L^{dp}_{2},L^{dp}_{5},L^{dp}_{6}\\})}$ is defined by:

  * (1)

$[x\to(y\to x)]\to x\approx x$

  * (2)

$x\lor x^{\prime}\approx 1$.

###### Corollary 13.23

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dp}_{1},L^{dp}_{2},L^{dp}_{5},L^{dp}_{6}\\})$ is
defined, relative to $\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * (1a)

$[\\{\alpha\to(\beta\to\alpha)\\}\to\alpha]\to_{H}\alpha$

  * (1b)

$\alpha\to_{H}[\\{\alpha\to(\beta\to\alpha)\\}\to\alpha]$.

  * (2a)

$\alpha\lor\alpha^{\prime}$.

###### Theorem 13.24

$\mathbf{V(\\{L^{dp}_{1},L^{dp}_{2},L^{dp}_{5},L^{dp}_{6}\\})}$ is defined by:

  * (1)

$[x\to(y\to x)]\to x\approx x$

  * (2)

$x^{\prime\prime}\approx x$.

###### Corollary 13.25

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dp}_{1},L^{dp}_{2},L^{dp}_{5},L^{dp}_{6}\\})$ is
defined, relative to $\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * (1a)

$[\\{\alpha\to(\beta\to\alpha)\\}\to\alpha]\to_{H}\alpha$,

  * (1b)

$\alpha\to_{H}[\\{\alpha\to(\beta\to\alpha)\\}\to\alpha]$,

  * (2a)

$\alpha^{\prime\prime}\to_{H}\alpha$,

  * (2b)

$\alpha\to_{H}\alpha^{\prime\prime}$.

Recall that $x^{+}:=x^{\prime}{{}^{*}}^{\prime}$.

###### Theorem 13.26

The variety
$\mathbf{V(\\{L^{dm}_{1},L^{dp}_{1},L^{dm}_{2},L^{dp}_{2},L^{dm}_{3},L^{dp}_{3},L^{dm}_{4},L^{dp}_{4},L^{dm}_{5},L^{dm}_{6},L^{dm}_{7},L^{dm}_{8}},D_{2},D_{3}\\})$
is defined by the identity:

  * $(0\to 1)^{+}\to(0\to 1)^{\prime}\approx 0\to 1$.

###### Corollary 13.27

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{1},L^{dp}_{1},L^{dm}_{2},L^{dp}_{2},L^{dm}_{3},L^{dp}_{3},L^{dm}_{4},L^{dp}_{4},L^{dm}_{5},L^{dm}_{6},L^{dm}_{7},L^{dm}_{8}$
$\mathbf{D_{2},D_{3}\\}))}$ is defined, relative to $\mathcal{RDQDS}{\rm
t}\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]\to_{H}(\bot\to\top)$,

  * (1b)

$(\bot\to\top)\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]$.

###### Theorem 13.28

The variety $\mathbf{V(\\{L^{dp}_{1},L^{dp}_{2},L^{dp}_{3},L^{dp}_{4}\\})}$ is
defined by the identities:

  * (1)

$(0\to 1)^{+}\to(0\to 1)^{\prime}\approx 0\to 1$

  * (2)

$x\lor x^{\prime}\approx 1$.

###### Corollary 13.29

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dp}_{1},L^{dp}_{2},L^{dp}_{3},L^{dp}_{4}\\}))$
is defined, relative to $\mathcal{RDQDS}{\rm t}\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]\to_{H}(\bot\to\top)$,

  * (1b)

$(\bot\to\top)\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]$,

  * (2a)

$\alpha\lor\alpha^{\prime}$.

###### Theorem 13.30

The variety
$\mathbf{V(\\{L^{dm}_{1},L^{dm}_{2},L^{dm}_{3},L^{dm}_{4},L^{dm}_{5},L^{dm}_{6},L^{dm}_{7},L^{dm}_{8}},$
$\mathbf{D_{2},D_{3}\\})}$ is defined by the identities:

  * (1)

$(0\to 1)^{+}\to(0\to 1)^{\prime}\approx 0\to 1$

  * (2)

$x^{\prime\prime}\approx x$.

###### Corollary 13.31

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{1},L^{dm}_{2},L^{dm}_{3},L^{dm}_{4},L^{dm}_{5},L^{dm}_{6},L^{dm}_{7},L^{dm}_{8},$
$\mathbf{D_{2},D_{3}\\}))}$ is defined, relative to
$\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]\to_{H}(\bot\to\top)$,

  * (1b)

$(\bot\to\top)\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]$,

  * (2a)

$\alpha^{\prime\prime}\to_{H}\alpha$,

  * (2b)

$\alpha\to_{H}\alpha^{\prime\prime}$.

###### Theorem 13.32

The variety $\mathbf{V(\\{L^{dm}_{5},L^{dm}_{6},L^{dm}_{7},L^{dm}_{8}},$
$\mathbf{D_{3}\\})}$ is defined by the identity:

  * $(0\to 1)^{+}\to(0\to 1)\approx(0\to 1)^{\prime}$.

###### Corollary 13.33

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{5},L^{dm}_{6},L^{dm}_{7},L^{dm}_{8},D_{2}\\})$
is defined, relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{+}\to(\bot\to\top)]\to_{H}(\bot\to\top)^{\prime}$,

  * (1b)

$(\bot\to\top)^{\prime}\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)]$.

$\mathbf{V(D_{3})}$ was axiomatized in [San11]. Here is another base for it.

###### Theorem 13.34

$\mathbf{V(D_{3})}$ is defined by the identities:

  * (1)

$(0\to 1)^{+}\to(0\to 1)\approx(0\to 1)^{\prime}$

  * (2)

$x\lor x^{*}\approx 1$.

###### Corollary 13.35

The logic $\mathcal{L}(\mathbb{V}(\\{D_{3}\\})$ is defined, relative to
$\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$(\bot\to\top)^{+}\to(\bot\to\top)\to_{H}(\bot\to\top)^{\prime}$,

  * (1b)

$(\bot\to\top)^{\prime}\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)]$.

  * (2a)

$\alpha\lor\alpha^{\prime}$.

###### Theorem 13.36

The variety generated by the algebras $\mathbf{L^{dm}_{1},L^{dm}_{2},}$
$\mathbf{L^{dm}_{3},L^{dm}_{4},}$ $\mathbf{D_{2},D_{3}}$ is defined by the
identities:

  * (1)

$(0\to 1)^{+}\to(0\to 1)^{\prime}\approx(0\to 1)$

  * (2)

$(0\to 1)^{+}\to(0\to 1){{}^{*}}^{\prime}{{}^{*}}\approx(0\to 1)$

  * (3)

$x^{\prime\prime}\approx x$.

###### Corollary 13.37

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{1},L^{dm}_{2},L^{dm}_{3},L^{dm}_{4},D_{2},D_{3}\\})$
is defined, relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}\to_{H}(\bot\to\top)$,

  * (1b)

$(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}\to_{H}(\bot\to\top)$,

  * (2a)

$[(\bot\to\top)^{+}\to(\bot\to\top){{}^{*}}^{\prime}{{}^{*}}]\to_{H}(\bot\to\top)$,

  * (2b)

$(\bot\to\top)\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top){{}^{*}}^{\prime}{{}^{*}}]$

  * (3a)

$\alpha^{\prime\prime}\to_{H}\alpha$,

  * (3b)

$\alpha\to_{H}\alpha^{\prime\prime}$.

###### Theorem 13.38

The variety generated by the algebras
$\mathbf{L^{dm}_{5},L^{dp}_{5},L^{dm}_{6},L^{dp}_{6},L^{dm}_{7},L^{dp}_{7},L^{dm}_{8},L^{dp}_{8},L^{dp}_{9},L^{dm}_{9},L^{dp}_{10},L^{dm}_{10}},$
$\mathbf{D_{1},D_{3}}$ is defined by the identity:

  * $(0\to 1)^{+}\to(0\to 1)^{\prime}\approx(0\to 1)^{\prime}$.

###### Corollary 13.39

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{5},L^{dp}_{5},L^{dm}_{6},L^{dp}_{6},L^{dm}_{7},L^{dp}_{7},L^{dm}_{8},L^{dp}_{8},\\\
L^{dm}_{9},L^{dp}_{9},L^{dm}_{10},L^{dp}_{10},D_{2},D_{3}\\})$ is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]\to_{H}(\bot\to\top)^{\prime}$,

  * (1b)

$(\bot\to\top)^{\prime}\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]$.

###### Theorem 13.40

The variety generated by the algebras
$\mathbf{L^{dp}_{5},L^{dp}_{6},L^{dp}_{7},L^{dp}_{8},L^{dp}_{9},L^{dp}_{10}}$
is defined by the identities:

  * (1)

$(0\to 1)^{+}\to(0\to 1)^{\prime}\approx(0\to 1)^{\prime}$

  * (2)

$x\lor x^{\prime}\approx 1$.

###### Corollary 13.41

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dp}_{5},L^{dp}_{6},L^{dp}_{7},L^{dp}_{8},L^{dp}_{9},L^{dp}_{10}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]\to_{H}(\bot\to\top)^{\prime}$,

  * (1b)

$(\bot\to\top)^{\prime}\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]$.

  * (2a)

$\alpha\lor\alpha^{\prime}$

###### Theorem 13.42

The variety generated by the algebras
$\mathbf{L^{dm}_{5},L^{dm}_{6},L^{dm}_{7},L^{dm}_{8},L^{dm}_{9},L^{dm}_{10}}$,
$\mathbf{D_{1},D_{3}}$ is defined by the identities:

  * (1)

$(0\to 1)^{+}\to(0\to 1)^{\prime}\approx(0\to 1)^{\prime}$.

  * (2)

$x^{\prime\prime}\approx x$.

###### Corollary 13.43

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{5},L^{dm}_{6},L^{dm}_{7},L^{dm}_{8},L^{dm}_{9},L^{dm}_{10},D_{1},D_{3}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]\to_{H}(\bot\to\top)^{\prime}$,

  * (1b)

$(\bot\to\top)^{\prime}\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]$.

  * (2a)

$\alpha^{\prime\prime}\to_{H}\alpha$,

  * (2b)

$\alpha\to_{H}\alpha^{\prime\prime}$.

###### Theorem 13.44

The variety generated by the algebras
$\mathbf{L^{dm}_{5},L^{dm}_{6},L^{dm}_{7},L^{dm}_{8}}$, $\mathbf{D_{3}}$ is
defined by the identities:

  * (1)

$(0\to 1)^{+}\to(0\to 1)^{\prime}\approx(0\to 1)^{\prime}$

  * (2)

$(0\to 1)^{+}\to(0\to 1)^{\prime}\approx(0\to 1)$.

It is also defined by

  * $(0\to 1)^{\prime}\approx 0\to 1$.

###### Corollary 13.45

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{5},L^{dm}_{6},L^{dm}_{7},L^{dm}_{8},D_{3}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]\to_{H}(\bot\to\top)^{\prime}$,

  * (1b)

$(\bot\to\top)^{\prime}\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]$.

  * (2a)

$[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]\to_{H}(\bot\to\top)$,

  * (2b)

$(\bot\to\top)\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]$.

###### Theorem 13.46

The variety generated by the algebras
$\mathbf{D_{1},D_{3}}$ is defined by the identities:

  * (1)

$(0\to 1)^{+}\to(0\to 1)^{\prime}\approx(0\to 1)^{\prime}$,

  * (2)

$x\lor x^{*}\approx 1$.

###### Corollary 13.47

The logic $\mathcal{L}(\mathbb{V}(\\{D_{1},D_{3}\\})$ is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]\to_{H}(\bot\to\top)^{\prime}$,

  * (1b)

$(\bot\to\top)^{\prime}\to_{H}[(\bot\to\top)^{+}\to(\bot\to\top)^{\prime}]$,

  * (2a)

$\alpha\lor\alpha^{\prime}$.

###### Theorem 13.48

The variety generated by the algebras
$\mathbf{L^{dm}_{1},L^{dp}_{1},L^{dm}_{2},L^{dp}_{2},L^{dm}_{3},L^{dp}_{3},L^{dm}_{4},L^{dp}_{4},L^{dp}_{5},\\\
L^{dp}_{6},L^{dp}_{7},L^{dp}_{8}}$,
$\mathbf{L^{dm}_{9},L^{dp}_{9},L^{dm}_{10},L^{dp}_{10},D_{1},D_{2}}$ is
defined by the identity:

  * $(0\to 1)^{\prime}\to(0\to 1)\approx 0\to 1$.

###### Corollary 13.49

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{1},L^{dp}_{1},L^{dm}_{2},L^{dp}_{2},L^{dm}_{3},L^{dp}_{3},L^{dm}_{4},L^{dp}_{4},L^{dp}_{5},L^{dp}_{6},L^{dp}_{7},L^{dp}_{8},\\\
L^{dm}_{9},L^{dp}_{9},L^{dm}_{10},L^{dp}_{10},D_{1},D_{2}\\})$ is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (a)

$[(\bot\to\top)^{\prime}\to(\bot\to\top)]\to_{H}(\bot\to\top)$,

  * (b)

$(\bot\to\top)\to_{H}[(\bot\to\top)^{\prime}\to(\bot\to\top)]$

###### Theorem 13.50

The variety generated by the algebras
$\mathbf{L^{dm}_{1},L^{dm}_{2},L^{dm}_{3},L^{dm}_{4},L^{dm}_{9},L^{dm}_{10}},$
$\mathbf{D_{1},D_{2}}$ is defined by the identities:

  * (1)

$(0\to 1)^{\prime}\to(0\to 1)\approx 0\to 1$.

  * (2)

$x^{\prime\prime}\approx x$.

###### Corollary 13.51

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{1},L^{dm}_{2},L^{dm}_{3},L^{dm}_{4},L^{dm}_{9},L^{dm}_{10},D_{1},D_{2}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{\prime}\to(\bot\to\top)]\to_{H}(\bot\to\top)$,

  * (1b)

$(\bot\to\top)\to_{H}[(\bot\to\top)^{\prime}\to(\bot\to\top)]$,

  * (2a)

$\alpha^{\prime\prime}\to_{H}\alpha$,

  * (2b)

$\alpha\to_{H}\alpha^{\prime\prime}$.

###### Theorem 13.52

The variety generated by the algebras
$\mathbf{D_{1},D_{2}}$ is defined by the identities:

  * (1)

$(0\to 1)^{\prime}\to(0\to 1)\approx 0\to 1$

  * (2)

$x\lor x^{*}\approx 1$.

###### Corollary 13.53

The logic $\mathcal{L}(\mathbb{V}(\\{D_{1},D_{2}\\})$ is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{\prime}\to(\bot\to\top)]\to_{H}(\bot\to\top)$,

  * (1b)

$(\bot\to\top)\to_{H}[(\bot\to\top)^{\prime}\to(\bot\to\top)]$,

  * (2a)

$\alpha\lor\alpha^{\prime}$

###### Theorem 13.54

The variety generated by the algebras
$\mathbf{L_{1}^{dm},L_{1}^{dp},L_{3}^{dm},L_{3}^{dp},L_{6}^{dm},L_{6}^{dp},L_{8}^{dm},L_{8}^{dp},D_{1},D_{2},D_{3}}$
is defined by the identity:

  * $x\lor[y\to(x\lor y)]\approx(0\to x)\lor x\lor y$.

###### Corollary 13.55

The logic
$\mathcal{L}(\mathbb{V}(\\{L_{1}^{dm},L_{1}^{dp},L_{3}^{dm},L_{3}^{dp},\\\
L_{6}^{dm},L_{6}^{dp},L_{8}^{dm},L_{8}^{dp},D_{1},D_{2},D_{3}\\})$ is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (a)

$[\alpha\lor\\{\beta\to(\alpha\lor\beta)\\}]\to_{H}[(\bot\to\top)\lor\alpha\lor\beta]$,

  * (b)

$[(\bot\to\top)\lor\alpha\lor\beta]\to_{H}[\alpha\lor\\{\beta\to(\alpha\lor\beta)\\}]$.

###### Theorem 13.56

The variety generated by the algebras
$\mathbf{L^{dm}_{2},L^{dp}_{2},L^{dm}_{5},L^{dp}_{5},D_{2}}$ is defined by the
identity:

  * $x\lor(y\to x)\approx[(x\to y)\to y]\to x$.

###### Corollary 13.57

The logic
$\mathcal{L}(\mathbb{V}(\\{L_{2}^{dm},L_{2}^{dp},L_{5}^{dm},L_{5}^{dp},D_{2}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor(\beta\to\alpha)]\to_{H}[\\{(\alpha\to\beta)\to\beta\\}\to\alpha]$,

  * (1b)

$[\\{(\alpha\to\beta)\to\beta\\}\to\alpha]\to_{H}[\alpha\lor(\beta\to\alpha)]$.

###### Theorem 13.58

The variety generated by the algebras
$\mathbf{L^{dm}_{3},L^{dp}_{3},L^{dm}_{4},L^{dp}_{4},D_{1},D_{2},D_{3}}$ is
defined by the identity:

  * $x\lor(x\to y)\approx x\to[x\lor(y\to 1)]$.

###### Corollary 13.59

The logic
$\mathcal{L}(\mathbb{V}(\\{L_{3}^{dm},L_{3}^{dp},L_{4}^{dm},L_{4}^{dp},D_{1},D_{2},D_{3}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor(\alpha\to\beta)]\to_{H}[\alpha\to\\{\alpha\lor(\beta\to\top)\\}]$,

  * (1b)

$[\alpha\to\\{\alpha\lor(\beta\to\top)\to_{H}[\alpha\lor(\alpha\to\beta)]\\}]$.

###### Theorem 13.60

The variety generated by the algebras
$\mathbf{L^{dm}_{5},L^{dp}_{6},L^{dm}_{7},L^{dp}_{8},D_{3}}$ is defined by the
identity:

  * $(0\to 1)^{*}\to(0\to 1)\approx(0\to 1)^{\prime}$.

###### Corollary 13.61

The logic
$\mathcal{L}(\mathbb{V}(\\{L_{5}^{dm},L_{6}^{dp},L_{7}^{dm},L_{8}^{dp},D_{3}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[(\bot\to\top)^{*}\to(\bot\to\top)]\to_{H}(\bot\to\top)^{\prime}$,

  * (1b)

$(\bot\to\top)^{\prime}\to_{H}[(\bot\to\top)^{*}\to(\bot\to\top)]$.

###### Theorem 13.62

The variety generated by the algebras
$\mathbf{L^{dm}_{1},L^{dp}_{1},L^{dm}_{2},L^{dp}_{2},L^{dm}_{3},L^{dp}_{3}}$,
$\mathbf{L^{dm}_{4},L^{dp}_{4},D_{2}}$ is defined by the
identity:

  * $0\to 1\approx 1$ (FTT identity).

###### Corollary 13.63

The logic
$\mathcal{L}(\mathbb{V}(\\{L_{1}^{dm},L_{1}^{dp},L_{2}^{dm},L_{2}^{dp},L_{3}^{dm},L_{3}^{dp},L_{4}^{dm},L_{4}^{dp},D_{2}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$\bot\to\top$ (FTT identity).

###### Theorem 13.64

The variety generated by the algebras
$\mathbf{L^{dm}_{1},L^{dp}_{1},L^{dm}_{3},L^{dp}_{3},L^{dm}_{6},L^{dp}_{6}}$,
$\mathbf{L^{dm}_{8},L^{dp}_{8},D_{1},D_{2},D_{3}}$ is
defined by the identity:

  * $x\lor(y\to x)\approx(x\lor y)\to x$.

###### Corollary 13.65

The logic
$\mathcal{L}(\mathbb{V}(\\{L_{1}^{dm},L_{1}^{dp},L_{3}^{dm},L_{3}^{dp},L_{6}^{dm},L_{6}^{dp},L_{8}^{dm},L_{8}^{dp},D_{1},D_{2},D_{3}\\})$
is defined, relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor(\beta\to\alpha)]\to_{H}[(\alpha\lor\beta)\to\alpha]$,

  * (1b)

$[(\alpha\lor\beta)\to\alpha]\to_{H}[\alpha\lor(\beta\to\alpha)]$.

###### Theorem 13.66

The variety generated by the algebras
$\mathbf{L^{dp}_{1},L^{dp}_{3},L^{dp}_{6}}$, $\mathbf{L^{dp}_{8}}$ is defined
by the identities:

  * (1)

$x\lor(y\to x)\approx(x\lor y)\to x$

  * (2)

$x\lor x^{\prime}\approx 1$.

###### Corollary 13.67

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dp}_{1},L^{dp}_{3},L^{dp}_{6},L^{dp}_{8}\\})$ is
defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor(\beta\to\alpha)]\to_{H}[(\alpha\lor\beta)\to\alpha]$,

  * (1b)

$[(\alpha\lor\beta)\to\alpha]\to_{H}[\alpha\lor(\beta\to\alpha)]$

  * (2a)

$\alpha\lor\alpha^{\prime}$.

###### Theorem 13.68

The variety generated by the algebras
$\mathbf{L^{dm}_{1},L^{dm}_{3},L^{dm}_{6}}$,
$\mathbf{L^{dm}_{8},D_{1},D_{2},D_{3}}$ is defined by the identities:

  * (1)

$x\lor(y\to x)\approx(x\lor y)\to x$.

  * (2)

$x^{\prime\prime}\approx x$.

###### Corollary 13.69

The logic
$\mathcal{L}(\mathbb{V}(\\{L_{1}^{dm},L_{3}^{dm},L_{6}^{dm},L_{8}^{dm},D_{1},D_{2},D_{2}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor(\beta\to\alpha)]\to_{H}[(\alpha\lor\beta)\to\alpha]$,

  * (1b)

$[(\alpha\lor\beta)\to\alpha]\to_{H}[\alpha\lor(\beta\to\alpha)]$

  * (2a)

$\alpha^{\prime\prime}\to_{H}\alpha$,

  * (2b)

$\alpha\to_{H}\alpha^{\prime\prime}$.

###### Theorem 13.70

The variety generated by the algebras
$\mathbf{L^{dm}_{1},L^{dp}_{1},L^{dm}_{2},L^{dp}_{2},L^{dm}_{5},L^{dp}_{5},L^{dm}_{6},L^{dp}_{6},L^{dm}_{9},L^{dp}_{9}}$,
$\mathbf{D_{1},D_{2},D_{3}}$ is defined by the identity:

  * $x^{*}\lor(x\to y)\approx(x\lor y)\to y$.

###### Corollary 13.71

The logic
$\mathcal{L}(\mathbb{V}(\\{L_{1}^{dm},L^{dp}_{1},L_{2}^{dm},L^{dp}_{2},L^{dm}_{5},L^{dp}_{5},\\\
L_{6}^{dm},L^{dp}_{6},L_{9}^{dm},L_{9}^{dp},D_{1},D_{2},D_{2}\\})$ is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha^{*}\lor(\alpha\to\beta)]\to_{H}[(\alpha\lor\beta)\to\beta]$,

  * (1b)

$[(\alpha\lor\beta)\to\beta]\to_{H}[\alpha^{*}\lor(\alpha\to\beta)]$.

###### Theorem 13.72

$\mathbf{V(\\{L^{dp}_{1},L^{dp}_{2},L^{dp}_{5},L^{dp}_{6},L^{dp}_{9}\\})}$ is
defined by the identity:

  * (1)

$x^{*}\lor(x\to y)\approx(x\lor y)\to y$

  * (2)

$x\lor x^{\prime}\approx 1$.

###### Corollary 13.73

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dp}_{1},L^{dp}_{2},L^{dp}_{5},L^{dp}_{6},L_{9}^{dp}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha^{*}\lor(\alpha\to\beta)]\to_{H}[(\alpha\lor\beta)\to\beta]$,

  * (1b)

$[(\alpha\lor\beta)\to\beta]\to_{H}[\alpha^{*}\lor(\alpha\to\beta)]$,

  * (2a)

$\alpha\lor\alpha^{\prime}$.

###### Theorem 13.74

The variety generated by the algebras
$\mathbf{L^{dm}_{1},L^{dm}_{2},L^{dm}_{5},L^{dm}_{6},L^{dm}_{9}}$,
$\mathbf{D_{1},D_{2},D_{3}}$ is defined by the identity:

  * (1)

$x^{*}\lor(x\to y)\approx(x\lor y)\to y$.

  * (2)

$x^{\prime\prime}\approx x$.

###### Corollary 13.75

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{1},L^{dm}_{2},L^{dm}_{5},L^{dm}_{6},L_{9}^{dm},D_{1},D_{2},D_{3}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha^{*}\lor(\alpha\to\beta)]\to_{H}[(\alpha\lor\beta)\to\beta]$,

  * (1b)

$[(\alpha\lor\beta)\to\beta]\to_{H}[\alpha^{*}\lor(\alpha\to\beta)]$,

  * (2a)

$\alpha^{\prime\prime}\to_{H}\alpha$,

  * (2b)

$\alpha\to_{H}\alpha^{\prime\prime}$.

###### Theorem 13.76

The variety generated by the algebras
$\mathbf{L^{dm}_{5},L^{dp}_{5},D_{2}}$ is defined by the identity:

  * $x\lor(0\to x)\lor(y\to 1)\approx x\lor[(x\to 1)\to(x\to y)]$.

###### Corollary 13.77

The logic $\mathcal{L}(\mathbb{V}(\\{L^{dm}_{5},L^{dp}_{5},D_{2}\\})$ is
defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor(\bot\to\alpha)\lor(\beta\to\top)]\to_{H}[\alpha\lor[(\alpha\to\top)\to(\alpha\to\beta)]$.

  * (1b)

$[\alpha\lor[(\alpha\to\top)\to(\alpha\to\beta)]\to_{H}[\alpha\lor(\bot\to\alpha)\lor(\beta\to\top)]$.

###### Theorem 13.78

The variety generated by the algebras
$\mathbf{L^{dm}_{6},L^{dp}_{6},D_{2}}$ defined by the identity:

  * $x\lor y\lor(x\to y)\approx x\lor[(x\to y)\to 1]$.

###### Corollary 13.79

The logic $\mathcal{L}(\mathbb{V}(\\{L^{dm}_{6},L^{dp}_{6},D_{2}\\})$ is
defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$\alpha\lor\beta\lor(\alpha\to\beta)\to_{H}\alpha\lor[(\alpha\to\beta)\to\top]$,

  * (1b)

$\alpha\lor\beta\lor(\alpha\to\beta)\to_{H}\alpha\lor[(\alpha\to\beta)\to\top]$.

###### Theorem 13.80

The variety generated by the algebras
$\mathbf{L^{dm}_{1},L^{dp}_{1},L^{dm}_{7},{L^{dp}_{7},D_{2}}}$ is defined by
the identity:

  * $x\lor[(0\to y)\to y]\approx x\lor[(x\to 1)\to y]$.

###### Corollary 13.81

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{1},L^{dp}_{1},L^{dm}_{7},L^{dp}_{7},D_{2}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor\\{(\bot\to\beta)\to\beta\\}]\to_{H}[\alpha\lor[(\alpha\to\top)\to\beta]$,

  * (1b)

$[\alpha\lor[(\alpha\to\top)\to\beta]\to_{H}[\alpha\lor\\{(\bot\to\beta)\to\beta\\}]$.

###### Theorem 13.82

The variety generated by the algebras
$\mathbf{L^{dm}_{7},L^{dp}_{7},L^{dm}_{8},L^{dp}_{8},D_{1},D_{2},D_{3}}$
defined by the identity:

  * $x\ \lor[x\to(y\land(0\to y))]\approx x\to[(x\to y)\to y]$.

###### Corollary 13.83

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{7},L^{dp}_{7},L^{dm}_{8},L^{dp}_{8},D_{1},D_{2},D_{3}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor[\alpha\to\\{\beta\land(\bot\to\beta)\\}]]\to_{H}[\alpha\to[(\alpha\to\beta)\to\beta]$,

  * (1b)

$[\alpha\to[(\alpha\to\beta)\to\beta]\to_{H}[\alpha\lor[\alpha\to\\{\beta\land(\bot\to\beta)\\}]]$.

###### Theorem 13.84

The variety generated by the algebras
$\mathbf{L^{dm}_{8},L^{dp}_{8},D_{1},D_{2},D_{3}}$ defined by the identity:

  * $x\lor y\lor[y\to(y\to x)]\approx x\to[x\lor(0\to y)]$.

It is also defined by the identity:

  * $x\lor[y\to(0\to(y\to x))]\approx x\lor y\lor(y\to x)$.

###### Corollary 13.85

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{8},L^{dp}_{8},D_{1},D_{2},D_{3}\\})$ is
defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor\beta\lor[\beta\to(\beta\to\alpha)]]\to_{H}[\alpha\to[\alpha\lor(0\to\beta)]]$,

  * (1b)

$[\alpha\to[\alpha\lor(0\to\beta)]]\to_{H}[\alpha\lor\beta\lor[\beta\to(\beta\to\alpha)]]$,

It is also defined by:

  * (2a)

$[\alpha\lor[\beta\to(0\to(\beta\to\alpha))]]\to_{H}[\alpha\lor\beta\lor(\beta\to\alpha)]$,

  * (2b)

${\alpha\lor\beta\lor(\beta\to\alpha)]\to_{H}[\alpha\lor\\{\beta\to(0\to(\beta\to\alpha))}]$.

###### Theorem 13.86

The variety generated by the algebras
$\mathbf{L^{dm}_{7},L^{dp}_{7},L^{dm}_{8},L^{dp}_{8},L^{dm}_{9},L^{dp}_{9}}$,
$\mathbf{L^{dm}_{10},L^{dp}_{10},D_{1},D_{2},D_{3}}$ is defined by the
identity:

  * $x\lor(x\to y)\approx x\lor[(x\to y)\to 1]$.

###### Corollary 13.87

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{7},L^{dp}_{7},L^{dm}_{8},L^{dp}_{8},L^{dm}_{9},L^{dp}_{9},L^{dm}_{10},L^{dp}_{10},D_{1},D_{2},D_{3}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor(\alpha\to\beta)]\to_{H}[\alpha\lor\\{(\alpha\to\beta)\to\top\\}]$.

  * (1b)

$[\alpha\lor\\{(\alpha\to\beta)\to\top\\}]\to_{H}[\alpha\lor(\alpha\to\beta)]$.

###### Theorem 13.88

The variety generated by the algebras
$\mathbf{2^{e},L^{dp}_{7},L^{dp}_{8},L^{dp}_{9}}$, $\mathbf{L^{dp}_{10}}$ is
defined by the identities:

  * (1)

$x\lor(x\to y)\approx x\lor[(x\to y)\to 1]$

  * (2)

$x\lor x^{\prime}\approx 1$.

###### Corollary 13.89

The logic
$\mathcal{L}(\mathbb{V}(\\{2^{e},L^{dp}_{7},L^{dp}_{8},L^{dp}_{10}\\})$ is
defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor(\alpha\to\beta)]\to_{H}[\alpha\lor\\{(\alpha\to\beta)\to\top\\}]$,

  * (1b)

$[\alpha\lor\\{(\alpha\to\beta)\to\top\\}]\to_{H}[\alpha\lor(\alpha\to\beta)]$,

  * (2a)

$\alpha\lor\alpha^{\prime}$.

###### Theorem 13.90

The variety generated by the algebras
$\mathbf{L^{dm}_{7},L^{dm}_{8},L^{dm}_{9},}$
$\mathbf{L^{dm}_{10},D_{1},D_{2},D_{3}}$ is defined by the identities:

  * (1)

$x\lor(x\to y)\approx x\lor[(x\to y)\to 1]$

  * (2)

$x^{\prime\prime}\approx x$.

###### Corollary 13.91

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{7},L^{dm}_{8},L^{dm}_{9},L^{dm}_{10},D_{1},D_{2},D_{3}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$[\alpha\lor(\alpha\to\beta)]\to_{H}[\alpha\lor\\{(\alpha\to\beta)\to\top\\}]$,

  * (1b)

$[\alpha\lor\\{(\alpha\to\beta)\to\top\\}]\to_{H}[\alpha\lor(\alpha\to\beta)]$,

  * (2a)

$\alpha^{\prime\prime}\to_{H}\alpha$,

  * (2b)

$\alpha\to_{H}\alpha^{\prime\prime}$.

###### Theorem 13.92

The variety generated by the algebras
$\mathbf{L^{dm}_{9},L^{dp}_{9},L^{dm}_{10},L^{dp}_{10},D_{1}}$ is defined by
the identity:

  * $0\to 1\approx 0$. (FTF identity)

###### Corollary 13.93

The logic
$\mathcal{L}(\mathbb{V}(\\{L^{dm}_{9},L^{dp}_{9},L^{dm}_{10},L^{dp}_{10},D_{1}\\})$
is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$(\bot\to\top)\to_{H}\bot$, (FTF identity).

###### Theorem 13.94

The variety generated by the algebras
$\mathbf{{L}^{dm}_{10},L^{dp}_{10},D_{1}}$ is defined by the identity:

  * $x\to y\approx y\to x$. (commutative identity)

###### Corollary 13.95

The logic $\mathcal{L}(\mathbb{V}(\\{L^{dm}_{10},L^{dp}_{10},D_{1}\\})$ is
defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * $(\alpha\to\beta)\to_{H}(\beta\to\alpha)$. (commutative identity)

A base for $\mathbf{V(C^{dp}_{10})}$ was given in [San11]. We give some new
ones below.

###### Theorem 13.96

The variety $\mathbf{V(C^{dp}_{10})}$ is defined by :

  * $x^{\prime}\land x^{\prime\prime}\approx 0$ (dual Stone identity).

It is also defined by :

  * $x^{\prime}\approx x^{\prime}{{}^{*}}^{\prime}$.

###### Corollary 13.97

The logic $\mathcal{L}(\mathbb{V}(L^{dp}_{10}\\})$ is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$\alpha^{\prime}\land\alpha^{\prime\prime}\to_{H}\bot$ (dual Stone identity).

It is also defined by :

  * (1a)

$\alpha^{\prime}\to_{H}\alpha^{\prime}{{}^{*}}^{\prime}$,

  * (1b)

$\alpha^{\prime}{{}^{*}}^{\prime}\to_{H}\alpha^{\prime}$.

A base for $\mathbf{V(C_{20})}$ was given in [San11] We give a new one below.

###### Theorem 13.98

The variety $\mathbb{V(C}_{20})$ is defined by :

  * $x^{*}\leq x^{\prime}$.

###### Corollary 13.99

The logic $\mathcal{L}(\mathbb{V}(C_{20})$ is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * $\alpha^{*}\to_{H}\alpha^{\prime}$.

A base for $\mathbf{V(2^{e},\bar{2}^{e})}$ was given in [San11]. We give a new
one below.

###### Theorem 13.100

The variety $\mathbb{V}(2^{e},\bar{2}^{e})$ is defined by :

  * $x^{*}\approx x^{\prime}$.

###### Corollary 13.101

The logic $\mathcal{L}(\mathbb{V}(2^{e},\bar{2}^{e})$ is defined,
relative to $\mathcal{RDQDS}t\mathcal{SH}_{1}$, by

  * (1a)

$\alpha^{*}\to_{H}\alpha^{\prime}$,

  * (1b)

$\alpha^{*}\to_{H}\alpha^{\prime}$.

###### Theorem 13.102

The variety generated by the algebras in