arXiv subjects

Hanamantagouda P. Sankappanavar

Publications and source records attributed to Hanamantagouda P. Sankappanavar.

17 recordsLinked to original sources

arXiv2025

Connexive logics and connexive semi-Heyting algebras

In this paper, we define and investigate a connexive logic, called 'Connexive semi-Heyting logic' (\mathcal{CSH} for short) and a new subvariety CSH of the variety SH of semi-Heyting algebras. It is shown that the logic \mathcal{CSH} is implicative in the sense of Rasiowa, and is algebraizable with CSH as an equivalent algebraic semantics (in the sense of Blok and Pigozzi). We also introduce the logics \mathcal{AT}i and \mathcal{BT}i, i = 1, 2, along with the subvarieties ATi and BTi, i = 1, 2, of SH. It is then shown that AT1 = AT2 and CSH = BT1 \subset BT2 \subset AT1. A 3-valued connexive semi-Heyting logic \mathcal{CSH}3 and its equivalent algebraic semantics CSH3 are introduced and axiomatized; and it is then shown that CSH3 is deductively equivalent to the 3-valued intuitionistic logic. New characterizations of anti-Boolean semi-Heyting algebras are given. We show that BT2 \cap SHc = V(2), and SHc \subset AT1, where SHc is defined by x \to y = y \to x. It is proved that the identity (AT1) is equivalent to the identity x* \to y* = y* \to x* (* being the pseudocomplement) in StSH and also is equivalent to 0 \to 1 = 0 in SH. We show that AT1 \cap EX \subset BT1, where EX is defined by x \to (y \to z) = y \to (x \to z). The paper concludes with some further remarks, mentions some open problems for future research and proposes two new principles to be considered as Connexive Theses.

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arXiv2025

Unorthodox Algebras and their associated Unorthodox Logics

This paper grew out of our investigation into a simple, but natural, question: Can 'F implies T' be distinct from F and T? To this end, we introduce five 'unorthodox' algebras that will play a major role, not only in providing a positive answer to the question, but also in their similarity to the 2-element Boolean algebra 2. Yet, they are remarkably dissimilar from 2 in many respects. In this paper, we will examine these five algebras both algebraically and logically. We define, and initiate an investigation into, a subvariety, called RUNO1, of the variety of De Morgan semi-Heyting algebras and show that RUNO1 is, in fact, the variety generated by the five algebras. Then several applications of this theorem are given. It is shown that RUNO1 is a discriminator variety and that all five algebras are primal. It is also shown that every subvariety of RUNO1 satisfies the Strong Amalgamation Property and the property that epimorphisms are surjective (ES). It is shown that the lattice of subvarieties of RUNO1 is a Boolean lattice of 32 elements. The bases for all the subvarieties of RUNO1 are also given. We introduce a new logic called mathcal{RUNO1} and show that it is algebraizable with the variety RUNO1 as its equivalent algebraic semantics. We then present axiomatizations for all 32 axiomatic extensions of mathcal{RUNO1} and deduce that all the axiomatic extensions are decidable. The paper ends with some open problems.

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arXiv2025

Johnson's axioms revisited: Bases for Boolean algebras containing identities of associative type. I

This paper is inspired by 1892 paper of Johnson, where he has given an axiomatization for the variety of Boolean algebras (equivalently, for classical propositional calculus). The fact that the axioms of Johnson include the associative law, the most well-known identity of associative type of length 3, led us naturally to the question as to whether there are axiom systems for Boolean algebras that include an identity of associative type, other than the associative law. It turns out that the answer to this question is positive in the strongest sense possible. In fact, corresponding to each of the 14 nonequivalent identities of associative type, there is a base containing that identity. Our goal, in this sequence of papers, of which the present paper is the first, is to describe bases for Boolean algebras, each of which contains at least one identity of associative type of length 3. In this paper, we prove, firstly, that the third axiom of Johnson is redundant. Secondly, we give several equational bases, some 2-bases and some 3-bases, for the variety of Boolean algebras, each containing an identity of associative type, other than the associative law. Each of these bases can be easily converted into an axiomatization for the classical propositional logic, as well.

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arXiv2024

Amalgamation Property in the subvarieties of Gautama and Almost Gautama algebras

Gautama algebras were introduced recently, as a common generalization of regular double Stone algebras and regular Kleene Stone algebras. Even more recently, Gautama algebras were further generalized to Almost Gautama algebras (AG for short). The main purpose of this paper is to investigate the Amalgamation Property (AP, for short) in the subvarieties of the variety AG. In fact, we show that, of the eight nontrivial subvarieties of AG, only four varieties, namely those of Boolean algebras, of regular double Stone algebras, of regular Kleene Stone algebras and of De Morgan Boolean algebras have the AP and the remaining four do not have the AP. We give several applications of this result; in particular, we examine the following properties for the subvarieties of AG: transferability property (TP), having enough injectives (EI), Embedding Property, Bounded Obstruction Property and having a model companion.

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arXiv2022

Regular Double $p$-Algebras: A converse to a Katriňák's Theorem, and Applications

In 1973, Katriňák proved that regular double $p$-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heying implication and its dual in terms of pseudocomplement and its dual. In this paper we prove a converse to the Katriňák's theorem, in the sense that in the variety RDPCH of regular dually pseudocomplemented Heyting algebras, the implication operation $\to$ satisfies the Katriňák's formula. As applications of this result together with the above-mentioned Katriňák's theorem, we show that the varieties RDBLP, RDPCH, RPCH$^d$ and RDBLH of regular double $p$-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties RDMP, RDMH, RDMDBLH, RDMDBLP of regular De Morgan $p$-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double $p$-algebras, respectively, are also term equivalent to each other. From these results and recent results of Adams, Sankappanavar and vaz de Carvalho, we deduce that the lattices of subvarieties of all these varieties have cardinality $2^{\aleph_0}$. We then define new logics, RDPCH, RPCHd, and RDMH, and show that they are algebraizable with RDPCH, RPCH$^d$ and RDMH, respectively as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality $2^{\aleph_0}$.

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arXiv2022

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

In this paper, we focus on the variety DHMSH of dually hemimorphic semi-Heyting algebras from a logical point of view. Firstly, we present a Hilbert-style axiomatization of a new logic called Dually hemimorphic semi-Heyting logic (DHMSH, for short), as an expansion of semi-intuitionistic logic SI (also called SH) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety DHMSH. It is deduced that the logic DHMSH is algebraizable in the sense of Blok and Pigozzi, with the variety DHMSH as its equivalent algebraic semantics and that the lattice of axiomatic extensions of DHMSH is dually isomorphic to the lattice of subvarieties of DHMSH. A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of DHMSH in which the Deduction Theorem holds. Thirdly, we present several new logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH. These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Lukasiewiczlogic. Surprisingly, many of these logics turn out to be connexive logics, a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan-Goedel logics and dually pseudocomplemented Goedel logics, Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems.

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arXiv2020

Semidistributivity and Whitman Property in Implication Zroupoids

In 2012, the second author introduced and studied the variety $\mathcal{I}$ of implication zroupoids that generalize De Morgan algebras and $\lor$-semilattices with $0$. An algebra $\mathbf A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an \emph{implication zroupoid} ($\mathcal{I}$-zroupoid, for short) if $\mathbf A$ satisfies: $(x \to y) \to z \approx [(z' \to x) \to (y \to z)']'$, where $x' : = x \to 0$, and $ 0'' \approx 0$. Let $\mathcal{I}$ denote the variety of implication zroupoids and $\mathbf A \in \mathcal{I}$. For $x,y \in \mathbf A$, let $x \land y := (x \to y')'$ and $x \lor y := (x' \land y')'$. In an earlier paper we had proved that if $\mathbf A \in \mathcal{I}$, then the algebra $\mathbf A_{mj} = \langle A, \lor, \land \rangle$ is a bisemigroup. In this paper we generalize the notion of semi-distributivity from lattices to bisemigroups and prove that, for every $\mathbf A \in \mathcal{I}$, the bisemigroup $\mathbf A_{mj}$ is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety $\mathcal{MEJ}$ of $\mathcal I$, defined by the identity: $x \land y \approx x \lor y$, satisfies the Whitman Property.

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arXiv2020

Implication Zroupoids and Birkhoff Systems

An algebra $A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an implication zroupoid (I-zroupoid, for short) if A satisfies the identities: $(x \to y) \to z \approx ((z' \to x) \to (y \to z)')'$, where $x' := x \to 0$, and $0'' \approx 0$. These algebras generalize De Morgan algebras and $\lor$-semilattices with zero. Let I denote the variety of implication zroupoids. For details on the motivation leading to these algebras, we refer the reader to [San12] (or the relevant papers mentioned at the end of this paper). The investigations into the structure of the lattice of subvarieties of I, begun in [San12], have continued in [CS16a, CS16b, CS17a, CS17b, CS18a, CS18b, CS19] and [GSV19]. The present paper is a sequel to this series of papers and is devoted to making further contributions to the theory of implication zroupoids. The identity (BR): $x \land (x \lor y) \approx x \lor (x \land y)$ is called the Birkhoff's identity. The main purpose of this paper is to prove that if A is an algebra in the variety I, then the derived algebra $A_{mj} := \langle A; \land, \lor \rangle$, where $a \land b := (a \to b')'$ and $a \lor b := (a' \land b')'$, satisfies the Birkhoff's identity. As a consequence, we characterize the implication zroupoids A whose derived algebras $A_{mj}$ are Birkhoff systems. It also follows from the main result that there are bisemigroups that are not bisemilattices but satisfy the Birkhoff's identity, which suggests a more general notion, than Birkhoff systems, of "Birkhoff bisemigroups" as bisemigroups satisfying the Birkhoff's identity. The paper concludes with an open problem on Birkhoff bisemigroups.

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arXiv2018

De Morgan Semi-Heyting and Heyting Algebras

The variety DMSH of semi-Heyting algebras with a De Morgan negation was introduced in [12] and an increasing sequence DMSHn of level n, n being a natural number, of its subvarieties was investigated in the series [12], [13], [14], [15], [16], and [17], of which the present paper is a sequel. In this paper, we prove two main results: Firstly, we prove that DMSH1-algebras of level 1 satisfy Stone identity, generalizing an earlier result that regular DMSH1-algebras of level 1 satisfy Stone identity. Secondly, we prove that the variety of DmsStSH of dually ms, Stone semi-Heyting algebras is at level 2. As an application, it is derived that the variety of De Morgan semi-Heyting algebras is also at level 2. It is also shown that these results are sharp.

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arXiv2017

Implication Zroupoids and Identities of Associative Type

An algebra $\mathbf A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an implication zroupoid ($\mathcal I$-zroupoid, for short) if $\mathbf A$ satisfies the identities: $(x \to y) \to z \approx [(z' \to x) \to (y \to z)']'$ and $ 0'' \approx 0$, where $x' : = x \to 0$, and $\mathcal I$ denotes the variety of all $\mathcal I$-zroupoids. An $\mathcal I$-zroupoid is symmetric if it satisfies $x'' \approx x$ and $(x \to y')' \approx (y \to x')'$. The variety of symmetric $\mathcal I$-zroupoids is denoted by $\mathcal S$. An identity $p \approx q$, in the groupoid language $\langle \to \rangle$, is called an identity of associative type of length $3$ if $p$ and $q$ have exactly 3 (distinct) variables, say x,y,z, and are grouped according to one of the two ways of grouping: (1) $\star \to (\star \to \star)$ and (2) $(\star \to \star) \to \star$, where $\star$ is a place holder for a variable. A subvariety of $\mathcal I$ is said to be of associative type of length $3$, if it is defined, relative to $\mathcal I$, by a single identity of associative type of length $3$. In this paper we give a complete analysis of the mutual relationships of all subvarieties of $\mathcal I$ of associative type of length $3$. We prove, in our main theorem, that there are exactly 8 such subvarieties of $\mathcal I$ that are distinct from each other and describe explicitly the poset formed by them under inclusion. As an application of the main theorem, we derive that there are three distinct subvarieties of the variety $\mathcal S$, each defined, relative to $\mathcal S$, by a single identity of associative type of length $3$.

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arXiv2017

Symmetric implication zroupoids and identities of Bol-Moufang type

An algebra $\mathbf A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an implication zroupoid ($\mathcal I$-zroupoid, for short) if $\mathbf A$ satisfies the identities: (I): $(x \to y) \to z \approx ((z' \to x) \to (y \to z)')'$, and (I$_{0}$): $ 0'' \approx 0$, where $x' : = x \to 0$. An implication zroupoid is symmetric if it satisfies the identities: $x'' \approx x$ and $(x \to y')' \approx (y \to x')'$. An identity is of Bol-Moufang type if it contains only one binary operation symbol, one of its three variables occurs twice on each side, each of the other two variables occurs once on each side, and the variables occur in the same (alphabetical) order on both sides of the identity. In this paper we make a systematic analysis of all $ 60$ identities of Bol-Moufang type in the variety $\mathcal S$ of symmetric $\mathcal I$-zroupoids. We show that $47$ of the subvarieties of $\mathcal S$, defined by the identities of Bol-Moufang type are equal to the variety $\mathcal{SL}$ of $\lor$-semilattices with the least element $0$ and, one of the others is equal to $\mathcal S$. Of the remaining 12, there are only $3$ distinct ones. We also give an explicit description of the poset of the (distinct) subvarieties of $\mathcal S$ of Bol-Moufang type.

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arXiv2017

Symmetric Implication Zroupoids and Weak Associative Laws

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arXiv2017

JI-distributive, dually quasi-De Morgan semi-Heyting and Heyting algebras

The variety DQD of semi-Heyting algebras with a weak negation, called dually quasi-De Morgan operation, and several of its subvarieties were investigated in a series of four papers. In this paper we define and investigate a new subvariety JID of DQD, called JI-distributive, dually quasi-De Morgan semi-Heyting algebras, as well as the variety DSt of dually Stone semi-Heyting algebras. We first prove that DSt and JID are discriminator varieties of level 1 and level 2, respectively. Secondly, we give a characterization of subdirectly irreducible algebras of the subvariety JID1 of level 1 of JID. As a first application of it, we derive that the variety JID1 is the join of the variety DSt and the variety of De Morgan Boolean semi-Heyting algebras. As a second application, we give a concrete description of the subdirectly irreducible algebras in the subvariety JIDL1 of JID1 defined by the linear identity, and deduce that the variety JIDL1 is the join of the variety DStHC generated by the dually Stone Heyting chains and the variety generated by the 4-element De Morgan Boolean Heyting algebra. Several applications of this result are also given, including a description of the lattice of subvarieties of JIDL1, equational bases of all subvarieties of JIDL1, and the amalgamation property of all subvarieties of DStHC.

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arXiv2016

On Implicator Groupoids

In a paper published in 2012, the second author extended the well-known fact that Boolean algebras can be defined using only implication and a constant, to De Morgan algebras-this result led him to introduce, and investigate (in the same paper), the variety I of algebras, there called implication zroupoids (I-zroupoids) and here called implicator gruopids (I- groupoids), that generalize De Morgan algebras. The present paper is a continuation of the paper mentioned above and is devoted to investigating the structure of the lattice of subvarieties of I, and also to making further contributions to the theory of implicator groupoids. Several new subvarieties of I are introduced and their relationship with each other, and with the subvarieties of I which were already investigated in the paper mentioned above, are explored.

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arXiv2015

Order in Implication Zroupoids

The variety $\mathbf{I}$ of implication zroupoids was defined and investigated by Sankappanavar ([7]) as a generalization of De Morgan algebras. Also, in [7], several new subvarieties of $\mathbf{I}$ were introduced, including the subvariety $\mathbf{I_{2,0}}$, defined by the identity: $x" \approx x$, which plays a crucial role in this paper. Several more new subvarieties of $\mathbf{I}$, including the subvariety $\mathbf{SL}$ of semilattices with a least element $0$, are studied in [3], and an explicit description of semisimple subvarieties of $\mathbf{I}$ is given in [5]. It is well known that the operation $\land$ induces a partial order ($\sqsubseteq$) in the variety $\mathbf{SL}$ and also in the variety $\mathbf{DM}$ of De Morgan algebras. As both $\mathbf{SL}$ and $\mathbf{DM}$ are subvarieties of $\mathbf{I}$ and the definition of partial order can be expressed in terms of the implication and the constant, it is but natural to ask whether the relation $\sqsubseteq$ (now defined) on $\mathbf{I}$ is actually a partial order in some (larger) subvariety of $\mathbf{I}$ that includes $\mathbf{SL}$ and $\mathbf{DM}$. The purpose of the present paper is two-fold: Firstly, a complete answer is given to the above mentioned problem. Indeed, our first main theorem shows that the variety $\mathbf{I_{2,0}}$ is a maximal subvariety of $\mathbf{I}$ with respect to the property that the relation $\sqsubseteq$ is a partial order on its members. In view of this result, one is then naturally led to consider the problem of determining the number of non-isomorphic algebras in $\mathbf{I_{2,0}}$ that can be defined on an $n$-element chain (herein called $\mathbf{I_{2,0}}$-chains), $n$ being a natural number. Secondly, we answer this problem in our second main theorem, which says that, for each $n \in \mathbb{N}$, there are exactly $n$ nonisomorphic $\mathbf{I_{2,0}}$-chains of size $n$.

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arXiv2015

Semisimple Varieties of Implication Zroupoids

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In 2012, this result was extended to De Morgan algebras in [8] which led Sankappanavar to introduce, and investigate, the variety I of implication zroupoids generalizing De Morgan algebras. His investigations were continued in [3] and [4] in which several new subvarieties of I were introduced and their relationships with each other and with the varieties of [8] were explored. The present paper is a continuation of [8] and [3]. The main purpose of this paper is to determine the simple algebras in I. It is shown that there are exactly five simple algebras in I. From this description we deduce that the semisimple subvarieties of I are precisely the subvarieties of the variety generated by these 5 simple I-zroupoids and are locally finite. It also follows that the lattice of semisimple subvarieties of I is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain.

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arXiv2014

A note on regular De Morgan semi-Heyting algebras

The purpose of this note is two-fold. Firstly, we prove that the variety RDMSH1 of regular De Morgan semi-Heyting algebras of level 1 satisfies Stone identity and present (equational) axiomatizations for several subvarieties of RDMSH1. Secondly, we give a concrete description of the lattice of subvarieties of the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras that contains RDMSH1 . Furthermore, we prove that every subvariety of RDQDStSH1, and hence of RDMSH1, has Amalgamation Property. The note concludes with some open problems for further investigation.

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