Hanamantagouda P Sankappanavar

State University of New York at New Paltz | SUNY New Paltz · Department of Mathematics

The variety $\mathbb{DHMSH}$ of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety $\mathbb{DHMSH}$ from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the...

The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to th...

This chapter consists of two parts. PART I presents a few historical glimpses into the fascinating interplay between algebra and logic that essentially started in the middle of the 19th century with Boole’s work. It is mostly non-technical and highlights a few significant examples of this interplay. PART I includes a brief discussion of Boole’s alg...

In 1973, Katri\v{n}\'{a}k 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 Katri\v{n}\'{a}k's theorem, in the sense that in the variety RDPCH...

It was proved by the authors that the quasivariety of quasi-Stone algebras Q1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Q}_{\mathbf {1,2}}$$\end{documen...

The variety DHMSH of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety DHMSH from a logical point of view. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heytin...

In 2012, the second author introduced, and initiated the investigations into, the variety 𝓘 of implication zroupoids that generalize De Morgan algebras and ∨-semilattices with 0. An algebra A = 〈 A , →, 0 〉, where → is binary and 0 is a constant, is called an implication zroupoid (𝓘-zroupoid, for short) if A satisfies: ( x → y ) → z ≈ [( z ′ → x )...

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: $(x \to y) \to z \approx [(z' \to x) \to (y \to z)']'$, where $x' : = x \to 0$, and $ 0'' \approx 0$. These algebras generalize De Morgan alg...

We determine the number of non‐isomorphic semi‐Heyting algebras on an n‐element chain, where n is a positive integer, using a recursive method. We then prove that the numbers obtained agree with those determined in [1]. We apply the formula to calculate the number of non‐isomorphic semi‐Heyting chains of a given size in some important subvarieties...

In this paper, we investigate the varieties Mn and Kn of regular pseudocomplemented de Morgan and Kleene algebras of range n, respectively. Priestley duality as it applies to pseudocomplemented de Morgan algebras is used. We characterise the dual spaces of the simple (equivalently, subdirectly irreducible) algebras in Mn and explicitly describe the...

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 $\m...

An implication semigroup is an algebra of type (2, 0) with a binary operation → and a 0-ary operation 0 satisfying the identities \((x\rightarrow y)\rightarrow z\approx x\rightarrow (y\rightarrow z)\), \((x\rightarrow y)\rightarrow z\approx \left [(z^{\prime }\rightarrow x)\rightarrow (y\rightarrow z)'\right ]'\) and \(0^{\prime \prime }\approx 0\)...

In this paper, we investigate the varieties $\mathbf M_n$ and $\mathbf K_n$ of regular pseudocomplemented de Morgan and Kleene algebras of range $n$, respectively. Priestley duality as it applies to pseudocomplemented de Morgan algebras is used. We characterise the dual spaces of the simple (equivalently, subdirectly irreducible) algebras in $\math...

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 wit...

An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfiesthe the identities : x ∧ (x → y) ≈ x∧y, x∧(y → z) ≈ x∧[(x∧y) → (x∧z)], and x → x ≈ 1. SH denotes the variety of semi-Heyting algebras. Semi-Heyting algebras were introduced by the second author as an abstraction from Heyting algebras...

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 DHMSH of dually hemimorphic semi-Heyting algebras obtained by...

An algebra A = A, →, 0, where → is binary and 0 is a constant, is called an implication zroupoid (I-zroupoid, for short) if A satisfies the identities: (x → y) → z ≈ ((z ′ → x) → (y → z) ′) ′ and 0 ′′ ≈ 0, where x ′ := x → 0. An implication zroupoid is symmetric if it satisfies x ′′ ≈ x and (x → y ′) ′ ≈ (y → x ′) ′. The variety of symmetric I-zrou...

In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDPn, n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP1 and show that this variety is locally finite. We also show that the lattice of subva...

In 2012, the second author introduced and examined a new type of algebras as a generalization of De Morgan algebras. These algebras are of type (2,0) with one binary and one nullary operation satisfying two certain specific identities. Such algebras are called implication zroupoids. They invesigated in a number of articles by the second author and...

An algebra \({\mathbf {A}} = \langle A, \rightarrow , 0 \rangle \), where \(\rightarrow \) 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 \rightarrow y) {\rightarrow } z \approx ((z' {\rightarrow } x) {\rightarrow } (y {\rightarrow }...

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 pro...

An algebra A = ⟨A, →, 0⟩, where → is binary and 0 is a constant, is called an implication zroupoid (I-zroupoid, for short) if A satisfies the identities: (x → y) → z ≈ [(z′ → x) → (y → z)′]′ and 0′′ ≈ 0, where x′ := x → 0, and I denotes the variety of all I-zroupoids. An I-zroupoid is symmetric if it satisfies x′′ ≈ x and (x → y′)′ ≈ (y → x′)′. The...

In 2012, the second author introduced and studied in Sankappanavar (Sci Math Jpn 75(1):21–50, 2012) the variety \({\mathcal {I}}\) of algebras, called implication zroupoids, that generalize De Morgan algebras. An algebra \({\mathbf {A}} = \langle A, \rightarrow , 0 \rangle \), where \(\rightarrow \) is binary and 0 is a constant, is called an impli...

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 the series [31], [32], [33], and [34]. In this paper we define and investigate a new subvari-ety JID of DQD, called "JI-distributive, dually quasi-De Morgan semi-Heyting algebras", defined by t...

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 \({\mathcal{I}}\) of algebras, there called implication zroupoids (I-zroupoids) and here cal...

The lattice of varieties of quasi-Stone algebras ordered by inclusion is an \({\omega+1}\) chain. It is shown that the variety \({\mathbf{Q_{2,2}}}\) (of height 13) is finite-to-finite universal (in the sense of Hedrlín and Pultr). Further, it is shown that this is sharp; namely, the variety \({\mathbf{Q_{3,1}}}\) (of height 12) is not finite-to-fi...

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...

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 se...

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. Sever...

The main purpose of this paper is to axiomatize the join of the variety DPCSHC of dually pseudocomplemented semi-Heyting algebras generated by chains and the variety generated by D2, the De Morgan expansion of the four element Boolean Heyting algebra. Toward this end, we first introduce the variety DQDLNSH of dually quasi-De Morgan linear semi-Heyt...

The main purpose of this paper is to axiomatize the join of the variety DPCSHC of dually pseudocomplemented semi-Heyting algebras generated by chains and the variety generated by D2, the De Morgan expansion of the four element Boolean Heyting algebra. Toward this end, we first introduce the variety DQDLNSH of dually quasi-De Morgan linear semi-Heyt...

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 [11] which led Sankappanavar to introduce, and investigate, the variety I of implication zroupoids (I- zroupoids) generalizing De Morgan algebras. The present paper, a continuation of [11], i...

In modern algebra it is well-known that one cannot, in general, apply ordinary equational reasoning when dealing with partial algebras. However Boole did not know this, and he took the opposite to be a fundamental truth, which he called the Principles of Symbolical Reasoning in his 1854 book Laws of Thought. Although Boole made no mention of it, hi...

In this paper we first describe the Priestley duality for pseudocomplemented De Morgan algebras by combining the known dualities of distributive p-algebras due to Priestley and for De Morgan algebras due to Cornish and Fowler. We then use it to characterize congruence-permutability, principal join property, and the property of having only principal...

This paper is the second of a two part series. In this Part, we prove, using the description of simples obtained in Part I, that the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element RDQDStSH1-chains and the variety of dually quasi-De Morgan Boole...

This paper is the first of a two part series. In this paper, we first prove that the variety of dually quasi-De Morgan Stone semi-Heyting algebras of level 1 satisfies the strongly blended ∨-De Morgan law introduced in [20]. Then, using this result and the results of [20], we prove our main result which gives an explicit description of simple algeb...

A rigorous, modern version of Boole's algebra of logic is presented, based partly on the 1890s treatment of Ernst Schroder.

A rigorous, modern version of Boole's algebra of logic is presented, based partly on the 1890s treatment of Ernst Schroder.

A rigorous treatment of Boole's algebra of logic is presented, based partly on the work of Hailperin (1976/1986).

This paper augments Hailperin's substantial efforts (1976/1986) to place Boole's algebra of logic on a solid footing. Namely Horn sentences are used to give a modern formulation of the principle that Boole adopted in 1854 as the foundation for his algebra of logic—we call this principle The Rule of 0 and 1.

It is well known that Boolean algebras can be defined using only the implication and the constant 0. It is, then, natural to ask whether De Morgan algebras can also be characterized using only a binary operation (implication) → and a constant 0. In this paper, we give an affirmative answer to this question by showing that the variety of De Morgan a...

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [48] and [50] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of...

The purpose of this paper is to define and investigate a new (equational) class of algebras called "Semi-Heyting Algebras" as an abstraction from Heyting algebras. We show that semi-Heyting algebras are distributive pseudocomplemented lattices, the congru-ences on these algebras are determined by filters, and the variety S H of semi-Heyting algebra...

The purpose of this paper is to define and investigate the new class of quasi-Stone algebras (QSA's). Among other things we characterize the class of simple QSA's and the class of subdirectly irreducible QSA's. It follows from this characterization that the subdirectly irreducible QSA's form an elementary class and that the variety of QSA's is loca...

A formula is given to express a principal congruence on a double demi-p-lattice as a join of countably many principal lattice congruences. It is then applied to show that the variety of double demi-p-lattices has the congruence extension property. As special cases one obtains some known results for distributive doublep-lattices due to T. Hecht and...

The purpose of this paper is to define and investigate a new (equational) class of algebras, which we call semi-De Morgan algebras, as a common abstraction of De Morgan algebras and distributive pseudocomplemented lattices. We were first led to this class of algebras in 1979 (in Brazil) as a result of our attempt to extend both the well-known theor...

This paper is a contribution toward developing a theory of (the variety H+ of) Heyting algebras with dual pseudocomplementation.

Investigations into the structure of the congruence lattices of pseudocomplemented semilattices (PCS's) were initiated in [10]. In this paper a we characterize the class of congruence-semim odular PCS's (i.e. PCS's with semimodular lattice of congruences) and the class of congruence-distr ibutive PCS's (i.e. with distributive congruence lattices)....

In this paper a characterization of principal congruences of De Morgan algebras is given and from it we derive that the variety of De Morgan algebras has DPC and CEP. The characterization is then applied to give a new proof of Kalman's characterization of subdirectly irreducibles in this variety and thus to obtain the representation theorem for DeM...

Principal congruences of pseudocomplemented semilattices are characterized and shown to be definable. This characterization is then applied to give a new proof of the fact that the variety of pseudocomplemented semilattices has the congruence extension property.