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.Comment: This paper, except the appendix, will appear in Algebra Universalis. 25 pages, 4 figures, a revised version with a new titl

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.Comment: 21 page

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 $n$.Comment: 35 page

Implication Zroupoids and Identities of Associative Type

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 variety of symmetric I-zroupoids is denoted by S. An identity p≈q, in the groupoid language ⟨→⟩, 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) ⋆→(⋆→⋆) and (2) (⋆→⋆)→⋆, where ⋆ is a place holder for a variable. A subvariety of I is said to be of associative type of length 3, if it is defined, relative to 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 I of associative type of length 3. We prove, in our main theorem, that there are exactly 8 such subvarieties of 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 S, each defined, relative to S, by a single identity of associative type of length 3.Fil: Cornejo, Juan Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Sankappanavar, Hanamantagouda P.. State University of New York; Estados Unido

‎Gautama and Almost Gautama Algebras and their associated logics

Recently, Gautama algebras were defined and investigated as a common generalization of the variety $\mathbb{RDBLS}\rm t$ of regular double Stone algebras and the variety $\mathbb{RKLS}\rm t$ of regular Kleene Stone algebras, both of which are, in turn, generalizations of Boolean algebras. Those algebras were named in honor and memory of the two founders of Indian Logic--{\bf Akshapada Gautama} and {\bf Medhatithi Gautama}. The purpose of this paper is to define and investigate a generalization of Gautama algebras, called ``Almost Gautama algebras ($\mathbb{AG}$, for short).'' More precisely, we give an explicit description of subdirectly irreducible Almost Gautama algebras. As consequences, explicit description of the lattice of subvarieties of $\mathbb{AG}$ and the equational bases for all its subvarieties are given. It is also shown that the variety $\mathbb{AG}$ is a discriminator variety. Next, we consider logicizing $\mathbb{AG}$; but the variety $\mathbb{AG}$ lacks an implication operation. We, therefore, introduce another variety of algebras called ``Almost Gautama Heyting algebras'' ($\mathbb{AGH}$, for short) and show that the variety $\mathbb{AGH}$ %of Almost Heyting algebras is term-equivalent to that of $\mathbb{AG}$. Next, a propositional logic, called $\mathcal{AG}$ (or $\mathcal{AGH}$), is defined and shown to be algebraizable (in the sense of Blok and Pigozzi) with the variety $\mathbb{AG}$, via $\mathbb{AGH},$ as its equivalent algebraic semantics (up to term equivalence). All axiomatic extensions of the logic $\mathcal{AG}$, corresponding to all the subvarieties of $\mathbb{AG}$ are given. They include the axiomatic extensions $\mathcal{RDBLS}t$, $\mathcal{RKLS}t$ and $\mathcal{G}$ of the logic $\mathcal{AG}$ corresponding to the varieties $\mathbb{RDBLS}\rm t$, $\mathbb{RKLS}\rm t$, and $\mathbb{G}$ (of Gautama algebras), respectively. It is also deduced that none of the axiomatic extensions of $\mathcal{AG}$ has the Disjunction Property. Finally, We revisit the classical logic with strong negation $\mathcal{CN}$ and classical Nelson algebras $\mathbb{CN}$ introduced by Vakarelov in 1977 and improve his results by showing that $\mathcal{CN}$ is algebraizable with $\mathbb{CN}$ as its algebraic semantics and that the logics $\mathcal{RKLS}\rm t$, $\mathcal{RKLS}\rm t\mathcal{H}$, 3-valued \L ukasivicz logic and the classical logic with strong negation are all equivalent.Fil: Cornejo, Juan Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Sankappanavar, Hanamantagouda P.. State University of New York. Department of Mathematics ; Estados Unido