‎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

Semi-Heyting Algebras and Identities of Associative Type

An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies 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. They share several important properties with Heyting algebras. An identity of associative type of length 3 is a groupoid identity, both sides of which contain the same three (distinct) variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of SH is of associative type of length 3 if it is defined by a single identity of associative type of length 3. In this paper we describe all the distinct subvarieties of the variety SH of asociative type of length 3. Our main result shows that there are 3 such subvarities of SH.Consejo Nacional de Investigaciones Cientificas y TecnicasUniversidad Nacional del Su

Contributions to the theory of Ockham algebras

In the first part of this thesis we consider particular ordered sets (connected and of small height) and determine the cardinality of the corresponding dual MS - algebra and of its set of fixed points. The remainder of the thesis is devoted to a study of congruences of Ockham algebras and a generalised variety K of Ockham algebras that contains all of the Berman varieties K[sub]p,[sub]q. In particular we consider the congruences [sub]i(i = 1, 2,...) defined on an Ockham algebra (L; f) by (x, y) ∊ [sub]i ⇔ fⁱ(x)= fⁱ(y) and show that (L; f) ∊ K is subdirectly irreducible if and only if the lattice of congruences of L reduces to the chain = ₀ ≤ ₁≤ ₂≤ … ≤< Where = ⌵ [sub]i≥0i. Finally we obtain a characterisation of the finite simple Ockham algebras