Gautama and Almost Gautama Algebras and their associated logics
Subject Areas : Transactions on Fuzzy Sets and Systems
Juan M. Cornejo
1
,
Hanamantagouda P. Sankappanavar
2
1 - Departamento de Matematica, Universidad Nacional del Sur, INMABB-CONICET, Bahia Blanca, Argentina.
2 - Department of Mathematics, State University of New York, New Paltz, NY, USA.
Keywords: Regular double Stone algebra, regular Kleene Stone algebra, Gautama algebra, Almost Gautama algebra, Almost Gautama Heyting algebra, subdirectly irreducible algebra, simple algebra, logic AG, logic G, logic RDBLSt, logic RKLSt.,
Abstract :
Recently, Gautama algebras were defined and investigated as a common generalization of the variety RDBLSt of regular double Stone algebras and the variety RKLSt 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–Akshapada Gautama and Medhatithi Gautama. The purpose of this paper is to define and investigate a generalization of Gautama algebras, called “Almost Gautama algebras (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 AG and the equational bases for all its subvarieties are given. It is also shown that the variety AG is a discriminator variety. Next, we consider logicizing AG; but the variety AG lacks an implication operation. We, therefore, introduce another variety of algebras called “Almost Gautama Heyting algebras” (AGH, for short) and show that the variety AGH is term-equivalent to that of AG. Next, a propositional logic, called AG (or AGH), is defined and shown to be algebraizable (in the sense of Blok and Pigozzi) with the variety AG, via AGH, as its equivalent algebraic semantics (up to term equivalence). All axiomatic extensions of the logic AG, corresponding to all the subvarieties of AG are given. They include the axiomatic extensions RDBLSt, RKLSt and G of the logic AG corresponding to the varieties RDBLSt, RKLSt, and G (of Gautama algebras), respectively. It is also deduced that none of the axiomatic extensions of AG has the Disjunction Property. Finally, We revisit the classical logic with strong negation CN and classical Nelson algebras CN introduced by Vakarelov in 1977 and improve his results by showing that CN is algebraizable with CN as its algebraic semantics and that the logics RKLSt, RKLStH, 3-valued Lukasivicz logic and the classical logic with strong negation are all equivalent.
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