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‎Gautama and Almost Gautama Algebras and their associated logics
Recently, Gautama algebras were defined and investigated as a common generalization of the variety of regular double Stone algebras and the variety 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 (, 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 and the equational bases for all its subvarieties are given. It is also shown that the variety is a discriminator variety. Next, we consider logicizing ; but the variety lacks an implication operation. We, therefore, introduce another variety of algebras called ``Almost Gautama Heyting algebras'' (, for short) and show that the variety %of Almost Heyting algebras is term-equivalent to that of . Next, a propositional logic, called (or ), is defined and shown to be algebraizable (in the sense of Blok and Pigozzi) with the variety , via as its equivalent algebraic semantics (up to term equivalence). All axiomatic extensions of the logic , corresponding to all the subvarieties of are given. They include the axiomatic extensions , and of the logic corresponding to the varieties , , and (of Gautama algebras), respectively. It is also deduced that none of the axiomatic extensions of has the Disjunction Property. Finally, We revisit the classical logic with strong negation and classical Nelson algebras introduced by Vakarelov in 1977 and improve his results by showing that is algebraizable with as its algebraic semantics and that the logics , , 3-valued \L ukasivicz logic and the classical logic with strong negation are all equivalent.Fil: Cornejo, Juan Manuel. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - BahÃa Blanca. Instituto de Matemática BahÃa Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática BahÃa Blanca; ArgentinaFil: Sankappanavar, Hanamantagouda P.. State University of New York. Department of Mathematics ; Estados Unido