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    ‎Gautama and Almost Gautama Algebras and their associated logics

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    Recently, Gautama algebras were defined and investigated as a common generalization of the variety RDBLSt\mathbb{RDBLS}\rm t of regular double Stone algebras and the variety RKLSt\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 (AG\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 AG\mathbb{AG} and the equational bases for all its subvarieties are given. It is also shown that the variety AG\mathbb{AG} is a discriminator variety. Next, we consider logicizing AG\mathbb{AG}; but the variety AG\mathbb{AG} lacks an implication operation. We, therefore, introduce another variety of algebras called ``Almost Gautama Heyting algebras'' (AGH\mathbb{AGH}, for short) and show that the variety AGH\mathbb{AGH} %of Almost Heyting algebras is term-equivalent to that of AG\mathbb{AG}. Next, a propositional logic, called AG\mathcal{AG} (or AGH\mathcal{AGH}), is defined and shown to be algebraizable (in the sense of Blok and Pigozzi) with the variety AG\mathbb{AG}, via AGH,\mathbb{AGH}, as its equivalent algebraic semantics (up to term equivalence). All axiomatic extensions of the logic AG\mathcal{AG}, corresponding to all the subvarieties of AG\mathbb{AG} are given. They include the axiomatic extensions RDBLSt\mathcal{RDBLS}t, RKLSt\mathcal{RKLS}t and G\mathcal{G} of the logic AG\mathcal{AG} corresponding to the varieties RDBLSt\mathbb{RDBLS}\rm t, RKLSt\mathbb{RKLS}\rm t, and G\mathbb{G} (of Gautama algebras), respectively. It is also deduced that none of the axiomatic extensions of AG\mathcal{AG} has the Disjunction Property. Finally, We revisit the classical logic with strong negation CN\mathcal{CN} and classical Nelson algebras CN\mathbb{CN} introduced by Vakarelov in 1977 and improve his results by showing that CN\mathcal{CN} is algebraizable with CN\mathbb{CN} as its algebraic semantics and that the logics RKLSt\mathcal{RKLS}\rm t, RKLStH\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
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