‎Gautama and Almost Gautama Algebras and their associated logics

Transactions on Fuzzy Sets and Systems

مقاله 6، دوره 2، شماره 2، بهمن 2023، صفحه 77-112   

نوع مقاله: Original Article

شناسه دیجیتال (DOI): 10.30495/tfss.2023.1983060.1068

نویسندگان

Juan M. Cornejo¹؛ Hanamantagouda P. Sankappanavar^{* 2}

¹Departamento de Matematica, Universidad Nacional del Sur INMABB-CONICET, Bahia Blanca, Argentina.

²Department of Mathematics, State University of New York, New Paltz, NY, USA.

چکیده

‎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.

کلیدواژه‌ها

‎Regular double Stone algebra‎؛ ‎regular Kleene Stone algebra‎؛ ‎Gautama algebra‎؛ ‎Almost Gautama algebra‎؛ ‎Almost Gautama Heyting algebra‎؛ ‎subdirectly irreducible algebra‎؛ ‎simple algebra‎؛ ‎logic $\mathcal{AG}$‎؛ ‎logic $\mathcal{G}$‎؛ ‎logic $\mathcal{RDBLS}t$‎؛ ‎logic $\mathcal{RKLS}t$‎

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