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Erschienen in:
11.06.2021 | Foundations
Symmetrical Heyting algebras of order \( {3\times 3}\)
Erschienen in: Soft Computing | Ausgabe 14/2021
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Abstract
The notion of \(n\times m\)-valued Łukasiewicz algebras with negation (or \(NS_{n \times m}\)-algebras) was introduced by C. Sanza in Notes on \(n\times m\)-valued Łukasiewicz algebras with negation, Logic J. of the IGPL 12, 6 (2004), 499–507. These algebras constitute a non-trivial generalization of n-valued Łukasiewicz–Moisil algebras and they are a particular case of matrix Łukasiewicz algebras, which were introduced by W. Suchoń in 1975. In this note, we focus on \(NS_{3 \times 3}\)-algebras. We prove that they are Heyting algebras and in case that they are centered we describe the Heyting implication in terms of their centers. We also establish a relationship between centered \(NS_{3 \times 3}\)-algebras and a class of symmetrical Heyting algebras with operators. Finally, we define symmetrical Heyting algebras of order \(3\times 3\) (or \(SH_{3 \times 3}\)-algebras) and we present a discrete duality for them.