Symmetrical Heyting algebras of order ${3\times 3}$

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.