Symmetrical Heyting algebras of order 3×3

The notion of n × m -valued Łukasiewicz algebras with negation (or N S n × m -algebras) was introduced by C. Sanza in Notes on n × 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 N S 3 × 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 N S 3 × 3 -algebras and a class of symmetrical Heyting algebras with operators. Finally, we define symmetrical Heyting algebras of order 3 × 3 (or S H 3 × 3 -algebras) and we present a discrete duality for them.