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.

的概念\（N \乘以m \） -valued卢卡西维奇代数与否定（或\（NS_ {N \乘以m} \） -代数）于引入由C.三匝在Notes \（N \乘以m \）带否定值的 Łukasiewicz 代数，IGPL 的 Logic J. 12, 6 (2004), 499–507。这些代数构成了n值 Łukasiewicz-Moisil 代数的非平凡推广，它们是矩阵 Łukasiewicz 代数的一个特例，由 W.suchoń 在 1975 年引入。在本笔记中，我们关注\(NS_{3 \次 3}\) -代数。我们证明它们是 Heyting 代数，如果它们是中心的，我们将根据它们的中心描述 Heyting 蕴涵。我们还建立了居中关系\(NS_{3 \times 3}\) -代数和一类带有运算符的对称 Heyting 代数。最后，我们定义了阶\(3\times 3\)（或\(SH_{3 \times 3}\) -algebras）的对称 Heyting 代数，并为它们提供了离散对偶性。