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Symmetrical Heyting algebras of order $$ {3\times 3}$$ 3 × 3
Soft Computing ( IF 3.1 ) Pub Date : 2021-06-11 , DOI: 10.1007/s00500-021-05905-z
Carlos Gallardo , Alicia Ziliani

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.



中文翻译:

$$ {3\times 3}$$ 3 × 3 阶对称 Heyting 代数

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。

更新日期:2021-06-11
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