On the variety of strong subresiduated lattices

Pub Date : 2023-07-11 DOI:10.1002/malq.202200067

Sergio Celani, Hernán J. San Martín

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Abstract

A subresiduated lattice is a pair $(A, D)$ , where A is a bounded distributive lattice, D is a bounded sublattice of A and for every $a, b \in A$ there exists the maximum of the set ${d \in D : a \land d \leq b}$ , which is denoted by $a \to b$ . This pair can be regarded as an algebra $(A, \land, \lor, \to, 0, 1)$ of type (2, 2, 2, 0, 0), where $D = {a \in A : 1 \to a = a}$ . The class of subresiduated lattices is a variety which properly contains the variety of Heyting algebras. In this paper we study the subvariety of subresiduated lattices, denoted by $S^{□}$ , whose members satisfy the equation $1 \to (a \lor b) = (1 \to a) \lor (1 \to b)$ . Inspired by the fact that in any subresiduated lattice whose order is total the previous equation and the condition $a \to b \in {1 \to b, 1}$ for every $a, b$ are satisfied, we also study the subvariety of $S^{□}$ generated by the class whose members satisfy that $a \to b \in {1 \to b, 1}$ for every $a, b$ .

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关于强次边值格的多样性

子边值格是一对（A，D）$（A，D）$，其中A是有界分配格，D是A的有界子格，并且对于每个A，b∈A$A，b\在A$中存在集合{d∈d:A∧d≤b}$\l轨道d\在d:A楔d\le b\l轨道$中的最大值，其由→ b$a\右箭头b$。这对可以看作是一个代数（A，∧，→ , 0，1）$（A，\wedge，\vee，\rightarrow，0,1）$的类型（2，2，0，0），其中D={A∈A:1→ a=a｝$D=\l在a:1\rightarrowa=a\rbrace$中竞速a\。次边值格类是一个适当包含Heyting代数的变种的变种。在本文中，我们研究了由S表示的次边值格的子变种□ $\mathrm｛S｝^｛\Box｝$，其成员满足等式1→ （a∧b）=（1→ a）∧（1→ b）$1\rightarrow（a\vee b）=（1\rightarrowa）\vee（1\rigightarrowb）$。受以下事实的启发：在阶为全的任何次边值格中，前面的方程和条件a→ b∈{1→ b，1｝$a\rightarrow b\in\lblase 1\rightarrowb，1\rbrace$对于每个a，b$a，b$都满足，我们还研究了S的亚变种□ $\mathrm｛S｝^｛\Box｝$由其成员满足→ b∈{1→ b，1｝$a\rightarrowb\in\lblase 1\rightarrow b，1\rbrace$对于每个a，b$a，b$。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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