On the variety of strong subresiduated lattices

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 $\{d\in D:a\wedge d\le b\}$, which is denoted by $a\to b$. This pair can be regarded as an algebra $(A,\wedge ,\vee ,\to ,0,1)$ 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 $S^{\square }$, whose members satisfy the equation $1\to (a\vee b)=(1\to a)\vee (1\to b)$. 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 $S^{\square }$ generated by the class whose members satisfy that $a\to b\in \{1\to b,1\}$ for every $a,b$.