On the variety of strong subresiduated lattices

Abstract

A subresiduated lattice is a pair (, ), where is a bounded distributive lattice, is a bounded sublattice of and for every , ∈ there exists the maximum of the set { ∈ ∶ ∧ ≤ }, which is denoted by →. This pair can be regarded as an algebra (, ∧, ∨, →, 0, 1) of type (2, 2, 2, 0, 0), where = { ∈ ∶ 1 → = }. 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 → ( ∨ ) = (1 → ) ∨ (1 → ). Inspired by the fact that in any subresiduated lattice whose order is total the previous equation and the condition → ∈ {1 → , 1} for every , are satisfied, we also study the subvariety of S□ generated by the class whose members satisfy that → ∈ {1 → , 1} for every , .