Dualities for 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 ∈ A there is c ∈ D such that for all d ∈ D, d∧a ≤ b if and only if d ≤ c. This c is denoted by a → b. This pair can be regarded as an algebra ⟨A,∧,∨,→,0,1⟩ of type (2,2,2,0,0) where D = {a ∈ A : 1 → a = a}. The class of subresiduated lattices is a variety which properly contains to the variety of Heyting algebras. In this paper we present dual equivalences for the algebraic category of subresiduated lattices. More precisely, we develop a spectral style duality and a bitopological style duality for this algebraic category. Finally we study the connections of these results with a known Priestley style duality for the algebraic category of subresiduated lattices.