Bounded distributive lattices with two subordinations

Abstract

In this paper we consider the notion of subordination on distributive lattices, equivalent to that of quasi-modal operator for distributive lattices introduced by Castro and Celani in 2004. We provide topological dualities for categories of distributive lattices with a subordination and then for some categories of distributive lattices with two subordinations, structures that we name bi-subordination lattices. We investigate three classes of bi-subordination lattices. In particular that of positive bi-subordination lattices.