Quasi covering dimension of finite distributive lattices

Abstract

In this paper, we study the quasi covering dimension dimq of finite distributive lattices. By the join-irreducible elements, we characterize the dense elements of a finite lattice. Based on the notion of the widths of posets, we prove that for every finite distributive lattice L, dimq(L)=max{width(↑a∩J(L))|a∈ Min(J(L))}-1, where Min(J(L)) is the set of all minimal elements of join-irreducible elements of L. Finally, we study the quasi covering
dimension of the linear sum and rectangular product of two finite distributive lattices.