Boolean Lifting Properties for Bounded Distributive Lattices

Abstract

In this paper, we introduce the lifting properties for the Boolean elements of bounded distributive lattices with respect to the congruences, filters and ideals, we establish how they relate to each other and to significant algebraic properties, and we determine important classes of bounded distributive lattices which satisfy these lifting properties.

Boolean lifting property in quantales

“…The lifting property introduced by previous definition generalizes the condition LIP from ring theory [26], as well as the other boolean lifting properties existing in literature [2], [4], [7], [9], [10], [11], [12], [15]. (2) If p is an m-prime element of A then one can prove that B([p) A ) = {0, 1}, therefore, by (1), p has LP.…”

“…If the reticulation L(A) is a chain then the quantale A has LP.Proof. By Corollary 4,[4] , the chain L(A) has Id-BLP, hence, by Proposition 5.8, it follows that A has BLP.…”

“…Any local quantale has LPProof. If A is local then L(A) is an Id-local bounded distributive lattice.According to Proposition 18,[4], the bounded distributive lattice L(A) has Id-BLP. Thus, by Proposition 5.8, the quantale A has LP.…”

Factor Congruence Lifting Property

“…If R is a bounded distributive lattice or a residuated lattice and φ ∈ Con(R), then: we say that φ fulfills the Boolean Lifting Property (abbreviated BLP) iff the Boolean morphism B(p φ ) : B(R) → B(R/φ) is surjective, and we say that R fulfills the Boolean Lifting Property (BLP) iff all congruences of R fulfill the BLP ( [6], [7], [13], [14]). Notice that, for any φ ∈ Con ( We recall that the filters of R are the non-empty subsets of R which are closed with respect to ⊙ and to upper bounds.…”

“…We say that L has the Boolean Lifting Property for filters (abbreviated Filt-BLP) iff all filters of L have the BLP. We say that L has the Boolean Lifting Property for ideals (abbreviated Id-BLP) iff all ideals of L have the BLP ( [6], [7], [14]). Clearly, if L has BLP, then L has Filt-BLP and Id-BLP.…”