Boolean lifting property for residuated lattices

Abstract

In this paper we define the Boolean Lifting Property (BLP) for residuated lattices to be the property that all Boolean elements can be lifted modulo every filter, and study residuated lattices with BLP. Boolean algebras, chains, local and hyperarchimedean residuated lattices have BLP. BLP behaves interestingly in direct products and involutive residuated lattices, and it is closely related to arithmetic properties involving Boolean elements, nilpotent elements and elements of the radical. When BLP is present, strong representation theorems for semilocal and maximal residuated lattices hold.

Boolean lifting property in quantales

“…The following proposition characterize the finite product of local quantales by using the properties LP and (*). It generalizes Proposition 6.13 of [11], Theorem 6.1 of [12], as well as Theorem 2.10 of [21]. Theorem 7.9.…”

“…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.…”

Factor Congruence Lifting Property

“…• in residuated lattices, BLP, CBLP and FCLP coincide; see, in [13] and [14], examples of residuated lattices without BLP, thus without CBLP or FCLP, as well as examples of residuated lattices with BLP, thus with CBLP and FCLP;…”

“…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.…”

Boolean Lifting Properties for Bounded Distributive Lattices

“…In ring theory, the Lifting Idempotents Property (LIP) ( [21], [1], [6], [16]), which is the property that the idempotent elements can be lifted modulo every left (respectively right) ideal of a (unitary) ring, is closely related to clean rings and exchange rings (in the commutative case, rings with LIP coincide to clean rings and to exchange rings). It is well known that the idempotent elements of a commutative unitary ring R form a Boolean algebra, called the Boolean center of R. Properties similar to LIP have been studied in other algebras which have a Boolean center, namely residuated structures: MV-algebras ( [10]), BL-algebras ( [9], [17]) and residuated lattices ( [11], [18], [12], [7]); the property studied in these algebras refers to the lifting of the Boolean elements modulo filters, and was called the Boolean Lifting Property (BLP). In [7], we have introduced a Boolean Lifting Property (BLP) modulo filters for bounded distributive lattices, we have studied the BLP for residuated lattices from the algebraic and the topological point of view, and we have proved that the BLP is transferrable, through the reticulation functor, between the class of residuated lattices and the class of bounded distributive lattices.…”