Boolean lifting property in quantales

Abstract

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings,maximal rings, etc. Inspired by LIP, there were defined lifting properties for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congruence distributive universal algebras,etc. In this paper we define a lifting property (LP) in quantales, structures that constitute a good abstraction of the lattices of ideals, filters or congruences. LP generalizes all the lifting properties existing in literature. The main tool in the study of LP in a quantale A is the reticulation of A, a bounded distributive lattice whose prime spectrum is homeomorphic to the prime spectrum os A. The principal results of the paper include a characterization theorem for quantales with LP, a structure theorem for semilocal quantales with LP and a charaterization theorem for hyperarhimedean quantales.

Flat topology on prime, maximal and minimal prime spectra of quantales

“…We remark that L(R) is isomorphic to the reticulation L(Id(R)) of the quantale Id(R). Lemma 3.2 [9] For all elements a, b ∈ K(A) the following properties hold: According to Lemma 3.3, one can consider the following order-preserving functions: u : Spec(A) → Spec Id (L(A)) and v : Spec Id (L(A)) → Spec(A), defined by u(p) = p * and v(P ) = P * , for all p ∈ Spec(A) and P ∈ Spec Id (L(A)). Sometimes the previous functions u and v will be denoted by u A and v A .…”

“…In this section we shall recall from [9], [17] the axiomatic definition of the reticulation of the coherent quantale and some of its basic properties. Let A be a coherent quantale and K(A) the set of its compact elements.…”

“…In [9], [17] there were proven the existence and the unicity of the reticulation for each coherent quantale A; this unique reticulation will be denoted by (L(A), λ A : K(A) → L(A)) or shortly L(A). The reticulation L(R) of a commutative ring R there was introduced by many authors, but the main references on this topic remain [40], [25].…”

“…We also introduce the Pierce spectrum Sp(A) of the quantale A. In Section 6 we continue the study of the hyperarchimedean quantales, initiated in [9]. The main result of the section presents new characterizations of these objects in terms of the flat and the patch topologies.…”

“…The new topological space M ax F (A) is Hausdorff and zero -dimensional. We use the flat topology on M ax(A) in order to obtain new properties that characterize the normal and the B-normal quantales [9], [20], [35], [41]. The normal quantales constitute an abstractization of the lattices of ideals in Gelfand rings [25], [27], [32] and in normal lattices [11], [37], [19], while the B-normal quantales generalize the lattices of ideals in clean rings [34], [23] and in B-normal lattices [10], [8].…”