Lifting Elements in Coherent Quantales

Abstract

An ideal I of a ring R is a lifting ideal if the idempotents of R can be lifted modulo I. A rich literature has been dedicated to lifting ideals. Recently, new algebraic and topological results on lifting ideals have been discovered. This paper aims to generalize some of these results to coherent quantales. We introduce the notion of lifting elements in a quantale and a lot of results about them are proven. Some properties and characterizations of a coherent quantale in which any element is a lifting element are obtained. The formulations and the proofs of our results use the transfer properties of reticulation, a construction that assigns to each coherent quantale a bounded distributive lattice. The abstract results on lifting elements can be applied to study some Boolean lifting properties in concrete algebraic structures: commutative rings, bounded distributive lattices, residuated lattices, MV-algebras, BL-algebras, abelian l-groups, some classes of universal algebras, etc.

Keywords

• coherent quantale
• reticulation of a quantale
• boolean lifting property
• lifting ideals