# Boolean lifting property in quantales

## TL;DR

This paper introduces a new lifting property (LP) in quantales, generalizing existing properties from various algebraic structures, and provides key theorems characterizing quantales with LP, including semilocal and hyperarhimedean cases.

## Contribution

It defines the LP in quantales, unifying and extending lifting properties across multiple algebraic structures, with characterization theorems and structural insights.

## Key findings

- Characterization of quantales with LP
- Structure theorem for semilocal quantales with LP
- Characterization of hyperarhimedean 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.