Boolean Lifting Property for Residuated Lattices

TL;DR

This paper introduces the Boolean Lifting Property (BLP) for residuated lattices, exploring its implications and characterizations, and establishing strong representation theorems for certain classes of these lattices.

Contribution

It defines BLP for residuated lattices, analyzes its behavior across various classes, and links it to key algebraic properties and representation theorems.

Findings

Boolean algebras, chains, local and hyperarchimedean residuated lattices have BLP

BLP interacts interestingly with direct products and involutive residuated lattices

Presence of BLP enables strong representation theorems for semilocal and maximal 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.