Congruence Boolean Lifting Property

TL;DR

This paper introduces the Congruence Boolean Lifting Property (CBLP) for congruence--distributive universal algebras, providing characterization and structure theorems, and exploring its relation to existing Boolean lifting properties across various algebraic classes.

Contribution

It defines CBLP, extends Boolean lifting properties to new algebraic contexts, and offers key theorems characterizing and structuring algebras with CBLP.

Findings

CBLP extends BLP to broader algebraic classes.

Characterization theorem for congruence--distributive algebras with CBLP.

Structure theorem for semilocal arithmetical algebras with CBLP.

Abstract

We introduce and study the Congruence Boolean Lifting Property (CBLP) for congruence--distributive universal algebras, as well as a property related to CBLP, which we have called $(\star )$. CBLP extends the so--called Boolean Lifting Properties (BLP) from MV--algebras, BL--algebras and residuated lattices, but differs from the BLP when particularized to bounded distributive lattices. Important classes of universal algebras, such as discriminator varieties, fulfill the CBLP. The main results of the present paper include a characterization theorem for congruence--distributive algebras with CBLP and a structure theorem for semilocal arithmetical algebras with CBLP. When we particularize the CBLP to the class of residuated lattices and to that of bounded distributive lattices and we study its relation to other Boolean Lifting Properties for these algebras, interesting results concerning the image of the reticulation functor between these classes are revealed.