New Results on Congruence Boolean Lifting Property

TL;DR

This paper extends recent results on the Congruence Boolean Lifting Property ($CBLP$) from commutative rings to a broader class of algebraic structures, providing characterizations and conditions for $CBLP$ in congruence modular algebras.

Contribution

It generalizes the concept of $CBLP$ to semidegenerate congruence modular algebras and offers new characterization theorems and conditions for $CBLP$ applicability.

Findings

Characterization theorem for congruences with $CBLP$

Conditions ensuring $CBLP$ in various algebraic structures

Application of reticulation in analyzing $CBLP$

Abstract

The Lifting Idempotent Property ($LIP$) of ideals in commutative rings inspired the study of Boolean lifting properties in the context of other concrete algebraic structures ($MV$-algebras, commutative l-groups, $BL$-algebras, bounded distributive lattices, residuated lattices,etc.), as well as for some types of universal algebras (C. Muresan and the author defined and studied the Congruence Boolean Lifting Property ($CBLP$) for congruence modular algebras). A lifting ideal of a ring $R$ is an ideal of $R$ fulfilling $LIP$. In a recent paper, Tarizadeh and Sharma obtained new results on lifting ideals in commutative rings. The aim of this paper is to extend an important part of their results to congruences with $CBLP$ in semidegenerate congruence modular algebras. The reticulation of such algebra will play an important role in our investigations (recall that the reticulation of a congruence modular algebra $A$ is a bounded distributive lattice $L(A)$ whose prime spectrum is homeomorphic with Agliano's prime spectrum of $A$). Almost all results regarding $CBLP$ are obtained in the setting of semidegenerate congruence modular algebras having the property that the reticulations preserve the Boolean center. The paper contains several properties of congruences with $CBLP$. Among the results we mention a characterization theorem of congruences with $CBLP$. We achieve various conditions that ensure $CBLP$. Our results can be applied to a lot of types of concrete structures: commutative rings, $l$-groups, distributive lattices, $MV$-algebras, $BL$-algebras, residuated lattices, etc.