New Results on Congruence Boolean Lifting Property (2109.12615v1)

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 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.