New results on Congruence Boolean Lifting Property

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 (M V -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 congruence 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 congruence 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, M V -algebras, BL-algebras, residuated lattices, etc.

Boolean lifting property in quantales

“…The two Boolean algebras Idp(A) and B(R(Id(A))) are isomorphic and the condition LIP can be expressed in terms of the frame R(Id(A)) (see [2]). Similar observations can be made in the case of the lifting properties for other concrete structures [4], [7], [11], [12], [23]. Thus the lifting property (LP) in a coherent quantale A will use the Boolean center B(A) of A.…”

“…The following proposition characterize the finite product of local quantales by using the properties LP and (*). It generalizes Proposition 6.13 of [11], Theorem 6.1 of [12], as well as Theorem 2.10 of [21].…”

“…The lifting property introduced by previous definition generalizes the condition LIP from ring theory [26], as well as the other boolean lifting properties existing in literature [2], [4], [7], [9], [10], [11], [12], [15]. Remark 5.7.…”

“…The notion of property (*) there was defined for residuated lattice in [11], for bounded distributive lattice in [4] and for congruence distributive universal algebras in [12]. The following definition proposes a notion of property (*) in the framework of the quantales.…”

“…In order to obtain a Boolean center of an algebra A we shall choose a Boolean subalgebra of the congruence lattice Con(A). In [12], [13] there are defined two notions of lifting property for a congruence distributive algebra A: Congruence Boolean Lifting Property (CBLP), whenever the Boolean center is the set B(Con(A)) of complemented congruences of A, and Factor Congruence Lifting Property (FCLP), whenever the Boolean center is the set FC(A) of factor congruences of A.…”