Clean unital ℓ-groups

Hager, Anthony W.; Kimber, Chawne M.; McGovern, Warren Wm.

doi:10.2478/s12175-013-0148-8

Cited by 10 publications

(10 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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]. (2) If p is an m-prime element of A then one can prove that B([p) A ) = {0, 1}, therefore, by (1), p has LP.…”

Section: A Lifting Propertymentioning

confidence: 84%

“…These rings have significant algebraic and topological properties and a whole literature has been dedicated to their study (see [16], [25], [22]). By analogy to LIP, there were introduced various lifting properties for other algebraic structures: MV-algebras [7] , commutative l-groups [15], BL-algebras [23], residuated lattices [14], bounded distributive lattices [4], etc. Both LIP definition and of other lifting properties assume the existence of Boolean centers, subsets of algebras endowed with a Boolean structure: the idempotent in the case of commutative rings and the complemented elements for the other mentioned algebras.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Boolean lifting property in quantales

Cheptea

Georgescu

2020

Soft Comput

View full text Add to dashboard Cite

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings,maximal rings, etc. Inspired by LIP, there were defined lifting properties for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congruence distributive universal algebras,etc. In this paper we define a lifting property (LP) in quantales, structures that constitute a good abstraction of the lattices of ideals, filters or congruences. LP generalizes all the lifting properties existing in literature. The main tool in the study of LP in a quantale A is the reticulation of A, a bounded distributive lattice whose prime spectrum is homeomorphic to the prime spectrum os A. The principal results of the paper include a characterization theorem for quantales with LP, a structure theorem for semilocal quantales with LP and a charaterization theorem for hyperarhimedean quantales.

show abstract

Section: A Lifting Propertymentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Boolean lifting property in quantales

Cheptea

Georgescu

2020

Soft Comput

View full text Add to dashboard Cite

show abstract

“…Inspired by the theory of rings with LIP , various lifting properties were defined for other algebraic structures (M V -algebras [15], commutative ℓ-groups [25], BL-algebras [14], pseudo BL-algebras [7], [8], bounded distributive lattices [12], residuated lattices [19], orthomodular lattices [31], etc.). A lifting property, named Congruence Boolean Lifting Property (CBLP ), was studied in a universal algebra framework: for congruence distributive algebras [22] and for semidegenerate congruence modular algebras [20].…”

Section: Introductionmentioning

confidence: 99%

New results on Congruence Boolean Lifting Property

Georgescu

2022

JAHLA

View full text Add to dashboard Cite

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.

show abstract

“…More about components can be found in [1,Section 1.4]. Following the recent work [5] (see also [4]), we say that E is clean if each element of E can be written as a sum of a strong unit and a component of E. In other words, E is clean if and only if, for every x ∈ E there exist a strong unit u of E and p ∈ C (E) such that x = u + p. We are in position now to give the characterization we were talking about. Proof.…”

Section: Preliminaries On Local Vector Latticesmentioning

confidence: 99%

An equivalence functor between local vector lattices and vector lattices

Boulabiar

2018

CGASA

View full text Add to dashboard Cite

We call a local vector lattice any vector lattice with a distinguished positive strong unit and having exactly one maximal ideal (its radical). We provide a short study of local vector lattices. In this regards, some characterizations of local vector lattices are given. For instance, we prove that a vector lattice with a distinguished strong unit is local if and only if it is clean with non no-trivial components. Nevertheless, our main purpose is to prove, via what we call the radical functor, that the category of all vector lattices and lattice homomorphisms is equivalent to the category of local vectors lattices and unital (i.e., unit preserving) lattice homomorphisms. 1 Introduction All vector lattices we consider in this paper are assumed to be real. Let L and M be vector lattices. Recall that a linear map L ω → M is called a lattice homomorphism if ω (|x|) = |ω (x)| , for all x ∈ L.

show abstract

Clean unital ℓ-groups

Cited by 10 publications

References 15 publications

Boolean lifting property in quantales

Boolean lifting property in quantales

New results on Congruence Boolean Lifting Property

An equivalence functor between local vector lattices and vector lattices

Contact Info

Product

Resources

About