The Commutator as Least Fixed Point of a Closure Operator

DeMeo, William

doi:10.48550/arxiv.1703.02764

Cited by 2 publications

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“…Of course, since each of the algebras M from the following examples is finite, we have ∇ M ∈ K(M ). We have used the method in [40] to calculate the commutators, excepting those in groups, where we have used the commutators on normal subgroups; recall that the variety of groups is congruence-modular [39]. Following [1], we say that A is:…”

Section: Some Examples Particular Cases and Preservation Of Finite Di...mentioning

confidence: 99%

The Reticulation of a Universal Algebr

Georgescu¹,

Mureșan²

2018

SACS

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The reticulation of an algebra A is a bounded distributive lattice L(A) whose prime spectrum of filters or ideals is homeomorphic to the prime spectrum of congruences of A, endowed with the Stone topologies. We have obtained a construction for the reticulation of any algebra A from a semi-degenerate congruencemodular variety C in the case when the commutator of A, applied to compact congruences of A, produces compact congruences, in particular when C has principal commutators; furthermore, it turns out that weaker conditions than the fact that A belongs to a congruence-modular variety are sufficient for A to have a reticulation. This construction generalizes the reticulation of a commutative unitary ring, as well as that of a residuated lattice, which in turn generalizes the reticulation of a BL-algebra and that of an MV-algebra. The purpose of constructing the reticulation for the algebras from C is that of transferring algebraic and topological properties between the variety of bounded distributive lattices and C, and a reticulation functor is particularily useful for this transfer. We have defined and studied a reticulation functor for our construction of the reticulation in this context of universal algebra.

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Section: Some Examples Particular Cases and Preservation Of Finite Di...mentioning

confidence: 99%

The Reticulation of a Universal Algebr

Georgescu¹,

Mureșan²

2018

SACS

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“…In the following examples, we have calculated the commutators using the method from [31]. Note that, by [1], the prime congruences of A are the meet-irreducible elements φ of Con(A) with the property that [α, α] A ⊆ φ implies α ⊆ φ for all α ∈ Con(A).…”

Section: Functoriality Of the Boolean Centermentioning

confidence: 99%

Functorial Properties of the Reticulation of a Universal Algebra

Georgescu,

Kwuida,

Mureşan

2019

Preprint

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The Commutator as Least Fixed Point of a Closure Operator

Abstract: We present a description of the (non-modular) commutator, inspired by that of Kearnes in [Kea95, p. 930], that provides a simple recipe for computing the commutator.

Cited by 2 publications

References 1 publication

The Reticulation of a Universal Algebr

The Reticulation of a Universal Algebr

Functorial Properties of the Reticulation of a Universal Algebra

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