Reticulation of Quasi-commutative Algebras

TL;DR

This paper introduces a generalized reticulation concept for quasi-commutative algebras within congruence-modular varieties, extending existing theories and connecting to Belluce's reticulation for rings.

Contribution

It defines and studies a new reticulation for quasi-commutative algebras, generalizing Belluce's reticulation for non-commutative rings, and provides characterization and transfer properties.

Findings

Reticulation for quasi-commutative algebras is well-defined and generalizes Belluce's reticulation.

The spectrum of the reticulation is homeomorphic to the prime spectrum of the algebra.

Characterization theorems and transfer properties are established for these algebras.

Abstract

The commutator operation in a congruence-modular variety $\mathcal{V}$ allows us to define the prime congruences of any algebra $A\in \mathcal{V}$ and the prime spectrum $Spec(A)$ of $A$. The first systematic study of this spectrum can be found in a paper by Agliano, published in Universal Algebra (1993). The reticulation of an algebra $A\in \mathcal{V}$ is a bounded distributive algebra $L(A)$, whose prime spectrum (endowed with the Stone topology) is homeomorphic to $Spec(A)$ (endowed with the topology defined by Agliano). In a recent paper, C. Mure\c{s}an and the author defined the reticulation for the algebras $A$ in a semidegenerate congruence-modular variety $\mathcal{V}$, satisfying the hypothesis $(H)$: the set $K(A)$ of compact congruences of $A$ is closed under commutators. This theory does not cover the Belluce reticulation for non-commutative rings. In this paper we shall introduce the quasi-commutative algebras in a semidegenerate congruence-modular variety $\mathcal{V}$ as a generalization of the Belluce quasi-commutative rings. We define and study a notion of reticulation for the quasi-commutative algebras such that the Belluce reticulation for the quasi-commutative rings can be obtained as a particular case. We prove a characterization theorem for the quasi-commutative algebras and some transfer properties by means of the reticulation