Reticulation of Quasi-commutative Algebras

The commutator theory, developed by Fresee and McKenzie in the framework of 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 (۱۹۹۳).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.

کلیدواژه ها:

commutator operation ، semidegenerate congruence - modular algebras ، reticulation ، spectral spaces