Remarks on special congruences

TL;DR

This paper explores special types of algebraic structures where non-trivial congruences have unique non-singleton classes, focusing on Rees and quasi-Rees algebras, and examines their properties and examples.

Contribution

It introduces and analyzes classes of algebras with unique non-singleton congruence classes, including Rees and quasi-Rees algebras, with new results on their properties and quotients.

Findings

Characterization of Rees and quasi-Rees algebras

Results on the behavior of these properties in quotient algebras

Examples illustrating the classes of algebras studied

Abstract

We study algebras and varieties where every non-trivial congruence has some class being a non-trivial subuniverse of the algebra in question. Then we focus on algebras where this non-trivial class is a unique non-singleton class of the congruence. In particular, we investigate Rees algebras, quasi-Rees algebras and algebras having the one-block-property. We also present results concerning these properties on quotient algebras. Several examples of such algebras and varieties are included.