Going Up and Lying Over in Congruence--modular Algebras

TL;DR

This paper generalizes the properties Going Up and Lying Over from ring theory to congruence--modular algebras, exploring their relationships, preservation under algebraic constructions, and characterizations.

Contribution

It extends classical properties to a broader algebraic context using prime congruences and provides new characterizations and examples.

Findings

Properties are preserved by finite direct products and quotients.

Many varieties always satisfy these properties.

Algebraic and topological characterizations are established.

Abstract

In this paper, we extend properties Going Up and Lying Over from ring theory to the general setting of congruence--modular equational classes, using the notion of prime congruence defined through the commutator. We show how these two properties relate to each other, prove that they are preserved by finite direct products and quotients and provide algebraic and topological characterizations for them. We also point out many kinds of varieties in which these properties always hold.