Absolute Retracts and Essential Extensions in Congruence Modular Varieties

TL;DR

This paper explores the structure of absolute retracts in congruence modular varieties, revealing their decomposition into products of simpler algebras under certain conditions.

Contribution

It characterizes absolute retracts with finite dimensional congruence lattices and in residually small varieties, showing their decomposition into products of specific algebra types.

Findings

Absolute retracts with finite dimensional congruence lattices are products of subdirectly irreducible algebras.

In residually small varieties, absolute retracts are products of an abelian and a centerless algebra.

Provides structural insights into absolute retracts in congruence modular varieties.

Abstract

This paper studies absolute retracts in congruence modular varieties of universal algebras. It is shown that every absolute retract with finite dimensional congruence lattice is a product of subdirectly irreducible algebras. Further, every absolute retract in a residually small variety is the product of an abelian algebra and a centerless algebra.