Finiteness properties of direct products of algebraic structures

TL;DR

This paper investigates how certain finiteness properties are preserved under direct products across various algebraic structures, identifying classes where these properties hold or fail.

Contribution

It provides a comprehensive analysis of the preservation of finiteness properties in direct products for multiple algebraic classes, extending known results and highlighting exceptions.

Findings

Preservation of properties in Mal'cev algebras and lattices

Failures outside specific algebraic classes

Broad classes where expected preservation holds

Abstract

We consider the preservation of properties of being finitely generated, being finitely presented and being residually finite under direct products in the context of different types of algebraic structures. The structures considered include Mal'cev algebras (including groups, rings and other classical algebras, as well as loops), idempotent algebras (including lattices), semigroups, and algebras in congruence modular varieties. We aim to identify as broad classes as possible in which the 'expected' preservation results (AxB satisfies property P if and only if A and B satisfy P) hold, and to exhibit ways in which they may fail outside those classes.