Generating subdirect products

TL;DR

This paper investigates conditions under which subdirect products of algebraic structures are finitely generated or presented, providing general results for various structures and highlighting complexities beyond congruence permutable varieties.

Contribution

It generalizes known results for groups to broader algebraic structures, establishing criteria for finite generation and presentation of subdirect products in diverse varieties.

Findings

Subdirect products of two factors are finitely generated if factors are finitely presented.

For multiple factors, finite generation depends on projections onto pairs and higher commutators.

In Noetherian K-algebras, finite generation and presentation of subdirect products are characterized by the properties of individual factors.

Abstract

We study conditions under which subdirect products of various types of algebraic structures are finitely generated or finitely presented. In the case of two factors, we prove general results for arbitrary congruence permutable varieties, which generalise previously known results for groups, and which apply to modules, rings, $K$-algebras and loops. For instance, if $C$ is a fiber product of $A$ and $B$ over a common quotient $D$, and if $A$, $B$ and $D$ are finitely presented, then $C$ is finitely generated. For subdirect products of more than two factors we establish a general connection with projections on pairs of factors and higher commutators. More detailed results are provided for groups, loops, rings and $K$-algebras. In particular, let $C$ be a subdirect product of $K$-algebras $A_1,\dots,A_n$ for a Noetherian ring $K$ such that the projection of $C$ onto any $A_i\times A_j$ has finite co-rank in $A_i\times A_j$. Then $C$ is finitely generated (resp. finitely presented) if and only if all $A_i$ are finitely generated (resp. finitely presented). Finally, examples of semigroups and lattices are provided which indicate further complications as one ventures beyond congruence permutable varieties.