On the number of countable subdirect powers of finite commutative semigroups

TL;DR

This paper characterizes finite commutative semigroups with countably many non-isomorphic countable subdirect powers, showing they are either finite abelian groups or null semigroups, extending previous group results.

Contribution

It extends the classification of finite algebraic structures with countably many subdirect powers from groups to commutative semigroups.

Findings

Finite abelian groups have countably many subdirect powers.

Null semigroups also have countably many subdirect powers.

Other finite commutative semigroups have uncountably many subdirect powers.

Abstract

In 1981/82, Hickin \& Plotkin and McKenzie both proved that a finite group has only countably many non-isomorphic subdirect powers if and only if it is abelian. In this paper, we prove that a finite commutative semigroup has only countably many non-isomorphic countable subdirect powers if and only if it is either a finite abelian group or a null semigroup.