On the number of countable subdirect powers of unary algebras

TL;DR

This paper characterizes when finite unary algebras have only countably many countable subdirect powers, showing it occurs precisely when all operations are permutations or constants.

Contribution

It provides a complete characterization of unary algebras with countably many subdirect powers based on the nature of their operations.

Findings

Countably many subdirect powers occur only when all operations are permutations or constants.

The characterization is both necessary and sufficient.

The result clarifies the structure of unary algebras with limited subdirect powers.

Abstract

A finite unary algebra $(A,F)$ has only countably many countable subdirect powers if and only if every operation $f\in F$ is either a permutation or a constant mapping.