Residual Finiteness and Related Properties in Monounary Algebras and their Direct Products

TL;DR

This paper characterizes how residual finiteness and related properties in monounary algebras relate to their direct products, providing graphical criteria and identifying conditions under which these properties are preserved.

Contribution

It introduces graphical criteria for residual finiteness and related properties in monounary algebras and characterizes when these properties are preserved in direct products.

Findings

A graphical criterion $ ext{C}_ ext{P}$ for each property $ ext{P}$ determines property satisfaction.

Direct product $ ext{A} imes ext{B}$ has property $ ext{P}$ if both factors do, or if one is backwards-bounded.

Backwards-boundedness ensures all properties hold in direct products.

Abstract

In this paper we discuss the relationship between direct products of monounary algebras and their components, with respect to the properties of residual finiteness, strong/weak subalgebra separability, and complete separability. For each of these properties $\mathcal{P}$, we give a graphical criterion $\mathcal{C_P}$ such that a monounary algebra $A$ has property $\mathcal{P}$ if and only if it satisfies $\mathcal{C_P}$. We also show that for a direct product $A\times B$ of monounary algebras, $A\times B$ has property $\mathcal{P}$ if and only if one of the following is true: either both $A$ and $B$ have property $\mathcal{P}$, or at least one of $A$ or $B$ are backwards-bounded, a special property which dominates direct products and which guarantees all $\mathcal{P}$ hold.