Clones with finitely many relative R-classes

TL;DR

This paper investigates clones on a set A that have finitely many relative R-classes, exploring an equivalence relation based on substitution instances within the clone.

Contribution

It introduces and analyzes the concept of finitely many relative R-classes in clones, extending Green's R-relation to this context.

Findings

Identification of conditions for finitely many relative R-classes

Characterization of clones with finite relative R-classes

Insights into the structure of clones via relative R-classes

Abstract

For each clone C on a set A there is an associated equivalence relation analogous to Green's R-relation, which relates two operations on A iff each one is a substitution instance of the other using operations from C. We study the clones for which there are only finitely many relative R-classes.