The submaximal clones on the three-element set with finitely many relative R-classes

TL;DR

This paper investigates the structure of submaximal clones on a three-element set, focusing on those with finitely many relative R-classes, an equivalence relation based on substitution instances within the clone.

Contribution

It characterizes which submaximal clones on a three-element set have finitely many relative R-classes, extending understanding of clone structures and Green's relations.

Findings

Identifies submaximal clones with finitely many relative R-classes

Provides a classification of these clones on a three-element set

Enhances the understanding of clone and Green's R-relation structures

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 if and only if each one is a substitution instance of the other using operations from C. We study the maximal and submaximal clones on a three-element set and determine which of them have only finitely many relative R-classes.