Meet-reducible submaximal clones determined by nontrivial equivalence relations

TL;DR

This paper classifies certain submaximal clones on finite sets, specifically those formed by intersections of clones associated with nontrivial equivalence relations and Rosenberg's relations, to better understand the complex lattice structure.

Contribution

It provides a classification of relations for which the intersection of clones is submaximal within a clone generated by a nontrivial equivalence relation.

Findings

Identifies conditions under which the intersection clone is submaximal

Classifies relations on finite sets related to Rosenberg's list

Enhances understanding of the clone lattice structure

Abstract

The structure of the lattice of clones on a finite set has been proven to be very complex. To better understand the top of this lattice, it is important to provide a characterization of submaximal clones in the lattice of clones. It is known that the clones $Pol(\theta)$ and $Pol(\rho)$ (where $\theta$ is a nontrivial equivalence relation on $E_k = \{0,\dots, k-1\}$, and $\rho$ is among the six types of relations which characterize maximal clones) are maximal clones. In this paper, we provide a classification of relations (of Rosenberg's List) on $E_k$ such that the clone $Pol(\theta) \cap Pol(\rho)$ is maximal in $Pol(\theta)$.