Key (critical) relations preserved by a weak near-unanimity function

TL;DR

This paper introduces key relations, explores their properties, and characterizes those preserved by weak near-unanimity functions, revealing their structure and limitations on finite sets.

Contribution

It provides a new notion of key relations, characterizes those preserved by weak near-unanimity functions, and describes their structure on finite sets.

Findings

All clones on finite sets are defined by key relations.

Key relations on two elements are disjunctions of linear equations.

Not all key relations have simple descriptions, but those preserved by weak near-unanimity functions do.

Abstract

In the paper we introduce a notion of a key relation, which is similar to the notion of a critical relation introduced by Keith A.Kearnes and \'Agnes Szendrei. All clones on finite sets can be defined by only key relations. In addition there is a nice description of all key relations on 2 elements. These are exactly the relations that can be defined as a disjunction of linear equations. In the paper we show that, in general key relations do not have such a nice description. Nevertheless, we obtain a nice characterization of all key relations preserved by a weak near-unanimity function. This characterization is presented in the paper.