Permutation clones that preserve relations

TL;DR

This paper studies permutation clones defined by relations, exploring their structure, classification on two-element sets, and their relation to dual weight mappings, expanding understanding of clone theory in finite sets.

Contribution

It provides structural results on relationally defined permutation clones, classifies all such clones on a two-element set, and links permutation clones to dual weight mappings.

Findings

All relationally defined permutation clones on two elements are classified (13 in total).

Maximal borrow closed permutation clones are either relationally defined or cancellatively defined.

Many infinite clone classes collapse when viewed as permutation clones.

Abstract

Permutation clones generalise permutation groups and clone theory. We investigate permutation clones defined by relations, or equivalently, the automorphism groups of powers of relations. We find many structural results on the lattice of all relationally defined permutation clones on a finite set. We find all relationally defined permutation clones on two element set. We show that all maximal borrow closed permutation clones are either relationally defined or cancellatively defined. Permutation clones generalise clones to permutations of $A^n$. Emil Je\v{r}\'{a}bek found the dual structure to be weight mappings $A^k\rightarrow M$ to a commutative monoid, generalising relations. We investigate the case when the dual object is precisely a relation, equivalently, that $M={\mathbb B}$, calling these relationally defined permutation clones. We determine the number of relationally defined permutation clones on two elements (13). We note that many infinite classes of clones collapse when looked at as permutation clones.