Unique inclusions of maximal C-clones in maximal clones

TL;DR

This paper characterizes the relationship between maximal C-clones and maximal clones on finite domains, revealing a unique maximal clone containment for each C-clone and providing a completeness criterion.

Contribution

It provides a complete description of how maximal C-clones relate to maximal clones, including a unique containment and a new completeness criterion.

Findings

Each maximal C-clone is contained in exactly one maximal clone.

A precise description of the unique maximal clone for each C-clone is given.

A completeness criterion for C-clones is established.

Abstract

$\mathit{C}$-clones are polymorphism sets of so-called clausal relations, a special type of relations on a finite domain, which first appeared in connection with constraint satisfaction problems in [Creignou et al. 2008]. We completely describe the relationship w.r.t. set inclusion between maximal $\mathit{C}$-clones and maximal clones. As a main result we obtain that for every maximal $\mathit{C}$-clone there exists exactly one maximal clone in which it is contained. A precise description of this unique maximal clone, as well as a corresponding completeness criterion for $\mathit{C}$-clones is given.