Unique inclusions of maximal C-clones in maximal clones

Abstract

Abstract. 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 work by Creignou et al. from 2008. We completely describe the relationship regarding set inclusion between maximal C-clones and maximal clones. As a main result we obtain that for every maximal 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 C-clones is given.Mathematics Subject Classification. 08A40, 08A02, 08A99.

The Number of Clones Determined by Disjunctions of Unary Relations

“…Combining Corollary 21 and Proposition 22 we can pinpoint the exact number of clones determined by disjunctions of non-trivial unary relations from a finite parameter set Γ. This answers, in particular, the question regarding the number of clausal clones on finite sets that was stated to be open in [Var11a,BVG18].…”

“…These are called clausal relations (over chains) and have been studied more extensively in [Var10,BV10,BVG18,Var11b,Var11a].…”

“…This problem is in fact asking for a description of all clones (with tractable CSP) associated with clausal relations, called clausal clones in [Var10]. Since there is a continuum of clones on finite, at least three-element sets, a first necessary step to ensure the feasibility of answering such a question is to determine the cardinality of the lattice of all clausal clones, which is another open problem from [Var11a], see also [BVG18, Section 1].…”