Abstract
The critical relations are the building blocks of the relational clone of a relational structure with respect to the relational operations intersection and direct product. In this paper we describe the critical relations of crowns. As a consequence, we obtain that the subpower membership problem for any crown is polynomial-time solvable.
Highlights
The relational structures in this note are assumed to be always finite
Following the terminology in [8], we call the posets of the form C where C is a crown locked crowns, see Fig. 2
Theorem 3 Let K be a class of models in a fixed language of relational structures such that K is closed under finite power
Summary
The relational structures in this note are assumed to be always finite. A crown is a height 1 poset whose comparabilty graph is a cycle, see Fig. 1. From results of the third author in [13] it follows that the clone of C is non-finitely generated if C is the four element crown The proof of this fact in [13] is reduced to Tardos’s original proof for the poset T. When studying finite generability for the clone of a finite poset P, often an other kind of extendibility question occurs: given a power Pn of P and a partial map f from Pn to P, decide whether f extends to a monotone total map from Pn to P We call this problem the restricted extendibility problem for P and denote it by RExt(P). We do not know the answer to the question if there is a finite poset P for which RExt(P) is NP-complete
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