Homomorphic Image Orders on Combinatorial Structures

Abstract

Combinatorial structures have been considered under various orders, including substructure order and homomorphism order. In this paper, we investigate the homomorphic image order, corresponding to the existence of a surjective homomorphism between two structures. We distinguish between strong and induced forms of the order and explore how they behave in the context of different common combinatorial structures. We focus on three aspects: antichains and partial well-order, the joint preimage property and the dual amalgamation property. The two latter properties are natural analogues of the well-known joint embedding property and amalgamation property, and are investigated here for the first time.

Well quasi-order in combinatorics: embeddings and homomorphisms

“…The collection of posets is not wqo, in either the reflexive or non-reflexive representations, as witnessed for example by the family illustrated in Figure 10. All this leaves trees as yet again the boundary class, and here we prove: Theorem 6.2 (Huczynska, Ruškuc [34]…”

“…Similar descriptions can be given for nearly all common combinatorial structures. We will not list all these descriptions here, but refer the reader to [34] for a fairly comprehensive treatment.…”

“…Furthermore, permutations themselves are fairly restricted combinatorial structures, with no obvious variations available, and with essentially all homomorphism-related orders reducing to the subpermutation ordering; see [34]. Thus the overall body of results is considerably smaller than in the case of graphs.…”

“…The homomorphism order itself has been studied, but the kind of questions we have been exploring here have received surprisingly little attention. As for the homomorphic image order, it has been almost entirely unexplored until a recent paper of the current authors ( [34]). …”

“…In [34] the present authors attempt to redress this balance, by focussing on the homomorphic image order in the combinatorial setting, and considering the properties of this order in the realms of digraphs, graphs, words, permutations, posets, and a range of other combinatorial objects. In fact, as outlined in Section 2, we are led to define three homomorphic image orders, corresponding to the three flavours of homomorphism: standard, strong (or induced) and M-strong; let us denote these orderings as ≤ H , ≤ IH , ≤ MH respectively.…”