Generalised dualities and maximal finite antichains in the homomorphism order of relational structures

Abstract

The motivation for this paper is threefold. First, we study the connectivity properties of the homomorphism order of directed graphs, and more generally for relational structures. As opposed to the homomorphism order of undirected graphs (which has no non-trivial finite maximal antichains), the order of directed graphs has finite maximal * This research was partially supported by the EU Research Training Network COMB-STRU. † The third author's research is supported by grants from NSERC and ARP. ‡ The Institute for Theoretical Computer Science is supported as project 1M0021620808 by the Ministry of Education of the Czech Republic. antichains of any size. In this paper, we characterise explicitely all maximal antichains in the homomorphism order of directed graphs. Quite surprisingly, these maximal antichains correspond to generalised dualities. The notion of generalised duality is defined here in the full generality as an extension of the notion of finitary duality, investigated in [17]. Building upon the results of the cited paper, we fully characterise the generalised dualities. It appears that these dualities are determined by forbidding homomorphisms from a finite set of forests (rather than trees). Finally, in the spirit of [1], [12], [4] we shall characterise "generalised" Constraint Satisfaction Problems (defined also here) problems that are first order definable. These are again just generalised dualities corresponding to finite maximal antichains in the homomorphism order.

Database Constraints and Homomorphism Dualities

“…In this section, we will present a characterization of the finite sets of structures F that are an obstruction set for a finite set of structures D. For the case of relational signatures without constant symbols, an elegant characterization of such sets F was established in [8]. Our main contribution in this section is to show that the characterization can be extended in a natural way to structures with constant symbols.…”

“…Known Results for Structures without Constant Symbols Consider signatures consisting only of relation symbols. The main result from [8] states that a finite set of homomorphically incomparable core structures F for such a signature is an obstruction set for a finite set of structures D if and only if each structure from F obeys a certain acyclicity condition, which we will now define.…”

“…This result provides the aforementioned necessary and sufficient condition for unique characterizability of GAV schema mappings; however, it does not yield immediately an algorithm for testing whether or not a given schema mapping is uniquely characterizable. In [8], it was shown that a finite set F of homomorphically incomparable core structures is an obstruction set for some finite set D if and only if every structure in F obeys a certain acyclicity condition. This result holds for structures over a signature with relation symbols but no constant symbols.…”

“…Let → be the existence-of-a-homomorphism relation between structures over the same schema, i.e., B → C means that there is a homomorphism from B to C. Assume that F and D are two collections of structures over the same schema. Following [8], we say that the pair (F; D) is a homomorphism duality if for every structure A, there exists a structure F ∈ F such that F → A if and only if there is no structure…”

Homomorphic Image Orders on Combinatorial Structures

“…Somewhat surprisingly, this order has received very little attention in the literature. In comparison, the homomorphism order, corresponding to the existence of any homomorphism between two structures, is much-studied ( [5], [10]), as are various related concepts such as the core of a graph ( [3]). …”