On Finite Maximal Antichains in the Homomorphism Order

The relation of existence of a homomorphism on the class of all relational structures of a fixed type is reflexive and transitive; it is a quasiorder. There are standard ways to transform a quasiorder into a partial order – by identifying equivalent objects, or by choosing a particular representative for each equivalence class. The resulting partial order is identical in both cases. Properties of this partial order (the homomorphism order) have been intensively studied in algebraic, category theory, random and combinatorial context, see [5]. Particular interest has been paid to density and universality. Here, we are interested in the characterisation of all finite maximal antichains in the homomorphism order. We show that for structures with at most two relations all finite maximal antichains correspond to what is known as finite homomorphism dualities (see [4, 8]). In addition, we examine the splitting property of finite maximal antichains in the homomorphism order (see [2]). We derive a structural condition which implies that most finite maximal antichains split. This was previously known for digraphs [3] and structures with at most one relation [4].