Comparable binary relations and the amalgamation property

Abstract

The theory of two binary relations has the strong amalgamation property when the first relation is assumed to be coarser than the second relation, and each relation satisfies a chosen set of properties from the following list: transitivity, reflexivity, symmetry, antireflexivity and antisymmetry. The amalgamation property is maintained when we add families of unary operations preserving all the relations. As a consequence, we get the existence of Fra\"\i ss\'e limits for classes of finite structures. The results fail, for general comparability conditions, when three or more binary relations are taken into account, or when we add an operation preserving just one relation.