Comparable binary relations and the amalgamation property

TL;DR

This paper investigates the conditions under which the theory of two binary relations exhibits the strong amalgamation property, especially when relations satisfy certain properties and are preserved by unary operations, leading to Fraïssé limits.

Contribution

It characterizes when the strong amalgamation property holds for two binary relations with specific properties and unary operations, and explores limitations with more relations or fewer preserved properties.

Findings

Strong amalgamation property holds for two relations under certain conditions.

Fraïssé limits exist for classes of finite structures with these properties.

The property fails with three or more relations or fewer preserved properties.

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.