Tukey morphisms between finite relations

TL;DR

This paper studies Tukey morphisms between finite binary relations, classifies small relations up to bimorphism, and constructs relations with specified dominating and dual dominating numbers.

Contribution

It introduces fundamental lemmas for Tukey morphisms, classifies all relations with up to six points, and provides a method to construct relations with arbitrary dominating properties.

Findings

Classified all binary relations with at most 6 points up to bimorphism.

Established fundamental lemmas for Tukey morphisms.

Constructed relations with any given dominating and dual dominating numbers.

Abstract

We investigate Tukey morphisms between binary relations, establishing several fundamental lemmas. We then specialize to finite binary relations, using computational methods to classify all binary relations with at most $6$ points in the domain and codomain up to bimorphism. Finally we give a construction of finite binary relations with arbitrary dominating number and dual dominating number.