Finite irreflexive homomorphism-homogeneous binary relational systems

TL;DR

This paper characterizes all finite irreflexive binary relational systems that are homomorphism-homogeneous, meaning every homomorphism between finite substructures extends to an endomorphism, thus extending the classical notion of homogeneity.

Contribution

It provides a complete classification of finite irreflexive binary relational systems that satisfy the homomorphism-homogeneous property.

Findings

Complete characterization of finite irreflexive homomorphism-homogeneous systems

Extension of classical homogeneity to homomorphism-homogeneity in finite structures

New insights into the structure of relational systems with binary relations

Abstract

A structure is called homogeneous if every isomorphism between finite substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Ne\v{s}et\v{r}il introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we characterize all finite homomorphism-homogeneous relational systems with one irreflexive binary relation.