Some more notions of homomorphism-homogeneity

TL;DR

This paper broadens the concept of homomorphism-homogeneity to include more types of maps, explores the relationships between these notions, and provides examples and detailed analysis of related structures.

Contribution

It introduces extended notions of homomorphism-homogeneity and investigates their interrelations with new examples and detailed structural insights.

Findings

Extended the class of maps considered in homomorphism-homogeneity

Established relations between different notions of homomorphism-homogeneity

Provided examples illustrating the extended concepts

Abstract

We extend the notion of 'homomorphism-homogeneity' to a wider class of kinds of maps than previously studied, and we investigate the relations between the resulting notions of homomorphism-homogeneity, giving several examples. We also give further details on related work reported in [Deborah Lockett and John K Truss, Generic endomorphisms of homogeneous structures, in 'Groups and model theory', Contemporary Mathematics 576, ed Strungmann, Droste, Fuchs, Tent, American Mathematical Society, 2012, 217-237] about the endomorphisms of chains and generic endomorphisms of tree.