Permutation monoids and MB-homogeneity for graphs and relational structures

TL;DR

This paper explores the relationship between infinite permutation monoids and MB-homogeneous structures, providing characterizations, classifications, and constructions of such structures and their automorphism groups.

Contribution

It introduces the concept of MB-homogeneity, characterizes closed permutation monoids, and constructs numerous non-isomorphic MB-homogeneous graphs with specified automorphism groups.

Findings

Characterization of closed permutation monoids

Construction of 2^{ - countable MB-homogeneous graphs

Any finite group can be realized as automorphism group of an MB-homogeneous graph

Abstract

In this paper, we investigate the connection between infinite permutation monoids and bimorphism monoids of first-order structures. Taking our lead from the study of automorphism groups of structures as infinite permutation groups and the more recent developments in the field of homomorphism-homogeneous structures, we establish a series of results that underline this connection. Of particular interest is the idea of MB-homogeneity; a relational structure $\mathcal{M}$ is MB-homogeneous if every monomorphism between finite substructures of $\mathcal{M}$ extends to a bimorphism of $\mathcal{M}$. The results in question include a characterisation of closed permutation monoids, a Fra\"{i}ss\'{e}-like theorem for MB-homogeneous structures, and the construction of $2^{\aleph_0}$ pairwise non-isomorphic countable MB-homogeneous graphs. We prove that any finite group arises as the automorphism group of some MB-homogeneous graph and use this to construct oligomorphic permutation monoids with any given finite group of units. We also consider MB-homogeneity for various well-known examples of homogeneous structures and in particular give a complete classification of countable homogeneous undirected graphs that are also MB-homogeneous.