Reversibility of Extreme Relational Structures

TL;DR

This paper investigates the properties of reversible relational structures, characterizing when bijective homomorphisms are isomorphisms, and identifies classes of such structures including countable ultrahomogeneous graphs.

Contribution

It introduces syntactical conditions under which interpretations of a relational language have extreme elements, aiding in the detection of reversible structures.

Findings

Characterization of reversible countable ultrahomogeneous graphs

Identification of syntactical conditions for extreme elements in interpretations

Detection of classes of reversible structures using $L_{ }$-theory

Abstract

A relational structure $\mathbb{X}$ is called reversible iff each bijective homomorphism from $\mathbb{X}$ onto $\mathbb{X}$ is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language $L$ is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations of the language $L$ on a fixed domain $X$ determine reversible structures. We isolate certain syntactical conditions providing that a consistent $L_{\infty \omega }$-theory defines a class of interpretations having extreme elements on a fixed domain and detect several classes of reversible structures. In particular, we characterize the reversible countable ultrahomogeneous graphs.