Reversibility of Disconnected Structures

TL;DR

This paper investigates the property of reversibility in disconnected relational structures, providing characterizations and conditions under which such structures are reversible, with applications to linear orders and posets.

Contribution

It offers new characterizations of reversibility in disconnected structures, including criteria involving condensations and sequences of components, extending previous understanding.

Findings

Reversibility in disconnected structures is characterized by non-mergeability via condensations.

A structure with reversible components cannot have condensations that merge different components.

Disjoint unions of CSB linear orders of a limit type are reversible iff their order types form a finite-to-one sequence.

Abstract

A relational structure is called reversible iff every bijective endomorphism of that structure is an automorphism. We give several equivalents of that property in the class of disconnected binary structures and some its subclasses. For example, roughly speaking and denoting the set of integers by ${\mathbb Z}$, a structure having reversible components is reversible iff its components can not be "merged" by condensations (bijective homomorphisms) and each ${\mathbb Z}$-sequence of condensations between different components must be, in fact, a sequence of isomorphisms. We also give equivalents of reversibility in some special classes of structures. For example, we characterize CSB linear orders of a limit type and show that a disjoint union of such linear orders is a reversible poset iff the corresponding sequence of order types is finite-to-one.