Reversible Sequences of Cardinals, Reversible Equivalence Relations, and Similar Structures

TL;DR

This paper characterizes when certain relational structures and sequences of cardinals are reversible, providing criteria for reversibility in equivalence relations, posets, and related structures, with implications for topology and combinatorics.

Contribution

It introduces a comprehensive characterization of reversible sequences of cardinals and applies this to classify reversible equivalence relations, posets, and other structures.

Findings

Reversible sequences are either finite-to-one or have a specific gcd property.

Reversible structures include certain equivalence relations and posets with specific connectivity properties.

The paper identifies broad classes of reversible posets and topological spaces based on their component sequences.

Abstract

A relational structure ${\mathbb X}$ is said to be reversible iff every bijective endomorphism $f:X\rightarrow X$ is an automorphism. We define a sequence of non-zero cardinals $\langle \kappa_i :i\in I\rangle$ to be reversible iff each surjection $f :I\rightarrow I$ such that $\kappa_j =\sum_{i\in f^{-1}[\{ j \}]}\kappa_i$, for all $j\in I $, is a bijection, and characterize such sequences: either $\langle \kappa_i :i\in I\rangle$ is a finite-to-one sequence, or $\kappa_i\in {\mathbb N}$, for all $i\in I$, $K:=\{ m\in {\mathbb N} : \kappa_i =m $, for infinitely many $i\in I \}$ is a non-empty independent set, and $\gcd (K)$ divides at most finitely many elements of the set $\{ \kappa_i :i\in I \}$. We isolate a class of binary structures such that a structure from the class is reversible iff the sequence of cardinalities of its connectivity components is reversible. In particular, we characterize reversible equivalence relations, reversible posets which are disjoint unions of cardinals $\leq \omega$, and some similar structures. In addition, we show that a poset with linearly ordered connectivity components is reversible, if the corresponding sequence of cardinalities is reversible and, using this fact, detect a wide class of examples of reversible posets and topological spaces.