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

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 $\langle κ_i :i\in I\rangle$ to be reversible iff each surjection $f :I\rightarrow I$ such that $κ_j =\sum_{i\in f^{-1}[\{ j \}]}κ_i$, for all $j\in I $, is a bijection, and characterize such sequences: either $\langle κ_i :i\in I\rangle$ is a finite-to-one sequence, or $κ_i\in {\mathbb N}$, for all $i\in I$, $K:=\{ m\in {\mathbb N} : κ_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 $\{ κ_i :i\in I \}$. 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 $\leq ω$, 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.