Different Similarities

TL;DR

This paper explores the hierarchy of twelve equivalence relations on relational structures, analyzing their interrelations and how they differ across finite, unary infinite, and non-unary infinite structures.

Contribution

It establishes a detailed hierarchy among various similarities and equivalences on relational structures, revealing how these relations differ across structure classes.

Findings

The number of similarities varies across structure classes.

Finite structures have a different similarity hierarchy than infinite structures.

Infinite non-unary structures exhibit all similarities as distinct.

Abstract

We establish the hierarchy among twelve equivalence relations (similarities) on the class of relational structures: the equality, the isomorphism, the equimorphism, the full relation, four similarities of structures induced by similarities of their self-embedding monoids and intersections of these equivalence relations. In particular, fixing a language $L$ and a cardinal $\kappa$, we consider the interplay between the restrictions of these similarities to the class $Mod _L (\kappa )$ of all $L$-structures of size $\kappa$. It turns out that, concerning the number of different similarities and the shape of the corresponding Hasse diagram, the class of all structures naturally splits into three parts: finite structures, infinite structures of unary languages, and infinite structures of non-unary languages (where all these similarities are different).