Computer-supported Exploration of a Categorical Axiomatization of Modeloids

TL;DR

This paper generalizes the concept of modeloids, which are sets of partial automorphisms, to inverse semigroups and categories, providing an algebraic framework for understanding Ehrenfeucht-Fra"issé games.

Contribution

It introduces a categorical axiomatization of modeloids, extending their algebraic structure and linking them to Ehrenfeucht-Fra"issé games through a new formal approach.

Findings

Generalization of modeloids to inverse semigroups and categories

Algebraic perspective on Ehrenfeucht-Fra"issé games

Framework enables new insights into automorphism structures

Abstract

A modeloid, a certain set of partial bijections, emerges from the idea to abstract from a structure to the set of its partial automorphisms. It comes with an operation, called the derivative, which is inspired by Ehrenfeucht-Fra\"iss\'e games. In this paper we develop a generalization of a modeloid first to an inverse semigroup and then to an inverse category using an axiomatic approach to category theory. We then show that this formulation enables a purely algebraic view on Ehrenfeucht-Fra\"iss\'e games.