# On monoids in the category of sets and relations

## TL;DR

This paper explores the structure of monoids within the category of sets and relations, revealing connections to hypergroups, partial monoids, and quantum logics through a 2-categorical framework.

## Contribution

It introduces a 2-categorical perspective on monoids in Rel, unifying various classes like hypergroups and effect algebras and clarifying their foundational relations.

## Key findings

- Characterizes monoids in Rel as including hypergroups and partial monoids.
- Shows how 2-categorical structure explains congruence relations in effect algebras.
- Connects categorical structures to quantum logics and small categories.

## Abstract

The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every homset $\mathbf{Rel}(A,B)$ is a poset with respect to inclusion. We examine the 2-category of monoids $\mathbf{RelMon}$ in this category. The morphism we use are lax. This category includes, as subcategories, various interesting classes: hypergroups, partial monoids (which include various types of quantum logics, for example effect algebras) and small categories. We show how the 2-categorical structure gives rise to several previously defined notions in these categories, for example certain types of congruence relations on generalized effect algebras. This explains where these definitions come from.