Internal monoids and groups in the category of commutative cancellative medial magmas

TL;DR

This paper explores the structure of commutative medial magmas with cancellation, focusing on internal monoids and groups, and establishing conditions for their existence and properties within this algebraic framework.

Contribution

It characterizes when internal monoids and groups exist in commutative medial magmas with cancellation and analyzes internal relations and congruences.

Findings

Each object admits at most one internal monoid structure for a given unit.

Conditions for the existence of internal monoids and groups are established.

Criteria for internal reflexive relations to be congruences are provided.

Abstract

This article considers the category of commutative medial magmas with cancellation, a structure that generalizes midpoint algebras and commutative semigroups with cancellation. In this category each object admits at most one internal monoid structure for any given unit. Conditions for the existence of internal monoids and internal groups, as well as conditions under which an internal reflexive relation is a congruence, are studied.