# On Group-Like Magmoids

## TL;DR

This paper explores various generalizations of group-like structures using partial binary operations, introducing new classes of magmoids and establishing foundational results and analogues of classical theorems.

## Contribution

It defines and analyzes multiple new classes of magmoids with generalized associativity, identities, and inverses, extending the theory of group-like structures.

## Key findings

- Introduction of poloids, groupoids, and skew variants.
- Derivation of basic properties and connections between classes.
- Proofs of analogues to classical theorems like Ehresmann-Schein-Nampooribad.

## Abstract

A magmoid is a non-empty set with a partial binary operation; group-like magmoids generalize group-like magmas such as semigroups, monoids and groups. In this article, we first consider the many ways in which the notions of associative multiplication, identities and inverses can be generalized when the total binary operation is replaced by a partial binary operation. Poloids, groupoids, skew-poloids, skew-groupoids, prepoloids, pregroupoids, skew-prepoloids and skew-pregroupoids are then defined in terms of generalized associativity, generalized identities and generalized inverses. Some basic results about these magmoids are derived, and connections between poloid-like and prepoloid-like magmoids, in particular semigroups, are described. Notably, analogues of the Ehresmann-Schein-Nampooribad theorem are proved.