Right, left and double division in semigroups that are semilattices of groups

TL;DR

This paper explores the algebraic structures formed by division operations in semigroups that are semilattices of groups, revealing a one-to-one correspondence with certain groupoid structures and extending known group-quasigroup relationships.

Contribution

It establishes a novel correspondence between division-based groupoid structures and semigroups that are semilattices of groups, generalizing existing algebraic concepts.

Findings

Binary division operations form interesting groupoid structures.

There is a one-to-one correspondence with semigroups that are semilattices of groups.

Extends the known relationship between groups and Ward quasigroups.

Abstract

The binary products of right, left or double division in semigroups that are semilattices of groups give interesting groupoid structures that are in one to one correspondence with semigroups that are semilattices of groups. This work is inspired by the known one to one correspondence between groups and Ward quasigroups.