Almost groupoids and their substructures

TL;DR

This work develops a substructure theory for almost groupoids, a generalization of Brandt groupoids introduced by Mihai Ivan. It defines and analyzes almost subgroupoids, isotropy, centralizers, centers, and generation operations, extending classical Brandt-groupoid results to the almost groupoid setting. The paper also establishes constructive procedures such as disjoint unions, direct products, and semidirect products of almost groupoids, and provides concrete finite and matrix-based examples to illustrate the theory. By detailing closures and normality properties, it lays foundational tools for further structural study and potential applications in algebraic contexts where partial multiplication interacts with unit maps and inverses.

Abstract

The aim of this paper is to present the main constructions of the substructures of an almost groupoid and to discuss their basic properties. The definitions and properties concerning these new algebraic constructions extend to almost groupoids, the corresponding well-known results for groupoids.