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On three generalizations of the group concept: groupoid, generalized group and almost groupoid

Gheorghe Ivan

TL;DR

This paper surveys three generalizations of the group concept—groupoid, generalized group, and almost groupoid—outlining their precise definitions, core properties, and interrelations. It develops foundational theory (structure maps, isotropy, morphisms), provides key constructions (disjoint unions, direct and semidirect products) and concrete examples, and situates these notions within historical contexts (Brandt and Ehresmann) while clarifying how groups embed into these broader frameworks. The work contributes to understanding how these generalizations extend algebraic structures with applications to topology, geometry, and beyond, highlighting the hierarchical relationships among groups, almost groupoids, and groupoids. Overall, it lays out a unified view of how generalized algebraic systems can be built, compared, and manipulated through explicit constructions and morphisms.

Abstract

The aim of this paper is to describe the definitions and main properties of three generalizations of the group concept, namely: groupoid, generalized group and almost groupoid. Some constructions of these algebraic structures and corresponding examples are presented.

On three generalizations of the group concept: groupoid, generalized group and almost groupoid

TL;DR

This paper surveys three generalizations of the group concept—groupoid, generalized group, and almost groupoid—outlining their precise definitions, core properties, and interrelations. It develops foundational theory (structure maps, isotropy, morphisms), provides key constructions (disjoint unions, direct and semidirect products) and concrete examples, and situates these notions within historical contexts (Brandt and Ehresmann) while clarifying how groups embed into these broader frameworks. The work contributes to understanding how these generalizations extend algebraic structures with applications to topology, geometry, and beyond, highlighting the hierarchical relationships among groups, almost groupoids, and groupoids. Overall, it lays out a unified view of how generalized algebraic systems can be built, compared, and manipulated through explicit constructions and morphisms.

Abstract

The aim of this paper is to describe the definitions and main properties of three generalizations of the group concept, namely: groupoid, generalized group and almost groupoid. Some constructions of these algebraic structures and corresponding examples are presented.

Paper Structure

This paper contains 4 sections, 2 theorems.

Table of Contents

  1. Introduction
  2. Groupoids: definitions and basic results
  3. Generalized groups: definitions, generalized subgroup and generalized homomorphism
  4. Almost groupoids

Key Result

Theorem 2.1

(giv1b) Every groupoid $~(\Gamma, \Gamma_{0})~$ is isomorphic to a subgroupoid of the symmetric groupoid $~({\cal S}(\Gamma), {\cal S}_{0}(\Gamma)).~$

Theorems & Definitions (2)

  • Theorem 2.1
  • Theorem 4.1