Almost groupoids and their substructures

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.

💡 Research Summary

The paper introduces the notion of an “almost groupoid,” a specialization of the classical groupoid in which the source and target maps coincide. Formally, an almost groupoid is a quintuple (G, θ, m, ι, G₀) where θ : G → G₀ is a surjection, the partial binary operation m is defined only for pairs (x, y) with θ(x)=θ(y), and ι provides inverses. The authors adopt three axioms—associativity (AG1), existence of units (AG2), and existence of inverses (AG3)—mirroring the standard groupoid axioms but adapted to the single‑map setting.

After recalling Brandt and Ehresmann groupoids, the paper establishes basic properties of almost groupoids. Proposition 3.1 shows that units are idempotent, that θ is idempotent on G, and that multiplication preserves the unit map (θ(x·y)=θ(x)). Proposition 3.2 proves uniqueness of units and inverses, left/right cancellation, and the usual inverse identities (x⁻¹·(x·y)=y, (x·y)·y⁻¹=x). Proposition 3.3 derives θ∘ι=θ and ι∘ι=Id_G, while Proposition 3.4 demonstrates that each fiber G(u)=θ⁻¹(u) is a genuine group with unit u. Consequently, an almost groupoid can be viewed as a family of groups indexed by the unit set G₀.

The authors define abelian almost groupoids as those whose fibers are abelian groups, and they present three concrete examples: (i) the trivial almost groupoid on a single element, (ii) a “null” almost groupoid where G₀ itself forms the structure with identity maps, and (iii) a matrix‑valued almost groupoid where the unit map normalizes the (1,1) entry to 1 and multiplication is defined only for matrices sharing the same second diagonal entry. Each example is verified against the axioms.

The paper also sketches notions of sub‑almost‑groupoids, wide sub‑almost‑groupoids, and morphisms between almost groupoids, showing that these concepts parallel those in ordinary groupoid theory. However, the treatment of normal sub‑structures, quotient constructions, centers, and strong morphisms is omitted.

Overall, the work provides a clear foundational framework for almost groupoids, extending many elementary groupoid results to this restricted setting. To deepen its impact, future work should explore richer structural aspects (normality, quotients, cohomology), connections to applications such as dynamical systems or categorical constructions, and a more thorough comparison with recent literature on partial algebraic structures.