Weak non associative objects.

Quasigroupoids and weak Hopf quasigroups are non-associative generalizations of groupoids and weak Hopf algebras. In this talk, we will establish their main properties and an equivalence between the category of finite quasigroupoids and that of pointed cosemisimple weak Hopf quasigroups over a field K. As an immediate consequence, we obtain a categorical equivalence between quasigroups (in the sense of Klim and Majid, i.e., loops with the inverse property) and pointed cosemisimple Hopf quasigroups over K. Additionally, we introduce the notions of exact factorization of a quasigroupoid and of a matched pair of quasigroupoids with a common base. We prove that, given a matched pair (A, H), one can construct a new quasigroupoid called their double cross product. Furthermore, we show that if a quasigroupoid B admits an exact factorization, then there exists a matched pair (A, H) such that the double cross product of A and H is isomorphic to B. Finally, we demonstrate that every matched pair of quasigroupoids (A, H) induces, via the quasigroupoid magma construction, a corresponding pair of weak Hopf quasigroups (K[A], K[H]). Moreover, the double cross product of these weak Hopf quasigroups is isomorphic, as a weak Hopf quasigroup, to the quasigroupoid magma of the double cross product of the quasigroupoids A and H.

References:

Alonso Álvarez, J.N., Fernández Vilaboa, J.M., González Rodríguez, R.: Weak Hopf quasigroups, Asian Journal of Mathematics 20, No. 4, 665-694 (2016). arXiv:1410.2180

Alonso Álvarez, J.N., Fernández Vilaboa, J.M., González Rodríguez, R.: A characterization of weak Hopf (co)quasigroups Mediterranean Journal of Mathematics 13, No. 5, 3747–3764 (2016). arXiv:1506.07664

Alonso Álvarez, J.N., Fernández Vilaboa, J.M., González Rodríguez, R.: Quasigroupoids and weak Hopf quasigroups Journal of Algebra 568, 408-436 (2021) Link

González Rodríguez, R.: Weak Hopf quasigroups and matched pairs of quasigroupoids, Journal of Algebra and its Applications (2026). arXiv:2403.09231