Abstract: Quasigroupoids and weak Hopf quasigroups are non-associative generalizations of groupoids and weak Hopf algebras. In this talk, we will show that the category of finite quasigroupoids is equivalent to the one of pointed cosemisimple weak Hopf quasigroups over a given field K. As a consequence, we obtain a categorical equivalence between the categories of quasigroups, in the sense of Klim and Majid (i.e., loops with the inverse property), and the category of pointed cosemisimple Hopf quasigroups over K. On the other hand, in this talk, we introduce the notion of exact factorization of a quasigroupoid and the notion of a matched pair of quasigroupoids with a common base. We prove that if (A, H) is a matched pair of quasigroupoids, it is possible to construct a new quasigroupoid called the double cross product of A and H. Moreover, we show that if a quasigroupoid B admits an exact factorization, there exists a matched pair of quasigroupoids (A, H) and an isomorphism of quasigroupoids between the double cross product of A and H and B. Finally, we show that every matched pair of quasigroupoids (A, H) induces, thanks to the quasigroupoid magma construction, a pair (K[A], K[H]) of weak Hopf quasigroups and a double crossed product of weak Hopf quasigroups isomorphic as weak Hopf quasigroups to the quasigroupoid magma of the double cross product groupoid of A and H.
Abstract: In this talk, we discuss the problem of automatic additivity for multiplicative maps between rings, following the classical results of Martindale and Jodeit-Lam. We present a complete description of Jordan multiplicative self-maps on full matrix algebras M_n(F), where F is a field of characteristic not equal to 2. Without imposing additional assumptions, we show that such maps are either constant (and equal to a fixed idempotent), or additive (and hence Jordan monomorphisms). This result is joint work with Ilja Gogić.