Weak Hopf quasigroups and matched pairs of quasigroupoids

Abstract

In this paper we introduce the notion of exact factorization of a quasigroupoid and the notion of matched pair of quasigroupoids with common base. We prove that if $(\mathsf{A},\mathsf{H})$ is a matched pair of quasigroupoids it is posible to construct a new quasigroupoid $\mathsf{A}\bowtie \mathsf{H}$ called the double cross product of $\mathsf{A}$ and $\mathsf{H}$. Also, we show that, if a quasigroupoid $\mathsf{B}$ admits an exact factorization, there exists a matched pair of quasigroupoids $(\mathsf{A},\mathsf{H})$ and an isomorphism of quasigroupoids between $\mathsf{A}\bowtie \mathsf{H}$ and $\mathsf{B}$. Finally, if $\mathbb{K}$ is a field, we show that every matched pair of quasigroupoids $(\mathsf{A},\mathsf{H})$ induce, thanks to the quasigroupoid magma construction, a pair $(\mathbb{K}[\mathsf{A}],\mathbb{K}[\mathsf{H}])$ of weak Hopf quasigroups and a double crossed product weak Hopf quasigroup $\mathbb{K}[\mathsf{A}]\bowtie \mathbb{K}[\mathsf{H}]$ isomorphic to $\mathbb{K}[\mathsf{A}\bowtie \mathsf{H}]$ as weak Hopf quasigroups.