A characterization of weak Hopf (co)quasigroups

TL;DR

This paper characterizes weak Hopf (co)quasigroups using a Galois-type condition, generalizing previous notions and establishing fundamental theorems and characterizations for these algebraic structures.

Contribution

It introduces a Galois-type characterization for weak Hopf (co)quasigroups, extending the understanding of their structure and relations to Hopf (co)quasigroups and weak Hopf algebras.

Findings

First fundamental theorem for Hopf (co)quasigroups

Characterization of weak Hopf algebras via bijective Galois morphism

Generalization of existing algebraic structures

Abstract

In this paper we show that weak Hopf (co)quasigroups can be characterized by a Galois-type condition. Taking into account that this notion generalizes the ones of Hopf (co)quasigroup and weak Hopf algebra, we obtain as a consequence the first fundamental theorem for Hopf (co)quasigroups and a characterization of weak Hopf algebras in terms of bijectivity of a Galois-type morphism (also called fusion morphism).