Multiplier Hopf coquasigroups with faithful integrals

TL;DR

This paper investigates the properties of multiplier Hopf coquasigroups with faithful integrals, establishing their uniqueness, duality relations, and biduality theorems, extending to certain Hopf quasigroups.

Contribution

It proves the uniqueness of faithful integrals, demonstrates duality between multiplier Hopf coquasigroups and Hopf quasigroups, and establishes a biduality theorem for these structures.

Findings

Faithful integrals are unique up to scalar.

The integral duality of a discrete type multiplier Hopf coquasigroup is a Hopf quasigroup.

The biduality $\,\, ext{and}\,\, ext{isomorphism}$ hold for these structures.

Abstract

Let $A$ be a multiplier Hopf coquasigroup. If the faithful integrals exist, then they are unique up to scalar. Furthermore, if $A$ is of discrete type, then its integral duality $\widehat{A}$ is a Hopf quasigroup, and the biduality $\widehat{\widehat{A}}$ is isomorphic to the original $A$ as multiplier Hopf coquasigroups. This biduality theorem also holds for a class of Hopf quasigroups with faithful integrals.