Constructing biquandles

TL;DR

This paper introduces biquandle structures on quandles, explores their automorphism groups, and applies these concepts to Alexander and dihedral biquandles, advancing the algebraic understanding of these structures.

Contribution

It establishes a framework for constructing biquandles from quandles and relates their automorphism groups, providing new insights into their algebraic properties.

Findings

Every biquandle can be obtained from a biquandle structure on its underlying quandle.

Automorphism groups of biquandles are related to those of their underlying quandles.

Automorphism groups of Alexander and dihedral biquandles are explicitly determined.

Abstract

We define biquandle structures on a given quandle, and show that any biquandle is given by some biquandle structure on its underlying quandle. By determining when two biquandle structures yield isomorphic biquandles, we obtain a relationship between the automorphism group of a biquandle and the automorphism group of its underlying quandle. As an application, we determine the automorphism groups of Alexander and dihedral biquandles. We also discuss product biquandles and describe their automorphism groups.