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Constructing biquandles
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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.
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General constructions of biquandles and their symmetries
All word pairs over two free generators that make every group a birack or biquandle are classified, and new union, product, holomorph, and covering constructions for biquandles are introduced with their automorphism groups.
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