Biracks: a notational proposal and applications

TL;DR

This paper introduces a new notation for biracks that encapsulates their underlying rack structure, enabling generalizations and clearer understanding of twisting constructions in algebraic solutions.

Contribution

It proposes a novel notation for biracks that integrates the associated rack structure, facilitating extensions of involutive results and improved conceptual clarity.

Findings

Generalizes involutive case results to non-involutive solutions

Provides a transparent view of twisting constructions

Demonstrates applications of the notation in algebraic contexts

Abstract

I propose a notation for biracks that includes from the begining the knowledege of the associated (or underlying, or derived) rack structure. Motivated by results of Rump in the involutive case, this notation allows to generalize some results from involutive case to the non necessarily involutive solutions, and also to view some twisting constructions and its relation to the underlying rack structure in a more transparent way. Two applications are given.