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The retraction relation for biracks. (English) Zbl 1411.16032

Summary: In [Duke Math. J. 100, No. 2, 169–209 (1999; Zbl 0969.81030)] P. Etingof et al. introduced, for each non-degenerate involutive set-theoretical solution \((X, \sigma, \tau)\) of the Yang-Baxter equation, the equivalence relation \(\sim\) defined on the set \(X\) and they considered a new non-degenerate involutive induced retraction solution defined on the quotient set \(X^\sim\). It is well known that translating set-theoretical non-degenerate solutions of the Yang-Baxter equation into the universal algebra language we obtain an algebra called a birack. In the paper we introduce the generalized retraction relation \(\approx\) on a birack, which is equal to \(\sim\) in an involutive case. We present a complete algebraic proof that the relation \(\approx\) is a congruence of the birack. Thus we show that the retraction of a set-theoretical non-degenerate solution is well defined not only in the involutive case but also in the case of all non-involutive solutions.

MSC:

16T25 Yang-Baxter equations
08A30 Subalgebras, congruence relations
20N02 Sets with a single binary operation (groupoids)
08A62 Finitary algebras

Citations:

Zbl 0969.81030

Software:

Mace4; Prover9
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Full Text: DOI arXiv

References:

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