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On the orbit decomposition of finite quandles. (English) Zbl 1105.57006

A quandle \((Q,\triangleright)\) is a pair of a set \(Q\) and a binary operation \(\triangleright\) on \(Q\) satisfying (i) \(x\triangleright x=x\), (ii) for every \(x,y\in Q\), there exists a unique element \(z\in Q\) such that \(x=z\triangleright y\), and (iii) \((x\triangleright y)\triangleright z=(x\triangleright z)\triangleright(y\triangleright z)\) for every \(x,y,z\in Q\). A subset \(X\subset Q\) is called a subquandle of the quandle \((Q,\triangleright)\) if \((X,\triangleright)\) is a quandle. If \(Q\setminus X\) is again a subquandle of \((Q,\triangleright)\), we say that \(X\) is \(Q\)-complemented.
In the paper under review the authors prove that for a finite quandle \((Q,\triangleright)\), \(Q\) can be decomposed into a disjoint union \(Q_{1}\sqcup Q_{2}\sqcup\cdots\sqcup Q_{n}\) such that \(Q_{i}\) is \(Q\)-complemented for every \(i\), and that no proper subquandle of \((Q_{j},\triangleright)\) is \(Q\)-complemented for every \(j\). Moreover the decomposition above is unique.
They also describe how to obtain a quandle from finite quandles.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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