Groupoids and the associative law. II: Groupoids with small semigroup distance. (English) Zbl 0809.20056

This paper is a continuation of the first paper of the authors [ibid. 33, 69-86 (1992; Zbl 0791.20084)]. Here, groupoids with small semigroup distance are investigated. The main results are Theorem: Let \(G\) be a semigroup. Then \(G[a, b, c]\) is associative for all \(a, b, c \in G\) iff \(\text{card}(G) \leq 2\) and \(G\) is a semilattice. Theorem: Let \(G\) be a finite groupoid with \(n\) elements and such that \(\text{sdist} (G) = 1\). Then \(1 \leq \text{ns} (G) \leq 2n(n - 1)\) and \(n^ 3 - 2n^ 2 + 2 \leq \text{as}(G) \leq n^ 3 - 1\). Moreover, if \(\text{ns}(G) = 2n(n - 1)\), then \(G\) is isomorphic to one of the groupoids \(R_ n(*)\), \(S_{n,1}(*)\), \(S_{n,2}(*)\), (to \(R_ 2(*)\) if \(n = 2\)). This part seems to be new. Not much is known about the semigroup distance of (finite) groupoids and this topic would deserve a more detailed study.


20N02 Sets with a single binary operation (groupoids)


Zbl 0791.20084
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