## 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.

### MSC:

 20N02 Sets with a single binary operation (groupoids)

### Keywords:

groupoids; semigroup distance; semilattice; finite groupoid

Zbl 0791.20084
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