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Groupoids with non-associative triples on the diagonal. (English) Zbl 0596.20065
For a groupoid G, let T(G) denote the set of ordered triples (a,b,c) of elements from G such that a.bc$$\neq ab.c$$. It is known that if S is a non- empty finite set and T is a subset of $$S^ 3$$ such that card $$T\leq (card S-2)/4$$, then there exists a binary operation defined on S such that $$T=T(S)$$ for the corresponding groupoid S. On the other hand, if G is an arbitrary groupoid, then never T(G) is the diagonal of $$G^ 3$$. Now, it is natural to consider the class $${\mathcal S}$$ of groupoids G such that T(G) is a subset of the diagonal. In the paper, the variety of groupoids generated by $${\mathcal S}$$ is described and some estimates for card T(G), $$G\in {\mathcal S}$$, are given.
Reviewer: T.Kepka
##### MSC:
 20N99 Other generalizations of groups 05C30 Enumeration in graph theory 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms) 05C20 Directed graphs (digraphs), tournaments
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##### References:
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