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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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