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Associativity in monoids and categories. (English) Zbl 1038.18002

Summary: Given a nonempty set \(A\) we consider the possible groupoids \((A,\cdot)\) with base set \(A\). If there is no proper subset \(T\) of \(A^3\) such that the satisfaction of \((xy)z=x(yz)\) for all \((x,y,z) \in T^3\) implies that \((A,\cdot)\) is a semigroup then we say that the associativity conditions are independent over the set A. G. Szász [Acta Sci. Math. 15, 20–28 (1953; Zbl 0051.25203)] showed that this is the case iff \(| A|\geq 4\). In this note the analogous problem is considered for categories and, as particular cases, for monoids. It is proved that if for all objects \(A\) and \(B\), \(\text{Hom} (A,B)\) is either empty or has at least five elements then the associativity conditions are independent. The bound “five” is shown to be sharp.

MSC:

18A05 Definitions and generalizations in theory of categories
20M99 Semigroups

Citations:

Zbl 0051.25203
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References:

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