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.


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


Zbl 0051.25203
Full Text: EuDML


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