The independence of the distributivity conditions in groupoids. (English) Zbl 0624.20053

In 1953 G. Szász [Acta Sci. Math. 15, 20-28 (1953; Zbl 0051.252)] proved that the associativity laws are independent on any set A of at least four elements in the following sense: If one chooses an arbitrary triple \((\bar x,\bar y,\bar z)\in A^ 3\) then there always exists a binary operation on A such that the associativity laws \((xy)z=x(yz)\) are fulfilled for all triples \((x,y,z)\in A^ 3\) with the only exception of \((\bar x,\bar y,\bar z)\). Since then several other laws of a variety have been investigated in this way, for instance by R. Wiegandt and the reviewer [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 58, 819-822 (1975; Zbl 0346.08006)].
In the present paper the identities \(x(yz)=(xy)(xz)\) and \((xy)z=(xz)(yz)\) in a groupoid \(<A,\cdot>\) are considered. \(<A,\cdot>\) is called distributive if both laws hold and semidistributive if only one of these laws is valid in \(<A,\cdot>\). It is proven that the distributivity conditions are independent on a set A if and only if A consists of at least four elements, while the semidistributivity conditions are independent on A exactly when A consists of at least three elements.
Reviewer: J.Wiesenbauer


20N99 Other generalizations of groups
20A05 Axiomatics and elementary properties of groups
39B52 Functional equations for functions with more general domains and/or ranges