On a theorem of Schauffler
Let \((Q,\Sigma)\) be an algebra, where \(\Sigma \subseteq \Omega\) and \(\Omega\) is the set of all binary operations on \(Q\). Assume also that for every \(X, Y \in \Sigma\), there are \(Z, U \in \Sigma\), such that for all \(x, y, z \in Q\) the following equality holds: (1) \(X(Y(x,y),z) = Z(x,U(y,z))\) [(2) \(Z(U(x,y),z) = X(x,Y(y,z))\)]. Then: 1) \((Q,\Sigma)\) is said to be left globally associative (it is an LA-algebra or LA-system) iff (1) holds: 2) \((Q,\Sigma)\) is right globally associative (an RA- algebra, RA-system) iff (2) holds; and 3) \((Q,\Sigma)\) is said to be globally associative (or A-algebra, A-system) iff it is an LA- and RA- algebra [\textit{V. D. Belousov}, Mat. Sb., Nov. Ser. 55(97), 221-236 (1961; Zbl 0124.25603)]. The first result about A-algebras is \textit{R. Schauffler's} theorem [Math. Z. 67, 428-435 (1957; Zbl 0077.24703)]: Let \(\Sigma\) be defined in the following way: \(A \in \Sigma\) iff \((Q,A)\) is a quasigroup. Then \((Q,\Sigma)\) is an A-algebra iff \(| Q| \leq 3\). Note that this is one of the first results on \(\forall \exists (\forall)\)-identities [the author, Superidentities and supervarieties in algebras [in Russian], Izd. Erevan Gos. Univ., Erevan (1990; Zbl 0728.08013)]. In the present paper, in the first place, several modifications of Schauffler's theorem are formulated, including as a new result the following two laws: (3) \(X(X(x,y),z) = Y(x,Z(y,z))\); and (4) \(X(x,X(y,z)) = Y(Z,(x,y),z)\). A part of the main result (including the above mentioned modifications of Schauffler's theorem) is the following: Let \(\Omega\) be the set of all binary operations on \(Q\) and \(\Omega_ k\) the set of all quasigroup operations on \(Q\). Then the following statements are equivalent: 1) for all \(X, Y \in \Omega_ k\) there are \(Z, U \in \Omega\) such that law (1) holds; 2) for all \(X, Y \subseteq \Omega_ k\) there are \(Z, U \in \Omega\) such that law (2) holds; 3) for every \(X \in \Omega_ k\) there are \(Y, Z \in \Omega\) such that law (3) is satisfied; 4) for every \(X \in \Omega_ k\) there are \(Y, Z \in \Omega\) such that law (4) holds; and 5) the set \(Q\) is infinite or \(| Q| \leq 3\).
scientific article; zbMATH DE number 567876
scientific article