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On a theorem of Schauffler. (English. Russian original) Zbl 0816.20066
Math. Notes 53, No. 2, 172-179 (1993); translation from Mat. Zametki 53, No. 2, 84-93 (1993).
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 [V. D. Belousov, Mat. Sb., Nov. Ser. 55(97), 221-236 (1961; Zbl 0124.25603)]. The first result about A-algebras is 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 $$| Q| \leq 3$$.

##### MSC:
 20N05 Loops, quasigroups 20M20 Semigroups of transformations, relations, partitions, etc. 08A62 Finitary algebras
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##### References:
 [1] R. Schauffler, ?Die Assoziativität im Ganzen besonders bei Quasigruppen,? Math. Z.,67, No. 5, 428-435 (1957). · Zbl 0077.24703 [2] V. D. Belousov, ?Systems of quasigroups with generalized identities,? Usp. Mat. Nauk,20, No. 1, 75-146 (1965). · Zbl 0135.03503 [3] Yu. M. Movsisyan, Superidentitites and Supervarieties in Algebras [in Russian], Izd. Erevan. Gos. Univ., Erevan (1990). [4] J. Usan, ?Globally associative systems of ternary quasigroups. A ternary analogue of a theorem of Schauffler,? Math. Balkanica,1, 273-281 (1971). [5] J. Usan and M. R. ?i?ovic, ?An n-ary analogue of Schauffler’s theorem,? Publ. Inst. Math. (Beograd),19, 167-172 (1975). [6] A. Krape?, ?Generalized associativity on groupoids,? Publ. Inst. Math. (Beograd),28, 105-112 (1980). · Zbl 0503.20022 [7] S. MacLane, Homology, Academic Press, New York, and Springer-Verlag, Berlin (1963). [8] R. H. Bruck, ?Some results in the theory of quasigroups,? Trans. Amer. Math. Soc.,55, 19-52 (1944). · Zbl 0063.00635
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