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 \(| Q| \leq 3\).

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 \(| Q| \leq 3\).

Reviewer: J.J.Ušan (Novi Sad)

##### MSC:

20N05 | Loops, quasigroups |

20M20 | Semigroups of transformations, relations, partitions, etc. |

08A62 | Finitary algebras |

##### Keywords:

Schauffler’s theorem; identities; binary operations; left globally associative algebras; LA-algebras; LA-systems; right globally associative algebras; RA-algebras; RA-systems; globally associative algebras; quasigroup operations
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\textit{Yu. M. Movsisyan}, Math. Notes 53, No. 2, 84--93 (1993; Zbl 0816.20066); translation from Mat. Zametki 53, No. 2, 84--93 (1993)

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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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