# zbMATH — the first resource for mathematics

Selfdistributive groupoids of small orders. (English) Zbl 0901.20051
By a groupoid is meant a nonempty set together with a binary, multiplicatively denoted operation. A groupoid is called left distributive if it satisfies the identity $$x(yz)=(xy)(xz)$$. Right distributive groupoids are defined dually, and distributive groupoids are groupoids that are both left and right distributive. A groupoid is called medial if it satisfies the identity $$(xy)(uv)=(xu)(yv)$$.
The authors prove the following theorem. If $$G$$ is a non-medial distributive groupoid such that every proper subgroupoid of $$G$$ is medial, then $$G$$ is a quasigroup. As a consequence it follows that every non-medial distributive groupoid contains at least 81 elements.
Reviewer: M.Novotný (Brno)

##### MSC:
 20N02 Sets with a single binary operation (groupoids) 20N05 Loops, quasigroups
Full Text:
##### References:
 [1] R. El. Bashir and A. Drápal: Quasitrivial left distributive groupoids. Commentationes Math. Univ. Carolinae 35 (1994), 597-606. · Zbl 0843.20051 [2] G. Bol: Gewebe und Gruppen. Math. Ann. 114 (1937), 414-431. · Zbl 0016.22603 [3] A. Drápal, T. Kepka and M. Musílek: Group conjugation has non-trivial LD-identities. Commentationes Math. Univ. Carolinae 35 (1994), 219-222. · Zbl 0810.20053 [4] M. Hall, Jr.: Automorphisms of Steiner triple systems. IBM J. of Research Development 4 (1960), 460-471. · Zbl 0100.01803 [5] J. Ježek and T. Kepka: Atoms in the lattice of varieties of distributive groupoids. Colloquia Math. Soc. J. Bolyai 14. Lattice Theory, Szeged, 1974, pp. 185-194. [6] J. Ježek, T. Kepka and P. Němec: Distributive groupoids. Rozpravy ČSAV, Řada mat. a přír. věd 91, Academia, Praha, 1981. · Zbl 0566.20043 [7] T. Kepka: Math. Nachr. 87 (1979), Distributive division groupoids 103-107. · Zbl 0444.20067 [8] T. Kepka: Notes on quasimodules. Commentationes Math. Univ. Carolinae 20 (1979), 229-247. · Zbl 0413.20054 [9] T. Kepka and P. Němec: Czech. Math. J. 31 (1981), Commutative Moufang loops and distributive groupoids of small orders 633-669. · Zbl 0573.20065 [10] J. P. Soublin: Étude algébrique de la notion de moyenne. J. Math. Pures et Appl. 50 (1971), 53-264. · Zbl 0215.40401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.