zbMATH — the first resource for mathematics

Sets of associative triples. (English) Zbl 0612.05003
Authors’ summary: ”A subset \(R\subseteq S^ 3\) will be called (associatively) admissible if there exists a binary operation * defined on S such that \(x*(y*z)=(x*y)*z\) iff (x,y,z)\(\in R\). If S is finite, \(card(S)=n\), \(R\subseteq S^ 3\), \(card(R)=r\) and \(r\leq n/4-3/4\) or \(n^ 3-n/4+1/2\leq r\), then R is admissible. There exists an admissible subset for any \(0\leq r\leq n^ 3\) and a non-admissible subset for 3n\(\leq r\leq n^ 3-n+2.\)”
Reviewer: K.Burian

05A05 Permutations, words, matrices
Full Text: DOI
[1] Climescu, A.C., Études sur la théorie des systémes multiplicatifs uniformes. I. ĺ indice de non-associativité, Bull. école polytech. jassy, 2, 347-371, (1947)
[2] Climescu, A.C., Ĺindépendence des conditions d’associativité, Bull. inst. polytech. jassy, 1, 1-9, (1955) · Zbl 0068.02901
[3] Drápal, A., Quasigroups rich in associative triples, Discrete math., 44, 251-265, (1983) · Zbl 0509.05019
[4] Drápal, A.; Kepka, T., A note on the number of associative triples in quasigroups isotopic to groups, Commet. math. univ. carolinae, 22, 735-743, (1981) · Zbl 0489.20053
[5] Drápal, A.; Kepka, T., Group modifications of some partial groupoids, Annals discr. math., 18, 319-332, (1983) · Zbl 0519.20051
[6] Hájek, P., Die száschen gruppoide, Mat-fyz. časopis, 15, 15-42, (1965) · Zbl 0128.25002
[7] Kepka, T., A note on associative triples of elements in cancellation groupoids, Comment. math. univ. carolinae, 21, 479-487, (1980) · Zbl 0444.20069
[8] Kepka, T., Notes on associative triples of elements in commutative groupoids, Acta univ. carolinae math. phys., 22, 2, 39-47, (1981) · Zbl 0515.20053
[9] Kepka, T., A note on the number of associative triples in finite commutative Moufang loops, Commet. math. univ. carolinae, 22, 745-753, (1981) · Zbl 0482.20046
[10] Norton, D.A., A note on associativity, Pacific J. math., 10, 591-595, (1960) · Zbl 0093.01901
[11] Szász, G., Die unabhängigheit der assoziativitäts-bedingungen, Actas sci. math. Szeged, 15, 20-28, (1953)
[12] Szasz, G., Überdie unabhängigkeitder assoziativitäts-bedingungen kommutativer multiplikativen strukturen, Act sci. math. Szeged, 15, 130-142, (1954)
[13] Wagner, A., On the associative law of groups, Rend. math. appl., 21, 60-76, (1962) · Zbl 0286.20090
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.