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On the foundations of quasigroups. (English) Zbl 0079.02402

Keywords:
Group Theory
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[1] Harris F. MacNeish, Euler squares, Ann. of Math. (2) 23 (1922), no. 3, 221 – 227. · JFM 49.0041.05 · doi:10.2307/1967920 · doi.org
[2] R. A. Fisher, The arrangement of field experiments, J. Minis. Ag. vol. 3 (1936) pp. 503-513.
[3] Anton Suschkewitsch, On a generalization of the associative law, Trans. Amer. Math. Soc. 31 (1929), no. 1, 204 – 214. · JFM 55.0085.01
[4] C. Burstin and W. Mayer, Distributive Gruppen, J. Reine Angew. Math. vol. 160 (1929) pp. 111-130. · JFM 55.0082.06
[5] Morgan Ward, Postulates for the inverse operations in a group, Trans. Amer. Math. Soc. 32 (1930), no. 3, 520 – 526. · JFM 56.0136.02
[6] M. Zorn, Theorie der alternitiven Ringe, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (1930) pp. 123-147. · JFM 56.0140.01
[7] Anton Suschkewitsch, Über ein Elementen-system, für welche zwei Distributivgesetze gelten, Kharkovski Matematichnovo Tobarictba, series 4, VII (1934) pp. 29-32.
[8] A. Adrian Albert, On a certain algebra of quantum mechanics, Ann. of Math. (2) 35 (1934), no. 1, 65 – 73. · Zbl 0008.42104 · doi:10.2307/1968118 · doi.org
[9] David G. Rabinow, Independent Sets of Postulates for Abelian Groups and Fields in Terms of the Inverse Operations, Amer. J. Math. 59 (1937), no. 1, 211 – 224. · Zbl 0016.01403 · doi:10.2307/2371573 · doi.org
[10] B. A. Hausmann and Oystein Ore, Theory of Quasi-Groups, Amer. J. Math. 59 (1937), no. 4, 983 – 1004. · Zbl 0017.39102 · doi:10.2307/2371362 · doi.org
[11] Herbert Boggs and G. Y. Rainich, Note on group postulates, Bull. Amer. Math. Soc. 43 (1937), no. 2, 81 – 84. · Zbl 0016.15505
[12] Alfred Tarski, Ein Beitrag zur Axiomatik der Abelschen Gruppen, Fund. Math. XXX (1938) pp. 253-256. · Zbl 0019.05204
[13] H. Howard Frisinger, Raj Chandra Bose: universal mathematician, Gaá\textonesuperior \?ita Bhāratī 4 (1982), no. 3-4, 83 – 89. · Zbl 0529.01009
[14] B. A. Bernstein, Postulates for abelian groups and fields in terms of non-associative operations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 1 – 6. · Zbl 0018.10103
[15] D. C. Murdoch, Quasi-Groups Which Satisfy Certain Generalized Associative Laws, Amer. J. Math. 61 (1939), no. 2, 509 – 522. · Zbl 0020.34702 · doi:10.2307/2371517 · doi.org
[16] H. W. Norton, The 7 x 7 squares, Ann. Eugenics 9 (1939), 269 – 307. · Zbl 0022.11102
[17] W. L. Stevens, The completely orthogonalized Latin squares, Annals of Eugenics vol. 9 (1939) pp. 82-93. · Zbl 0020.29402
[18] Emil L. Post, Polyadic groups, Trans. Amer. Math. Soc. 48 (1940), 208 – 350. · Zbl 0025.01201
[19] G. N. Garrison, Quasi-groups, Ann. of Math. (2) 41 (1940), 474 – 487. · Zbl 0024.15005 · doi:10.2307/1968729 · doi.org
[20] D. C. Murdoch, Note on normality in quasi-groups, Bull. Amer. Math. Soc. 47 (1941), 134 – 138. · Zbl 0025.10002
[21] D. C. Murdoch, Structure of abelian quasi-groups, Trans. Amer. Math. Soc. 49 (1941), 392 – 409. · Zbl 0025.10003
[22] Kôshichi Toyoda, On axioms of linear functions, Proc. Imp. Acad. Tokyo 17 (1941), 221 – 227. · Zbl 0061.02403
[23] Solomon Lefschetz, Algebraic Topology, American Mathematical Society Colloquium Publications, v. 27, American Mathematical Society, New York, 1942. · Zbl 0061.39302
[24] I. M. H. Etherington, Non-associative arithmetics, Proc. Roy. Soc. Edinburgh. Sect. A. 62 (1949), 442 – 453. · Zbl 0033.01102
[25] Takasaki Mituhisa, Abstraction of symmetric transformations, Tôhoku Math. J. 49 (1943), 145 – 207 (Japanese).
[26] Richard H. Bruck, Some results in the theory of quasigroups, Trans. Amer. Math. Soc. 55 (1944), 19 – 52. · Zbl 0063.00635
[27] G. N. Garrison, Note on invariant complexes of a quasigroup, Ann. of Math. (2) 47 (1946), 50 – 55. · Zbl 0063.01546 · doi:10.2307/1969034 · doi.org
[28] R. H. Bruck, Contributions to the theory of loops, Trans. Amer. Math. Soc. 60 (1946), 245 – 354. · Zbl 0061.02201
[29] Grace E. Bates, Free loops and nets and their generalizations, Amer. J. Math. 69 (1947), 499 – 550. · Zbl 0035.01202 · doi:10.2307/2371882 · doi.org
[30] Fred Kiokemeister, A theory of normality for quasigroups, Amer. J. Math. 70 (1948), 99 – 106. · Zbl 0035.01103 · doi:10.2307/2371934 · doi.org
[31] J. Aczél, On mean values, Bull. Amer. Math. Soc. 54 (1948), 392 – 400. · Zbl 0030.02702
[32] Grace E. Bates and Fred Kiokemeister, A note on homomorphic mappings of quasigroups into multiplicative systems, Bull. Amer. Math. Soc. 54 (1948), 1180 – 1185. · Zbl 0034.29801
[33] A. C. Choudhury, Quasi-groups and nonassociative systems. I, Bull. Calcutta Math. Soc. 40 (1948), 183 – 194. · Zbl 0031.24701
[34] T. Evans, Homomorphisms of non-associative systems, J. London Math. Soc. 24 (1949), 254 – 260. · Zbl 0034.01304 · doi:10.1112/jlms/s1-24.4.254 · doi.org
[35] Marlow Sholander, On the existence of the inverse operations, Amer. J. Math. vol. 59 (1949) pp. 211-224. · Zbl 0034.29802
[36] B. Knaster, Sur une équivalence pour les fonctions, Colloquium Math. 2 (1949), 1 – 4 (French). · Zbl 0038.04203
[37] T. Evans, A note on the associative law, J. London Math. Soc. 25 (1950), 196 – 201. · Zbl 0039.01501 · doi:10.1112/jlms/s1-25.3.196 · doi.org
[38] Albert Sade, Quasigroups, Boulevard du Jardin-Zoologique 14, Marseilles, France, 1950.
[39] Bourbaki, IV, premier partie, Les structures fondamentales de l’analyse, Livre II, Algèbre, Chapitre I, Structures algébriques, (1951).
[40] Graham Higman and B. H. Neumann, Groups as groupoids with one law, Publ. Math. Debrecen 2 (1952), 215 – 221. · Zbl 0050.25303
[41] M. Hosszu, On the functional equation of autodistributivity, Publ. Math. Debrecen 3 (1953), 83 – 86 (1954). · Zbl 0055.34701
[42] Paul Dubreil, Les relations d’équivalence et leurs principales applications, Les conférences du Palais de la Découverte, série A, no. 194, Université de Paris, Paris, 1954 (French). · Zbl 0059.01605
[43] M. Hosszü, Some functional equations related with the associative law, Publ. Math. Debrecen 3 (1954), 205 – 214 (1955). · Zbl 0058.10803
[44] Orrin Frink, Symmetric and self-distributive systems, Amer. Math. Monthly 62 (1955), 697 – 707. · Zbl 0065.26501 · doi:10.2307/2307074 · doi.org
[45] Harry Furstenberg, The inverse operation in groups, Proc. Amer. Math. Soc. 6 (1955), 991 – 997. · Zbl 0066.01104
[46] D. A. Norton and Sherman K. Stein, An integer associated with Latin squares, Proc. Amer. Math. Soc. 7 (1956), 331 – 334. · Zbl 0071.01401
[47] D. A. Norton and Sherman K. Stein, Cycles in algebraic systems, Proc. Amer. Math. Soc. 7 (1956), 999 – 1004. · Zbl 0077.02605
[48] Sherman K. Stein, Foundations of quasigroups, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 545 – 546. · Zbl 0071.01804
[49] Miklós Hosszú, On the functional equation of transitivity, Acta Sci. Math. Szeged 15 (1954), 203 – 208. · Zbl 0056.11601
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