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On the foundations of quasigroups
Author: Sherman K. Stein
Journal: Trans. Amer. Math. Soc. 85 (1957), 228-256
MSC: Primary 20.00
DOI: https://doi.org/10.1090/S0002-9947-1957-0094404-6
MathSciNet review: 0094404
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Harris F. MacNeish, Euler squares, Ann. of Math. (2) vol. 23 (1921-1922) pp. 221-227. R. A. Fisher, The arrangement of field experiments, J. Minis. Ag. vol. 3 (1936) pp. 503-513. Anton Suschkewitsch, On a generalization of the associative law, Trans. Amer. Math. Soc. vol. 31 (1929) pp. 204-214. C. Burstin and W. Mayer, Distributive Gruppen, J. Reine Angew. Math. vol. 160 (1929) pp. 111-130. Morgan Ward, Postulates for the inverse operations in a group, Trans. Amer. Math. Soc. vol. 32 (1930) pp. 520-526. M. Zorn, Theorie der alternitiven Ringe, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (1930) pp. 123-147. Anton Suschkewitsch, Über ein Elementen-system, für welche zwei Distributivgesetze gelten, Kharkovski Matematichnovo Tobarictba, series 4, VII (1934) pp. 29-32. A. Adrian Albert, On a certain algebra of quantum mechanics, Ann. of Math. vol. 35 (1934) pp. 65-73. David G. Rabinow, Independent sets of postulates for abelian groups and fields in terms of the inverse operations, Amer. J. Math. vol. 59 (1937) pp. 211-224. B. A. Hausmann and Oystein Ore, Theory of quasigroups, Amer. J. Math. vol. 59 (1937) pp. 983-1004. Herbert Boggs and G. Y. Rainich, Note on group postulates, Bull. Amer. Math. Soc. vol. 43 (1937) pp. 81-84. Alfred Tarski, Ein Beitrag zur Axiomatik der Abelschen Gruppen, Fund. Math. XXX (1938) pp. 253-256. Raj Chandra Bose, On the application of the properties of Galois Field to the problem of construction of hyper graeco Latin squares, Sankhya, vol. 3 pt. 4 (1938) pp. 323-338. B. A. Bernstein, Postulates for abelian groups and fields in terms of non-associative operations, Trans. Amer. Math. Soc. vol. 43 (1938) pp. 1-6. D. C. Murdoch, Quasigroups which satisfy certain generalized associative laws, Amer. J. Math. vol. 61 (1939) pp. 509-522. H. W. Norton, The $7 \times 7$ squares, Annals of Eugenics vol. 9 (1939) pp. 269-307. W. L. Stevens, The completely orthogonalized Latin squares, Annals of Eugenics vol. 9 (1939) pp. 82-93. E. L. Post, Polyadic groups, Trans. Amer. Math. Soc. vol. 48 (1940) pp. 208-350. G. N. Garrison, Quasigroups, Ann. of Math. vol. 41 (1940) pp. 474-487. D. C. Murdoch, Note on normality in quasigroups, Bull. Amer. Math. Soc. vol. 47 (1941) pp. 134-138. ---, Structure of abelian quasigroups, Trans. Amer. Math. Soc. vol. 49 (1941) pp. 392-409. Koshichi Toyoda, On axioms of linear functions, Proc. Imp. Acad. Tokyo XVII (1941) pp. 221-227. S. Lefschetz, Algebraic topology, Amer. Math. Soc. Colloquium Publications, vol. 27, New York, 1942. I. M. H. Etherinton, Non-associative arithmetics, Proceedings of the Royal Society of Edinburgh vol. 62 (1943-1944) pp. 442-453. Takasaki Mituhisa, Abstractions of symmetric functions, Tôhoku Math. J. vol. 49 (1943) pp. 145-207 (in Japanese). R. H. Bruck, Some results in the theory of quasigroups, Trans. Amer. Math. Soc. vol. 55 (1944) pp. 19-52. G. N. Garrison, A note on invariant complexes of a quasigroup, Ann. of Math. vol. 47 (1946) pp. 50-55. R. H. Bruck, Contributions to the theory of loops, Trans. Amer. Math. Soc. vol. 60 (1946) pp. 245-354. Grace E. Bates, Free loops and nets and their generalizations, Amer. J. Math. vol. 69 (1947) pp. 495-550. Fred Kiokemeister, A theory of normality for quasigroups, Amer. J. Math. vol. 70 (1948) pp. 99-106. J. Aczel, On mean values, Bull. Amer. Math. Soc. vol. 54 (1948) pp. 392-400. Grace E. Bates and Fred Kiokemeister, A note on homomorphic mappings of quasigroups into multiplicative systems, Bull. Amer. Math. Soc. vol. 54 (1948) A. C. Choudhury, Quasigroups and nonassociative systems. I, Bulletin of the Calcutta Mathematical Society (1948) pp. 183-194. T. Evans, Homomorphisms of non-associative systems, J. London Math. Soc. vol. 24 (1949) pp. 254-260. Marlow Sholander, On the existence of the inverse operations, Amer. J. Math. vol. 59 (1949) pp. 211-224. B. Knaster, Sur une equivalence pour les fonctions, Colloquium Mathematicum, vol. II, Fasc. 1, 1949, pp. 1-4. T. Evans, A note on the associative law, J. London Math. Soc. vol. 25 (1950) pp. 196-201. Albert Sade, Quasigroups, Boulevard du Jardin-Zoologique 14, Marseilles, France, 1950. Bourbaki, IV, premier partie, Les structures fondamentales de l'analyse, Livre II, Algèbre, Chapitre I, Structures algébriques, (1951). G. Higman and B. H. Neumann, Groups as groupoids with one law, Publicationes Mathematicae, Tomus 2, Fasc. 3-4 1952. M. Hosszu, On the functional equation of autodistributivity, Publicationes Mathematicae, Tomas 3, Fasc. 1-2, Debrecen, 1953, pp. 83-86. Paul Dubreil, Algèbre, Tome I, Paris, 1954. M. Hosszu, Some functional equations related with the associative law, Publicationes Mathematicae, vol. 3, Fasc. 3-4, Debrecen, 1954, pp. 205-214. Orrin Frink, Symmetric and self-distributive systems, Amer. Math. Monthly vol. 62 (1955) pp. 697-707. Harry Furstenberg, The inverse operation in groups, Proc. Amer. Math. Soc. vol. 6 (1955) pp. 991-997. D. A. Norton and Sherman K. Stein, An integer associated with Latin squares, Proc. Amer. Math. Soc. vol. 7 (1956) pp. 331-334. ---, Cycles in algebraic systems, Proc. Amer. Math. Soc. vol. 7 (1956) pp. 999-1004. S. K. Stein, Foundations of quasigroups, Proc. Nat. Acad. Sci. U. S. A. vol. 42 (1956) pp. 545-546. M. Hosszu, On the functional equation of transitivity, Acta Sci. Math. vol. 15 (1954) pp. 203-208.
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