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Quasigroups and cubic curves

Published online by Cambridge University Press:  20 January 2009

I. M. H. Etherington
Affiliation:
Mathematical Institute, The University, Edinburg
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A note on this subject was read to the Edinburgh Mathematical Society in June 1951. Subsequently I had the benefit of conversations and correspondence with J. G. Brennan and I. R. Porteous, and substantial contributions from them are incorporated here, more than are acknowledged in detail below.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1965

References

REFERENCES

(1) Aczél, J.On mean values, Bull. Amer. Math. Soc. 54 (1948), 392400; MR 9, 501.CrossRefGoogle Scholar
(2) Aczél, J.Some general methods for functional equations, Uspehi Mat. Nauk (N.S.), 11 (1956), No. 3 (69), 368 (Russian); MR 18, 807.Google Scholar
(3) Aczél, J., Belousov, V. D. and Hossztj, M.Generalized associativity and bisymmetry on quasigroups, Ada Math. Acad. Sci. Hungar. 11 (1960), 127136; MR 25 # 4018.CrossRefGoogle Scholar
(4) Albert, A. A.Quasigroups I, Trans. Amer. Math. Soc. 54 (1943), 507519; MR 20 # 229.CrossRefGoogle Scholar
(5) Belousov, V. D.Transitive distributive quasigroupsa, Ukrain. Mat. Ž. 10. (1958), No. 1, 1322 (Russian); MR 20 # 2390.Google Scholar
(6) Bruck, R. H.Some results in the theory of quasigroups, Trans. Amer. Math. Soc. 55 (1944), 1952; MR 5, 229.CrossRefGoogle Scholar
(7) Bruck, R. H. reviewing HALL and SWIFT, MR 18 (1957), 192.Google Scholar
(8) Bruck, R. H.A Survey of Binary Systems (Berlin, 1958); MR 20 # 76.CrossRefGoogle Scholar
(9) Carmichael, R. D.Introduction to the Theory of Groups of Finite Order (Boston, U.S.A., 1937).Google Scholar
(10a) Etherington, I. M. H.Transposed algebras, Proc. Edin. Math. Soc. (2) 7 (1945), 104121; MR 7, 4.CrossRefGoogle Scholar
(10) Etherington, I. M. H.Non-associative arithmetics, Proc. Roy. Soc. Edin. A, 62 (1949), 442453; MR 10, 677.Google Scholar
(11) Etherington, I. M. H.Groupoids with additive endomorphisms, Amer. Math. Monthly 65 (1958), 596601; MR 20 # 5816.CrossRefGoogle Scholar
(12) Etherington, I. M. H.Note on quasigroups and trees, Proc. Edin. Math. Soc. (2) 13 (1963), 219222; MR 28 # 157.CrossRefGoogle Scholar
(13) Evans, T.Properties of algebras almost equivalent to identities, J. London Math. Soc. 37 (1962), 5359; MR 24 # A3225.CrossRefGoogle Scholar
(14) Frtnk, O.Symmetric and self-distributive systems, Amer. Math. Monthly 62 (1955), 697707; MR 17, 458.Google Scholar
(15) Hesse, O.Ober Curven dritter Ordnung und die Kegelschnitte, welche diese Curven in drei verschiedenen Puncten beriihren, J. Reine Angew. Math. 36 (1847), 143176.Google Scholar
(16) Minc, H.The free commutative entropic logarithmetic, Proc. Roy. Soc. Edin. A, 65 (1959), 177192; MR 22 # 11018.Google Scholar
(17) Murdoch, D. C.Quasi-groups which satisfy certain generalized associative laws, Amer. J. Math. 61 (1939), 509522.CrossRefGoogle Scholar
(18) Murdoch, D. C.Structure of abelian quasi-groups, Trans. Amer. Math. Soc. 49 (1941), 392409; MR 2, 218.CrossRefGoogle Scholar
(19) Netto, E.Lehrbuch der Combinatorik (Leipzig, 1901; 2nd ed., 1927).Google Scholar
(20) Marshall Osborn, J.New loops from old geometries, Amer. Math. Monthly 68 (1961), 103107; MR 23 # A1686.CrossRefGoogle Scholar
(21) Sade, A.Quasigroupes, published by the author (Marseille, 1950); MR 13, 203.Google Scholar
(22) Sade, A.Quasigroupes obeissant a certaines lois, Rev. Fac. Sci. Univ. Istanbul A, 22 (1957), 151184; MR 21 # 4987.Google Scholar
(23) Sade, A.Quasigroupes automorphes par la groupe lindaire et giometrie finie, J. Reine Angew. Math. 199 (1958), 100120; MR 20 # 78.CrossRefGoogle Scholar
(24) Salmon, G.Higher Plane Curves (3rd ed., Dublin, 1879).Google Scholar
(25) Semple, J. G. and Roth, L.Introduction to Algebraic Geometry (Oxford, 1949); MR 11, 535.Google Scholar
(26) Severi, F.Trattato di Geometrica Algebrica (Bologna, 1926).Google Scholar
(27) Sholander, M.On the existence of the inverse operation in alternation groupoids, Bull. Amer. Math. Soc. 55 (1949), 746757; MR 11, 159.CrossRefGoogle Scholar
(28) Stein, S. K.On the foundations of quasigroups, Trans. Amer. Math. Soc. 85 (1957), 228256; MR 20 # 922.CrossRefGoogle Scholar
(29) Steiner, J.Geometrische Lehrsatze, J. Reine Angew. Math. 32 (1846), 182184 = Ges. Werke II, 369–373.Google Scholar
(30) Steiner, J.Combinatorische Aufgabe, J. Reine Angew. Math. 45 (1853), 181182 = Ges. Werke II, 435–436.Google Scholar
(31) Takasaki, M.Abstraction of symmetric transformations, Tôohoku Math. J. 49 (1943), 145207 (Japanese); MR 9, 8.Google Scholar
(32) Toyoda, K.On axioms of mean transformations and automorphic transformations of abelian groups, Tohoku Math. J. 46 (1940), 239251; MR 2, 6.Google Scholar
(33) Walker, R. J.Algebraic Curves (Princeton, 1950); MR 11, 387.Google Scholar