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A historical perspective of the theory of isotopisms. (English) Zbl 1423.17018

Summary: In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.

MSC:

17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
05B15 Orthogonal arrays, Latin squares, Room squares

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[1] Albert, A.A.; Non-Associative Algebras: I. Fundamental Concepts and Isotopy; Ann. Math.: 1942; Volume 43 ,685-707. · Zbl 0061.04807
[2] Albert, A.A.; Quasigroups I; Trans. Am. Math. Soc.: 1943; Volume 54 ,507-519. · Zbl 0063.00039
[3] Albert, A.A.; Quasigroups II; Trans. Am. Math. Soc.: 1944; Volume 55 ,401-419. · Zbl 0063.00042
[4] Bruck, R.H.; Some results in the theory of quasigroups; Trans. Am. Math. Soc.: 1944; Volume 55 ,19-52. · Zbl 0063.00635
[5] Poincaré, H.; Analysis Situs; Journal de l’École Polytechnique: 1895; Volume 1 ,1-121. · JFM 26.0541.07
[6] Poincaré, H.; Fifth supplement to Analysis Situs; Rend. Circ. Matem. Palermo: 1904; Volume 18 ,45-110. · JFM 35.0504.13
[7] Dehn, M.; Heegaard, P.; Analysis Situs; Enzyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen: Leipzig, Germany 1907; ,153-220. · JFM 38.0510.14
[8] Tietze, H.; Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten; Monatsh. Math. Phys.: 1908; Volume 19 ,1-118. · JFM 39.0720.05
[9] Brouwer, L.E.J.; Beweis der Invarianz der Dimensionenzahl; Math. Ann.: 1911; Volume 70 ,161-165. · JFM 42.0416.02
[10] Brouwer, L.E.J.; Continuous one-one transformations of surfaces in themselves; V Proceedings of Koninklijke Nederlandse Akademie van Wetenschappen: Amsterdam, The Netherlands 1912; ,352-361.
[11] Veblen, O.; Theory on Plane Curves in Non-metrical Analysis Situs; Trans. Am. Math. Soc.: 1905; Volume 6 ,83-98. · JFM 36.0530.02
[12] Veblen, O.; Analysis Situs; The Cambridge Colloquium 1916: New York, NY, USA 1922; . · JFM 48.0647.10
[13] McLarty, C.; Emmy Noether’s “set theoretic” topology, From Dedekind to the first functors; The Architecture of Modern Mathematics, Essays in History and Philosophy: Oxford, UK 2006; ,187-208. · Zbl 1129.01010
[14] Vietoris, L.; Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen; Math. Ann.: 1927; Volume 97 ,454-472. · JFM 53.0552.01
[15] Hopf, H.; Eine Verallgemeinerung der Euler-Poincaréschen Formel; Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: Göttingen, Germany 1928; ,127-136. · JFM 54.0610.02
[16] Mayer, W.; Über abstrakte Topologie; Monatshefte für Mathematik und Physik: 1929; Volume 36 ,1-42. · JFM 55.0963.02
[17] Alexandroff, P.; Untersuchungen Über Gestalt und Lage Abgeschlossener Mengen Beliebiger Dimension; Ann. Math.: 1929; Volume 30 ,101-187. · JFM 54.0609.02
[18] Čech, E.; Höherdimensionale Homotopiegruppen; Verhandlungen des InternationalenMathematiker-Kongresses Zürich 1932: Zürich, Switzerland 1932; ,203. · JFM 58.0646.06
[19] Čech, E.; Théorie générale de l’homologie dans un espace quelconque; Fund. Math.: 1932; Volume 19 ,149-183. · JFM 58.0622.02
[20] Hurewicz, W.; Beiträge zur Topologie der Deformationen I. Höherdimensionale Homotopiegruppen; Proc. Koninkl. Akad. Amst.: 1935; Volume 38 ,112-119. · Zbl 0010.37801
[21] Hurewicz, W.; Beiträge zur Topologie der Deformationen II. Homotopie- und Homologiegruppen; Proc. Koninkl. Akad. Amst.: 1935; Volume 38 ,521-528. · Zbl 0011.37101
[22] Hurewicz, W.; Beiträge zur Topologie der Deformationen III. Klassen und Homologietypen von Abbidungen; Proc. Koninkl. Akad. Amst.: 1936; Volume 39 ,117-126. · Zbl 0013.22903
[23] Hurewicz, W.; Beiträge zur Topologie der Deformationen IV. Asphärische Räumen; Proc. Koninkl. Akad. Amst.: 1936; Volume 39 ,215-224. · Zbl 0013.28303
[24] Hopf, H.; Ein topologischer Beitrag zur reellen Algebra; Comment. Math. Helvetici: 1940; Volume 13 ,219-239. · Zbl 0024.36002
[25] Albert, A.A.; A determination of all normal division algebras in sixteen units; Trans. Am. Math. Soc.: 1929; Volume 31 ,253-260. · JFM 55.0090.04
[26] Wedderburn, J.H.M.; On hypercomplex numbers; Proc. Lond. Math. Soc.: 1908; Volume VI ,77-118. · JFM 39.0139.01
[27] Frobenius, F.G.; Über lineare Substitutionen und bilineare Formen; Journal Für Die Reine und Angewandte Mathematik: 1878; Volume 84 ,1-63. · JFM 09.0085.02
[28] Hurwitz, A.; Über die Composition der quadratischen Formen von beliebig vielen Variabeln; Goett. Nachr.: 1898; Volume 1898 ,309-316. · JFM 29.0177.01
[29] Wedderburn, J.H.M.; A theorem on finite algebras; Trans. Am. Math. Soc.: 1905; Volume 6 ,349-352. · JFM 36.0139.01
[30] Slaught, H.E.; The April meeting of the Chicago Section; Bull. Am. Math. Soc.: 1906; Volume 12 ,434-446. · JFM 37.0036.09
[31] Dickson, L.E.; On commutative linear algebras in which division is always uniquely possible; Trans. Am. Math. Soc.: 1906; Volume 7 ,514-522. · JFM 37.0112.01
[32] Dickson, L.E.; Linear algebras in which division is always uniquely possible; Trans. Am. Math. Soc.: 1906; Volume 7 ,370-390. · JFM 37.0111.06
[33] Dickson, L.E.; On triple algebras and ternary cubic forms; Bull. Am. Math. Soc.: 1908; Volume 14 ,160-169. · JFM 39.0138.03
[34] Dickson, L.E.; Linear algebras with associativity not assumed; Duke Math. J.: 1935; Volume 1 ,113-125. · Zbl 0012.14801
[35] Dickson, L.E.; Linear associative algebras and abelian equations; Trans. Am. Math. Soc.: 1914; Volume 15 ,31-46. · JFM 45.0189.03
[36] Wedderburn, J.H.M.; A type of primitive algebra; Trans. Am. Math. Soc.: 1914; Volume 15 ,162-166. · JFM 45.0189.04
[37] Wedderburn, J.H.M.; On division algebras; Trans. Am. Math. Soc.: 1921; Volume 22 ,129-135. · JFM 48.0126.01
[38] Dickson, L.E.; ; Algebras and their Arithmetics: Chicago, IL, USA 1923; . · JFM 49.0079.01
[39] Dickson, L.E.; ; Algebren und Ihre Zahlentheorie: Zurich, Switzerland 1927; .
[40] Hurwitz, A.; Über die Komposition der quadratischen Formen; Math. Ann.: 1923; Volume 88 ,1-25. · JFM 48.1164.03
[41] Roquette, P.J.; ; The Brauer-Hasse-Noether Theorem in Historical Perspective: Berlin/Heidelberg, Germany 2006; . · Zbl 1060.01009
[42] Cecioni, F.; Sopra un tipo di algebre prive di divisori dello zero; Rend. Circ. Matem. Palermo: 1923; Volume XLVII ,209-254. · JFM 49.0088.02
[43] Brauer, R.; Über Systeme hyperkomplexer Zahlen; Math. Z.: 1929; Volume 30 ,79-107. · JFM 55.0088.02
[44] Albert, A.A.; New results in the theory of normal division algebras; Trans. Am. Math. Soc.: 1930; Volume 32 ,171-195. · JFM 56.0146.01
[45] Hasse, H.; Über ℘-adische Schiefkörpe und ihre Bedeutung für die Arithmetik hyperkomplexer Zahlsysteme; Math. Ann.: 1931; Volume 104 ,495-534. · Zbl 0001.19805
[46] Albert, A.A.; A construction of non-cyclic normal division algebras; Bull. Am. Math. Soc.: 1932; Volume 38 ,449-456. · Zbl 0005.00603
[47] Brauer, R.; Untersuchungen über die arithmetischen Eigenschaften von Gruppen linearer Substitutionen, Zweite Mitteilung; Math. Z.: 1930; Volume 31 ,733-747. · JFM 56.0865.04
[48] Albert, A.A.; Normal Division Algebras of Degree 4 Over F of Characteristic 2; Am. J. Math.: 1934; Volume 56 ,75-86. · Zbl 0008.24202
[49] Albert, A.A.; Normal division algebras over a modular field; Trans. Am. Math. Soc.: 1934; Volume 36 ,388-394. · JFM 60.0105.02
[50] Brauer, R.; Hasse, H.; Noether, E.; Beweis eines Hauptsatzes in der Theorie der Algebren; J. Reine Angew. Math.: 1932; Volume 167 ,399-404. · JFM 58.0142.03
[51] Albert, A.A.; Hasse, H.; A determination of all normal division algebras over an algebraic number field; Trans. Am. Math. Soc.: 1932; Volume 34 ,722-726. · Zbl 0005.05003
[52] Fenster, D.D.; Schwermer, J.; A delicate collaboration, Adrian Albert and Helmut Hasse and the principal theorem in division algebras in the early 1930’s; Arch. Hist. Exact Sci.: 2005; Volume 59 ,349-379. · Zbl 1066.01026
[53] Albert, A.A.; Division algebras over an algebraic field; Bull. Am. Math. Soc.: 1931; Volume 37 ,777-784. · JFM 57.0161.01
[54] Hasse, H.; Theory of cyclic algebras over an algebraic number field; Trans. Am. Math. Soc.: 1932; Volume 34 ,171-214. · JFM 58.1034.02
[55] Albert, A.A.; On direct products; Trans. Am. Math. Soc.: 1931; Volume 33 ,690-711. · JFM 57.0159.01
[56] Albert, A.A.; ; Structure of Algebras: New York, NY, USA 1939; . · Zbl 0023.19901
[57] Bruck, R.H.; Some results in the theory of linear non-associative algebras; Trans. Am. Math. Soc.: 1944; Volume 56 ,141-199. · Zbl 0061.05202
[58] Hausmann, B.A.; Ore, Ø; Theory of Quasi-Groups; Am. J. Math.: 1937; Volume 59 ,983-1004. · Zbl 0017.39102
[59] Suschkewitsch, A.; On a generalization of the associative law; Trans. Am. Math. Soc.: 1929; Volume 31 ,204-214. · JFM 55.0085.01
[60] Frobenius, F.G.; ; Über Endliche Gruppen: Berlin, Germany 1895; .
[61] Moore, E.H.; A definition of abstract groups; Trans. Am. Math. Soc.: 1902; Volume 3 ,485-492. · JFM 33.0142.01
[62] Dickson, L.E.; Definitions of a group and a field by independent postulates; Trans. Am. Math. Soc.: 1905; Volume 6 ,198-204. · JFM 36.0207.01
[63] Moufang, R.; Zur Struktur von Alternativkörpern; Math. Ann.: 1935; Volume 110 ,416-430. · Zbl 0010.00403
[64] Murdoch, D.C.; Structure of abelian quasi-groups; Trans. Am. Math. Soc.: 1941; Volume 49 ,392-409. · Zbl 0025.10003
[65] Garrison, G.N.; Quasi-groups; Ann. Math.: 1940; Volume 41 ,474-487. · JFM 66.0094.04
[66] Griffin, H.; The abelian quasi-group; Am. J. Math.: 1940; Volume 62 ,725-737. · JFM 66.0095.01
[67] Bruck, R.H.; Contributions to the theory of loops; Trans. Am. Math. Soc.: 1946; Volume 60 ,245-354. · Zbl 0061.02201
[68] Bruck, R.H.; Some theorems on Moufang loops; Math. Z.: 1960; Volume 73 ,59-78. · Zbl 0099.25103
[69] Fisher, R.A.; Yates, F.; The 6 × 6 Latin squares; Proc. Camb. Philos. Soc.: 1934; Volume 30 ,492-507. · Zbl 0010.10301
[70] Norton, H.W.; The 7 × 7 squares; Ann. Eugen.: 1939; Volume 9 ,269-307. · Zbl 0022.11102
[71] Shaw, J.B.; On parastrophic algebras; Trans. Am. Math. Soc.: 1915; Volume 16 ,361-370. · JFM 45.0190.01
[72] Etherington, I.M.H.; Transposed algebras; Proc. Edinb. Math. Soc.: 1945; Volume 7 ,104-121. · Zbl 0060.08012
[73] Albert, A.A.; Finite division algebras and finite planes; Proc. Sympos. Appl. Math.: 1960; Volume 10 ,53-70. · Zbl 0096.15003
[74] Hughes, D.R.; Collineation Groups of Non-Desarguesian Planes, I, The Hall Veblen-Wedderburn Systems; Am. J. Math.: 1959; Volume 81 ,921-938. · Zbl 0091.32504
[75] Sandler, R.; Autotopism groups of some finite non-associative algebras; Am. J. Math.: 1962; Volume 84 ,239-264. · Zbl 0156.26904
[76] Oehmke, R.H.; Sandler, R.; The collineation groups of division ring planes. I. Jordan division algebras; J. Reine Angew. Math.: 1964; Volume 216 ,67-87. · Zbl 0139.03302
[77] Petersson, H.P.; Quasi composition algebras; Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg: 1971; Volume 35 ,215-222. · Zbl 0217.06701
[78] Benkart, G.; Osborn, J.M.; Britten, D.; On applications of isotopy to real division algebras; Hadron. J.: 1981; Volume 4 ,497-529. · Zbl 0451.17002
[79] Darpö, E.; Dieterich, E.; Real commutative division algebras; Algebr. Represent. Theory: 2007; Volume 10 ,179-196. · Zbl 1147.17002
[80] Hall, M.; Projective planes; Trans. Am. Math. Soc.: 1943; Volume 54 ,229-277. · Zbl 0060.32209
[81] Knuth, D.E.; Finite Semifields and Projective Planes; J. Algebra: 1965; Volume 2 ,182-217. · Zbl 0128.25604
[82] Ganley, M.J.; Polarities in translation planes; Geometriae Dedicata: 1972; Volume 1 ,103-116. · Zbl 0241.50024
[83] Soubeyran, B.; Sur certains semi-corps dérivables et une généralisation des systèmes de Hall et de Kirkpatrick; C. R. Acad. Sci. Paris Sér. A: 1974; Volume 278 ,851-854. · Zbl 0297.12104
[84] Ball, S.; Lavrauw, M.; On the Hughes-Kleinfeld and Knuth’s semifields two-dimensional over a weak nucleus; Des. Codes Cryptogr.: 2007; Volume 44 ,63-67. · Zbl 1123.51009
[85] Menichetti, G.; On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field; J. Algebra: 1977; Volume 47 ,400-410. · Zbl 0362.17002
[86] Albert, A.A.; On nonassociative division algebras; Trans. Am. Math. Soc.: 1952; Volume 72 ,296-309. · Zbl 0046.03601
[87] Spille, B.; Pieper-Seier, I.; On strong isotopy of Dickson semifields and geometric implications; Results Math.: 1998; Volume 33 ,364-373. · Zbl 0944.17002
[88] Jha, V.; Local Schur’s lemma and commutative semifields; Des. Codes Cryptogr.: 2005; Volume 36 ,203-216. · Zbl 1082.51001
[89] Coulter, R.S.; Henderson, M.; Commutative presemifields and semifields; Adv. Math.: 2008; Volume 217 ,282-304. · Zbl 1194.12007
[90] Schafer, R.D.; Alternative algebras over an arbitrary field; Bull. Am. Math. Soc.: 1943; Volume 49 ,549-555. · Zbl 0061.05201
[91] McCrimmon, K.; Homotopes of alternative algebras; Math. Ann.: 1971; Volume 191 ,253-262. · Zbl 0203.33802
[92] Allison, B.N.; Conjugate inversion and conjugate isotopes of alternative algebras with involution; Algebras Groups Geom.: 1986; Volume 3 ,361-385. · Zbl 0635.17013
[93] Babikov, M.; Isotopy and identities in alternative algebras; Proc. Am. Math. Soc.: 1997; Volume 125 ,1571-1575. · Zbl 0874.17034
[94] Pchelintsev, S.V.; Isotopes of prime (−1,1)-and Jordan algebras; Algebra Logika: 2010; Volume 49 ,388-423. · Zbl 1241.17035
[95] Petersson, H.P.; Isotopisms of Jordan algebras; Proc. Am. Math. Soc.: 1969; Volume 20 ,477-482. · Zbl 0181.04502
[96] Jacobson, N.; A coordinatization theorem for Jordan algebras; Proc. Natl. Acad. Sci. USA: 1962; Volume 48 ,1154-1160. · Zbl 0115.02703
[97] Petersson, H.P.; The structure group of an alternative algebra; Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg: 2002; Volume 72 ,165-186. · Zbl 1014.17004
[98] McCrimmon, K.; The generic norm of an isotope of a Jordan algebra; Scripta Math.: 1973; Volume 29 ,229-236. · Zbl 0288.17010
[99] Loos, O.; Generically algebraic Jordan algebras over commutative rings; J. Algebra: 2006; Volume 297 ,474-529. · Zbl 1088.17013
[100] McCrimmon, K.; Inner ideals in quadratic Jordan algebras; Trans. Am. Math. Soc.: 1971; Volume 159 ,445-468. · Zbl 0224.17011
[101] Petersson, H.P.; The isotopy problem for Jordan matrix algebras; Trans. Am. Math. Soc.: 1978; Volume 244 ,185-197. · Zbl 0392.17012
[102] Petersson, H.P.; Racine, M.L.; Springer forms and the first Tits construction of exceptional Jordan division algebras; Manuscr. Math.: 1984; Volume 45 ,249-272. · Zbl 0536.17009
[103] Thakur, M.L.; Isotopy and invariants of Albert algebras; Comment. Math. Helv.: 1999; Volume 74 ,297-305. · Zbl 0931.17021
[104] Jiménez-Gestal, C.; Pérez-Izquierdo, J.M.; Ternary derivations of finite-dimensional real division algebras; Linear Algebra Appl.: 2008; Volume 428 ,2192-2219. · Zbl 1205.17004
[105] Falcón, O.J.; Falcón, R.M.; Núñez, J.; Isotopism and isomorphism classes of certain Lie algebras over finite fields; Results Math.: 2015; Volume 71 ,167-183. · Zbl 1400.17010
[106] Falcón, O.J.; Falcón, R.M.; Núñez, J.; Isomorphism and isotopism classes of filiform Lie algebras of dimension up to seven; Results Math.: 2017; Volume 71 ,1151-1166. · Zbl 1400.17011
[107] Malcev, A.I.; Analytic loops; Mat. Sb.: 1955; Volume 36 ,569-576. · Zbl 0065.00702
[108] Sagle, A.A.; Malcev algebras; Trans. Am. Math. Soc.: 1961; Volume 101 ,426-458. · Zbl 0101.02302
[109] Falcón, O.J.; Falcón, R.M.; Núñez, J.; A computational algebraic geometry approach to enumerate Malcev magma algebras over finite fields; Math. Meth. Appl. Sci.: 2016; Volume 39 ,4901-4913. · Zbl 1401.17024
[110] Etherington, I.M.H.; Genetic algebras; Proc. R. Soc. Edinb.: 1939; Volume 59 ,242-258. · JFM 66.1209.01
[111] Bertrand, M.; Algèbres non associatives et algèbres génétiques; Mémorial des Sciences Mathématiques 162: Paris, France 1966; . · Zbl 0147.28401
[112] Holgate, P.; Genetic algebras associated with polyploidy; Proc. Edinb. Math. Soc.: 1966; Volume 15 ,1-9. · Zbl 0144.27202
[113] Ringwood, G.A.; Hypergeometric algebras and Mendelian genetics; Nieuw Arch. Wisk.: 1985; Volume 3 ,69-83. · Zbl 0565.17014
[114] Campos, T.M.M.; Holgate, P.; Algebraic Isotopy in Genetics; IMA J. Math. Appl. Med. Biol.: 1987; Volume 4 ,215-222. · Zbl 0628.92018
[115] Tian, J.P.; Evolution Algebras and their Applications; Lecture Notes in Mathematics 1921: Berlin, Germany 2008; .
[116] Tian, J.P.; Vojtechovsky, P.; Mathematical concepts of evolution algebras in non-mendelian genetics; Quasigr. Relat. Syst.: 2006; Volume 14 ,111-122. · Zbl 1112.17001
[117] Research on Evolution Algebras and Coalgebras; ; .
[118] Falcón, O.J.; Falcón, R.M.; Núñez, J.; Classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field; J. Algebra: 2017; Volume 472 ,573-593. · Zbl 1407.17041
[119] Falcón, O.J.; Falcón, R.M.; Núñez, J.; Algebraic computation of genetic patterns related to three-dimensional evolution algebras; Appl. Math. Comput.: 2018; Volume 319 ,510-517. · Zbl 1430.13048
[120] Sade, A.; Quasigroupes demi-symétriques. Isotopies préservant la demi-symétrie; Math. Nachr.: 1967; Volume 33 ,177-188. · Zbl 0143.03002
[121] Sade, A.; Autotopies des quasigroupes et des systèmes associatifs; Arch. Math.: 1968; Volume 4 ,1-23. · Zbl 0211.33201
[122] Sade, A.; Critères d’isotopie d’un quasigroupe avec un quasigroupe demi-symétrique; Univ. Lisboa Revista Fac. Ci.: 1964/1965; Volume 11 ,121-136. · Zbl 0149.02303
[123] Belousov, V.D.; Crossed isotopies of quasigroups; Mat. Issled.: 1990; Volume 113 ,14-20. · Zbl 0728.20058
[124] Belousov, V.D.; Onoĭ, V.I.; Loops that are isotopic to left-distributive quasigroups; Mat. Issled.: 1972; Volume 7 ,135-152. · Zbl 0282.20059
[125] Evans, T.; On multiplicative systems defined by generators and relations. II. Monogenic loops; Proc. Camb. Philos. Soc.: 1953; Volume 49 ,579-589. · Zbl 0051.01102
[126] Falconer, E.; Isotopes of some special quasigroup varieties; Acta Math. Acad. Sci. Hungar.: 1971; Volume 22 ,73-79. · Zbl 0226.20078
[127] Osborn, J.M.; Loops with the weak inverse property; Pac. J. Math.: 1960; Volume 10 ,295-304. · Zbl 0091.02101
[128] Robinson, D.A.; A Bol loop isomorphic to all loop isotopes; Proc. Am. Math. Soc.: 1968; Volume 19 ,671-672. · Zbl 0157.04801
[129] Stein, S.K.; On the foundations of quasigroups; Trans. Am. Math. Soc.: 1957; Volume 85 ,228-256. · Zbl 0079.02402
[130] Artzy, R.; Isotopy and parastrophy of quasigroups; Proc. Am. Math. Soc.: 1963; Volume 14 ,429-431. · Zbl 0108.25801
[131] Aczél, J.; Quasigroups, nets and nomograms; Adv. Math.: 1965; Volume 1 ,383-450. · Zbl 0135.03601
[132] Lindner, C.C.; Steedley, D.; On the number of conjugates of a quasigroup; Algebra Universalis: 1975; Volume 5 ,191-196. · Zbl 0324.20078
[133] Smith, J.D.H.; Finite distributive quasigroups; Math. Proc. Camb. Philos. Soc.: 1976; Volume 80 ,37-41. · Zbl 0338.20098
[134] Drápal, A.; Valent, V.; Few associative triples, isotopisms and groups; Des. Codes Cryptogr.: 2018; Volume 86 ,555-568. · Zbl 1434.20051
[135] Deriyenko, I.I.; Autotopisms of some quasigroups; Quasigr. Relat. Syst.: 2015; Volume 23 ,217-220. · Zbl 1338.20064
[136] McKay, B.D.; Wanless, I.M.; Zhang, X.; The order of automorphisms of quasigroups; J. Comb. Des.: 2015; Volume 23 ,275-288. · Zbl 1327.05043
[137] Dudek, W.A.; Parastrophes of quasigroups; Quasigr. Relat. Syst.: 2015; Volume 23 ,221-230. · Zbl 1343.20069
[138] Grošek, O.; Sýs, M.; Isotopy of Latin squares in cryptography; Tatra Mt. Math. Publ.: 2010; Volume 45 ,27-36. · Zbl 1274.94074
[139] Jayéolá, T.G.; On middle universal m-inverse quasigroups and their applications to cryptography; An. Univ. Vest Timiş. Ser. Mat. Inform.: 2011; Volume 49 ,69-87. · Zbl 1274.20090
[140] Dénes, J.; Keedwell, A.D.; ; Latin Squares and Their Applications: New York, NY, USA 1974; . · Zbl 0283.05014
[141] Wanless, I.M.; Ihrig, E.C.; Symmetries that Latin squares inherit from 1-factorizations; J. Comb. Des.: 2005; Volume 13 ,157-172. · Zbl 1067.05061
[142] Ihrig, E.C.; Ihrig, B.M.; The recognition of symmetric Latin squares; J. Comb. Des.: 2008; Volume 16 ,291-300. · Zbl 1146.05010
[143] McKay, B.D.; Meynert, A.; Myrvold, W.; Small Latin squares, quasigroups, and loops; J. Comb. Des.: 2007; Volume 15 ,98-119. · Zbl 1112.05018
[144] Hulpke, A.; Kaski, P.; Östergård, P.R.J.; The number of Latin squares of order 11; Math. Comput.: 2011; Volume 80 ,1197-1219. · Zbl 1210.05017
[145] Kotlar, D.; Parity types, cycle structures and autotopisms of Latin squares; Electron. J. Comb.: 2012; Volume 19 ,1-17. · Zbl 1253.05048
[146] Kotlar, D.; Computing the autotopy group of a Latin square by cycle structure; Discret. Math.: 2014; Volume 331 ,74-82. · Zbl 1297.05039
[147] Mendis, M.J.L.; Wanless, I.M.; Autoparatopisms of quasigroups and Latin squares; J. Comb. Des.: 2016; Volume 25 ,51-74. · Zbl 1362.05027
[148] Stones, D.S.; Symmetries of partial Latin squares; Eur. J. Comb.: 2013; Volume 34 ,1092-1107. · Zbl 1292.05065
[149] Dawson, E.; Donovan, D.; Offer, A.; Quasigroups, isotopisms and authentication schemes; Australas. J. Comb.: 1996; Volume 13 ,75-88. · Zbl 0858.94014
[150] Falcón, R.M.; Latin squares associated to principal autotopisms of long cycles. Application in Cryptography; Proceedings of Transgressive Computing 2006, a Conference in Honor of Jean Della Dora: Granada, Spain 2006; ,213-230. · Zbl 1203.05022
[151] Stones, R.J.; Su, M.; Liu, X.; Wang, G.; Lin, S.; A Latin square autotopism secret sharing scheme; Des. Codes Cryptogr.: 2015; Volume 80 ,635-650. · Zbl 1347.05023
[152] Artamonov, V.A.; Chakrabarti, S.; Pal, S.K.; Characterization of polynomially complete quasigroups based on Latin squares for cryptographic transformations; Discret. Appl. Math.: 2016; Volume 200 ,5-17. · Zbl 1338.20063
[153] Andres, S.D.; Falcón, R.M.; Colouring games based on autotopisms of Latin hyper-rectangles; Quaest. Math.: ; . · Zbl 1420.05114
[154] Andres, S.D.; Bergold, H.; Falcón, R.M.; Autoparatopism stabilized colouring games on rook’s graphs; Electron. Notes Discret. Math.: 2018; Volume 68C ,233-238. · Zbl 1397.05115
[155] Bailey, R.A.; Enumeration of totally symmetric Latin squares; Util. Math.: 1979; Volume 15 ,193-216. · Zbl 0415.05015
[156] Falcón, R.M.; The set of autotopisms of partial Latin squares; Discret. Math.: 2013; Volume 313 ,1150-1161. · Zbl 1277.05024
[157] Falcón, R.M.; Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method; Eur. J. Comb.: 2015; Volume 48 ,215-223. · Zbl 1315.05023
[158] Falcón, R.M.; Stones, R.J.; Classifying partial Latin rectangles; Electron. Notes Discret. Math.: 2015; Volume 49 ,765-771. · Zbl 1346.05021
[159] Falcón, R.M.; Falcón, O.J.; Núñez, J.; Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers; Math. Methods Appl. Sci.: 2018; . · Zbl 1409.05039
[160] Dietrich, H.; Wanless, I.M.; Small partial Latin squares that embed in an infinite group but not into any finite group; J. Symb. Comput.: 2018; Volume 86 ,142-152. · Zbl 1380.05019
[161] Wanless, I.M.; Webb, B.S.; Small partial Latin squares that cannot be embedded in a Cayley table; Australas. J. Comb.: 2017; Volume 67 ,352-363. · Zbl 1375.05041
[162] Eiran, D.; Falcón, R.M.; Kotlar, D.; Marbach, T.G.; Stones, R.J.; Two-line graphs of partial Latin rectangles; Electron. Notes Discret. Math.: 2018; Volume 68 ,53-58. · Zbl 1397.05027
[163] Falcón, R.M.; Stones, R.J.; Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups; Discret. Math.: 2017; Volume 340 ,1242-1260. · Zbl 1422.05027
[164] Falcón, R.M.; Cycle structures of autotopisms of the Latin squares of order up to 11; Ars Comb.: 2012; Volume 103 ,239-256. · Zbl 1265.05077
[165] Stones, D.S.; Vojtěchovský, P.; Wanless, I.M.; Cycle structure of autotopisms of quasigroups and Latin squares; J. Comb. Des.: 2012; Volume 20 ,227-263. · Zbl 1248.05023
[166] Falcón, R.M.; Martín-Morales, J.; Gröbner bases and the number of Latin squares related to autotopisms of order ≤7; J. Symb. Comput.: 2007; Volume 42 ,1142-1154. · Zbl 1129.05007
[167] Evans, T.; Embedding incomplete Latin squares; Am. Math. Soc.: 1960; Volume 67 ,958-961. · Zbl 0100.25601
[168] Bryant, D.; Buchanan, M.; Embedding partial totally symmetric quasigroups; J. Comb. Theory A: 2007; Volume 114 ,1046-1088. · Zbl 1126.20047
[169] Lindner, C.C.; Cruse, A.B.; Small embeddings for partial semisymmetric and totally symmetric quasigroups; J. Lond. Math. Soc.: 1976; Volume 12 ,479-484. · Zbl 0338.20099
[170] Raines, M.E.; Rodger, C.A.; Embedding partial extended triple systems and totally symmetric quasigroups; Discret. Math.: 1997; Volume 176 ,211-222. · Zbl 0901.05011
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