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Title: Identities and the group of isostrophisms (English)
Author: Drápal, Aleš
Author: Shcherbacov, Victor
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 53
Issue: 3
Year: 2012
Pages: 347-374
Summary lang: English
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Category: math
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Summary: In this paper we reexamine the concept of isostrophy. We connect it to the notion of term equivalence, and describe the action of dihedral groups that are associated with loops by means of isostrophy. We also use it to prove and present in a new way some well known facts on $m$-inverse loops and middle Bol loops. (English)
Keyword: isostrophe
Keyword: isostrophism
Keyword: paratope
Keyword: paratopism
Keyword: middle Bol
MSC: 15A30
MSC: 20N05
idZBL: Zbl 1258.20056
idMR: MR3017835
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Date available: 2012-08-31T11:35:43Z
Last updated: 2014-10-06
Stable URL: http://hdl.handle.net/10338.dmlcz/142929
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Reference: [1] Artzy R.: On loops with a special property.Proc. Amer. Math. Soc. 6 (1955), 448–453. Zbl 0066.27101, MR 0069804, 10.1090/S0002-9939-1955-0069804-4
Reference: [2] Artzy R.: Crossed-inverse and related loops.Trans. Amer. Math. Soc. 91 (1959), 480–492. Zbl 0086.24702, MR 0106958, 10.1090/S0002-9947-1959-0106958-3
Reference: [3] Artzy R.: Relations between loop identities.Proc. Amer. Math. Soc. 11 (1960), 847–851. Zbl 0101.25802, MR 0117295, 10.1090/S0002-9939-1960-0117295-9
Reference: [4] Artzy R.: Net motions and loops.Arch. Math. (Basel) 14 (1963), 95–101. Zbl 0228.20068, MR 0148786, 10.1007/BF01234928
Reference: [5] Basarab A.S., Belioglo A.I.: Universalno-avtomorfno-inversnye $\mathcal G$-lupy.Mat. Issled. 51 (1979), 3–7. MR 0544326
Reference: [6] Belousov V.D.: Osnovy teorii kvazigrupp i lup.Nauka, Moskva, 1967. MR 0218483
Reference: [7] Belousov V.D.: Vzaimmobratnye kvazigruppy i lupy.Izvestiia Akademii nauk Moldavskoi SSR 1963, issue 11, 3–10.
Reference: [8] Csörgö P., Drápal A., Kinyon M.K.: Buchsteiner loops.Internat. J. Algebra Comput. 19 (2009), 1049–1088. MR 2603718, 10.1142/S0218196709005482
Reference: [9] Evans T.: On multiplicative systems defined by generators and relations, I. Normal form theorems.Proc. Cambridge Philos. Soc. 47 (1951), 637–649. Zbl 0043.02001, MR 0043764
Reference: [10] Evans T.: On multiplicative systems defined by generators and relations, II. Monogenic loops.Proc. Cambridge Philos. Soc. 49 (1953), 579–589. Zbl 0051.01102, MR 0057865
Reference: [11] Gvaramija A.A.: Ob odnom klasse lup.Moskov. Gos. Ped. Inst. Učen. Zap. No. 375 (1971), 25–34. MR 0323936
Reference: [12] Greer M., Kinyon M.: Pseudoautomorphisms of Bruck loops and their generalizations.Comment. Math. Univ. Carolin. 53 (2012), no. 3, 383–389.
Reference: [13] Johnson K.W., Sharma B.L.: A variety of loops.Ann. Soc. Sci. Bruxelles Sér. I 92 (1975), 25–41. Zbl 0381.20056, MR 0498926
Reference: [14] Karkliňš B.B., Karkliň V.B.: Inversnyje lupy.Mat. Issled. 39 (1976), 87–101.
Reference: [15] Karkliňš B.B., Karkliň V.B.: Universalno-avtomorfno-inversnyje lupy.Mat. Issled. 39 (1976), 82–86.
Reference: [16] Keedwell A.D.: Crossed-inverse quasigroups with long inverse cycles and applications to cryptography.Australas. J. Combin. 20 (1999), 241–250. Zbl 0935.20061, MR 1723878
Reference: [17] Keedwell A.D., Shcherbacov V.A.: On $m$-inverse loops and quasigroups with a long inverse cycle.Australas. J. Combin. 26 (2002), 99–119. Zbl 1020.20041, MR 1918146
Reference: [18] Keedwell A.D., Shcherbacov V.A.: Construction and properties of $(r,s,t)$-inverse quasigroups. I.Discrete Math. 266 (2003), 275–291. Zbl 1070.20077, MR 1991723, 10.1016/S0012-365X(02)00814-2
Reference: [19] Keedwell A.D., Shcherbacov V.A.: Construction and properties of $(r,s,t)$-inverse quasigroups. II.Discrete Math. 288 (2004), 61–71. Zbl 1071.20058, MR 2101117, 10.1016/j.disc.2004.06.020
Reference: [20] Onoĭ V.I.: Solution of a problem on inverse loops (Russian).General algebra and discrete geometry 161 (1980), 53–58. MR 0647646
Reference: [21] Osborn J.M.: Loops with weak inverse properties.Pacific J. Math. 10 (1960), 295–304. MR 0111800, 10.2140/pjm.1960.10.295
Reference: [22] Pflugfelder H.O.: Quasigroups and Loops: Introduction.Helderman, Berlin, 1990. Zbl 0715.20043, MR 1125767
Reference: [23] Robinson D.A.: Bol loops.Trans. Amer. Math. Soc. 123 (1966), 341–354. Zbl 0803.20053, MR 0194545, 10.1090/S0002-9947-1966-0194545-4
Reference: [24] Sade A.: Paratopie et autoparatopie des quasigroupes.Ann. Soc. Sci. Bruxelles Sér. I 76 (1962), 88–96. Zbl 0105.01502, MR 0152600
Reference: [25] Syrbu P.: Loops with universal elasticity.Quasigroups Related Systems 1 (1994), 57–65. Zbl 0948.20047, MR 1327946
Reference: [26] Syrbu P.: On middle Bol loops.ROMAI J. 6 (2010), 229–236. MR 2786138
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