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Quasigroup associativity and biased expansion graphs
Author:
Thomas Zaslavsky
Journal:
Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 13-18
MSC (2000):
Primary 05C22, 20N05; Secondary 05B15, 05B35
Posted:
February 10, 2006
MathSciNet review:
2200950
Full-text PDF Free Access
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Abstract: We present new criteria for a multary (or polyadic) quasigroup to be isotopic to an iterated group operation. The criteria are consequences of a structural analysis of biased expansion graphs. We mention applications to transversal designs and generalized Dowling geometries.
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9 (1959), 51–56 (Hungarian). MR 0105457
(21 #4198)
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no. 1, 17–88. MR 1328292
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Thomas Zaslavsky, Associativity in multary quasigroups: The way of biased expansions, submitted.
- 1.
- J. Aczél, V. D. Belousov, and M. Hosszú, Generalized associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hungar. 11 (1960), 127-136. MR 0140600 (25 #4018)
- 2.
- Maks A. Akivis and Vladislav V. Goldberg, Solution of Belousov's problem, Discuss. Math. Gen. Algebra Appl. 21 (2001), no. 1, 93-103. MR 1868620 (2002h:20098)
- 3.
- V. D. Belousov, Associative systems of quasigroups, Uspekhi Mat. Nauk 13 (1958), 243. (Russian)
- 4.
- V. D. Belousov and M. D. Sandik,
 -ary quasi-groups and loops, Sibirsk. Mat. Z. 7 (1966), 31-54; English transl., Siberian Math. J. 7 (1966), 24-42. MR 0204564 (34 #4403)
- 5.
- T. A. Dowling, A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B 14 (1973), 61-86. Erratum, J. Combin. Theory Ser. B 15 (1973), 211. MR 0307951 (46 #7066); MR 0319828 (47 #8369)
- 6.
- Wieslaw A. Dudek, Personal communication, 2 October 2003.
- 7.
- B. R. Frenkin, Reducibility and uniform reducibility in certain classes of
 -groupoids. II, Mat. Issled. 7 (1972), no. 1(23), 150-162. (Russian) MR 0294554 (45 #3624)
- 8.
- Miklós Hosszú, A theorem of Belousov and some of its applications, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 9 (1959), 51-56. (Hungarian) MR 0105457 (21 #4198)
- 9.
- Jeff Kahn and Joseph P. S. Kung, Varieties of combinatorial geometries, Trans. Amer. Math. Soc. 271 (1982), 485-499. MR 0654846 (84j:05043)
- 10.
- W. T. Tutte, Graph Theory, Encyc. Math. Appl., Vol. 21, Addison-Wesley, Reading, Mass., 1984. MR 0746795 (87c:05001)
- 11.
- Thomas Zaslavsky, Biased graphs. I. Bias, balance, and gains, II. The three matroids, III. Chromatic and dichromatic invariants, V. Group and biased expansions, J. Combin. Theory Ser. B 47 (1989), 32-52; 51 (1991), 46-72; 64 (1995), 17-88; in preparation. MR 1007712 (90k:05138); MR 1088626 (91m:05056); MR 1328292 (96g:05139)
- 12.
- Thomas Zaslavsky, Associativity in multary quasigroups: The way of biased expansions, submitted.
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Additional Information
Thomas Zaslavsky
Affiliation:
Binghamton University, Binghamton, New York 13902-6000
Email:
zaslav@math.binghamton.edu
DOI:
http://dx.doi.org/10.1090/S1079-6762-06-00155-7
PII:
S 1079-6762(06)00155-7
Keywords:
Multary quasigroup,
polyadic quasigroup,
factorization graph,
generalized associativity,
biased expansion graph,
transversal design,
Dowling geometry
Received by editor(s):
September 15, 2004
Posted:
February 10, 2006
Additional Notes:
Research partially assisted by grant DMS-0070729 from the National Science Foundation.
Communicated by:
Efim Zelmanov
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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