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
DOI:
https://doi.org/10.1090/S1079-6762-06-00155-7
Published electronically:
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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AMQ Thomas Zaslavsky, Associativity in multary quasigroups: The way of biased expansions, submitted.
ABH J. Aczél, V. D. Belousov, and M. Hosszú, Generalized associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hungar. 11 (1960), 127–136. (25 #4018)
AG Maks A. Akivis and Vladislav V. Goldberg, Solution of Belousov’s problem, Discuss. Math. Gen. Algebra Appl. 21 (2001), no. 1, 93–103. (2002h:20098)
ASQ V. D. Belousov, Associative systems of quasigroups, Uspekhi Mat. Nauk 13 (1958), 243. (Russian)
BelSand V. D. Belousov and M. D. Sandik, $n$-ary quasi-groups and loops, Sibirsk. Mat. Ž. 7 (1966), 31–54; English transl., Siberian Math. J. 7 (1966), 24–42. (34 #4403)
CGL 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. (46 #7066); (47 #8369)
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Frenkin B. R. Frenkin, Reducibility and uniform reducibility in certain classes of $n$-groupoids. II, Mat. Issled. 7 (1972), no. 1(23), 150–162. (Russian) (45 #3624)
HosszuA Miklós Hosszú, A theorem of Belousov and some of its applications, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 9 (1959), 51–56. (Hungarian) (21 #4198)
VCG Jeff Kahn and Joseph P. S. Kung, Varieties of combinatorial geometries, Trans. Amer. Math. Soc. 271 (1982), 485–499. (84j:05043)
Tbook W. T. Tutte, Graph Theory, Encyc. Math. Appl., Vol. 21, Addison-Wesley, Reading, Mass., 1984. (87c:05001)
BG 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. (90k:05138); (91m:05056); (96g:05139)
AMQ 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
Keywords:
Multary quasigroup,
polyadic quasigroup,
factorization graph,
generalized associativity,
biased expansion graph,
transversal design,
Dowling geometry
Received by editor(s):
September 15, 2004
Published electronically:
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.
