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

Published electronically:
February 10, 2006

MathSciNet review:
2200950

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**J. Aczél, V. D. Belousov, and M. Hosszú,*Generalized associativity and bisymmetry on quasigroups*, Acta Math. Acad. Sci. Hungar.**11**(1960), 127–136 (English, with Russian summary). MR**0140600****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**, 10.7151/dmgaa.1030**3.**V. D. Belousov,*Associative systems of quasigroups*, Uspekhi Mat. Nauk**13**(1958), 243. (Russian)**4.**V. D. Belousov and M. D. Sandik,*𝑁-ary quasi-groups and loops*, Sibirsk. Mat. Ž.**7**(1966), 31–54 (Russian). MR**0204564****5.**T. A. Dowling,*A class of geometric lattices based on finite groups*, J. Combinatorial Theory Ser. B**14**(1973), 61–86. MR**0307951**

T. A. Dowling,*Erratum: “A class of geometric lattices based on finite groups” (J. Combinatorial Theory Ser. B 14 (1973), 61–86)*, J. Combinatorial Theory Ser. B**15**(1973), 211. MR**0319828****6.**Wieslaw A. Dudek, Personal communication, 2 October 2003.**7.**B. R. Frenkin,*Reducibility and uniform reducibility in certain classes of 𝑛-groupoids. II*, Mat. Issled.**7**(1972), no. 1(23), 150–162 (Russian). MR**0294554****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****9.**J. Kahn and J. P. S. Kung,*Varieties of combinatorial geometries*, Trans. Amer. Math. Soc.**271**(1982), no. 2, 485–499. MR**654846**, 10.1090/S0002-9947-1982-0654846-9**10.**W. T. Tutte,*Graph theory*, Encyclopedia of Mathematics and its Applications, vol. 21, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. With a foreword by C. St. J. A. Nash-Williams. MR**746795****11.**Thomas Zaslavsky,*Biased graphs. I. Bias, balance, and gains*, J. Combin. Theory Ser. B**47**(1989), no. 1, 32–52. MR**1007712**, 10.1016/0095-8956(89)90063-4

Thomas Zaslavsky,*Biased graphs. II. The three matroids*, J. Combin. Theory Ser. B**51**(1991), no. 1, 46–72. MR**1088626**, 10.1016/0095-8956(91)90005-5

Thomas Zaslavsky,*Biased graphs. III. Chromatic and dichromatic invariants*, J. Combin. Theory Ser. B**64**(1995), no. 1, 17–88. MR**1328292**, 10.1006/jctb.1995.1025**12.**Thomas Zaslavsky,*Associativity in multary quasigroups: The way of biased expansions*, submitted.

Retrieve articles in *Electronic Research Announcements of the American Mathematical Society*
with MSC (2000):
05C22,
20N05,
05B15,
05B35

Retrieve articles in all journals with MSC (2000): 05C22, 20N05, 05B15, 05B35

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

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.