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ArticleOriginal scientific text

Title

Axiomatization of quasigroups

Authors 1

Affiliations

  1. Department of Mathematics, Iowa State University Ames, Iowa 50011-2064, U.S.A.

Abstract

Quasigroups were originally described combinatorially, in terms of existence and uniqueness conditions on the solutions to certain equations. Evans introduced a universal-algebraic characterization, as algebras with three binary operations satisfying four identities. Now, quasigroups are redefined as heterogeneous algebras, satisfying just two conditions respectively known as hypercommutativity and hypercancellativity.

Keywords

ENG
quasigroup, heterogeneous algebra, hyperidentity

Bibliography

  1. L. Euler, Recherches sur une nouvelle espèce de carrés magiques, Mém. de la Societé de Vlissingue 9 (1779), pp.85 ff.
  2. T. Evans, Homomorphisms of non-associative systems, J. London Math. Soc. 24 (1949), 254-260.
  3. I.M. James, Quasigroups and topology, Math. Zeitschr. 84 (1964), 329-342.
  4. H.O. Pflugfelder, Quasigroups and Loops: Introduction, Heldermann, Berlin, 1990.
  5. A. Sade, Quasigroupes obéissant à certaines lois, Rev. Fac. Sci. Univ. Istanbul, Ser. A 22 (1957), 151-184.
  6. J.D.H. Smith and A.B. Romanowska, Post-Modern Algebra, Wiley, New York, NY, 1999.
  7. W. Taylor, Hyperidentities and hypervarieties. Aequationes Math. 23 (1981), 30-49.
Discussiones Mathematicae - General Algebra and Applications
2007/27/1
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Main language of publication
English
Received
2006-02-27
Accepted
2006-03-31
Published
2006

DOI: 10.7151/dmgaa.1116

Exact and natural sciences