Axiomatization of quasigroups

Jonathan D.H. Smith

Discussiones Mathematicae - General Algebra and Applications (2006)

  • Volume: 27, Issue: 1, page 21-33
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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Jonathan D.H. Smith. "Axiomatization of quasigroups." Discussiones Mathematicae - General Algebra and Applications 27.1 (2006): 21-33. <http://eudml.org/doc/276951>.

@article{JonathanD2006,
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.},
author = {Jonathan D.H. Smith},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {quasigroup; heterogeneous algebra; hyperidentity; quasigroups; heterogeneous algebras; hyperidentities; hyperquasigroups; identities; hypercommutativity; hypercancellativity; axioms},
language = {eng},
number = {1},
pages = {21-33},
title = {Axiomatization of quasigroups},
url = {http://eudml.org/doc/276951},
volume = {27},
year = {2006},
}

TY - JOUR
AU - Jonathan D.H. Smith
TI - Axiomatization of quasigroups
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2006
VL - 27
IS - 1
SP - 21
EP - 33
AB - 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.
LA - eng
KW - quasigroup; heterogeneous algebra; hyperidentity; quasigroups; heterogeneous algebras; hyperidentities; hyperquasigroups; identities; hypercommutativity; hypercancellativity; axioms
UR - http://eudml.org/doc/276951
ER -

References

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  1. [1] L. Euler, Recherches sur une nouvelle espèce de carrés magiques, Mém. de la Societé de Vlissingue 9 (1779), pp.85 ff. 
  2. [2] T. Evans, Homomorphisms of non-associative systems, J. London Math. Soc. 24 (1949), 254-260. Zbl0034.01304
  3. [3] I.M. James, Quasigroups and topology, Math. Zeitschr. 84 (1964), 329-342. Zbl0124.16104
  4. [4] H.O. Pflugfelder, Quasigroups and Loops: Introduction, Heldermann, Berlin, 1990. Zbl0715.20043
  5. [5] A. Sade, Quasigroupes obéissant à certaines lois, Rev. Fac. Sci. Univ. Istanbul, Ser. A 22 (1957), 151-184. Zbl0086.02101
  6. [6] J.D.H. Smith and A.B. Romanowska, Post-Modern Algebra, Wiley, New York, NY, 1999. 
  7. [7] W. Taylor, Hyperidentities and hypervarieties. Aequationes Math. 23 (1981), 30-49. Zbl0491.08009

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