# Axiomatization of quasigroups

Discussiones Mathematicae - General Algebra and Applications (2006)

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

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topJonathan 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

top- [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. Zbl0034.01304
- [3] I.M. James, Quasigroups and topology, Math. Zeitschr. 84 (1964), 329-342. Zbl0124.16104
- [4] H.O. Pflugfelder, Quasigroups and Loops: Introduction, Heldermann, Berlin, 1990. Zbl0715.20043
- [5] A. Sade, Quasigroupes obéissant à certaines lois, Rev. Fac. Sci. Univ. Istanbul, Ser. A 22 (1957), 151-184. Zbl0086.02101
- [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. Zbl0491.08009

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