A scoop from groups: equational foundations for loops

Phillips, J. D.; Petr Vojtěchovský

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 2, page 279-290
  • ISSN: 0010-2628

Abstract

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Groups are usually axiomatized as algebras with an associative binary operation, a two-sided neutral element, and with two-sided inverses. We show in this note that the same simplicity of axioms can be achieved for some of the most important varieties of loops. In particular, we investigate loops of Bol-Moufang type in the underlying variety of magmas with two-sided inverses, and obtain ``group-like'' equational bases for Moufang, Bol and C-loops. We also discuss the case when the inverses are only one-sided and/or the neutral element is only one-sided.

How to cite

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Phillips, J. D., and Vojtěchovský, Petr. "A scoop from groups: equational foundations for loops." Commentationes Mathematicae Universitatis Carolinae 49.2 (2008): 279-290. <http://eudml.org/doc/250498>.

@article{Phillips2008,
abstract = {Groups are usually axiomatized as algebras with an associative binary operation, a two-sided neutral element, and with two-sided inverses. We show in this note that the same simplicity of axioms can be achieved for some of the most important varieties of loops. In particular, we investigate loops of Bol-Moufang type in the underlying variety of magmas with two-sided inverses, and obtain ``group-like'' equational bases for Moufang, Bol and C-loops. We also discuss the case when the inverses are only one-sided and/or the neutral element is only one-sided.},
author = {Phillips, J. D., Vojtěchovský, Petr},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {inverse property loop; Bol loop; Moufang loop; C-loop; equational basis; magma with inverses; axioms; varieties of loops; inverse property loops; Bol loops; Moufang loops; C-loops; equational bases; magmas with inverses},
language = {eng},
number = {2},
pages = {279-290},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A scoop from groups: equational foundations for loops},
url = {http://eudml.org/doc/250498},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Phillips, J. D.
AU - Vojtěchovský, Petr
TI - A scoop from groups: equational foundations for loops
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 2
SP - 279
EP - 290
AB - Groups are usually axiomatized as algebras with an associative binary operation, a two-sided neutral element, and with two-sided inverses. We show in this note that the same simplicity of axioms can be achieved for some of the most important varieties of loops. In particular, we investigate loops of Bol-Moufang type in the underlying variety of magmas with two-sided inverses, and obtain ``group-like'' equational bases for Moufang, Bol and C-loops. We also discuss the case when the inverses are only one-sided and/or the neutral element is only one-sided.
LA - eng
KW - inverse property loop; Bol loop; Moufang loop; C-loop; equational basis; magma with inverses; axioms; varieties of loops; inverse property loops; Bol loops; Moufang loops; C-loops; equational bases; magmas with inverses
UR - http://eudml.org/doc/250498
ER -

References

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