Symmetric linear operator identities in quasigroups

Reza Akhtar

Commentationes Mathematicae Universitatis Carolinae (2017)

  • Volume: 58, Issue: 4, page 401-417
  • ISSN: 0010-2628

Abstract

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Let G be a quasigroup. Associativity of the operation on G can be expressed by the symbolic identity R x L y = L y R x of left and right multiplication maps; likewise, commutativity can be expressed by the identity L x = R x . In this article, we investigate symmetric linear identities: these are identities in left and right multiplication symbols in which every indeterminate appears exactly once on each side, and whose sides are mirror images of each other. We determine precisely which identities imply associativity and which imply commutativity, providing counterexamples as appropriate. We apply our results to show that there are exactly eight varieties of quasigroups satisfying such identities, and determine all inclusion relations among them.

How to cite

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Akhtar, Reza. "Symmetric linear operator identities in quasigroups." Commentationes Mathematicae Universitatis Carolinae 58.4 (2017): 401-417. <http://eudml.org/doc/294378>.

@article{Akhtar2017,
abstract = {Let $G$ be a quasigroup. Associativity of the operation on $G$ can be expressed by the symbolic identity $R_x L_y = L_y R_x$ of left and right multiplication maps; likewise, commutativity can be expressed by the identity $L_x=R_x$. In this article, we investigate symmetric linear identities: these are identities in left and right multiplication symbols in which every indeterminate appears exactly once on each side, and whose sides are mirror images of each other. We determine precisely which identities imply associativity and which imply commutativity, providing counterexamples as appropriate. We apply our results to show that there are exactly eight varieties of quasigroups satisfying such identities, and determine all inclusion relations among them.},
author = {Akhtar, Reza},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasigroup; linear identity; associativity; commutativity},
language = {eng},
number = {4},
pages = {401-417},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Symmetric linear operator identities in quasigroups},
url = {http://eudml.org/doc/294378},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Akhtar, Reza
TI - Symmetric linear operator identities in quasigroups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 4
SP - 401
EP - 417
AB - Let $G$ be a quasigroup. Associativity of the operation on $G$ can be expressed by the symbolic identity $R_x L_y = L_y R_x$ of left and right multiplication maps; likewise, commutativity can be expressed by the identity $L_x=R_x$. In this article, we investigate symmetric linear identities: these are identities in left and right multiplication symbols in which every indeterminate appears exactly once on each side, and whose sides are mirror images of each other. We determine precisely which identities imply associativity and which imply commutativity, providing counterexamples as appropriate. We apply our results to show that there are exactly eight varieties of quasigroups satisfying such identities, and determine all inclusion relations among them.
LA - eng
KW - quasigroup; linear identity; associativity; commutativity
UR - http://eudml.org/doc/294378
ER -

References

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  8. Phillips J.D., Vojtěchovský P., 10.1007/s00012-005-1941-1, Algebra Universalis 54 (2005), no. 3, 259–271. Zbl1102.20054MR2219409DOI10.1007/s00012-005-1941-1
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  10. Pflugfelder H., Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics, 7, Heldermann, Berlin, 1990. Zbl0715.20043MR1125767

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