Commutators and associators in Catalan loops

Jan M. Raasch

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 2, page 333-347
  • ISSN: 0010-2628

Abstract

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Various commutators and associators may be defined in one-sided loops. In this paper, we approximate and compare these objects in the left and right loop reducts of a Catalan loop. To within a certain order of approximation, they turn out to be quite symmetrical. Using the general analysis of commutators and associators, we investigate the structure of a specific Catalan loop which is non-commutative, but associative, that appears in the original number-theoretic application of Catalan loops.

How to cite

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Raasch, Jan M.. "Commutators and associators in Catalan loops." Commentationes Mathematicae Universitatis Carolinae 51.2 (2010): 333-347. <http://eudml.org/doc/37764>.

@article{Raasch2010,
abstract = {Various commutators and associators may be defined in one-sided loops. In this paper, we approximate and compare these objects in the left and right loop reducts of a Catalan loop. To within a certain order of approximation, they turn out to be quite symmetrical. Using the general analysis of commutators and associators, we investigate the structure of a specific Catalan loop which is non-commutative, but associative, that appears in the original number-theoretic application of Catalan loops.},
author = {Raasch, Jan M.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {loop; right loop; left loop; commutator; associator; Catalan loops; commutators; associators; right loops; left loops},
language = {eng},
number = {2},
pages = {333-347},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Commutators and associators in Catalan loops},
url = {http://eudml.org/doc/37764},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Raasch, Jan M.
TI - Commutators and associators in Catalan loops
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 2
SP - 333
EP - 347
AB - Various commutators and associators may be defined in one-sided loops. In this paper, we approximate and compare these objects in the left and right loop reducts of a Catalan loop. To within a certain order of approximation, they turn out to be quite symmetrical. Using the general analysis of commutators and associators, we investigate the structure of a specific Catalan loop which is non-commutative, but associative, that appears in the original number-theoretic application of Catalan loops.
LA - eng
KW - loop; right loop; left loop; commutator; associator; Catalan loops; commutators; associators; right loops; left loops
UR - http://eudml.org/doc/37764
ER -

References

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  1. Balcerzyk S., Józefiak T., Commutative Noetherian and Krull Rings, PWN, Warsaw, 1989. MR1130610
  2. Bruck R.H., A Survey of Binary Systems, Springer, Berlin, 1958. Zbl0141.01401MR0093552
  3. Coxeter H.S.M., Moser W.O.J., Generators and Relations for Discrete Groups, Springer, Berlin, 1957. Zbl0487.20023MR0088489
  4. Long L., Smith J.D.H., Catalan loops, preprint, 2009. 
  5. Mac Lane S., Homology, Springer, Berlin, 1963. Zbl0818.18001

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