Bol loops with a large left nucleus

Orin Chein; Edgar G. Goodaire

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 2, page 171-196
  • ISSN: 0010-2628

Abstract

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Possession of a unique nonidentity commutator/associator is a property which dominates the theory of loops whose loop rings, while not associative, nevertheless satisfy an ``interesting'' identity. Indeed, until now, with the exception of some ad hoc examples, the only known class of Bol loops whose loop rings satisfy the right Bol identity have this property. In this paper, we identify another class of loops whose loop rings are ``strongly right alternative'' and present various constructions of these loops.

How to cite

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Chein, Orin, and Goodaire, Edgar G.. "Bol loops with a large left nucleus." Commentationes Mathematicae Universitatis Carolinae 49.2 (2008): 171-196. <http://eudml.org/doc/250454>.

@article{Chein2008,
abstract = {Possession of a unique nonidentity commutator/associator is a property which dominates the theory of loops whose loop rings, while not associative, nevertheless satisfy an ``interesting'' identity. Indeed, until now, with the exception of some ad hoc examples, the only known class of Bol loops whose loop rings satisfy the right Bol identity have this property. In this paper, we identify another class of loops whose loop rings are ``strongly right alternative'' and present various constructions of these loops.},
author = {Chein, Orin, Goodaire, Edgar G.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Bol loop; left nucleus; centre; nonassociative Bol loops; left nuclei; centre; commutators; associators; strongly right alternative rings; loop rings; power associative loops},
language = {eng},
number = {2},
pages = {171-196},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Bol loops with a large left nucleus},
url = {http://eudml.org/doc/250454},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Chein, Orin
AU - Goodaire, Edgar G.
TI - Bol loops with a large left nucleus
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 2
SP - 171
EP - 196
AB - Possession of a unique nonidentity commutator/associator is a property which dominates the theory of loops whose loop rings, while not associative, nevertheless satisfy an ``interesting'' identity. Indeed, until now, with the exception of some ad hoc examples, the only known class of Bol loops whose loop rings satisfy the right Bol identity have this property. In this paper, we identify another class of loops whose loop rings are ``strongly right alternative'' and present various constructions of these loops.
LA - eng
KW - Bol loop; left nucleus; centre; nonassociative Bol loops; left nuclei; centre; commutators; associators; strongly right alternative rings; loop rings; power associative loops
UR - http://eudml.org/doc/250454
ER -

References

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  2. Chein O., Goodaire E.G., 10.1016/0021-8693(90)90088-6, J. Algebra 130 (1990), 2 385-387. (1990) MR1051309DOI10.1016/0021-8693(90)90088-6
  3. Goodaire E.G., Jespers E., Polcino Milies C., Alternative Loop Rings, North-Holland Math. Studies, vol. 184, Elsevier, Amsterdam, 1996. Zbl0878.17029MR1433590
  4. Goodaire E.G., Alternative loop rings, Publ. Math. Debrecen 30 (1983), 31-38. (1983) Zbl0537.17006MR0733069
  5. Goodaire E.G., Robinson D.A., Loops which are cyclic extensions of their nuclei, Compositio Math. 45 (1982), 341-356. (1982) Zbl0488.20057MR0656610
  6. Goodaire E.G., Robinson D.A., 10.1080/00927879408825150, Comm. Algebra 22 (1995), 14 5623-5634. (1995) MR1298738DOI10.1080/00927879408825150
  7. Kunen K., 10.1080/00927879808826147, Comm. Algebra 26 (1998), 557-564. (1998) Zbl0895.20053MR1604107DOI10.1080/00927879808826147
  8. Moorhouse G.E., http://everest.uwyo.edu/ ∼ moorhous/pub/bol.html, . 
  9. Paige L.J., 10.1090/S0002-9939-1955-0068529-9, Proc. Amer. Math. Soc. 6 (1955), 279-280. (1955) Zbl0064.02903MR0068529DOI10.1090/S0002-9939-1955-0068529-9
  10. Pflugfelder H.O., Quasigroups and Loops: Introduction, Heldermann Verlag, Berlin, 1990. Zbl0715.20043MR1125767
  11. Vojtěchovský P., A class of Bol loops with a subgroup of index two, Comment. Math. Univ. Carolin. 45 (2004), 371-381. (2004) Zbl1101.20048MR2075284

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