Semisymmetrization and Mendelsohn quasigroups

Jonathan D. H. Smith

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 4, page 553-566
  • ISSN: 0010-2628

Abstract

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The semisymmetrization of an arbitrary quasigroup builds a semisymmetric quasigroup structure on the cube of the underlying set of the quasigroup. It serves to reduce homotopies to homomorphisms. An alternative semisymmetrization on the square of the underlying set was recently introduced by A. Krapež and Z. Petrić. Their construction in fact yields a Mendelsohn quasigroup, which is idempotent as well as semisymmetric. We describe it as the Mendelsohnization of the original quasigroup. For quasigroups isotopic to an abelian group, the relation between the semisymmetrization and the Mendelsohnization is studied. It is shown that the semisymmetrization is the total space for an action of the Mendelsohnization on the abelian group. The Mendelsohnization of an abelian group isotope is then identified as the idempotent replica of its semisymmetrization, with fibers isomorphic to the abelian group.

How to cite

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Smith, Jonathan D. H.. "Semisymmetrization and Mendelsohn quasigroups." Commentationes Mathematicae Universitatis Carolinae 61.4 (2020): 553-566. <http://eudml.org/doc/297116>.

@article{Smith2020,
abstract = {The semisymmetrization of an arbitrary quasigroup builds a semisymmetric quasigroup structure on the cube of the underlying set of the quasigroup. It serves to reduce homotopies to homomorphisms. An alternative semisymmetrization on the square of the underlying set was recently introduced by A. Krapež and Z. Petrić. Their construction in fact yields a Mendelsohn quasigroup, which is idempotent as well as semisymmetric. We describe it as the Mendelsohnization of the original quasigroup. For quasigroups isotopic to an abelian group, the relation between the semisymmetrization and the Mendelsohnization is studied. It is shown that the semisymmetrization is the total space for an action of the Mendelsohnization on the abelian group. The Mendelsohnization of an abelian group isotope is then identified as the idempotent replica of its semisymmetrization, with fibers isomorphic to the abelian group.},
author = {Smith, Jonathan D. H.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semisymmetric; quasigroup; Mendelsohn triple system},
language = {eng},
number = {4},
pages = {553-566},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Semisymmetrization and Mendelsohn quasigroups},
url = {http://eudml.org/doc/297116},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Smith, Jonathan D. H.
TI - Semisymmetrization and Mendelsohn quasigroups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 4
SP - 553
EP - 566
AB - The semisymmetrization of an arbitrary quasigroup builds a semisymmetric quasigroup structure on the cube of the underlying set of the quasigroup. It serves to reduce homotopies to homomorphisms. An alternative semisymmetrization on the square of the underlying set was recently introduced by A. Krapež and Z. Petrić. Their construction in fact yields a Mendelsohn quasigroup, which is idempotent as well as semisymmetric. We describe it as the Mendelsohnization of the original quasigroup. For quasigroups isotopic to an abelian group, the relation between the semisymmetrization and the Mendelsohnization is studied. It is shown that the semisymmetrization is the total space for an action of the Mendelsohnization on the abelian group. The Mendelsohnization of an abelian group isotope is then identified as the idempotent replica of its semisymmetrization, with fibers isomorphic to the abelian group.
LA - eng
KW - semisymmetric; quasigroup; Mendelsohn triple system
UR - http://eudml.org/doc/297116
ER -

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