On universal quasigroup identities

Elena Brožíková

Mathematica Bohemica (1992)

  • Volume: 117, Issue: 1, page 20-32
  • ISSN: 0862-7959

Abstract

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The paper deals with quasigroup identities under isotopies. The terminology is taken from [2], [3] and [4]. Stimulated by geometric illustrations, V. D. Belousov in [2] has presented two important identity properties and posed a question for which identities these properties are necessary and sufficient for the identity to be invariant under isotopies. Inspired by V. D. Belousov, G. Monoszová investigated in [6] one special kind of identities for which both Belousov’s properties give necessary and sufficient conditions for the identity to be invariant under isotopies. Our purpose is to amend Belousov’s properties to such ones which guarantee the identity invariance under isotopies for general identities. We also show a close connection between quasigroup identities invariant under isotopies and 3 -basic quasigroup identities.

How to cite

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Brožíková, Elena. "On universal quasigroup identities." Mathematica Bohemica 117.1 (1992): 20-32. <http://eudml.org/doc/29391>.

@article{Brožíková1992,
abstract = {The paper deals with quasigroup identities under isotopies. The terminology is taken from [2], [3] and [4]. Stimulated by geometric illustrations, V. D. Belousov in [2] has presented two important identity properties and posed a question for which identities these properties are necessary and sufficient for the identity to be invariant under isotopies. Inspired by V. D. Belousov, G. Monoszová investigated in [6] one special kind of identities for which both Belousov’s properties give necessary and sufficient conditions for the identity to be invariant under isotopies. Our purpose is to amend Belousov’s properties to such ones which guarantee the identity invariance under isotopies for general identities. We also show a close connection between quasigroup identities invariant under isotopies and $3$-basic quasigroup identities.},
author = {Brožíková, Elena},
journal = {Mathematica Bohemica},
keywords = {derived operations; isotopy; quasigroups; position condition; regular universal identity; identities; $3$-basic quasigroups; universal identity; $3$-basic quasigroup identity; derived operations; isotopy; quasigroups; position condition; regular universal identity; identities; 3-basic quasigroups},
language = {eng},
number = {1},
pages = {20-32},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On universal quasigroup identities},
url = {http://eudml.org/doc/29391},
volume = {117},
year = {1992},
}

TY - JOUR
AU - Brožíková, Elena
TI - On universal quasigroup identities
JO - Mathematica Bohemica
PY - 1992
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 117
IS - 1
SP - 20
EP - 32
AB - The paper deals with quasigroup identities under isotopies. The terminology is taken from [2], [3] and [4]. Stimulated by geometric illustrations, V. D. Belousov in [2] has presented two important identity properties and posed a question for which identities these properties are necessary and sufficient for the identity to be invariant under isotopies. Inspired by V. D. Belousov, G. Monoszová investigated in [6] one special kind of identities for which both Belousov’s properties give necessary and sufficient conditions for the identity to be invariant under isotopies. Our purpose is to amend Belousov’s properties to such ones which guarantee the identity invariance under isotopies for general identities. We also show a close connection between quasigroup identities invariant under isotopies and $3$-basic quasigroup identities.
LA - eng
KW - derived operations; isotopy; quasigroups; position condition; regular universal identity; identities; $3$-basic quasigroups; universal identity; $3$-basic quasigroup identity; derived operations; isotopy; quasigroups; position condition; regular universal identity; identities; 3-basic quasigroups
UR - http://eudml.org/doc/29391
ER -

References

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  1. V. D. Belousov, Foundations of the theory of quasigroups and loops, Moskva, 1967. (In Russian.) (1967) MR0218483
  2. V. D. Belousov, Algebraic nets and quasigroups, Kishinev, 1971. (In Russian.) (1971) 
  3. E. Brožíková, On 3-basic quasigroups and their congruences, Časopis pro pěst. matematiky 115, no. 1 (1990), 38-47. (1990) MR1044012
  4. T. Evans, 10.1112/jlms/s1-26.1.64, J. London Math. Soc. 26 (1951), 64-71. (1951) Zbl0042.03303MR0038958DOI10.1112/jlms/s1-26.1.64
  5. T. Evans, On multiplicative systems defìned by generators and relations, Proc. Cambridge Philos. Soc. 47 (1951), 637-649. (1951) Zbl0043.02001MR0043764
  6. G. Monoszová, Some quasigroup identities invariant under quasigroup isotopies, Matem. issledovanija, Kishinev no. 102 (1988), 80-91. (In Russian.) (1988) MR0947461

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