The dimension of a variety

Ewa Graczyńska; Dietmar Schweigert

Discussiones Mathematicae - General Algebra and Applications (2007)

  • Volume: 27, Issue: 1, page 35-47
  • ISSN: 1509-9415

Abstract

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Derived varieties were invented by P. Cohn in [4]. Derived varieties of a given type were invented by the authors in [10]. In the paper we deal with the derived variety V σ of a given variety, by a fixed hypersubstitution σ. We introduce the notion of the dimension of a variety as the cardinality κ of the set of all proper derived varieties of V included in V. We examine dimensions of some varieties in the lattice of all varieties of a given type τ. Dimensions of varieties of lattices and all subvarieties of regular bands are determined.

How to cite

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Ewa Graczyńska, and Dietmar Schweigert. "The dimension of a variety." Discussiones Mathematicae - General Algebra and Applications 27.1 (2007): 35-47. <http://eudml.org/doc/276936>.

@article{EwaGraczyńska2007,
abstract = {Derived varieties were invented by P. Cohn in [4]. Derived varieties of a given type were invented by the authors in [10]. In the paper we deal with the derived variety $V_\{σ\}$ of a given variety, by a fixed hypersubstitution σ. We introduce the notion of the dimension of a variety as the cardinality κ of the set of all proper derived varieties of V included in V. We examine dimensions of some varieties in the lattice of all varieties of a given type τ. Dimensions of varieties of lattices and all subvarieties of regular bands are determined.},
author = {Ewa Graczyńska, Dietmar Schweigert},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {derived algebras; derived varieties; the dimension of a variety; variety; hypersubstitution; derived algebra; derived variety; dimension; lattice; idempotent semigroup},
language = {eng},
number = {1},
pages = {35-47},
title = {The dimension of a variety},
url = {http://eudml.org/doc/276936},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Ewa Graczyńska
AU - Dietmar Schweigert
TI - The dimension of a variety
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2007
VL - 27
IS - 1
SP - 35
EP - 47
AB - Derived varieties were invented by P. Cohn in [4]. Derived varieties of a given type were invented by the authors in [10]. In the paper we deal with the derived variety $V_{σ}$ of a given variety, by a fixed hypersubstitution σ. We introduce the notion of the dimension of a variety as the cardinality κ of the set of all proper derived varieties of V included in V. We examine dimensions of some varieties in the lattice of all varieties of a given type τ. Dimensions of varieties of lattices and all subvarieties of regular bands are determined.
LA - eng
KW - derived algebras; derived varieties; the dimension of a variety; variety; hypersubstitution; derived algebra; derived variety; dimension; lattice; idempotent semigroup
UR - http://eudml.org/doc/276936
ER -

References

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  3. [3] K. Denecke and J. Koppitz, M-solid varieties of algebras, Advances in Mathematics, Vol. 10, Springer 2006. Zbl1094.08001
  4. [4] K. Denecke and S.L.Wismath, Hyperidentities and Clones, Gordon & Breach, 2000, ISBN 90-5699-235-X. ISSN 1041-5394. 
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  6. [6] Ch.F. Fennemore, All varieties of bands, Ph.D. dissertation, Pensylvania State University 1969. 
  7. [7] Ch.F. Fennemore, All varieties of bands I, Mathematische Nachrichten 48 (1971), 237-252. Zbl0194.02703
  8. [8] J.A. Gerhard, The lattice of equational classes of idempotent semigroups, J. of Algebra 15 (1970), 195-224. Zbl0194.02701
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  10. [10] E. Graczyńska and D. Schweigert, Hyperidentities of a given type, Algebra Universalis 27 (1990), 305-318. Zbl0715.08002
  11. [11] E. Graczyńska and D. Schweigert, Derived and fluid varieties, in print. Zbl1174.08308
  12. [12] G. Grätzer, Universal Algebra. 2nd ed., Springer, New York 1979. 
  13. [13] R. McKenzie, G.F. McNulty and W. Taylor, Algebras, Lattices, Varieties, vol. I, 1987, ISBN 0-534-07651-3. Zbl0611.08001
  14. [14] J. Płonka, On equational classes of abstract algebras defined by regular equations, Fund. Math. 64 (1969), 241-247. Zbl0187.28702
  15. [15] J. Płonka, Proper and inner hypersubstitutions of varieties, pp. 106-116 in: 'Proceedings of the International Conference Summer School on General Algebra and Ordered Sets', Olomouc 1994. Zbl0828.08003
  16. [16] D. Schweigert, Hyperidentities, pp. 405-506 in: Algebras and Orders, I.G. Rosenberg and G. Sabidussi, Kluwer Academic Publishers, 1993, ISBN 0-7923-2143-X. 
  17. [17] D. Schweigert, On derived varieties, Discussiones Mathematicae Algebra and Stochastic Methods 18 (1998), 17-26. Zbl0916.08006

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