# 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

## Access Full Article

top## Abstract

top## How to cite

topEwa 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

top- [1] G. Birkhoff, On the structure of abstract algebras, J. Proc. Cambridge Phil. Soc. 31 (1935), 433-454. Zbl0013.00105
- [2] P.M. Cohn, Universal Algebra, Reidel, 1981 Dordreht.
- [3] K. Denecke and J. Koppitz, M-solid varieties of algebras, Advances in Mathematics, Vol. 10, Springer 2006. Zbl1094.08001
- [4] K. Denecke and S.L.Wismath, Hyperidentities and Clones, Gordon & Breach, 2000, ISBN 90-5699-235-X. ISSN 1041-5394.
- [5] T. Evans, The lattice of semigroups varieties, Semigroup Forum 2 (1971), 1-43. Zbl0225.20043
- [6] Ch.F. Fennemore, All varieties of bands, Ph.D. dissertation, Pensylvania State University 1969.
- [7] Ch.F. Fennemore, All varieties of bands I, Mathematische Nachrichten 48 (1971), 237-252. Zbl0194.02703
- [8] J.A. Gerhard, The lattice of equational classes of idempotent semigroups, J. of Algebra 15 (1970), 195-224. Zbl0194.02701
- [9] E. Graczyńska, Universal algebra via tree operads, Opole 2000, ISSN 1429-6063, ISBN 83-88492-75-6. Zbl1234.08001
- [10] E. Graczyńska and D. Schweigert, Hyperidentities of a given type, Algebra Universalis 27 (1990), 305-318. Zbl0715.08002
- [11] E. Graczyńska and D. Schweigert, Derived and fluid varieties, in print. Zbl1174.08308
- [12] G. Grätzer, Universal Algebra. 2nd ed., Springer, New York 1979.
- [13] R. McKenzie, G.F. McNulty and W. Taylor, Algebras, Lattices, Varieties, vol. I, 1987, ISBN 0-534-07651-3. Zbl0611.08001
- [14] J. Płonka, On equational classes of abstract algebras defined by regular equations, Fund. Math. 64 (1969), 241-247. Zbl0187.28702
- [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] 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] D. Schweigert, On derived varieties, Discussiones Mathematicae Algebra and Stochastic Methods 18 (1998), 17-26. Zbl0916.08006

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.