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The dimension of a variety

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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.
Twórcy
  • Opole University of Technology, Institute of Mathematics, Luboszycka 3, 45-036 Opole, Poland
  • Technische Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany
Bibliografia
  • [1] G. Birkhoff, On the structure of abstract algebras, J. Proc. Cambridge Phil. Soc. 31 (1935), 433-454.
  • [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.
  • [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.
  • [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.
  • [8] J.A. Gerhard, The lattice of equational classes of idempotent semigroups, J. of Algebra 15 (1970), 195-224.
  • [9] E. Graczyńska, Universal algebra via tree operads, Opole 2000, ISSN 1429-6063, ISBN 83-88492-75-6.
  • [10] E. Graczyńska and D. Schweigert, Hyperidentities of a given type, Algebra Universalis 27 (1990), 305-318.
  • [11] E. Graczyńska and D. Schweigert, Derived and fluid varieties, in print.
  • [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.
  • [14] J. Płonka, On equational classes of abstract algebras defined by regular equations, Fund. Math. 64 (1969), 241-247.
  • [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.
  • [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.
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1117
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