The dimension of a variety

• Autorzy: Ewa Graczyńska , Dietmar Schweigert
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

derived algebras; derived varieties; the dimension of a variety

Kategorie tematyczne

• 08A40: Operations, polynomials, primal algebras
• 08B05: Equational logic, Mal\cprime cev (Mal\cprime tsev) conditions
• 08B15: Lattices of varieties
• 08B99: None of the above, but in this section

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2007
• Tom: 27
• Numer: 1
• Strony: 35-47

Daty

wydano

2007

otrzymano

2006-02-23

poprawiono

2006-06-10

Twórcy

autor

Ewa Graczyńska

• Opole University of Technology, Institute of Mathematics, Luboszycka 3, 45-036 Opole, Poland

autor

Dietmar Schweigert

• 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.

Identyfikatory

DOI

10.7151/dmgaa.1117