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2005 | 25 | 1 | 89-101
Tytuł artykułu

T-Varieties and Clones of T-terms

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Abstrakty
EN
The aim of this paper is to describe how varieties of algebras of type τ can be classified by using the form of the terms which build the (defining) identities of the variety. There are several possibilities to do so. In [3], [19], [15] normal identities were considered, i.e. identities which have the form x ≈ x or s ≈ t, where s and t contain at least one operation symbol. This was generalized in [14] to k-normal identities and in [4] to P-compatible identities. More generally, we select a subset T of $W_{τ}(X)$, the set of all terms of type τ, and consider identities from T×T. Since any variety can be described by one heterogenous algebra, its clone, we are also interested in the corresponding clone-like structure. Identities of the clone of a variety V correspond to M-hyperidentities for certain monoids M of hypersubstitutions. Therefore we will also investigate these monoids and the corresponding M-hyperidentities.
Twórcy
  • University of Potsdam, Institute of Mathematics, Am Neuen Palais, 14415 Potsdam, Germany
  • KhonKaen University, Department of Mathematics, KhonKaen, 40002 Thailand
Bibliografia
  • [1] G. Birkhoff and J. D. Lipson, Heterogeneous algebra, J. Combin. Theory 8 (1970), 115-133.
  • [2] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York 1981.
  • [3] I. Chajda, Normally presented varieties, Algebra Universalis 34 (1995), 327-335.
  • [4] I. Chajda, K. Denecke and S. L. Wismath, A characterization of P-compatible varieties, Preprint 2004.
  • [5] W. Chromik, Externally compatible identities of algebras, Demonstratio Math. 23 (1990), 345-355.
  • [6] K. Denecke and L. Freiberg, The algebra of full terms, preprint 2003.
  • [7] K. Denecke and K. Ha kowska, P-compatible hypersubstitutions and MP-solid varieties, Studia Logica 64 (2000), 355-363.
  • [8] K. Denecke, P. Jampachon, Clones of N-full terms, Algebra and Discrete Math. (2004), no. 4, 1-11.
  • [9] K. Denecke, P. Jampachon, N-Full varieties and clones of n-full terms, Southeast Asian Bull. Math. 25 (2005), 1-14.
  • [10] K. Denecke, P. Jampachon and S. L. Wismath, Clones of n-ary algebras, J. Appl. Algebra Discrete Struct. 1 (2003), 141-158.
  • [11] K. Denecke, M. Reichel, Monoids of hypersubstitutions and M-solid varieties, Contributions to General Algebra 9 (1995), 117-126.
  • [12] K. Denecke and S. L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C. 2002.
  • [13] K. Denecke and S. L. Wismath, Galois connections and complete sublattices, 'Galois Connections and Applications', Kluwer Academic Publ., Dordrecht 2004, 211-230.
  • [14] K. Denecke and S. L. Wismath, A characterization of k-normal varieties, Algebra Universalis 51 (2004), 395-409.
  • [15] E. Graczyńska, On Normal and regular identities and hyperidentities, ' Universal and Applied Algebra', World Scientific Publ. Co., Singapore 1989, 107-135.
  • [16] P.J. Higgins, Algebras with a scheme of operators, Math. Nachr. 27 (1963), 115-132.
  • [17] A. Hiller, P-Compatible Hypersubstitutionen und Hyperidentitäten, Diplomarbeit, Potsdam 1996.
  • [18] H.-J. Hoehnke and J. Schreckenberger, Partial Algebras and their Theories, Manuscript 2005.
  • [19] I.I. Melnik, Nilpotent shifts of varieties, (Russian), Mat. Zametki 14 (1973), 703-712, (English translation in: Math. Notes 14 (1973), 962-966).
  • [20] J. Płonka, On varieties of algebras defined by identities of special forms, Houston Math. J. 14 (1988), 253-263.
  • [21] J. Płonka, P-compatible identities and their applications to classical algebras, Math. Slovaca 40 (1990), 21-30.
  • [22] B. Schein and V. S. Trochimenko, Algebras of multiplace functions, Semigroup Forum 17 (1979), 1-64.
  • [23] W. Taylor, Hyperidentities and Hypervarieties, Aequat. Math. 23 (1981), 111-127.
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1093
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