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The Galois correspondence between subvariety lattices and monoids of hpersubstitutions

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Abstrakty
EN
Denecke and Reichel have described a method of studying the lattice of all varieties of a given type by using monoids of hypersubstitutions. In this paper we develop a Galois correspondence between monoids of hypersubstitutions of a given type and lattices of subvarieties of a given variety of that type. We then apply the results obtained to the lattice of varieties of bands (idempotent semigroups), and study the complete sublattices of this lattice obtained through the Galois correspondence.
Twórcy
  • Universität Potsdam, Institut für Mathematik, Am Neuen Palais, D-14415 Potsdam, Germany
  • Dept. of Mathematics, University of Northern British Columbia, Prince George, B.C., Canada
  • Dept. of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, Canada T1K 3M4
Bibliografia
  • [1] J. Aczel, Proof of a theorem of distributive type hyperidentities, Algebra Universalis 1 (1971), 1-6.
  • [2] V.D. Belousov, Systems of quasigroups with generalized identities, (Russian) Uspekhi Mat. Nauk. 20 (1965) 75-146 (English translation: Russian Math. Surveys 20 (1965) 75-143).
  • [3] P.A. Birjukov, Varieties of idempotent semigroups (Russian), Algebra i Logika 9 (1970), 255-273.
  • [4] K. Denecke, Pre-solid varieties, Demonstratio Math. 27 (1994), 741-750.
  • [5] K. Denecke and J. Koppitz, Hyperassociative varieties of semigroups, Semigroup Forum 49 (1994), 41-48.
  • [6] K. Denecke and J. Koppitz, Presolid varieties of semigroups, Arch. Math. (Brno) 31 (1995), 171-181.
  • [7] K. Denecke and J. Koppitz, M-solid varieties of semigroups, Discuss. Math.- Algebra and Stochastic Methods 15 (1995), 23-41.
  • [8] K. Denecke and J. Koppitz, Finite monoids of hypersubstitutions of type τ = (2), Semigroup Forum 56 (1998), 265-275.
  • [9] K. Denecke, D. Lau, R. Pöschel and D. Schweigert, Hyperidentities, hyperequational classes and clone congruences, Contributions to General Algebra 7 (1991), 97-118.
  • [10] K. Denecke and M. Reichel, Monoids of hypersubstitutions and M-solid varieties, Contributions to General Algebra 9 (1995), 117-126.
  • [11] K. Denecke and S.L. Wismath, Solid varieties of semigroups, Semigroup Forum 48 (1994), 219-234.
  • [12] K. Denecke and S.L. Wismath, Hyperidentities and Clones, Gordon & Breach Sci. Publ, London 2000.
  • [13] C. Fennemore, All varieties of bands. I and II, Math. Nachr. 48 (1971), 237-252 and 253-262.
  • [14] J.A. Gerhard, The lattice of equational classes of idempotent semigroups, J. Algebra 15 (1970), 195-224.
  • [15] J.A. Gerhard and M. Petrich, Varieties of bands revisited, Proc. London Math. Soc. (3) 58 (1989), 323-350.
  • [16] E. Graczyńska and D. Schweigert, Hypervarieties of a given type, Algebra Universalis 27 (1990), 305-318.
  • [17] J. P onka, Proper and inner hypersubstitutions of varieties, General Algebra and Ordered Sets, Palacký Univ., Olomouc 1994, 106-115.
  • [18] L. Polák, On hyperassociativity, Algebra Universalis 36 (1996), 363-378.
  • [19] D. Schweigert, Hyperidentities, Algebras and Orders, Kluwer Acad. Publ., Dordrecht 1993, 405-505.
  • [20] W. Taylor, Hyperidentities and hypervarieties, Aequationes Math. 23 (1981), 111-127.
  • [21] S.L. Wismath, Hyperidentities for some varieties of semigroups, Algebra Universalis 27 (1990), 111-127.
  • [22] S.L. Wismath, Hyperidentities for some varieties of commutative semigroups, Algebra Universalis 28 (1991), 245-273.
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1002
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