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Structural diversity in the lattice of equational theories

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Abstract

This paper is principally concerned with conditions under which various partition lattices are isomorphic to intervals in either the lattice of equational theories extending a given equational theory or the lattice of subtheories of a given equational theory.

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References

  1. K. Baker,Equational classes of modular lattices, Pacific J. Math.28 (1969), 9–15.

    MATH  MathSciNet  Google Scholar 

  2. K. Baker,Equational axioms for classes of lattices, Bull. A.M.S.71 (1971), 97–102.

    Google Scholar 

  3. D. Bean, A. Ehrenfeucht, andG. McNulty,Avoidable patterns in strings of symbols, to appear in the Pacific J Mathematics.

  4. A. Birjukov,Varities of idempotent semigroups, Algebra i Logika9 (1970) 255–273.

    MATH  MathSciNet  Google Scholar 

  5. G. Birkhoff,Lattice Theory 3rd edition, Amer. Math. Soc. Providence, 1967.

    MATH  Google Scholar 

  6. A. Bol'bot,Varities of Ω-algebras, Algebra i Logika13 (1970) 406–414.

    MathSciNet  Google Scholar 

  7. S. Burris,On the structure of the lattice of equational classes of L(τ), Algebra Universalis1 (1971) 39–45.

    MATH  MathSciNet  Google Scholar 

  8. S. Burris andE. Nelson, Embedding the dual ofΠ m in the lattice of equational classes of commutative semigroups, Proc. Amer. Math. Soc.30 (1971) 37–39.

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Burris andE. Nelson, Embedding the dual ofΠ w in the lattice of equational classes of semigroups, Algebra Universalis1 (1971) 248–254.

    MATH  MathSciNet  Google Scholar 

  10. S. Burris andH. Sankappanavar,Lattice-theoretic decision problems in universal algebra, Algebra Universalis5 (1975), 163–177.

    MATH  MathSciNet  Google Scholar 

  11. C. Chang andH. Keisler,Model Theory, North-Holland Publishing Company, Amsterdam, 1973.

    MATH  Google Scholar 

  12. T. Evans,The lattice of semigroup varieties, Semigroup Forum2 (1971) 1–43.

    MATH  MathSciNet  Google Scholar 

  13. C. Fennemore,All varieties of bands, Semigroup Forum1 (1970) 172–179.

    MATH  MathSciNet  Google Scholar 

  14. R. Freese,The structure of modular lattices of width four with applications to the varieties of lattices, Mem. Amer. Math. Soc. Vol. 9 no. 181 (1977) pp. 91.

    MathSciNet  Google Scholar 

  15. J. Gerhard,The lattice of equational classes of idempotent semigroups, J. Algebra15 (1970) 195–224.

    Article  MATH  MathSciNet  Google Scholar 

  16. G. Gratzer,Universal Algebra, second edition, Springer-Verlag, New York, 1979.

    Google Scholar 

  17. E. Jacobs andR. Schwabauer,The lattice of equational classes of algebras with one unary operation, Amer. Math. Monthly,71 (1964) 151–155.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Jezek,Primitive classes of algebras with unary and nullary operations, Colloq. Math.20 (1969) 159–179.

    MATH  MathSciNet  Google Scholar 

  19. J. Jezek,On atoms in lattices of primitive classes, Comment. Math. Univ. Carolina11 (1970) 515–532.

    MATH  MathSciNet  Google Scholar 

  20. J. Jezek,Intervals in the lattice of varieties, Algebra Universalis,6 (1976).

  21. B. Josson,Sums of finitely based lattice varieties, Advances in Math.14 (1976) 454–468.

    Article  Google Scholar 

  22. —,The variety covering the variety of all modular lattices, Math. Scand.41 (1977) 5–14.

    MATH  MathSciNet  Google Scholar 

  23. B. Jonsson, G. McNulty, andR. Quackenbush,Ascending and descending varietal chains of a variety, Canad. J. Math.27 (1975) 25–32.

    MATH  MathSciNet  Google Scholar 

  24. B. Jonsson andE. Nelson,Relatively free products in regular varieties, Algebra Universalis4 (1974) 14–19.

    MATH  MathSciNet  Google Scholar 

  25. J. Kalicki,The number of equationally complete classes of equations, Nederl. Akad. Wetensch. Proc. Ser. A. 58 (1955) 660–662.

    MATH  MathSciNet  Google Scholar 

  26. A. Mal'cev,On the general theory of algebraic systems (in Russian) Math. Sbornik (N.S.)35 (77) (1954) 3–20.

    MATH  MathSciNet  Google Scholar 

  27. R. McKenzie,Equational bases for lattice theories, Math. Scand.27 (1970) 24–38.

    MATH  MathSciNet  Google Scholar 

  28. —,Equational bases for lattice theories, Math. Scand.3 (1971) 197–237.

    MATH  MathSciNet  Google Scholar 

  29. G. McNulty,Structural diversity in the lattice of equational theories, Notices Amer. Math. Soc.20 (1973) A-33.

    Google Scholar 

  30. —,Structural diversity in the lattice of equational theories, II, Notices Amer. Math. Soc.20 (1973) A-448.

    Google Scholar 

  31. —,Structural diversity in the lattice of equational theories, III, Notices Amer. Math. Soc.23 (1976) A-401.

    Google Scholar 

  32. —,The decision problem for equational bases of algebras, Annals of Math. Logic11 (1976) 193–259.

    Article  MathSciNet  Google Scholar 

  33. —,Undecidable properties of finite sets of equations, J. Symbolic Logic11 (1976) 589–604.

    MathSciNet  Google Scholar 

  34. E. Nelson,The lattice of equational classes of semigroups with zero, Canad. Math. Bull.14 (1971) 531–535

    MATH  MathSciNet  Google Scholar 

  35. —,The lattice of equational classes of commutative semigroups, Canad. J. Math.23 (1971) 875–595.

    MATH  MathSciNet  Google Scholar 

  36. H. Neumann,Varieties of Groups, Springer-Verlag, Berlin, 1967.

    MATH  Google Scholar 

  37. P. Perkins,Bases for equational theories of semigroups, J. Algebra11 (1969) 298–314.

    Article  MATH  MathSciNet  Google Scholar 

  38. D. Pigozzi,Equational logic and equational theories of algebras, technical report, Purdue University 1970, reissued at Iowa State University 1975.

  39. A. Pixley,Local Mal'cev conditions, Canad. Math. Bull.15 (1972) 559–568.

    MATH  MathSciNet  Google Scholar 

  40. P. Pudlak andJ. Tuma,Every finite lattice can be embedded in the lattice of all equivalence relations over a finite set, Algebra Universalis, to appear.

  41. D. Sachs,Identities in finite partition lattices, Proc. Amer. Math. Soc.12 (1961) 944–945.

    Article  MATH  MathSciNet  Google Scholar 

  42. D. Scott,Equationally complete extensions of finite algebras, Nederl. Akad. Wetensch. (Ser. A)59 (1956) 35–38.

    MATH  Google Scholar 

  43. A. Tarski,Equational logic and equational theories of algebras, 275–288 in:H. A. Schmidt,K. Schütte, andH. J. Thiele, eds.,Contributions to Mathematical Logic, North-Holland, Amsterdam, 1968.

    Google Scholar 

  44. W. Taylor,Characterizing Mal'cev conditions, Algebra Universalis3 (1973) 351–397.

    Article  MATH  MathSciNet  Google Scholar 

  45. W. Taylor,Equational logic, Houston Journal of Math., (survey issue, 1979).

  46. A. Trahtman,Covering elements in the lattice of varieties of algebras, Mat. Zametki15 (1974) 307–312.

    MATH  MathSciNet  Google Scholar 

  47. P. Whitman,Lattices, equivalence relations, and subgroups, Bull. Amer. Math. Soc.52 (1946) 507–522.

    Article  MATH  MathSciNet  Google Scholar 

  48. R. Wille,Kongruenzklassengeometrien, Lecture Note in Math. no. 113, Springer-Verlag, Berlin, 1970.

    MATH  Google Scholar 

  49. G. McNulty,Covering in the lattice of equational theories and some properties of term finite theories, to appear.

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Authors and Affiliations

  1. University of South Carolina, Columbia, South Carolina, USA

    George F. McNulty

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  1. George F. McNulty
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This paper is for Elizabeth Eldridge.

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McNulty, G.F. Structural diversity in the lattice of equational theories. Algebra Universalis 13, 271–292 (1981). https://doi.org/10.1007/BF02483841

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