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Inverse Semigroups and their Lattices of Inverse Subsemigroups

  • Peter R. Jones

Abstract

For any class C of algebras it is natural to wonder how well algebras from C are determined by their lattices of subalgebras. This topic has a long history, beginning with subgroup lattices of groups (see §3). The subspace lattice of a vector space was shown to be intimately connected with projective geometry by R. Baer [1]. An interesting historical perspective may be found in the introduction to [33]. In my talk I will consider this topic from the point of view of inverse semigroups.

Keywords

Maximal Subgroup Inverse Semigroup Regular Semigroup Subgroup Lattice Lattice Isomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    R. Baer, Linear Algebra and Projective Geometry, Academic Press, New York, 1952.zbMATHGoogle Scholar
  2. 2.
    A. A. Borisov, On lattice isomorphisms of semigroups of partial transformations, Leningrad Pedogog. Inst., Sovreman Algebra, 6 (1977), 33–52, (in Russian).Google Scholar
  3. 3.
    A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. II, Amer. Math. Soc, Providence, RI, 1967.Google Scholar
  4. 4.
    T.I. Ershova, Inverse semigroups with certain types of inverse subsemigroups, Mat Zap. Ural. Univ., Sverdlovsk, 7 (1969), 67–76, (in Russian).Google Scholar
  5. 5.
    —, Determinability of monogenic inverse semigroups by the lattice of inverse subsemigroups, Mat. Zap. Ural. Univ., Sverdlovsk, 8 (1971), 34–49, (in Russian).zbMATHGoogle Scholar
  6. 6.
    —, On the lattice of inverse subsemigroups of monogenic inverse semigroups, Reports of XI All-Union Algebraic College, Kishinev, (1971), 195, (in Russian).Google Scholar
  7. 7.
    —, Lattice isomorphisms of inverse semigroups, Izv. Vysh. Uch. Zav. Matematika, 9 (1972), 25–32, (in Russian).Google Scholar
  8. 8.
    —, Projeciivities of inverse semigroups, Reports of All-Union Algebraic Symp., Homel, (1975), 212, (in Russian).Google Scholar
  9. 9.
    —, Inverse semigroups with certain finiteness conditions, Izv. Vysh. Uch. Zav. Matematika, 11, (1977), 7–14, (in Russian).Google Scholar
  10. 10.
    —, Projectivities of inverse semigroups, Reports of All-Union Algebraic Conf., Leningrad, (1981), part 2, page 54, (in Russian).Google Scholar
  11. 11.
    —, Projectivities of Brandt semigroups, Ural. Gos. Univ. Mat. Zap., 13 (1982), 27–39, (in Russian).MathSciNetGoogle Scholar
  12. 12.
    S. M. Goberstein, Inverse semigroups with isomorphic partial isomorphism semigroups, J. Austral. Math. Soc. (to appear).Google Scholar
  13. 13.
    J. M. Howie, An Introduction to Semigroup Theory, Academic Press, London, 1975.Google Scholar
  14. 14.
    K. G. Johnston, Lattice isomorphisms of modular inverse semigroups, Proc. Edinburgh Math. Soc. 31 (1988), 441–446.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    K. G. Johnston and P. R. Jones, The lattice of full regular subsemigroups of a regular semigroup, Proc. Roy. Soc. Edinburgh 98A (1984), 203–214.MathSciNetCrossRefGoogle Scholar
  16. 16.
    P. R. Jones —, Modular inverse semigroups, J. Austral. Math. Soc. 43 (1987), 47–63.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    P. R. Jones —, Inverse semigroups whose lattices of full inverse subsemigroups are modular, in Semigroups and Their Applications, S. Goberstein and P. Higgins, editor, Reidel, Boston, 1987.Google Scholar
  18. 18.
    P. R. Jones, The lattice of inverse subsemigroups of a reduced inverse semigroup, Glasgow Math. J. 17 (1976), 161–172.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    —, Lattice isomorphisms of inverse semigroups, Proc. Edinburgh Math. Soc. 17 (1978), 149–157.CrossRefGoogle Scholar
  20. 20.
    —, Semimodular inverse semigroups, J. London Math. Soc. 17 (1978), 446–456.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    —, Distributive inverse semigroups, J. London Math. Soc. 17 (1978), 457–466.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    —, Lattice isomorphisms of distributive inverse semigroups, Quart. J. Math. Oxford 30 (1979), 301–314.CrossRefzbMATHGoogle Scholar
  23. 23.
    —, Inverse semigroups determined by their lattices of inverse subsemigroups, J. Austral. Math. Soc. 30 (1981), 321–346.CrossRefzbMATHGoogle Scholar
  24. 24.
    —, Inverse semigroups whose inverse subsemigroups form a chain, Glasgow Math. J. 18 (1981), 159–165.CrossRefGoogle Scholar
  25. 25.
    —, Lattice isomorphisms of free products of inverse semigroups, J. Algebra 89 (1984), 280–290.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    —, Free products of inverse semigroups, Trans. Amer. Math. Soc. 282 (1984), 293–318.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    S. Li, On the subgroup lattice characterization of finite simple groups of Lie type, Chin. Ann. Math. 4B (1983), 165–169.Google Scholar
  28. 28.
    M. Petrich, Inverse Semigroups, Wiley, N.Y., 1984.zbMATHGoogle Scholar
  29. 29.
    G. B. Preston, Inverse semigroups: some open questions, in Proceedings, Symp. on Inverse Semigroups and Their Generalizations, Northern Illinois University, 1973.Google Scholar
  30. 30.
    L. N. Shevrin, Basic problems in the theory of projectivities of semilattices, Mat. Sbornik, 66 (1965), 568–597, (in Russian).Google Scholar
  31. 31.
    —, Lattice properties of idempotent semigroups II, Sib Mat. J. 7 (1966), 437–454.Google Scholar
  32. 32.
    —, Basic problems in the theory of projectivities of semilattices II, Mat. Zap. Ural. Univ., Sverdlovsk, 5 (1966), 107–122, (in Russian).zbMATHGoogle Scholar
  33. 33.
    L. N. Shevrin and A. J. Ovsyannikov, Semigroups and their subsemigroup lattices, Semigroup Forum 27 (1983), 1–154.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    M. Suzuki, Structure of a Group and the Structure of its Lattice of Subgroups, Springer, Berlin, 1956.CrossRefzbMATHGoogle Scholar
  35. 35.
    H. Volkein, On the lattice isomorphisms of the finite Chevalley groups, Indagationes Math. 48 (1986), 213–228.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Peter R. Jones
    • 1
  1. 1.Department of Mathematics, Statistics and Computer ScienceMarquette UniversityMilwaukeeUSA

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