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Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups

  • Lev N. Shevrin
  • Alexander J. Ovsyannikov
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 379)

Abstract

In the preceding chapter we have seen that sometimes imposing natural restrictions on the lattice SubiS of all inverse subsemigroups of an inverse semigroup S turns out to be a rather strong requirement. One may say that, from the point of view of such restrictions, the lattice SubiS is “too rich”. However, there exists, so to say, a less rich lattice which is naturally associated with an inverse semigroup S and influences substantially the structure of S (and in the group case transforming into the subgroup lattice); this is the lattice of all full inverse subsemigroups. An inverse subsemigroup A of an inverse semigroup S is called full if \(A\, \supseteq \,{E_s}\). We denote the lattice of all full inverse subsemigroups of an inverse semigroup S by SubfiS. This chapter contains basic information about inverse semigroups S with restrictions on the lattice SubfiS. The lattice SubfiS is a complete sublattice in SubS and coincides with the interval [E, S] in the lattice SubiS. Obviously, the equality SubfiS = (SubiS)\{ø} holds if and only if S is a group.

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Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Lev N. Shevrin
    • 1
  • Alexander J. Ovsyannikov
    • 1
  1. 1.Department of MathematicsUral State UniversityEkatarinburgRussia

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