Advertisement

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

As we mentioned in the Preface, one can treat inverse semigroups as unary semi-groups, i.e. as algebraic systems with two operations: the binary operation of multiplication and the unary operation of taking the inverse element. From this point of view it is natural to associate with any inverse semigroup S the lattice of all subsystems, i.e. unary subsemigroups of S. But a unary subsemigroup of an inverse semigroup is none other than an inverse subsemigroup. So we shall consider the lattice SubiS of all inverse subsemigroups of S (of course, again the empty set is treated as an inverse subsemigroup). In view of the identity (xy)−1 = y −1 x −1, which is valid in any inverse semigroup, the join of two inverse subsemigroups of an inverse semigroup S is an inverse subsemigroup; therefore SubiS is a sublattice of SubS. It is easy to see that SubS = SubiS if and only if S is a periodic Clifford semigroup. This observation will be used several times below without explicit reference. For an arbitrary group G, the lattice SubiG coincides with the lattice of all subgroups of G augmented by the adjoined zero. It follows that, in any general considerations of lattices of inverse subsemigroups, the case when we deal with the subgroup lattice of a group is inevitably involved. As in the case of the usual subsemigroup lattices, in examinations of lattices of all inverse subsemigroups one succeeds not infrequently in realizing a reduction to groups.

Keywords

Maximal Subgroup Inverse Semigroup Simple Semigroup Finite Width Subgroup Lattice 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations