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Semigroups and Their Subsemigroup Lattices

  • Lev N. Shevrin
  • Alexander J. Ovsyannikov

Part of the Mathematics and Its Applications book series (MAIA, volume 379)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Semigroups with Certain Types of Subsemigroup Lattices

    1. Front Matter
      Pages 1-1
    2. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 3-24
    3. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 25-39
    4. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 40-62
    5. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 63-104
    6. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 105-126
    7. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 127-150
  3. Properties of Subsemigroup Lattices

    1. Front Matter
      Pages 151-151
    2. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 153-170
    3. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 171-198
  4. Lattice Isomorphisms

    1. Front Matter
      Pages 199-199
    2. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 201-214
    3. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 215-242
    4. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 243-273
    5. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 274-293
    6. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 294-325
    7. Lev N. Shevrin, Alexander J. Ovsyannikov
      Pages 326-352
  5. Back Matter
    Pages 353-380

About this book

Introduction

0.1. General remarks. For any algebraic system A, the set SubA of all subsystems of A partially ordered by inclusion forms a lattice. This is the subsystem lattice of A. (In certain cases, such as that of semigroups, in order to have the right always to say that SubA is a lattice, we have to treat the empty set as a subsystem.) The study of various inter-relationships between systems and their subsystem lattices is a rather large field of investigation developed over many years. This trend was formed first in group theory; basic relevant information up to the early seventies is contained in the book [Suz] and the surveys [K Pek St], [Sad 2], [Ar Sad], there is also a quite recent book [Schm 2]. As another inspiring source, one should point out a branch of mathematics to which the book [Baer] was devoted. One of the key objects of examination in this branch is the subspace lattice of a vector space over a skew field. A more general approach deals with modules and their submodule lattices. Examining subsystem lattices for the case of modules as well as for rings and algebras (both associative and non-associative, in particular, Lie algebras) began more than thirty years ago; there are results on this subject also for lattices, Boolean algebras and some other types of algebraic systems, both concrete and general. A lot of works including several surveys have been published here.

Keywords

Algebraic structure Group theory Lattice commutative property mathematical logic

Authors and affiliations

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

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-015-8751-8
  • Copyright Information Springer Science+Business Media B.V. 1996
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4749-6
  • Online ISBN 978-94-015-8751-8
  • Buy this book on publisher's site