Semigroups and Their Subsemigroup Lattices

1. Front Matter

   Pages i-xi

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2. Semigroups with Certain Types of Subsemigroup Lattices

   1. Front Matter

      Pages 1-1

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   2. Preliminaries

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 3-24

   3. Semigroups with Modular or Semimodular Subsemigroup Lattices

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 25-39

   4. Semigroups with Complementable Subsemigroups

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 40-62

   5. Finiteness Conditions

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 63-104

   6. Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 105-126

   7. Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 127-150

3. Properties of Subsemigroup Lattices

   1. Front Matter

      Pages 151-151

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   2. Lattice Characteristics of Classes of Semigroups

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 153-170

   3. Embedding Lattices in Subsemigroup Lattices

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 171-198

4. Lattice Isomorphisms

   1. Front Matter

      Pages 199-199

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   2. Preliminaries on Lattice Isomorphisms

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 201-214

   3. Cancellative Semigroups

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 215-242

   4. Commutative Semigroups

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 243-273

   5. Semigroups Decomposable into Rectangular Bands

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 274-293

   6. Semigroups Defined by Certain Presentations

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 294-325

   7. Inverse Semigroups

      • Lev N. Shevrin, Alexander J. Ovsyannikov

      Pages 326-352

5. Back Matter

   Pages 353-380

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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.

• Algebraic structure
• Group theory
• Lattice
• commutative property
• mathematical logic

• Department of Mathematics, Ural State University, Ekatarinburg, Russia

  Lev N. Shevrin, Alexander J. Ovsyannikov

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