Semigroups and Their Subsemigroup Lattices

• Lev N. Shevrin
• Alexander J. Ovsyannikov
Book

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

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

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