Overview
- Authors:
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Lev N. Shevrin
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Department of Mathematics, Ural State University, Ekatarinburg, Russia
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Alexander J. Ovsyannikov
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Department of Mathematics, Ural State University, Ekatarinburg, Russia
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Table of contents (14 chapters)
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Semigroups with Certain Types of Subsemigroup Lattices
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 3-24
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 25-39
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 40-62
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 63-104
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 105-126
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 127-150
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Properties of Subsemigroup Lattices
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Front Matter
Pages 151-151
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 153-170
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 171-198
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Lattice Isomorphisms
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Front Matter
Pages 199-199
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 201-214
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 215-242
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 243-273
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 274-293
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 294-325
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- Lev N. Shevrin, Alexander J. Ovsyannikov
Pages 326-352
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Back Matter
Pages 353-380
About this book
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.
