Ebook: Semigroups and Their Subsemigroup Lattices

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.

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The study of various interrelations between algebraic systems and their subsystem lattices is an area of modern algebra which has enjoyed much progress in the recent past. Investigations are concerned with different types of algebraic systems such as groups, rings, modules, etc. In semigroup theory, research devoted to subsemigroup lattices has developed over more than four decades, so that much diverse material has accumulated.
This volume aims to present a comprehensive presentation of this material, which is divided into three parts. Part A treats semigroups with certain types of subsemigroup lattices, while Part B is concerned with properties of subsemigroup lattices. In Part C lattice isomorphisms are discussed. Each chapter gives references and exercises, and the volume is completed with an extensive Bibliography.
Audience: This book will be of interest to algebraists whose work includes group theory, order, lattices, ordered algebraic structures, general mathematical systems, or mathematical logic.

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The study of various interrelations between algebraic systems and their subsystem lattices is an area of modern algebra which has enjoyed much progress in the recent past. Investigations are concerned with different types of algebraic systems such as groups, rings, modules, etc. In semigroup theory, research devoted to subsemigroup lattices has developed over more than four decades, so that much diverse material has accumulated.
This volume aims to present a comprehensive presentation of this material, which is divided into three parts. Part A treats semigroups with certain types of subsemigroup lattices, while Part B is concerned with properties of subsemigroup lattices. In Part C lattice isomorphisms are discussed. Each chapter gives references and exercises, and the volume is completed with an extensive Bibliography.
Audience: This book will be of interest to algebraists whose work includes group theory, order, lattices, ordered algebraic structures, general mathematical systems, or mathematical logic.
Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Preliminaries....Pages 3-24
Semigroups with Modular or Semimodular Subsemigroup Lattices....Pages 25-39
Semigroups with Complementable Subsemigroups....Pages 40-62
Finiteness Conditions....Pages 63-104
Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups....Pages 105-126
Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups....Pages 127-150
Front Matter....Pages 151-151
Lattice Characteristics of Classes of Semigroups....Pages 153-170
Embedding Lattices in Subsemigroup Lattices....Pages 171-198
Front Matter....Pages 199-199
Preliminaries on Lattice Isomorphisms....Pages 201-214
Cancellative Semigroups....Pages 215-242
Commutative Semigroups....Pages 243-273
Semigroups Decomposable into Rectangular Bands....Pages 274-293
Semigroups Defined by Certain Presentations....Pages 294-325
Inverse Semigroups....Pages 326-352
Back Matter....Pages 353-380

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The study of various interrelations between algebraic systems and their subsystem lattices is an area of modern algebra which has enjoyed much progress in the recent past. Investigations are concerned with different types of algebraic systems such as groups, rings, modules, etc. In semigroup theory, research devoted to subsemigroup lattices has developed over more than four decades, so that much diverse material has accumulated.
This volume aims to present a comprehensive presentation of this material, which is divided into three parts. Part A treats semigroups with certain types of subsemigroup lattices, while Part B is concerned with properties of subsemigroup lattices. In Part C lattice isomorphisms are discussed. Each chapter gives references and exercises, and the volume is completed with an extensive Bibliography.
Audience: This book will be of interest to algebraists whose work includes group theory, order, lattices, ordered algebraic structures, general mathematical systems, or mathematical logic.
Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Preliminaries....Pages 3-24
Semigroups with Modular or Semimodular Subsemigroup Lattices....Pages 25-39
Semigroups with Complementable Subsemigroups....Pages 40-62
Finiteness Conditions....Pages 63-104
Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups....Pages 105-126
Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups....Pages 127-150
Front Matter....Pages 151-151
Lattice Characteristics of Classes of Semigroups....Pages 153-170
Embedding Lattices in Subsemigroup Lattices....Pages 171-198
Front Matter....Pages 199-199
Preliminaries on Lattice Isomorphisms....Pages 201-214
Cancellative Semigroups....Pages 215-242
Commutative Semigroups....Pages 243-273
Semigroups Decomposable into Rectangular Bands....Pages 274-293
Semigroups Defined by Certain Presentations....Pages 294-325
Inverse Semigroups....Pages 326-352
Back Matter....Pages 353-380
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