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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.
Preface. Part A: Semigroups with Certain Types of Subsemigroup Lattices. I. Preliminaries. II. Semigroups with Modular or Semimodular Subsemigroup Lattices. III. Semigroups with Complementable Subsemigroups. IV. Finiteness Conditions. V. Inverse Semigroups with Certain Types of Lattices of Inverse Subsemigroups. VI. Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups. Part B: Properties of Subsemigroup Lattices. VII. Lattice Characteristics of Classes of Semigroups. VIII. Embedding Lattices in Subsemigroup Lattices. Part C: Lattice Isomorphisms. IX. Preliminaries on Lattice Isomorphisms. X. Cancellative Semigroups. XI. Commutative Semigroups. XII. Semigroups Decomposable into Rectangular Bands. XIII. Semigroups Defined by Certain Presentations. XIV. Inverse Semigroups. Bibliography. Index. List of Notations. List of Subsections Containing Unsolved Problems or Open Questions.