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.

Inhalt
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.- B. Properties of Subsemigroup Lattices.- VII. Lattice Characteristics of Classes of Semigroups.- VIII. Embedding Lattices in Subsemigroup Lattices.- 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.- List of Notations.- List of Subsections Containing Unsolved Problems or Open Questions.