## Semigroups and Their Subsemigroup Lattices0.1. 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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### Contents

Semigroups with Modular or Semimodular Subsemigroup Lattices25 | 25 |

Semigroups with Complementable Subsemigroups | 40 |

Finiteness Conditions | 63 |

Inverse Semigroups with Certain Types of Lattices | 105 |

Inverse Semigroups with Certain Types of Lattices | 127 |

Lattice Characteristics of Classes of Semigroups | 153 |

Embedding Lattices in Subsemigroup Lattices | 171 |

Preliminaries on Lattice Isomorphisms | 201 |

Commutative Semigroups | 243 |

Semigroups Decomposable into Rectangular Bands | 274 |

Semigroups Defined by Certain Presentations | 294 |

Inverse Semigroups | 326 |

353 | |

371 | |

List of Subsections Containing Unsolved Problems or Open Questions | 378 |

Cancellative Semigroups | 215 |

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0-simple abelian anti-isomorphism antichain archimedean archimedean semigroups Assume bicyclic semigroup bijection Brandt semigroup cancellative semigroup chain Chapter classes of semigroups commutative semigroup completely simple semigroups completing the proof congruence consider contradiction Corollary cyclic group cyclic semigroup decomposable definition denote direct product element of infinite embeddable epigroup equality following conditions free semigroup group G H-class holds homomorphism hypothesis ideal idempotent idempotent semigroup identity implies inclusion induced infinite cyclic infinite order inverse subsemigroups K-semigroup lattice isomorphic lattice SubS Lemma mapping maximal subgroups modular monogenic inverse semigroup natural number nilpotent nilsemigroup non-trivial Observation obtain obviously ordinal components ordinal sum p-group p(xy pair periodic group Proposition prove quasicyclic group result of Exercise satisfies Section semilattice semimodular single-covering element strictly lattice-determined Sub’S SubfiS subgroup lattice SubiS Subsection subsemigroup lattices subsemilattice Suppose unipotent variety verify view of Lemma weak isomorphism whence zero