Embedding Lattices in Subsemigroup Lattices
Recall that for a class A of semigroups SubA denotes the class of all lattices which are isomorphic to lattices SubS with S in A. It is remarkable that, as the material of Part A demonstrates, imposing many popular lattice conditions on the subsemigroup lattices leads to structural characterizations of the corresponding semigroups formulated in terms of current notions and common constructions, and to characterizations which are, moreover, generally quite transparent. As to the “reverse direction”, that is, from a class of semigroups A to the class SubA, this is not the case: the material of Chapter VII shows that lattice characterizations (which are, by the way, relative, i.e. describe SubA not as an abstract class but within the class SubS, where S is the class of all semigroups) for many important classes A turn out to be of a special kind and are sometimes rather complicated. However, there is another approach to exploring the class SubA for classes of semigroups A under consideration, namely, to take an interest in lattices (in particular, finite lattices) which are embeddable in lattices of SubA. This trend may be of interest from the lattice-theoretic point of view. Let us denote by LatA [by LatfinA] the class of all [all finite] lattices embeddable in lattices of SubA. One may consider the classes LatA and LatfinA as, so to speak, the “size of local complexity” of lattices of SubA.
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