Finiteness Conditions

Abstract

We recall that, given a class of algebraic systems, by a finiteness condition is meant any property which is possessed by all finite systems of this class. Imposing finiteness conditions is a classical approach in investigation of algebraic systems of different kinds. Many such conditions are formulated in terms of subsystem lattices; the most important examples are the minimal and the maximal conditions (which are equivalent to the descending and the ascending chain conditions respectively). This chapter contains fundamental information concerning semigroups whose subsemigroup lattices satisfy certain lattice-theoretic finiteness conditions. Since the subsemigroup lattice of a finite semigroup is finite, imposing lattice finiteness conditions on SubS leads to semigroup finiteness conditions for a semigroup S. In the extremal case when SubS is finite, we have simply finiteness of S (see the assertion f) in Proposition 3.2). So, to describe semigroups S with a non-trivial finiteness condition for SubS, we should clarify, so to say, a character and a degree of “deviations” from the property of being a finite semigroup. Such deviations will almost always take place in maximal subgroups of semigroups under consideration. Thus reduction to groups in this chapter is as typical as in the preceding two chapters. For the most of the conditions investigated here, groups with these conditions have been considered in monographs, and in such cases we give the references; for other conditions relevant group-theoretic material is given with proofs. Section 13 is specially devoted to the group case. Central results of the chapter are given in Section 12. The overwhelming majority of conditions under consideration here imply periodicity of a semigroup; so appropriate information about epigroups turns out to be useful. Furthermore, these conditions are hereditary for subsemigroups. Therefore it is important to determine “forbidden” types of subsemigroups, whose lack is a key factor when the corresponding semigroups are described. The role of such subsemigroups is played by semigroups with a unique infinite basis. Epigroups without such subsemigroups are the subject of Section 11, whose material is of considerable independent interest. For most of the conditions considered in Section 12, a “projection” of related results onto the commutative case is straightforward, but for two of these conditions the complete solution needs additional effort; the main part of Section 14 is devoted to this work.