Some lattice-theoretical properties of groups and semi-groups

M. N. Arshinov ; L. E. Sadovskii

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 27 (1972) no. 6, pp. 149-191

Résumé

During the period following the publication of the survey [5] a number of new papers appeared in which connections between the structure of an algebraic system (a group, a semigroup or a topological group) and the lattice of its subsystems (subgroups, subsemigroups, closed subgroups) are studied. In a sense the present article is a continuation of [5], although its style differs somewhat in that it includes fragments of proofs of the most interesting facts. It also considers other lattices similar to the subgroup lattice of a discrete group. Accordingly it contains five sections studying the subgroup lattice of infinite groups (§ 1), the subsemigroup lattice of these groups (§ 2), the subsemigroup lattice of a semigroup (§ 3), the subgroup lattice in groups with various finiteness conditions (§ 4), and finally the lattice of closed subgroups of a topological group (§ 5). All the definitions necessary for an understanding of the new results are given here. Definitions of other concepts that are already known well-enough can be found in [5] or in Kurosh's book [4]. The authors have tried to examine all the available relevant literature; this is listed at the end of the article. Titles cited in [5] are repeated here only when they are directly referred to in the text in connection with new results not mentioned in [5].