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 Uspekhi Mat. Nauk, 1972, Volume 27, Issue 6(168), Pages 139–180 (Mi umn5141)

Some lattice-theoretical properties of groups and semi-groups

M. N. Arshinov, L. E. Sadovskii

Abstract: 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].

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English version:
Russian Mathematical Surveys, 1972, 27:6, 149–191

Bibliographic databases:

UDC: 519.4
MSC: 22A26, 20E15, 20D30

Citation: M. N. Arshinov, L. E. Sadovskii, “Some lattice-theoretical properties of groups and semi-groups”, Uspekhi Mat. Nauk, 27:6(168) (1972), 139–180; Russian Math. Surveys, 27:6 (1972), 149–191

Citation in format AMSBIB
\Bibitem{ArsSad72} \by M.~N.~Arshinov, L.~E.~Sadovskii \paper Some lattice-theoretical properties of groups and semi-groups \jour Uspekhi Mat. Nauk \yr 1972 \vol 27 \issue 6(168) \pages 139--180 \mathnet{http://mi.mathnet.ru/umn5141} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=404400} \zmath{https://zbmath.org/?q=an:0278.20030} \transl \jour Russian Math. Surveys \yr 1972 \vol 27 \issue 6 \pages 149--191 \crossref{https://doi.org/10.1070/RM1972v027n06ABEH001394}