ΠΡΠΈΡΠΊΠΎ Π·Π° ΡΠ°Π·ΠΈ Π΅Π»Π΅ΠΊΡΡΠΎΠ½Π½Π° ΠΊΠ½ΠΈΠ³Π°
The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous.
In the second part, Chapters 6 through 9, the Stone-CΓͺch compactification Ξ²G of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then Ξ²G contains no nontrivial finite groups. Also the ideal structure of Ξ²G is investigated. In particular, one shows that for every infinite Abelian group G, Ξ²G contains 22|G| minimal right ideals.
In the third part, using the semigroup Ξ²G, almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized. Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely Ο-resolvable, and consequently, can be partitioned into Ο subsets such that every coset modulo infinite subgroup meets each subset of the partition.
The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas.
ΠΡΠ΅Π½ΠΊΠΈ ΠΈ ΠΎΡΠ·ΠΈΠ²ΠΈ
ΠΠ° Π°Π²ΡΠΎΡΠ°
Yevhen G. Zelenyuk, University of the Witwatersrand, Johannesburg, South Africa.
