Algebra in the Stone-Čech Compactification

Selective survey on Subset Combinatorics of Groups

“…It follows from the second step of the extension that, for every q ∈ βG, the mapping p → pq is continuous, so the semigroup βG is right topological. For the structure of compact right topological semigroup βG and its combinatorial applications see [22,41]. For application of βG to topologies on groups see [45] and [I. V. Protasov.…”

“…This is the main question in the Ramsey Group Theory (see [22,Chapters 5,14,16,18]). Most results of this theory state that at least one cell of partition has rich algebraic structure or is "large" by the combinatorial size, so following Taras Banakh, we may say that a full chaos is impossible.…”

“…It means that P S(A) ⊆ A i . Much more delicate statements follow from the Hindman theorem (see [22,Chapter 5]): there exist i ∈ {1, . .…”

“…We leave outside (or touch a little bit) the other aspects of the Subset Combinatorics of Groups, in particular, the Additive Combinatorics [52] and the Ramsey Group Theory [22,41]. By the combinatorial size of a subset A of a group G we mean some (mainly, cardinal) characteristic which reflects an arrangement of A in G. The main results under consideration concern the groups, however, for our goal it is more convenient to use the terminology of G-spaces and, sometimes, much more general context of balleans.…”

“…The thin subsets play an important role in algebra of the Stone-Čech compactifications [22]. By means of thin subsets, C. Chou [11] proved that every infinite amenable group G admits 2 2 |G| distinct Banach measures.…”