The semigroup of ultrafilters near 0. (English) Zbl 0942.22003

Hindman, Neil; Leader, Imre

Semigroup Forum 59, No. 1, 33-55 (1999).

In this interesting paper the authors investigate the semigroup \(0^+\). The set \(0^+\) of ultrafilters on \((0,1)\) that converge to \(0\) is a semigroup under the restriction of the usual operation \(+\) on \(\beta{\mathbb R}_d\), the Stone-Čech compactification of the discrete semigroup \((\mathbb R_d, +)\). In Section 2 the authors characterize the Stone-Čech compactifications of subsemigroups of \((\mathbb R, +)\). For example, Theorem 2.3. Let \(S=(0, \infty)\). There is an extension of \(+\) to \(\beta S\) such that \((\beta S, +)\) is a right topological semigroup. In Section 3 the authors characterize the members of the smallest ideal of \((0^+, +)\) and its closure. They also describe those subsets of \(\mathbb R\) that have idempotents in \((0^+, +)\) in their closure. Let \(S\) be a dense subsemigroup of \(((0, \infty), +)\) and \(K\) be the smallest ideal of \((0^+, +)\); then a set \(A\subset S\) is called central near \(0\) if and only if there is some idempotent \(p\in K\) such that \(A\in p\). In Section 4 sets that are central near \(0\) are described. The authors use the collectionwise piecewise syndetic characterization [N. Hindman, A. Maleki and D. Strauss, J. Comb. Theory, Ser. A 74, 188-208 (1996; Zbl 0858.22003)] in the classification of sets central near \(0\). In Section 5 the authors give some interesting combinatorial applications of the obtained results to Ramsey theory.

Reviewer: Oleg V.Gutik (Lviv)

MSC:

22A15

Structure of topological semigroups

Keywords:

semigroup of ultrafilters; Stone-Čech compactification; subsemigroup of \(\beta S\); idempotent; smallest ideal; piecewise syndetic set near \(0\); collectionwise piecewise syndetic family near \(0\); central set near \(0\); dense subsemigroup; Ramsey theory

Citations:

Zbl 0858.22003