Combined algebraic properties of central* sets. (English) Zbl 1156.22003

Let \((\beta,\beta \mathbb N)\) be the universal right topological semigroup compactification of the discrete additive semigroup \((\mathbb N,+)\) of natural numbers. A subset \(A\) of \(\mathbb N\) is a central* set if \(\overline{\beta(A)}\) contains every minimal idempotent of \(\beta\mathbb N\). Let \(\langle x_n\rangle\) be a sequence of natural numbers. We denote by \(FS(\langle x_n\rangle)\), respectively, \(FP(\langle x_n\rangle)\) the set of all finite sums, respectively, finite products of elements of the sequence \(\langle x_n\rangle\). A sequence \(\langle y_n\rangle\) is said to be a sum subsystem of \(\langle x_n\rangle\) provided there is a sequence \(\langle H_n\rangle\) of nonempty finite subsets of \(\mathbb N\) such that \(\max H_n<\min H_{n+1}\) and \(y_n= \sum_{m\in H_n} x_m\) for every \(n\in \mathbb N\).
The sequence \(\langle x_n\rangle\) is called minimal if the intersection \(\bigcap_{m\in\mathbb N}\overline{\beta(FP(\langle x_n\rangle_{n\geq m}))}\) meets the minimal ideal of \(\beta \mathbb N\). The main result of the paper states that if \(A\) is a central\(^*\) set in \(\mathbb N\) and if \(\langle x_n\rangle\) is a minimal sequence then there exists a sum subsystem \(\langle y_n\rangle\) of \(\langle x_n\rangle\) such that \(FS(\langle y_n\rangle)\cup FP(\langle y_n\rangle)\subseteq A\).


22A15 Structure of topological semigroups
05A17 Combinatorial aspects of partitions of integers
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