## 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 $$FS(\langle y_n\rangle)\cup FP(\langle y_n\rangle)\subseteq A$$.

### MSC:

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