Image partition regular matrices and concepts of largeness. (English) Zbl 1434.05152
Summary: We show that for several notions of largeness in a semigroup, if \(u,v \in \mathbb{N}\), \(A\) is a \(u \times v\) matrix satisfying restrictions that vary with the notion of largeness, and if \(C\) is a large subset of \(\mathbb{N}\), then \(\{\vec{x} \in \mathbb{N}^v:A\vec{x} \in C^u\}\) is large in \(\mathbb{N}^v\). We show that in most cases the restrictions on \(A\) are necessary. Several other results, including some generalizations, are also obtained. Included is a simple proof that if \(u > 1\), then \(\beta(\mathbb{N}^v)\) is not isomorphic to \((\beta\mathbb{N})^u\).
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