Image partition regular matrices and concepts of largeness, II - Hindman, Neil; Strauss, Dona (Topology Proceedings, 2026)
Abstract. Let $u$ and $v$ be positive integers and let $A$ be a $u × v$ matrix with rational entries. We determine several characterizations of the property that whenever $B$ is a piecewise syndetic subset of the set $\mathbb{N}$ of positive integers, $\left\{\vec{x} \in \mathbb{N}^v: A \vec{x} \in B^u\right\}$ is piecewise syndetic in $\mathbb{N}^v$ as well as the corresponding property with $\mathbb{Z}$ replacing $\mathbb{N}$. We investigate related phenomena for several other notions of largeness in a semigroup.