Preservation of notion of large sets near zero over reals

Abstract

The study of the size of subsets in a semigroup have shown that many of these subsets have strong combinatorial properties and contribute richly to the algebraic structure of the Stone-Cech compactification of a discrete semigroup. N. Hindman and D. Strauss have proved that if u, v $\in \mathbb{N}$, M is a u \times v matrix satisfying restrictions that vary with the notion of largeness and if $Ψ$ is a notion of large sets in $\mathbb{N}$ then $\{\vec{x} \in \mathbb{N}^v: M\vec{x} \in Ψ^u\}$ is large set in $\mathbb{N}^v$. In this article, we investigate the above result for various notions of largeness near zero in $\mathbb{R}^+$.