C-image partition regularity near zero

TL;DR

This paper explores the concept of C-image partition regular matrices near zero within dense subsemigroups of positive real numbers, extending the theory of image partition regularity close to zero.

Contribution

It introduces the notion of C-image partition regular matrices near zero, expanding the understanding of infinite matrix regularity in dense subsemigroups.

Findings

Defined C-image partition regularity near zero.

Extended the theory to dense subsemigroups of positive reals.

Provided new insights into infinite matrix regularity near zero.

Abstract

In \cite{dehind1}, the concept of image partition regularity near zero was first instigated. In contrast to the finite case , infinite image partition regular matrices near zero are very fascinating to analyze. In this regard the abstraction of Centrally image partition regular matrices near zero was introduced in \cite{biswaspaul}. In this paper we propose the notion of matrices that are C-image partition regular near zero for dense subsemigropus of $((0,\infty),+)$.