Combined Algebraic Properties in Gaussian and Quaternion Ring

TL;DR

This paper extends combinatorial algebraic properties related to IP*, central*, and C*-sets from natural numbers to Gaussian integers and integer quaternions, demonstrating similar sum and product set inclusions.

Contribution

It introduces analogous results for Gaussian and quaternion rings, broadening the scope of algebraic combinatorics in these algebraic structures.

Findings

Results for Gaussian integers analogous to natural numbers.

Results for integer quaternions similar to natural numbers.

Establishment of sum and product set inclusions in new rings.

Abstract

It is known that for an IP^{*} set A in (\mathbb{N},+) and a sequence \left\langle x_{n}\right\rangle _{n=1}^{\infty} in \mathbb{N}, there exists a sum subsystem \left\langle y_{n}\right\rangle _{n=1}^{\infty} of \left\langle x_{n}\right\rangle _{n=1}^{\infty} such that FS\left(\left\langle y_{n}\right\rangle _{n=1}^{\infty}\right)\cup FP\left(\left\langle y_{n}\right\rangle _{n=1}^{\infty}\right)\subseteq A. Similar types of results have also been proved for central^{*} sets and C^{*}-sets where the sequences have been considered from the class of minimal sequences and almost minimal sequences. In this present work, our aim to establish the similar type of results for the ring of Gaussian integers and the ring of integer quaternions.