Finding Product and Sum Patterns in non-commutative settings

Hindman conjectured that any finite partition of $\mathbb{N}$ has a monochromatic $\{x,y,x+y,xy\}$. Recently, Bowen proved the result for all 2-partition. In this paper, we extend Bowen’s result to any semiring $(S,+,\cdot )$ such that $Ss$ is piecewise syndetic for all $s\in S$. As a method, we gave a combinatorial proof for a piecewise syndetic version of Bergerson and Glasscock’s IP${}_r^{\ast }$ Szemerédi Theorem, and discussed the case when the operation is not commutative.