Finding Product and Sum Patterns in non-commutative settings

Hindman conjectured that any finite partition of N has a monochromatic ,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,+,·) such that $Ss$ is piecewise syndetic for all s∈ S. As a method, we gave a combinatorial proof for a piecewise syndetic version of Bergerson and Glasscock's IPᵣ^* Szemer\'edi Theorem, and discussed the case when the operation is not commutative.