Elementary characterization of essential F-sets and its combinatorial consequences

Abstract: There is a long history of studying Ramsey theory using the algebraic structure of the Stone-\v{C}ech compactification of discrete semigroup. It has been shown that various Ramsey theoretic structures are contained in different algebraic large sets. In this article we will deduce the combinatorial characterization of certain sets, that are the member of the idempotent ultrafilters of the closed subsemigroup of $\beta S$, arising from certain Ramsey family. In a special case when $S=\mathbb{N}$, we will deduce that sets which are the members of all idempotent ultrafilters of those semigroups contain certain additive-multiplicative structures. Later we will generalize this result for weak rings, where we will show a non-commutative version of the additive-multiplicative structure.