Elementary characterization of essential $${\mathscr {F}}$$ F -sets and its combinatorial consequences

There is a long history of studying Ramsey theory using the algebraic structure of the Stone–Čech compactification $$\beta S$$ of a discrete semigroup S. It has been shown that various Ramsey theoretic structures are contained in different algebraically large sets. In this article we deduce combinatorial characterizations of certain sets that are members of idempotent ultrafilters of closed subsemigroups of $$\beta S$$ , arising from certain Ramsey families. In the special case when $$S={\mathbb {N}}$$ , we deduce that sets which are members of all idempotent ultrafilters in these semigroups contain certain additive and multiplicative structures. We generalize this result for weak rings where we establish a non-commutative version of the additive and multiplicative structure.