Two new algebraic structures more general than weak rings

Abstract

In 1994, Bergelson and Hindman established a remarkable result for IP∗sets: for any IP∗ set A and any sequence l in N, there exists a sum  subsystem l ′ of l such that all finite sums and finite products of l ′ are contained in A. There are many studies about this result,  where Hindman and Strauss extended it to general weak  rings. In this paper, we introduce two algebraic structures - adequate partial  weak rings and adequate poor rings, which are more general than weak rings. And we establish that result in both structures. Moreover,  we also obtain Hindman theorem with two operations in these two structures.