Density in Arbitrary Semigroups

We introduce some notions of density in an arbitrary semigroup S which extend the usual notions in countable left amenable semigroups in which density is based on Folner sequences. The new notions are based on nets of finite sets. We show that under certain conditions on the nets and on S these notions relate nicely to some established notions of size in S such as central, syndetic, and piecewise syndetic. And we investigate the conditions under which these notions have other desirable properties such as translation invariance. We obtain new information about the algebraic structure of the Stone-Cech compactification β S of S and derive generalizations of some known Ramsey Theoretic results, including Bergelson's density version of Schur's Theorem.