In any semigroup S satisfying the Strong Følner Condition, there are three natural notions of density for a subset A of S: Følner density d(A), Banach density \(d^*(A)\) , and translation density \(d_t(A)\) . If S is commutative or left cancellative, it is known that these three notions coincide. We shall show that these notions coincide for every semigroup S which satisfies the Strong Følner Condition. We solve a problem that has been open for decades, showing that if S is left amenable, the set of ultrafilters every member of which has positive Banach density is a two-sided ideal of \(\beta S\) . We investigate the density properties of subsets of S in the case in which the minimal left ideals of \(\beta S\) are singletons. This occurs in all semilattices and all semigroups which have a right zero. We show that this is equivalent to the statement that S satisfies SFC and that, for every subset A of S, \(d(A)\in \{0,1\}\) . We also examine the relation between the density properties of two semigroups when one is a quotient of the other. If S satisfies SFC, we show that an arbitrary Følner net in S determines the density of all of the subsets of S. And we prove that, if S and T are left amenable semigroups, then \(d^*(A\times B)=d^*(A)d^*(B)\) for every subset A of S and every subset B of T.
