F\o{}lner, Banach, and translation density are equal and other new results about density in left amenable semigroups

Abstract

In any semigroup $S$ satisfying the Strong Folner Condition, there are three natural notions of density for a subset $A$ of $S$: Folner 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 Folner Condition. Using this fact, we solve a problem that has been open for decades, showing that the set of ultrafilters every member of which has positive Folner density is a two sided ideal of $\beta S$. We also show that, if $S$ is a left amenable semigroup, then 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 the Stone-\v{C}ech compactification $\beta S$ are singletons. This occurs in many familiar examples, including all semilattices and all semigroups which have a right zero. We show that this is equivalent to the statement that $S$ satisfies the Strong Folner Condition 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. The Folner density of a subset of $S$ is always determined by some Folner net in $S$. We show that an arbitrary Folner 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 $T$.