2024/12/18 by Daniel Glasscock, Glasscock, Daniel, Neil Hindman +3 · 1 citation
Mathematics · Computer Science · #Advanced Banach Space Theory #semigroups and automata theory #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2412.14281
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 dt(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 βS. 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 βS. We investigate the density properties of subsets of S in the case in which the minimal left ideals of the Stone-Čech compactification βS 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)∈ \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× B)=d^*(A)d^*(B) for every subset A of S and every subset B of T.