Følner numbers and Følner type conditions for amenable semigroups (Q1088105)

On a semigroup S various Følner-type conditions have been studied in connection with left amenability of S, i.e., the existence of a left invariant mean on the space of bounded functions defined on S. These conditions are equivalent to amenability in the case of groups. Adopting a definition due to \textit{J. C. S. Wong} [On Følner conditions and Følner numbers for semigroups, to appear], the author defines the Følner number \(\phi\) (S) to be the infimum of all \(k\in]0,1]\) such that whenever \(s_ 1,...,s_ n\) are nonnecessarily distinct elements in S, there exists a nonvoid finite subset A in S for which \[ (1/n)\sum^{n}_{i=1}| A\setminus s_ iA| \leq k | A|. \] As the latter inequality holds trivially for \(k=1\), \(\phi\) (S) is well-defined. The author observes that a) \(\phi (S)<1\) expresses the weak Følner condition (WFC), b) \(\phi (S)<1/2\) expresses the strong Følner-Namioka condition (SNFC); moreover, \(\phi (S)=0\) is implied by the strong Følner condition (SFC): Given a finite subset F in S and \(\epsilon >0\), there exists a finite subset A in S such that \(| A\setminus sA| \leq \epsilon | A|\) whenever \(s\in F.\) Følner numbers are studied in detail. The author shows that in case S is finite, amenability is equivalent to (SFC), (SNFC). In case S is cancellative, the following dichotomy holds: a) \(\phi (S)=0\) and S is amenable, b) \(\phi (S)=1\) and S is nonamenable; a) is equivalent to (WFC). The author produces a left amenable semigroup S with Følner number equal to 1; so none of (SNFC), (WFC) is necessary for left amenability; nor is the weak Følner-Namioka condition (WNFC): There exists \(k\in]0,1[\) such that, whenever \(s_ 1,...,s_ n\), \(s_ 1',...,s_ n'\) are elements in S, there exists a finite subset A in S satisfying \[ (1/n)\sum^{n}_{i=1}| s_ iA\cap s_ i'A| \geq k | A|. \] The author also shows the existence of a semigroup S with a homomorphic image h(S) such that \(\phi (S)=0\), \(\phi (h(S))=1\).