Density in arbitrary semigroups
The authors introduce new numerical density invariants for subsets of arbitrary semigroups, extending the usual definitions based on Følner sequences, which are essentially limited to countable left amenable semigroups. They begin their exposition with a discussion of classical density notions for the semigroup \(\mathbb N\) of natural numbers, comparing them with notions of size such as thick, (piecewise) syndetic, \(\Delta\)-set, IP-set, and corresponding dual notions. They show how these notions of size and their mutual relationships extend to an arbitrary semigroup \(S\); the proofs use known equivalent descriptions in terms of the algebraic structure on the Stone-Čech compactification \(\beta S\). Next the upper density based on a Følner sequence in \(S\) is presented, and its limitations are explained: the existence of a Følner sequence implies that \(S\) is left amenable, and together with some weak right cancellation property it implies that \(S\) is countable. After this overview of the state of knowledge the authors introduce three density invariants depending on a given net \({\mathcal F} = \langle F_n\rangle_{n\in D}\) of finite subsets of \(S\); this net replaces the Følner sequence. One of these densities is \(\overline d_{\mathcal F}(A) = \sup \{\alpha:(\forall m \in D)(\exists n\geq m)(| A \cap F_n| \geq \alpha | F_n|)\}\). Various conditions are derived under which these invariants share the nice properties of the older ones. Some of the conditions refer to properties of the net, others are weak cancellation properties of \(S\). It is shown that under such conditions, the density relates nicely to algebraic properties of \(\beta S\), and to the notions of size considered earlier. This approach is carried further for two particular types of nets. One of these is obtained by forming finite products of a given set of generators of \(S\); it yields reasonable notions of density even in some nonamenable semigroups, e.g., in free semigroups. The other special type of nets, called Følner nets, is a direct generalization of Følner sequences. Følner nets exist precisely if \(S\) satisfies the strong Følner condition introduced by \textit{L. N. Argabright} and \textit{C. O. Wilde} [Proc.~Am.~Math.~Soc.~18, 587--591 (1967; Zbl 0152.33302)]. In this case, the supremum of the densities \(\overline d_{\mathcal F}\) over all Følner nets \(\mathcal F\) is again a density invariant, which has very nice properties and admits a natural direct definition. Finally, applications to Ramsey theory and to the structure of \(\beta S\) are given. In particular, a generalization to uncountable semigroups is obtained for the density version of Schur's theorem due to \textit{V. Bergelson} [J.~Comb.~Theory, Ser.~A 43, 338--343 (1986; Zbl 0607.10040)]. The generalization says the following: if \(S\) is infinite and \(\mathcal F\) is a Følner net in \(S\), let \(S= C_1 \cup \dots \cup C_r\) be a finite partition; then some \(C_i\) satisfies \(\overline d _{\mathcal F}\{s \in C_i : \overline d_{\mathcal F}(C_i\cap s^{-1}C_i)>0\}>0\). Remark: The announcement `the following properties' at the bottom of page 274 refers to the implications shown in Figure 1 on page 277; the figure has been shifted during the publication process.
scientific article; zbMATH DE number 5118908