F-finite embeddabilities of sets and ultrafilters

Let $S$ be a semigroup, let $n\in\mathbb{N}$ be a positive natural number, let $A,B\subseteq S$, let $\mathcal{U},\mathcal{V}\in\beta S$ and let let $\mathcal{F}\subseteq\{f:S^{n}\rightarrow S\}$. We say that $A$ is $\mathcal{F}$-finitely embeddable in $B$ if for every finite set $F\subseteq A$ there is a function $f\in\mathcal{F}$ such that $f\left(A^{n}\right)\subseteq B$, and we say that $\mathcal{U}$ is $\mathcal{F}$-finitely embeddable in $\mathcal{V}$ if for every set $B\in\mathcal{V}$ there is a set $A\in\mathcal{U}$ such that $A$ is $\mathcal{F}$-finitely embeddable in $B$. We show that $\mathcal{F}$-finite embeddabilities can be used to study certain combinatorial properties of sets and ultrafilters related with finite structures. We introduce the notions of set and of ultrafilter maximal for $\mathcal{F}$-finite embeddability, whose existence is proved under very mild assumptions. Different choices of $\mathcal{F}$ can be used to characterize many combinatorially interesting sets/ultrafilters as maximal sets/ultrafilters, for example thick sets, AP-rich sets, $\overline{K(\beta S)}$ and so on. The set of maximal ultrafilters for $\mathcal{F}$-finite embeddability can be characterized algebraically in terms of $\mathcal{F}$. This property can be used to give an algebraic characterization of certain interesting sets of ultrafilters, such as the ultrafilters whose elements contain, respectively, arbitrarily long arithmetic, geoarithmetic or polynomial progressions. As a consequence of the connection between sets and ultrafilters maximal for $\mathcal{F}$-finite embeddability we are able to prove a general result that entails, for example, that given a finite partition of a set that contains arbitrarily long geoarithmetic (resp. polynomial) progressions, one cell must contain arbitrarily long geoarithmetic (resp. polynomial) progressions.