  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.
