  In this paper, a construction of an infinite dimensional associative algebra,
which will be called a \emph{Surface Algebra}, is associated in a "canonical"
way to a dessin d'enfant, or more generally, a cellularly embedded graph in a
Riemann surface. Once the surface algebras are constructed we will see a
construction of what we call here the associated \emph{Dessin Order} or more
generally the \emph{Surface Order}. This provides a way of associating to every
algebraic curve $X$, with function field $k(X)$ (defined over an arbitrary
field $k$) the representation theory of its Surface Algebra and the lattices
over Surface Orders, which are defined as pullbacks of certain matrix algebras
over commutative $k$-algebras. We will then be able to prove that the center
and the (noncommutative) normalization of the surface orders are invariant
under the action of the absolute Galois group
$\mathcal{G}(\overline{\mathbb{Q}}/\mathbb{Q})$. We will see that the surface
algebras and surface orders are closely related to the fundamental group(oid)
of the Riemann surfaces and the associated monodromy group. A description of
the projective resolutions of the simple modules over the surface order is
given and it will be shown that one can completely recover the dessin with the
projective resolutions of the simple modules alone. In particular, the
projective resolutions of the simple modules encode all combinatorial and
topological data of the monodromy group (or cartographic group) of a dessin.
Finally, as a corollary we are able to say that classifying dessins in an orbit
of $\mathcal{G}(\overline{\mathbb{Q}}/\mathbb{Q})$ is equivalent to classifying
dessin orders with a given normalization. We end with some further examples of
surface algebras and surface orders related to the classical and geometric
version of the Langlands Program.
