Let $S$ be a finite set. Now $\mathop{End}(S)$ is a monoid, and we may build a ring $R$ by allowing formal sums of functions.

Preliminary questions, since $R$ is surely well-known: What is it called? In a general category, what is the name of the construction that builds a ring out of the endomorphisms of an object?

My real question is about $A = R \otimes \mathbb{C}$. Restricting to the subalgebra generated by the automorphisms of $S$, it is clear that understanding the representation theory of symmetric groups is a prerequisite for understanding $A$. Are the symmetric groups on sets of size smaller than $| S |$ responsible for all nontrivial properties of $A$? If so, what suitably strong formulation of inclusion/exclusion is at work?