Characterization of functional monoids I define a functional monoid to be a monoid that is isomorphic to the set of all functions from a set $S$ to itself under the operation of composition. I want to know useful necessary and sufficient conditions for a monoid to be functional. Any help would be appreciated.
 A: The more common name for what you call a functional monoid is the full transformation monoid on the given set. A transformation monoid is then by definition a submonoid of the full transformation monoid. 
Suppose now that $M$ is a transformation monoid on a set $X$. Clearly, $(m,x)\mapsto m(x)$ is a left action of $M$ on $X$. Moreover, this action is effective, in the sense that for all $m,m'\in M$ if $m(x)=m'(x)$ for all $x\in X$, then $m=m'$ (this property is just a monoid formulation of the definition of function). So, transformation monoids give rise to effective actions. 
Conversely, an effective action of $M$ on some set $S$, allows one to associate with $m\in M$ the function $T_m:S\to S$ given by $T_m(s)=ms$. It is easy to see then that the function $m\mapsto T_m$ is a monoid homomorphism which moreover is injective precisely if $M$ is a transformation monoid, evidently isomorphic to the image of this homomorphism. So, transformation monoid are obtained from effective actions. 
Now, you are interested in the full transformation monoid, which is severely restricted by size. In particular, there is, up to isomorphism, precisely one full transformation monoid of every cardinality of the form $k^k$. 
I hope all of this information will help you answer what you actually are trying to solve. 
