Where is the "relation" here? Looking at the definition of a monoid it says that:

A monoid is a set that is closed under an associative binary operation and 
  has an identity  element $I \in S$ such that for all $a \in S$, $I a = a I =a$

But what does $I a$ mean here? I mean it's just one element from the set, followed by a space and another element of the set. Is it assumed that this means binary function of some sort?
I mean when I write $0+x$, I don't write $0\ x$...
Thanks, any help in understanding this is appreciated.
 A: I'd say what is confusing about the definition above is that it doesn't make clear that the binary operation is part of the monoid - it only asserts the existence of the operation on the set.  For example, the above definition would make $\mathbb N$ a monoid because there exists an associative binary operation blah blah blah. But there are many such associative binary operations on $\mathbb N$.
It should really say, "A monoid is a pair $(M,\star)$ where $M$ is a set and $\star$ is an associative binary operation $\star:M\times M\rightarrow M$, such that there exists an $i\in M$  statisfying $i\star m = m\star i = m$ for all $m\in M$.
In particular, when the definition above says: $Ia=aI=a$, that is shorthand for the operation $I\star a = a\star I = a$.
Oh, and the only reason we tend to write monoids in a "multiplicative form," rather than more like addition, is that addition, in almost all instances, is commutative: $a+b=b+a$.  But multiplication in many instances is not - for instance, matrix multiplication is not commutative.  So we usually think of the monoid operation as being "like" multiplication.
A: It doesn't matter whether you use additive notation "$+$" or multiplicative notation "$\cdot$" to denote the group (or in this case: monoid) operation. The notation $Ia$ really is the lazy version of of writing $I \cdot a$.
But you could equally well write $I + a$. See for example here for notational conventions for abelian groups.
A: Perhaps it's helpful to look at some usual examples.


*

*The set of natural numbers including $0$, together with addition, forms a monoid: the binary operation is $\lambda (a,b).a\!+\!b$ and the identity element is $0$, for $$a+0 = 0+a = a.$$ In Haskell, this translates to the Sum instance of Monoid with mempty = Sum 0 and mappend a b = Sum (getSum a + getSum b).

*The same set, with multiplication as the relation and $1$ as the identity. Here, it is common in mathematics to just omit the multiplication sign,
$$
  a\cdot 1 = 1\cdot a = 1a = a.
$$
In Haskell, mempty = Product 1 and mappend a b = Product (getProduct a * getProduct b).

*The set of matrices on e.g. $\mathbb{R}^2$ together with matrix multiplication. Here, the identity is $(\begin{smallmatrix}1&0\\0&1\end{smallmatrix})$,
$$
  (\begin{smallmatrix}1&0\\0&1\end{smallmatrix})\cdot (\begin{smallmatrix}a&b\\c&d\end{smallmatrix}) = (\begin{smallmatrix}a&b\\c&d\end{smallmatrix})\cdot(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}) = (\begin{smallmatrix}a&b\\c&d\end{smallmatrix})
$$

*The set of lists of letters together with list concatenation. The identity is the empty list. That one's rather a bit un-mathematic, I shall again omit excplicitly writing the binary operator or the "promotion" of characters to single-element lists and write it this way:
$$
  {{}''}\mathrm{Word} = \mathrm{Word}'' = \mathrm{Word}
$$
(and also $
  = \mathrm{W{{}''}ord} = \mathrm{Wor{{}''}d} = \mathrm{{{}''}Wo{{}''}r{{}''}{{}''}{{}''}{{}''}d}$.)
In Haskell, it's the more general [] instance of Monoid, with mempty=[] (that is, for strings, mempty="") and mappend a b = a++b.

