Axioms of an algebraic structures I have started going over the axioms of semi-group, monoid, group.
I was wondering, can we say that associativity is an axiom of the operation and not of the set? maybe I am wrong, but are there axioms that depends just on the operation or the set?
 A: You are correct. The Group axioms are properties of the composition operation, and not of the 'naked' set elements. 
When we say '$\mathbb Z_2$ has an identity element' we really mean '$\mathbb Z_2$ has an identity element with respect to the standard + operation'. Of course that element is $0$. But we could use a different operation on the set $\mathbb Z_2 = \{0,1\}$ to make $1$ into the identity. Denote that operation by $++$. The new group structure is as follows.
$1$$++$$1=1$
$0$$++$$0=1$
$0$$++$$ 1=0$
$1$$++$$0=0$
We could go further again and put an operation $*$ on the set $\mathbb Z_2 = \{0,1\}$ to make it a monoid but not a group. That operation is just multiplication. The monoid operation is as follows.
$1*1=1$
$1*0=0$
$0*1=0$
$0*0=0$
In conclusion saying '$\mathbb Z_2$ is a group', '$\mathbb Z_2$ is a monoid' or 'This is the identity of $\mathbb Z_2$' are all dependent on what we use for the operation. 
However in practice when we mention 'the group $\mathbb Z_2$' we mean the set $\{0,1\}$ taken with modular addition. Usually when we want to use a different operation we will specify. And when we want to talk about the bare set we will write $\{0,1\}$ instead of $\mathbb Z_2$. 
Likewise for other familiar groups like $\mathbb Z_n , \mathbb R, \mathbb C, \mathbb R_{\ne 0}, \mathbb C _{\ne 0}$. However there is no obvious way to 'write out' the elements of the real line to make it clear we are not worried about addition.
In fact, given a group $(G,*)$ the only importance of the particular set $G$ is the cardinality. Given any other set $F$ with $|F|=|G|$, we can invent a group operation $\odot$ on $F$ such that the groups $(G,*)$ and $(F, \odot)$ are isomorphic.
