How many monoids, groups, etc are there defined on a finite set of size $n$? Suppose we have a set $X$. We can define a binary relation $\cdot :X\times X\to X$, and get an algebraic structure $(X,\cdot)$. 
There are $|X|^{|X|^2}$ such binary relations that can be defined. I.e. let $N=n^{n^2}$ where $n=|X|$. 
Now, if we introduce axioms on $\cdot$, the amount of binary relations we can define will of course shrink. Denote by $N_A$ the amount of possible binary relations that satisfy axioms $A$. Denote by $N_A^*$ the number of them that are unique up to isomorphism. $N_A$ and $N_A^*$ are obviously a functions of $n$.
I am wondering how much the different algebraic axioms constrain the size $N_A,N_A^*$, such as associativity, commutativity, existence of inverses, etc. How does $N_A,N_A^*$ depend on the different axioms?


*

*Are there precise formulas for $N_A$ and $N_A^*$, depending on the common axioms as a function of $n$? (Associativity, inverses, identity, commutativity,...)

*Are there interesting interaction effects between the axioms?

*Is there a name for the topic I’m pointing to in this question?

For groups this is a well studied problem with no easy answer. See finite simple groups classification for reference. If you take enough axioms though then this computation can become quite simple, for instance number of finite abelian groups of order $n$ can be easily described look at this question for example.
In summary it can go both ways.
