Is there a term for a magma with identity (only)? If we start from magmas and consider: associativity, identity, invertibility (divisibility). We will theoretically get $2^3=8$ structures by regarding whether such structure possess these properties. As is shown in the picture: 
https://en.wikipedia.org/wiki/Magma_(algebra)#/media/File:Magma_to_group2.svg
There miss two structures - associative, divisible magma and magma with identity. Well, we know associative quasigroup is a group. So, there is only one left. Is there a term for it? 
And it is necessary to differ between invertibility and divisibility?
 A: The term "unital magma" does get used, including by Bourbaki (Algebra I, p. 12):

A magma with an identity element is called a unital magma.
  Here, an identity element is defined to be two-sided.

Divisibility does not imply invertibility, so they are distinct.  One can have divisibility without an identity element; one needs both for invertibility.  Note: a left inverse and a right inverse might not be the same, but invertibility does not demand this.
A: Here's an example of such a structure.
$$\begin{array}{c|ccc}
\cdot & e & a & b \\
\hline
e & e & a & b\\
a & a & b & a\\
b & b & b & a\\
\end{array}$$
We see that $e$ is an identity element.  However, $(S, \cdot)$ is not associative, since $(a\cdot b)\cdot b = a\cdot b = a$, while $a\cdot (b\cdot b) = a \cdot a = b$.
Also, $(S, \cdot)$ does not have inverses.  There is no element $x\in S$ such that $a\cdot x = e$.
Also, though you didn't mention commutativity, $(S, \cdot)$ is not commutative: $a\cdot b = b$, while $b\cdot a = a$.  We actually could make it commutative, though:
$$\begin{array}{c|ccc}
\star & e & a & b \\
\hline
e & e & a & b\\
a & a & b & b\\
b & b & b & a\\
\end{array}$$
$(S, \star)$ still has an identity, still lacks inverses, and is now commutative.  And $(S, \star)$ is still non-associative, since $(a\star b)\star b = b\star b = a$, while $a \star (b\star b) = a \star a = b$.
I guess one way to make a magma with identity is to start with whatever magma you want, and then add another element, $e$, and extend your binary operation to make $e$ act as an identity.  So I think this might explain why the some people suggested in the comments that this type of structure might not be very interesting, and why it doesn't have its own name.
