TITLE: Is there a term for a magma with identity (only)?
QUESTION [11 upvotes]: 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?

REPLY [10 votes]: 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.