# Magma with inverse and identity yet not a quasigroup

What kind of non-associative structure is this? Only a subset satisfies the latin square criteria, and it also isn't Cancellative, but all elements have an inverse, and the identity element exists.

Wikipedia claims: "A unital magma in which all elements are invertible is called a loop."

But nLab says under the quasigroup entry: "Note that, in the absence of associativity, it is not enough (even for a loop) to say that every element has an inverse element (on either side); instead, you must say that division is always possible."

And it defines loop as a quasigroup with an identity element. The definitions seem to conflict, is it possible to be a loop without also being a quasigroup?

They also mention below that a magma where each element is invertible is considered a quasigroup, but they later also require the latin square criteria. http://www.cs.cas.cz/portal/AlgoMath/AlgebraicStructures/StructuresWithOneOperation/Groupoids/Groupoid.htm

Such an example already exists with 3 elements. Consider the set $$\{e, a, b\}$$ with e being the identity, and a and b being inverses of each other and also idempotent. This is not a quasigroup because $$aa=ea=a$$.