0
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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 $aa=ea=a$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.