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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

Jean-Baptiste
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  • The quoted sentence from the Wikipedia "Inverse element" article was incorrect. It was fixed about a year and a half after this question was asked. – Michael Kinyon Jul 31 '22 at 01:24

1 Answers1

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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 $aa=ea=a$.

MattAllegro
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Geoffrey Trang
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    Excellent example, here keeping the underlying set small really helps! – MattAllegro Apr 11 '20 at 17:01
  • The above reply is correct in what it says, and is even helpful, but I don't believe it answers the OP's question. The OP wanted to know what one should call a magma with an inverse and an identity which is not a quasigroup, and the above response doesn't name anything -- it just gives an example of something with the required properties. I would also be interested in an answer to the OP's question, were someone to provide one. – KesterKester Feb 26 '21 at 01:41