Can anyone point me toward a simple example of a non-associative algebraic loop (i.e. a quasigroup with an identity) for which at least one element has a left inverse which is not equal to its right inverse?
That's the end of the question, but here is a little extra detail in case it helps:
I can find plenty of non-associative loops for which the opposite condition is true, i.e. for which the left and right inverses for each element are identical to each other. The simplest such example I have come across had this Cayley table:
0 1 2 3 4 5 6 7
1 7 5 0 6 2 4 3
2 6 7 5 0 3 1 4
3 0 6 7 5 4 2 1
4 5 0 6 7 1 3 2
5 2 3 4 1 7 0 6
6 4 1 2 3 0 7 5
7 3 4 1 2 6 5 0
which I found here: http://ericmoorhouse.org/pub/bol/htmlfiles8/8_1_4_0.html This example is clearly a loop as it has (1) the Latin-square property required by quasigroups, and (2) an identity, which in this case is zero. It is also non-associative (which I require) since 3.(4.5)=3.1=0 while (3.4).5=5.5=7. However: all the zeros are spaced symmetrically about the leading diagonal so left and right inverses are identical. That's not what I want!
A good answer to my question would be a similar Cayley table which has both an identity element and the Latin square property and which is not associative, but which has the identity elements (i.e. the zeros) not distributed symmetrically about the leading diagonal.
Note that an identity (for the purposes of our question, and Loops in general) must be both a left and right identity. Therefore, the following Cayley table is not a valid answer (even though it is a Latin square with non-symmetrically placed zeros) because it demonstrates only a left identity not a right identity (and because it is probably also associative .. though I didn't check this):
0 2 1
1 0 2
2 1 0
The (not necessarily authoritative!) wikipedia page https://en.wikipedia.org/wiki/Quasigroup#Loops suggests that Loops of the form I am looking for do exist. But if wikipedia is wrong, and there are no non-associative Loops with differing left and right inverses, then a proof of that would also be acceptable as an answer!
P.S.:
My question is almost the opposite of a very old one which wanted to know about loops which had two-sided inverses: Examples of loops which have two-sided inverses.
That question strikes me as strange, in retrospect, because EVERY example of a non-associative loop I have found in the last hour would have answered that question — almost.
Comment added after this question was successfully answered:
The accepted answer referred to the following Cayley table. I copy it here so that future readers will not have to rely on the stability of a hyperlink in order to be able to read the answer.
∘ 0 1 2 3 4
0 0 1 2 3 4
1 1 4 0 2 3
2 2 3 4 1 0
3 3 0 1 4 2
4 4 2 3 0 1
Best Answer
Pointer: https://math.stackexchange.com/a/1444602/123905
The referenced MSE answer exhibits a five element example of a non-associative loop having at least one element with distinct left- and right-inverses.