In order to answer this, we have to remember what a quasigroup and a loop are. A quasigroup is a set with an operation * where, for any a and b in the set, the equations a * x = b and y * a = b each have exactly one solution. Another way to think about it is that every row and every column of the operation table has each element exactly once, just like a Sudoku. A loop is a quasigroup that also has an identity element e, meaning e * a = a and a * e = a for every a. So, we need a quasigroup that has no identity element.
A good example is the integers with subtraction, where a * b = a − b. Let check the two equations. For a − x = b, we solve for x by subtracting a on both sides and multiplying by −1 so it becomes:
x = a − b
For y − a = b, we add a on both sides so it becomes:
y = a + b
Both solutions are integers and there is only one of each, so the integers with subtraction is a quasigroup.
Now let check if there is an identity. For a − e = a to work for every a, we need:
e = 0
But we also need e − a = a, and plugging in e = 0 gives:
0 − a = −a
This only equals a when a = 0. For example, 0 − 5 = −5, not 5. So no element works on both sides, which means there is no identity element and it is not a loop.
If you want a finite example, you can use the same idea with the numbers 0, 1 and 2, where a * b = (a − b) mod 3. The table becomes:
* | 0 1 2
0 | 0 2 1
1 | 1 0 2
2 | 2 1 0
If you notice, every row and every column has 0, 1 and 2 exactly once, so it is a quasigroup. 0 works as an identity on the right since a − 0 = a, but none of the rows match 0, 1, 2 in order, so there is no identity on the left and it is not a loop either.
So, our final answer is the integers under subtraction, a * b = a − b, which is a quasigroup but not a loop because it has no identity element.
