Home.

A caution pertaining to Cayley tables. Below are two ordinary Latin squares, the same except that the rows have been rearranged. For most combinatorical purposes, the two squares would be regarded as essentially the same thing.

Latin square 1
acdb
dbac
bdca
cabd
 
Latin square 2
cabd
acdb
dbac
bdca

Below, each of these Latin squares has been installed as the body of the Cayley table for a binary quasigroup:

F = 4::91
from
Latin square 1
second input
abcd
first inputa ac db
b db ac
c bd ca
d ca bd
 
G = 4::292
from
Latin square 2
second input
abcd
first inputa ca bd
b ac db
c db ac
d bd ca

For all p, observe that F (p, p) = p, while by contrast G (p, p) ≠ p. With the loss of idempotence, F and G would be deemed non-isomorphic for virtually all algebraic purposes, even though they are based on equivalent Latin squares. What happened? The outputs of the operations were changed, but not the inputs, resulting in an inconsistency.

The same principle naturally applies to Latin cubes and ternary quasigroups.