isotope of a groupoid
is called an isotope of (or is isotopic to ) if there is an isotopy .
Some easy examples of isotopies:
If is an isotopy, then so is
for if and , then , so that
If and are isotopies, then so is
From the examples above, it is easy to see that “groupoids being isotopic” on the class of groupoids is an equivalence relation, and that an isomorphism class is contained in an isotopic class. In fact, the containment is strict. For an example of non-isomorphic isotopic groupoids, see the reference below. However, if is a groupoid with unity and is isotopic to a semigroup , then it is isomorphic to . Other conditions making isotopic groupoids isomorphic can be found in the reference below.
An isotopy of the form is called a principal isotopy, where is the identity function on . is called a principal isotope of . If is isotopic to , then is isomorphic to a principal isotope of .
Then is well-defined, since are bijective, for all pairs of elements of . Hence is a groupoid. Furthermore, is an isotopy by definition, so that is a principal isotope of . Finally, , showing that is a bijective homomorphism, and hence an isomorphism. ∎
Any isotope of a quasigroup is a quasigroup.
Suppose is an isotopy, and a quasigroup. Pick . Let be such that and . Let be such that . Set . Then . Similarly, there is such that . Hence is a quasigroup. ∎
On the other hand, an isotope of a loop may not be a loop. Nevertheless, we sometimes say that an isotope of a loop as a loop isotopic to .
- 1 R. H. Bruck: A Survey of Binary Systems. Springer-Verlag. New York (1966).
|Title||isotope of a groupoid|
|Date of creation||2013-03-22 18:35:54|
|Last modified on||2013-03-22 18:35:54|
|Last modified by||CWoo (3771)|