Universal Properties
We say that a loop property P is universal if it is isotopy invariant, that is, P holds for a loop L if and only if P holds for all loop isotopes of L. Clearly, it is enough to check if P holds for all principal isotopes of L.
For example, since the isotopes of a commutative loop need not be commutative, commutativity is not universal. However, associativity and being an abelian group are universal properties. In fact, every group is a G-loop.
Read more about this topic: Isotopy Of Loops
Famous quotes containing the words universal and/or properties:
“... the ordinary is simply the universal observed from the surface, that the direct approach to reality is not without, but within. Touch life anywhere ... and you will touch universality wherever you touch the earth.”
—Ellen Glasgow (1873–1945)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)