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:
“We call contrary to nature what happens contrary to custom; nothing is anything but according to nature, whatever it may be, Let this universal and natural reason drive out of us the error and astonishment that novelty brings us.”
—Michel de Montaigne (1533–1592)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)