Associativity is not to be confused with commutativity, which addresses whether or not the order of two operands changes the result.
42.
In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle.
43.
The resulting operations satisfy the axioms of a field except that the commutativity of multiplication requires Pappus's hexagon theorem.
44.
This corresponds to shifting commutativity of assumptions and derivations outside the sequent calculus, whereas LK embeds it within the system itself.
45.
It turns out that requiring the cancellative property in a monoid is not required to perform the Grothendieck construction commutativity is sufficient.
46.
There are, however, analogues of the nilradical defined for noncommutative rings, that coincide with the nilradical when commutativity is assumed.
47.
It must also satisfy certain criteria, such as commutativity under addition and rotational invariance ( are these the only ones ? ).
48.
Proof of commutativity can be seen by letting one summand shrink until it is very small and then pulling it along the other knot.
49.
Many of the other familiar properties of the derivative follow from this, such as multilinearity and commutativity of the higher-order derivatives.
50.
Rings that also satisfy commutativity for multiplication ( such as the ring of integers ) are called "'commutative rings " '.
