Topic summary
Pointwise order
Extracted from the Wikipedia article Pointwise.
Pointwise relations
In order theory it is common to define a pointwise partial order on functions. With A, Bposets, the set of functions A → B can be ordered by defining f ≤ g if (∀x ∈ A) f(x) ≤ g(x). Pointwise orders also inherit some properties of the underlying posets. For instance if A and B are continuous lattices, then so is the set of functions A → B with pointwise order. Using the pointwise order on functions one can concisely define other important notions, for instance: