Pointwise Relations
In order theory it is common to define a pointwise partial order on functions. With A, B posets, the set of functions A → B can be ordered by f ≤ g if and only 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:
- A closure operator c on a poset P is a monotone and idempotent self-map on P (i.e. a projection operator) with the additional property that idA ≤ c, where id is the identity function.
- Similarly, a projection operator k is called a kernel operator if and only if k ≤ idA.
An example of infinitary pointwise relation is pointwise convergence of functions — a sequence of functions
with
converges pointwise to a function if for each in
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