# Category:Pointwise Operations

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This category contains results about **Pointwise Operations**.

Definitions specific to this category can be found in Definitions/Pointwise Operations.

Let $S$ be a set.

Let $\struct {T, \circ}$ be an algebraic structure.

Let $T^S$ be the set of all mappings from $S$ to $T$.

Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings.

Then the operation $f \oplus g$ is defined on $T^S$ as follows:

- $f \oplus g: S \to T: \forall x \in S: \map {\paren {f \oplus g} } x = \map f x \circ \map g x$

The operation $\oplus$ is called the **pointwise operation on $T^S$ induced by $\circ$**.

### Induced Structure

The algebraic structure $\struct {T^S, \oplus}$ is called the **algebraic structure on $T^S$ induced by $\circ$**.

## Subcategories

This category has the following 6 subcategories, out of 6 total.

### E

### P

- Pointwise Scalar Multiplication (empty)

## Pages in category "Pointwise Operations"

The following 33 pages are in this category, out of 33 total.

### C

### E

### I

### P

- Pointwise Inverse in Induced Structure
- Pointwise Operation is Composite of Operation with Mapping to Cartesian Product
- Pointwise Operation on Distributive Structure is Distributive
- Pointwise Product of Simple Functions is Simple Function
- Pointwise Sum of Integrable Functions is Integrable Function
- Pointwise Sum of Simple Functions is Simple Function

### S

- Set of Endomorphisms on Entropic Structure is Closed in Induced Structure on Set of Self-Maps
- Set of Endomorphisms on Entropic Structure is Closed in Induced Structure on Set of Self-Maps/Converse
- Structure Induced by Abelian Group Operation is Abelian Group
- Structure Induced by Absorbing Operations is Absorbing
- Structure Induced by Associative Operation is Associative
- Structure Induced by Commutative Operation is Commutative
- Structure Induced by Distributive Operation is Distributive
- Structure Induced by Group Operation is Group
- Structure Induced by Idempotent Operation is Idempotent
- Structure Induced by Lattice Operations is Lattice
- Structure Induced by Left Distributive Operation is Left Distributive
- Structure Induced by Right Distributive Operation is Right Distributive
- Structure Induced by Semigroup Operation is Semigroup
- Structure Induced by Semilattice Operation is Semilattice
- Structure Induced on Set of All Mappings to Ordered Semigroup is Ordered Semigroup
- Structure Induced on Set of Self-Maps on Entropic Structure is Entropic