# Category:Increasing Mappings

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This category contains results about **Increasing Mappings**.

Definitions specific to this category can be found in Definitions/Increasing Mappings.

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

Let $\phi: S \to T$ be a mapping.

Then $\phi$ is **increasing** if and only if:

- $\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$

Note that this definition also holds if $S = T$.

## Also see

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Increasing Mappings"

The following 24 pages are in this category, out of 24 total.

### C

### I

### M

- Mapping at Element is Supremum implies Mapping is Increasing
- Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing
- Mapping is Constant iff Increasing and Decreasing
- Mapping on Integers is Endomorphism of Max or Min Operation iff Increasing

### O

### S

- Semilattice Homomorphism is Order-Preserving
- Strictly Increasing Mapping is Increasing
- Strictly Increasing Mapping on Well-Ordered Class
- Strictly Order-Preserving and Order-Reversing Mapping on Strictly Totally Ordered Set is Injection
- Suprema Preserving Mapping on Ideals is Increasing
- Supremum is Increasing relative to Product Ordering
- Supremum of Ideals is Increasing