# Category:Suprema

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This category contains results about Suprema in the context of Order Theory.

Definitions specific to this category can be found in Definitions/Suprema.

Let $\struct {S, \preceq}$ be an ordered set.

Let $T \subseteq S$.

An element $c \in S$ is the **supremum of $T$ in $S$** if and only if:

- $(1): \quad c$ is an upper bound of $T$ in $S$
- $(2): \quad c \preceq d$ for all upper bounds $d$ of $T$ in $S$.

If there exists a **supremum** of $T$ (in $S$), we say that:

**$T$ admits a supremum (in $S$)**or**$T$ has a supremum (in $S$)**.

## Subcategories

This category has the following 4 subcategories, out of 4 total.

### E

### I

### S

## Pages in category "Suprema"

The following 30 pages are in this category, out of 30 total.

### C

### F

### I

### S

- Set is Subset of Finite Suprema Set
- Set of Numbers of form n - 1 over n is Bounded Above
- Set of Rational Numbers Strictly between Zero and One has no Greatest or Least Element
- Sum of Indexed Suprema
- Suprema and Infima of Combined Bounded Functions
- Supremum and Infimum are Unique
- Supremum does not Precede Infimum
- Supremum is Dual to Infimum
- Supremum is not necessarily Greatest Element
- Supremum of Absolute Value of Difference equals Supremum of Difference
- Supremum of Empty Set is Smallest Element
- Supremum of Function is less than Supremum of Greater Function
- Supremum of Simple Order Product
- Supremum of Singleton
- Supremum of Subset of Real Numbers May or May Not be in Subset
- Supremum of Suprema over Overlapping Domains
- Supremum of Union of Bounded Above Sets of Real Numbers