# 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 $\struct {S, \preccurlyeq}$ 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 $S$
- $(2): \quad c \preccurlyeq d$ for all upper bounds $d$ of $T$ in $S$.

## Subcategories

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

## Pages in category "Suprema"

The following 41 pages are in this category, out of 41 total.

### C

### F

### I

### P

### S

- Semilattice has Unique Ordering such that Operation is Supremum
- 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
- Suprema in Ordered Group
- Supremum and Infimum are Unique
- Supremum does not Precede Infimum
- Supremum in Ordered Subset
- Supremum is Dual to Infimum
- Supremum is not necessarily Greatest Element
- Supremum of Absolute Value of Difference equals Supremum of Difference
- Supremum of Doubleton in Totally Ordered Set
- Supremum of Elements of Sublattice not necessarily Same as for Lattice
- 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 Subgroups in Lattice
- Supremum of Subset of Real Numbers May or May Not be in Subset
- Supremum of Subset Product in Ordered Group
- Supremum of Suprema over Overlapping Domains
- Supremum of Union of Bounded Above Sets of Real Numbers