# Category:Complete Lattices

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This category contains results about **Complete Lattices**.

Definitions specific to this category can be found in **Definitions/Complete Lattices**.

### Definition 1

Let $\struct {S, \preceq}$ be a lattice.

Then $\struct {S, \preceq}$ is a **complete lattice** if and only if:

### Definition 2

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

Then $\struct {S, \preceq}$ is a **complete lattice** if and only if:

- $\forall S' \subseteq S: \inf S', \sup S' \in S$

That is, if and only if all subsets of $S$ have both a supremum and an infimum.

## Subcategories

This category has the following 5 subcategories, out of 5 total.

## Pages in category "Complete Lattices"

The following 50 pages are in this category, out of 50 total.

### A

### C

- Characterization of Compact Element in Complete Lattice
- Characterization of Completely Prime Filter in Complete Lattice
- Characterization of Completely Prime Ideal in Complete Lattice
- Characterization of Locale
- Closed Interval in Complete Lattice is Complete Lattice
- Complete Join Semilattice is Dual to Complete Meet Semilattice
- Complete Lattice has Both Greatest Element and Smallest Element
- Complete Lattice is Bounded

### E

### I

- Ideals form Complete Lattice
- Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice
- Image under Increasing Mapping equal to Special Set is Complete Lattice
- Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice
- Infimum and Supremum of Subgroups
- Infimum of Subgroups in Lattice
- Infimum Precedes Coarser Infimum

### O

### R

### S

- Set of Division Subrings forms Complete Lattice
- Set of Ideals forms Complete Lattice
- Set of Subfields forms Complete Lattice
- Set of Subrings forms Complete Lattice
- Set of Subsets which contains Set and Intersection of Subsets is Complete Lattice
- Shift Mapping is Lower Adjoint iff Meet is Distributive with Supremum
- Superset of Order Generating is Order Generating
- Supremum of Subgroups in Lattice