# Category:Complete Lattices

Jump to navigation
Jump to search

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 2 subcategories, out of 2 total.

## Pages in category "Complete Lattices"

The following 43 pages are in this category, out of 43 total.

### A

### 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