Order, lattices, ordered algebraic structures Complete lattices and completions Concept
Complete lattice
Summary
A complete lattice is a partially ordered set in which every subset, finite or infinite, has a supremum and an infimum. It is the setting of many fixed-point theorems, such as the Knaster–Tarski theorem. Power sets, lattices of subgroups and lattices of closed sets of a topological space are complete lattices. Every partially ordered set embeds in a complete lattice, its Dedekind–MacNeille completion.
Where this leads
21 pages build on this one.
- Grätzer–Schmidt theorem
- Formal concept analysis
- Continuous geometry
- Compact element
- Frame
- Lattice of clones
- Conditionally complete lattice
- Subalgebra lattice
- Dedekind–MacNeille completion
- Birkhoff–Frink theorem
- Quantale
- Davis's converse of the Knaster–Tarski theorem
- Algebraic lattice
- Lattice of equational theories
- Knaster–Tarski theorem
- Complete Boolean algebra
- Completely distributive lattice
- Completeness in order theory
- Lattice of subvarieties
- Congruence lattice
- Continuous lattice
Further reading
About this page
| Tag | D7A0ZGZT5S |
|---|---|
| URI | m2://D7A0ZGZT5S |
| Kind | concept |
| Writing | W0 stub |
| MSC | 06B23 |
| Specializes | Lattice |
| Same as |
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Used by
- Grätzer–Schmidt theorem
- Formal concept analysis
- Continuous geometry
- Compact element
- Frame
- Lattice of clones
- Conditionally complete lattice
- Subalgebra lattice
- Dedekind–MacNeille completion
- Birkhoff–Frink theorem
- Quantale
- Davis's converse of the Knaster–Tarski theorem
- Algebraic lattice
- Lattice of equational theories
- Knaster–Tarski theorem
- Complete Boolean algebra
- Completely distributive lattice
- Completeness in order theory
- Lattice of subvarieties
- Congruence lattice
- Continuous lattice