General algebraic systems Congruences and quotients Concept
Congruence lattice
Summary
The congruence lattice of an algebra is the set of all its congruences, ordered by inclusion. It is an algebraic lattice, and it records all the quotient algebras of the algebra and how they relate. Many properties of algebras and varieties, such as simplicity, subdirect irreducibility and congruence modularity, are properties of this lattice. By a theorem of Grätzer and Schmidt, every algebraic lattice is isomorphic to the congruence lattice of some algebra.
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Further reading
About this page
| Tag | REE07CSXHR |
|---|---|
| URI | m2://REE07CSXHR |
| Kind | concept |
| Writing | W0 stub |
| MSC | 08A30, 06B15 |
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Used by
- Factor congruence
- Simple algebra
- Congruence-distributive variety
- Correspondence theorem for congruences
- Congruence-modular variety
- Permuting congruences
- Subdirectly irreducible algebra
- Finite lattice representation problem
- Congruence lattice problem
Results about this
- Correspondence theorem for congruences
- Finite lattice representation problem
- Congruence lattice problem