Core Structure Theory
The Congruence Extension Property
The condition that congruences on a subalgebra extend to the whole algebra, the varieties that satisfy it, and its role in transferring structural results.
Learning objectives
- State the congruence extension property precisely
- Identify varieties with and without CEP
- Explain how CEP simplifies arguments about subalgebras
The property
An algebra A has the CEP if for every subalgebra B of A and every congruence θ on B, there is a congruence φ on A whose restriction to B is θ — that is, φ ∩ (B × B) = θ.
A class of algebras has the CEP when every member does. The notation φ↾B is used for the restriction of φ to B.
The easy direction is automatic: restricting a congruence on A to a subalgebra B always yields a congruence on B. CEP asserts the converse — that every congruence downstairs is reached from upstairs.
Which varieties have it
| Variety | CEP? | Comment |
|---|---|---|
| Abelian groups | Yes | Subgroups are normal; every subgroup extends |
| Groups | No | A normal subgroup of a subgroup need not be normal above |
| Rings | No | An ideal of a subring need not extend |
| R-modules | Yes | Submodules always extend |
| Lattices | Yes | A standard result |
| Distributive lattices | Yes | |
| Boolean algebras | Yes | Filters extend |
| Semigroups | No | |
| Discriminator varieties | Yes | A defining structural strength |
Why groups fail
Take the symmetric group S3 and the subgroup B generated by a transposition, of order two. The congruence on B collapsing it entirely corresponds to the normal subgroup B of itself. But B is not normal in S3, and the congruences of S3 correspond to its normal subgroups — only the trivial group, the alternating group and the whole group. None restricts to the required congruence on B.
Why CEP is useful
With CEP, structural information passes freely downward. Statements about the congruence lattice of A constrain the congruence lattices of all its subalgebras, and constructions performed on subalgebras lift.
- Subdirect representations restrict. A subdirect decomposition of A induces one on any subalgebra.
- Simplicity is inherited in a controlled way. In a variety with CEP, subalgebras of simple algebras are constrained.
- Boolean product representations transfer. Chapter IV uses CEP repeatedly when moving representations between an algebra and its subalgebras.
- Amalgamation arguments simplify. CEP is one of the standard hypotheses in amalgamation and injectivity results.
Discriminator varieties — the central objects of Chapter IV §9 — have CEP, and this is one of the properties that makes them so well-behaved.
Relation to other conditions
CEP is logically independent of the congruence lattice conditions. Lattices are congruence-distributive and have CEP; groups are congruence-modular and congruence-permutable but lack CEP; and there are varieties with CEP that are neither modular nor distributive.
Frequently asked questions
Does CEP have a Mal'cev-style characterisation?
There are term conditions equivalent to CEP for varieties, though they are less clean than Mal'cev's condition for permutability. Day's and Gumm's work on congruence conditions supplies the relevant characterisations.
Is the extending congruence unique?
No. CEP asserts existence, not uniqueness. Typically there is a smallest extension and often many larger ones.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section II.5, book pages 44-46.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.