Core Structure Theory
The Correspondence Theorem for Algebras
The bijection between congruences above a fixed congruence and congruences on the quotient — a lattice isomorphism that makes Con of a quotient an interval in Con of the original.
Learning objectives
- State the correspondence theorem and its lattice-theoretic content
- Use it to compute congruence lattices of quotients
- Connect it to maximal congruences and simplicity
The theorem
Let θ be a congruence on A. The map φ ↦ φ/θ is a lattice isomorphism from the interval [θ, ∇] in Con A onto Con(A/θ).
So the congruence lattice of a quotient is not merely related to that of the original — it is an interval in it, up to isomorphism.
Con(A/θ) ≅ [θ, ∇] ⊆ Con A
To find all congruences on a quotient, no new computation is needed: read off the interval above θ in the congruence lattice already computed for A. The order relation and the lattice operations transfer unchanged.
Consequences
- Maximal congruences give simple quotients. A/θ is simple exactly when θ is a maximal element of Con A below ∇ — the interval [θ, ∇] then has just two elements.
- Meet-irreducible congruences give subdirectly irreducible quotients. This is the key to Birkhoff's subdirect representation theorem: A/θ is subdirectly irreducible exactly when θ is completely meet-irreducible.
- Congruence conditions are inherited by quotients. If Con A is distributive, so is every interval, hence so is Con of every quotient. Same for modularity.
- Chains of congruences give chains of quotients. Composition series arguments transfer directly.
Worked application
Congruences of a cyclic group quotient
Let G be cyclic of order 12. Its congruence lattice is isomorphic to the divisor lattice of 12: subgroups of orders 1, 2, 3, 4, 6, 12.
To find the congruences of G/N where N has order 2, take the interval above the corresponding congruence. That interval consists of the congruences corresponding to subgroups of order 2, 4, 6 and 12 — four elements, matching the divisor lattice of 6. And indeed G/N is cyclic of order 6.
For groups the correspondence theorem is normally stated as a bijection between subgroups containing N and subgroups of G/N. The algebraic version is stated for congruences, which for groups correspond to normal subgroups — so the classical statement about all subgroups is slightly stronger in that particular setting, and does not generalise.
Relation to the third isomorphism theorem
The correspondence theorem and the third isomorphism theorem are two views of the same fact. The correspondence theorem describes the bijection between congruences; the third isomorphism theorem identifies the quotients that bijection produces.
Together they say: quotienting twice, first by θ and then by φ/θ, is the same as quotienting once by φ. Neither requires any hypothesis beyond θ ⊆ φ.
Frequently asked questions
Does the correspondence preserve joins and meets?
Yes — it is a lattice isomorphism, not merely an order isomorphism. Joins and meets in the interval correspond exactly to joins and meets in Con(A/θ).
Can Con(A/θ) be larger than Con(A)?
No. It is isomorphic to an interval in Con(A), hence never larger in the sense of containing more elements than that interval. Quotienting can only lose congruences, never gain them.
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.6, book pages 52-54.
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.