Core Universal Algebra

Homomorphisms and the Isomorphism Theorems

The three isomorphism theorems and the correspondence theorem are proved once here, in arbitrary type, and never need proving again for any specific structure.

Engineering · Mathematics4 min readKV-MATH-0215

Learning objectives

Define homomorphism, embedding, epimorphism and isomorphism for arbitrary type.
Prove the first isomorphism theorem from the kernel correspondence.
State the second and third isomorphism theorems in congruence language.
Apply the correspondence theorem to intervals in Con A.
Recognise the congruence extension property and where it fails.

01Homomorphisms in arbitrary type

A map α : A → B between algebras of the same type is a homomorphism when it commutes with every operation.

α(f^A(a₁,…,aₙ)) = f^B(α(a₁),…,α(aₙ)) for every n-ary f ∈ F

For nullary f this reads α(c^A) = c^B: constants are preserved. This is why the type matters for what counts as a homomorphism.

Kinds of homomorphism
Name	Condition	Note
Homomorphism	commutes with operations	the base notion
Embedding	injective homomorphism	image is a subalgebra isomorphic to source
Epimorphism	surjective homomorphism	image is all of B
Isomorphism	bijective homomorphism	inverse is automatically a homomorphism
Endomorphism	homomorphism A → A	forms a monoid under composition
Automorphism	isomorphism A → A	forms a group

Unlike lattices under monotone maps, a bijective homomorphism of algebras is always an isomorphism: the inverse map automatically commutes with the operations. The lattice pathology arose because monotone maps are not homomorphisms.

02The first isomorphism theorem

ProcedureFrom a homomorphism to a factorisation

in: α → out: canonical factorisation α = β ∘ ν through A/ker(α)

input: homomorphism α : A → B
compute θ := ker(α) = { ⟨a,b⟩ : α(a) = α(b) } — a congruence on A
form the quotient A/θ and the natural map ν : A → A/θ
define β : A/θ → B by β(a/θ) := α(a)
well defined: a/θ = b/θ means α(a) = α(b)
β is an injective homomorphism with image α(A)
therefore A/ker(α) ≅ α(A), and α = β ∘ ν

Correctness: well-definedness is exactly the kernel condition, injectivity its converse. This single proof subsumes the first isomorphism theorem for groups, rings, modules, lattices and every other structure.

Key resultEvery homomorphism factors as surjection then injection

The factorisation is unique up to isomorphism and is the reason the congruence lattice controls the homomorphic images. Homomorphic images of A correspond exactly to congruences on A, which is the H in HSP made concrete.

03The second and third theorems

Second isomorphism theorem

Quotients of quotients

For θ ≤ φ in Con A, the relation φ/θ is a congruence on A/θ and (A/θ)/(φ/θ) ≅ A/φ. Collapsing in two stages equals collapsing once.

Third isomorphism theorem

Restriction to a subalgebra

For a subalgebra B ≤ A and θ ∈ Con A, the restriction θ|B is a congruence on B and B/(θ|B) embeds in A/θ, with image the subalgebra generated by the θ-classes meeting B.

The second theorem is the one that makes interval reasoning in Con A possible: it identifies the congruence lattice of A/θ with the interval [θ, ∇] in Con A. That identification is used constantly and is the content of the correspondence theorem.

04The correspondence theorem

Con(A/θ) ≅ [θ, ∇] ⊆ Con A

An order isomorphism between the congruence lattice of the quotient and the interval above θ in the congruence lattice of A.

Congruences on the quotient correspond to congruences above θ
A congruence on A/θ pulls back to a congruence on A containing θ, and pushes forward again to the original.
The correspondence is a lattice isomorphism
Meets and joins are preserved in both directions, so the interval is a faithful copy of Con(A/θ).
Consequence: simplicity of quotients
A/θ is simple exactly when θ is a coatom of Con A — a maximal congruence below ∇. This is how maximal congruences are recognised.

05The congruence extension property

An algebra has the congruence extension property (CEP) when every congruence on every subalgebra extends to a congruence on the whole algebra. A variety has CEP when all its members do.

CautionCEP is a genuine restriction, not a formality

Groups have it; semigroups do not. Where CEP fails, a congruence on a subalgebra can be an obstruction with no global counterpart, and arguments that pass congruences between a subalgebra and its parent break down. Check CEP before using such an argument.

CEP interacts with the classification: every congruence-distributive variety generated by a finite algebra has good behaviour here, and discriminator varieties have CEP outright. It is one of the standard hypotheses in the structure theory of the Boolean Constructions stream.

Frequently asked

Why is a bijective homomorphism automatically an isomorphism?

Because the inverse of a bijective homomorphism commutes with the operations: apply α to both sides of the required identity and use that α is injective and a homomorphism. The lattice-theoretic failure discussed elsewhere concerned monotone bijections, which are not homomorphisms.

Do the isomorphism theorems require any hypotheses on the type?

None at all. They hold for algebras of arbitrary type, which is precisely the point — proving them once here removes the need to prove them separately for every structure. This is the clearest single dividend of the universal-algebraic approach.

Does the correspondence theorem hold for subalgebras too?

Not in the same clean form. There is no general order isomorphism between Sub(A/θ) and an interval in Sub(A); the third isomorphism theorem gives an embedding rather than an isomorphism. The asymmetry between congruences and subalgebras runs throughout the subject.

Sources and further reading

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.

Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.