Core Structure Theory
Quotient Algebras and the Natural Map
The construction of the quotient algebra modulo a congruence, the natural surjection onto it, and the universal property that makes quotients the right notion.
Learning objectives
- Construct the quotient algebra and verify its operations are well defined
- Describe the natural map and its kernel
- State the universal property of the quotient
The construction
Let θ be a congruence on A. The quotient algebra A/θ has universe A/θ, the set of θ-classes, with operations defined by fA/θ(a1/θ,…,an/θ) = fA(a1,…,an)/θ.
Well-definedness is exactly the substitution property, and it is the only thing that needs checking. Once verified, A/θ is an algebra of the same type as A.
Quotients never change the type. Consequently, if A satisfies an identity then so does A/θ — identities are preserved by homomorphic images. This is one third of Birkhoff's theorem, and it follows directly from the construction.
The natural map
The map νθ: A → A/θ sending a to a/θ.
It is a surjective homomorphism by construction, and its kernel is θ itself. So every congruence is the kernel of a homomorphism — the converse of the fact that every homomorphism's kernel is a congruence.
The universal property
Let θ be a congruence on A and let α: A → B be a homomorphism with θ ⊆ ker(α). Then there is a unique homomorphism β: A/θ → B with α = β ∘ νθ.
The map β is forced: it must send a/θ to α(a), and the hypothesis θ ⊆ ker(α) is exactly what makes this well defined.
The universal property characterises A/θ up to unique isomorphism without reference to the construction. Every use of quotients in the later chapters is ultimately an application of it — the isomorphism theorems, the correspondence theorem and the construction of free algebras in a variety all reduce to this factorisation.
Extremes
| Congruence | Quotient | Natural map |
|---|---|---|
| Δ — the identity relation | A itself | The identity map; an isomorphism |
| ∇ — all pairs | The one-element algebra | The constant map |
Every other congruence gives a quotient strictly between these. An algebra is simple when Δ and ∇ are its only congruences, so the only quotients are itself and the trivial algebra.
Frequently asked questions
Is the quotient of a group by a congruence the same as the quotient by a normal subgroup?
Yes. The congruence corresponding to a normal subgroup N has classes the cosets of N, so A/θ is exactly G/N with its usual operations.
Does the quotient of an algebra in a variety stay in the variety?
Yes. Varieties are closed under homomorphic images by definition, and the quotient is a homomorphic image. This is why quotient constructions are always available within a variety.
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 39-40.
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.