Core Universal Algebra
Congruences and Quotient Algebras
Normal subgroups and ideals are coordinates for something more basic. The congruence is the object; the distinguished subobject is a convenience that most signatures do not provide.
- State the compatibility condition and verify it for a given relation.
- Construct the quotient algebra and check the operations are well defined.
- Explain why groups and rings admit a subobject coordinatisation and semigroups do not.
- Compute the congruence generated by a set of pairs.
- Recognise principal congruences and their role.
01The compatibility condition
An equivalence relation θ on A is a congruence when it is compatible with every operation: whenever corresponding arguments are θ-related, the resulting values are θ-related.
The reformulation deserves emphasis. A congruence on A is exactly an equivalence relation that is a subalgebra of A × A. Many facts about congruences follow immediately from facts about subalgebras once this is noticed.
02The quotient algebra
- input: algebra A of type F, congruence θ ∈ Con A
- universe: A/θ := { a/θ : a ∈ A }, the set of θ-classes
- for each n-ary f ∈ F, define:
- f^{A/θ}(a₁/θ, …, aₙ/θ) := f^A(a₁,…,aₙ)/θ
- well-definedness: if aᵢ/θ = bᵢ/θ for all i, then ⟨aᵢ,bᵢ⟩ ∈ θ,
- so ⟨f(a⃗), f(b⃗)⟩ ∈ θ by compatibility, so the classes agree
- output: algebra A/θ of the same type F
The natural map ν : A → A/θ sending a to a/θ is a surjective homomorphism with kernel θ. Together with the kernel construction in the other direction this gives the bijection between congruences and surjective homomorphic images.
03Why subobjects work in groups and fail elsewhere
A semigroup congruence is not determined by a single class. There is no distinguished element whose class carries the information, and the number of congruences on a finite semigroup can far exceed the number of subsemigroups. Any intuition transferred wholesale from group theory will mislead here, and this is the practical reason universal algebra insists on working with the relation itself.
The precise reason groups work is that they are congruence-permutable and have a constant in the type; the Mal'cev term x · y⁻¹ · z is what converts a congruence into its identity class and back. Where no such term exists, no coordinatisation exists.
04Generated congruences
Arbitrary intersections of congruences are congruences, so Con A is a closure system and the congruence generated by a set of pairs exists.
- input: algebra A, set X ⊆ A × A of pairs
- close X under the equivalence axioms: reflexivity, symmetry, transitivity
- close under compatibility: whenever ⟨aᵢ,bᵢ⟩ present and f n-ary,
- add ⟨f(a₁,…,aₙ), f(b₁,…,bₙ)⟩
- iterate both closures to a fixed point
- equivalently: ⟨c,d⟩ ∈ Θ(X) iff a finite chain of unary polynomial
- images of pairs from X connects c to d
- output: Θ(X), the least congruence containing X
The polynomial-chain description matters: it says that c and d are congruent under Θ(X) exactly when there is a finite sequence stepping from c to d, each step applying a unary polynomial to one of the generating pairs. Finiteness of the chain is what makes Θ finitary.
05Principal congruences
The congruence generated by a single pair, written Θ(a, b), is called principal. Principal congruences are the compact elements' building blocks and appear constantly.
Principal congruence formulas — first-order formulas expressing membership in Θ(a, b) uniformly — are a favourite tool of universal algebraists and are developed in the Model-Theoretic stream.
Frequently asked
Is every equivalence relation on a group a congruence?
No. Only those whose identity class is a normal subgroup. On the cyclic group of order four there are equivalence relations with two classes of size two that are not congruences, because the induced operation would be ill defined.
Can two different congruences give isomorphic quotients?
Yes, easily. On a set with a trivial operation many distinct congruences produce quotients of the same size and structure. The correspondence is between congruences and surjective homomorphisms, not between congruences and isomorphism classes of quotients.
Why is θ a subuniverse of A × A?
Because the compatibility condition says exactly that applying an operation coordinatewise to pairs in θ yields a pair in θ, which is the closure condition for the product algebra. This reformulation is genuinely useful: it converts congruence questions into subalgebra questions about A × A, where different tools apply.
- 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.