Varieties, Free Algebras and Equational Logic
Fully Invariant Congruences and Completeness
Fully invariant congruences on the term algebra, their correspondence with equational theories, and the lattice anti-isomorphism between theories and varieties.
Learning objectives
- Define fully invariant congruence and identify examples
- Establish the correspondence with equational theories
- Describe the anti-isomorphism between the theory lattice and the subvariety lattice
Fully invariant congruences
A congruence θ on an algebra A is fully invariant if it is preserved by every endomorphism: ⟨a, b⟩ ∈ θ implies ⟨α(a), α(b)⟩ ∈ θ for every endomorphism α of A.
The fully invariant congruences on A form a complete sublattice of Con A, written ConFI(A). The fully invariant congruence generated by a set S of pairs is written ΘFI(S).
An endomorphism of T(X) is determined by where it sends the variables, so it is exactly a substitution. Full invariance on the term algebra is therefore precisely closure under substitution — rule 5 of equational deduction.
The correspondence
The map sending a set Σ of identities over X to the relation {⟨p, q⟩ : Σ ⊢ p ≈ q} is a bijection between equational theories over X and fully invariant congruences on T(X).
Each rule of equational deduction corresponds to a closure property:
| Rules 1–3 | Rule 4 | Rule 5 |
|---|---|---|
| Equivalence relation | Compatible with operations — a congruence | Closed under substitution — fully invariant |
Identities are pairs of terms. Equational theories are fully invariant congruences on the term algebra. Provability is membership. Every question about equational logic becomes a question about a congruence lattice.
The anti-isomorphism
The lattice of equational theories of a given type is anti-isomorphic to the lattice of varieties of that type. Larger theories correspond to smaller varieties.
| Theory | Variety |
|---|---|
| The smallest theory — only derivable-from-nothing identities | All algebras of the type |
| The largest theory — every identity, including x ≈ y | Trivial algebras only |
| Theory of groups | The variety of groups |
| Theory of abelian groups (larger) | Variety of abelian groups (smaller) |
Free algebras revisited
The construction now closes on itself. The free algebra in a variety V over X is
FV(X) = T(X) / ΘFI(Σ)
where Σ is any equational basis for V. So free algebras, equational theories and fully invariant congruences are three descriptions of the same data.
Three equivalent views of a variety: as a class closed under H, S and P; as the models of a set of identities; and as the fully invariant congruence on the term algebra that those identities generate. Birkhoff's theorem connects the first two, and this section connects the second to the third.
Frequently asked questions
Why do endomorphisms rather than automorphisms appear?
Because substitution need not be invertible — a substitution may collapse two variables to one. Closure under all endomorphisms is the correct condition, and it is strictly stronger than closure under automorphisms.
Is every congruence on the term algebra fully invariant?
No. A congruence identifying x with y but not identifying all pairs of terms fails full invariance, since substituting arbitrary terms for x and y would force more identifications.
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.14, book pages 99-110.
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.