Varieties, Free Algebras and Equational Logic
Varieties and the Variety Generated by a Class
Varieties as classes closed under H, S and P, the variety generated by a class, and the lattice of subvarieties.
Learning objectives
- Define variety and verify closure in examples
- Compute V(K) = HSP(K)
- Describe the lattice of subvarieties of a given variety
Definition
A class of algebras of a fixed type closed under homomorphic images, subalgebras and direct products. Equivalently, a class K with H(K) = S(K) = P(K) = K.
The one-element algebra lies in every variety, as the empty product. So every variety is non-empty, and the smallest variety of a given type is the class of all trivial algebras.
Examples of varieties
| Variety | Generated by | Notes |
|---|---|---|
| Boolean algebras | 2 | One finite generator |
| Distributive lattices | The two-element chain | One finite generator |
| Abelian groups | The class of cyclic groups | Not finitely generated |
| All groups | — | Not generated by any set of finite algebras |
| K-vector spaces | K as a one-dimensional space | One generator |
| Semilattices | The two-element semilattice | One finite generator |
| Modular lattices | — | Not generated by finitely many finite lattices |
A variety generated by a finite algebra is locally finite: every finitely generated member is finite. This follows because the free algebra on n generators embeds in a power of the generating algebra and is therefore finite. Boolean algebras and distributive lattices are locally finite; groups are not.
The lattice of subvarieties
The subvarieties of a variety V, ordered by inclusion, form a complete lattice. Meets are intersections; joins are the varieties generated by unions.
| Variety | Lattice of subvarieties |
|---|---|
| Boolean algebras | Two elements: trivial and all |
| Distributive lattices | Three elements: trivial, one-element chains, all |
| Lattices | Continuum many subvarieties |
| Groups | A proper class-sized problem; extremely complicated |
| Abelian groups | Isomorphic to the divisibility lattice of the natural numbers extended by ∞ |
The subvarieties of abelian groups are exactly the classes defined by nx ≈ 0 for each n, together with the whole variety. The lattice is therefore the divisor lattice of the naturals with a top element added — a rare instance of a completely determined subvariety lattice.
Why varieties and not something else
The choice of H, S and P as the defining closures is not arbitrary. Birkhoff's theorem shows they characterise exactly the equationally definable classes, which is the strongest possible justification.
Each level up trades structural strength for generality. Varieties have free algebras and the full weight of the HSP machinery; quasivarieties retain free algebras but lose closure under quotients; elementary classes retain neither.
Frequently asked questions
Is the intersection of two varieties a variety?
Yes. Each closure condition is preserved by intersection, so arbitrary intersections of varieties are varieties — which is why the subvarieties form a complete lattice.
Can a variety be generated by a single infinite algebra?
Yes, and often only by one. The variety of all groups is generated by a single infinite group — for instance a free group of countable rank.
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.9, book pages 66-67.
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.