Core Structure Theory
A Catalogue of Algebras: Groups, Rings, Lattices
The standard algebraic structures presented uniformly as algebras, showing how each familiar definition translates into a type plus a set of identities.
Learning objectives
- Present groups, rings and lattices in the universal-algebraic format
- Identify which classes are varieties and which are not
- Recognise the pattern that decides whether a class is equationally definable
Groups
A group is an algebra ⟨G, ·, −1, e⟩ of type ⟨2, 1, 0⟩ satisfying:
- x · (y · z) ≈ (x · y) · z
- x · e ≈ x and e · x ≈ x
- x · x−1 ≈ e and x−1 · x ≈ e
Every axiom is an identity, so by Birkhoff's theorem the class of groups is closed under homomorphic images, subalgebras and direct products — all three of which are standard facts of group theory, here obtained at a stroke.
Abelian groups add commutativity and remain a variety. So do nilpotent groups of a fixed class, and solvable groups of a fixed derived length — each is defined by identities.
Rings and modules
A ring with unit is an algebra of type ⟨2, 2, 1, 0, 0⟩ whose additive part is an abelian group, whose multiplication is associative with unit, and in which multiplication distributes over addition on both sides. All axioms are identities, so rings form a variety.
For a fixed ring R, an R-module is an algebra with the abelian group operations plus one unary operation for each element of R, representing scalar multiplication by that element. The module axioms then become identities in this type.
Scalar multiplication is a map R × M → M, which is not an operation on M because one argument comes from outside. Splitting it into a family of unary operations — one per ring element — brings it inside the type. The type may then be infinite, which is permitted.
Lattices, Boolean algebras and semilattices
| Structure | Type | Variety? |
|---|---|---|
| Semilattice | ⟨2⟩ | Yes — commutative, associative, idempotent |
| Lattice | ⟨2, 2⟩ | Yes — L1–L4 |
| Bounded lattice | ⟨2, 2, 0, 0⟩ | Yes |
| Distributive lattice | ⟨2, 2⟩ | Yes |
| Modular lattice | ⟨2, 2⟩ | Yes |
| Boolean algebra | ⟨2, 2, 1, 0, 0⟩ | Yes |
| Heyting algebra | ⟨2, 2, 2, 0, 0⟩ | Yes |
Classes that are not varieties
Not every familiar class is a variety, and the reasons are instructive.
| Class | Fails closure under | Why |
|---|---|---|
| Fields | Products | A product of two fields has zero divisors |
| Integral domains | Products | Same reason |
| Simple groups | Products, subalgebras | Simplicity is not equational |
| Torsion-free abelian groups | Homomorphic images | A quotient can acquire torsion |
| Finite groups | Products | An infinite product of finite groups is infinite |
| Cancellative semigroups | Homomorphic images | Cancellation is a quasi-identity, not an identity |
The pattern is exact: a class is a variety precisely when it is closed under H, S and P. Failure of any one closure certifies that no set of identities can define the class, however it is axiomatised.
Classes like cancellative semigroups and torsion-free abelian groups are defined by quasi-identities — conditional equations of the form “if these equations hold then this one does”. Quasivarieties are closed under S and P but not H, and they form a well-behaved theory one level up from varieties.
Frequently asked questions
Is the class of all algebras of a fixed type a variety?
Yes, trivially — it is defined by the empty set of identities, and it is closed under H, S and P. It is the largest variety of that type.
Why are fields not a variety when they seem so well behaved?
Because the axiom that non-zero elements have inverses is not an identity — it is conditional on being non-zero. Concretely, the product of two fields contains elements like (1,0) with no inverse, so the class is not closed under products.
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.1, book pages 26-30.
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.