Orientation
Preliminaries: Sets, Classes, Relations and Notation
Universal algebra quantifies over classes that are provably not sets. The notational preliminaries are therefore load-bearing, not ceremonial.
- Explain why classes rather than sets are required to state the HSP theorem.
- Compose binary relations and compute converses in the convention used throughout.
- Distinguish the equality symbol = from the formal equation symbol ≈.
- Define the kernel of a function and recognise it as an equivalence relation.
- Read indexed-family notation and direct product notation without ambiguity.
01Sets, classes and why the distinction bites
A variety is closed under arbitrary direct products, so if a variety contained even one algebra with more than one element it would contain algebras of unbounded cardinality. No set can contain a member of every cardinality, by Cantor's theorem applied to the union. So a non-trivial variety is a proper class.
This is not a technicality that can be waved away by working inside a large enough set. The statement of the HSP theorem quantifies over all algebras of a given type, and the operators H, S and P are applied to classes. A naive theory of sets and classes suffices — von Neumann–Bernays–Gödel, or Grothendieck universes, or simply careful informal usage — but the distinction must be maintained.
The working convention is the usual one: a class of sets is often called a family; classes may be quantified over but not collected into further classes; and every set is a class. Where a construction would require the class of all algebras to be a set, the theory instead restricts to algebras generated by a fixed set of generators, which is a genuine set once the type is fixed.
02Relations, composition and converse
A binary relation on A is a subset of A × A. Three operations are used constantly and their conventions differ between texts, so they are fixed here.
| Notation | Meaning | Note |
|---|---|---|
| r ∘ s | ⟨a, c⟩ ∈ r ∘ s iff ⟨a, b⟩ ∈ r and ⟨b, c⟩ ∈ s for some b | Left-to-right composition. Some texts reverse this; the difference matters for permutability. |
| ř | the converse: ⟨a, b⟩ ∈ ř iff ⟨b, a⟩ ∈ r | Also written r⁻¹. |
| Δ | the identity relation {⟨a, a⟩ : a ∈ A} | The least equivalence relation on A. |
| ∇ | the all relation A × A | The greatest equivalence relation on A. |
| a/θ | the θ-class of a | Also written [a]θ elsewhere. |
| A/θ | the set of θ-classes | Carries an algebra structure when θ is a congruence. |
The Δ and ∇ notation is worth internalising early. Every congruence lattice has Δ as its bottom element and ∇ as its top, and a great many theorems are stated as conditions on where a congruence sits relative to those two — an algebra is simple exactly when Con A = {Δ, ∇}, and the centre of an algebra is characterised by asking when Z(A) = ∇.
03Permutability, and why composition order matters
Two equivalence relations θ and φ are said to permute when θ ∘ φ = φ ∘ θ. This innocuous-looking condition is one of the most consequential in the subject: the join of two permuting equivalence relations is simply their composite, whereas in general the join requires an infinite alternating union.
θ ∨ φ = θ ∘ φ when θ and φ permute
Groups, rings and modules all have permuting congruences, which is why quotient arguments in those settings feel frictionless. Lattices and semigroups do not. The Mal'cev condition for congruence permutability — the existence of a ternary term p with p(x, y, y) ≈ x and p(x, x, y) ≈ y — explains exactly why: the group term x·y⁻¹·z does the job, and no such term exists for lattices.
04Kernels and the two directions of the correspondence
For a function α : A → B, the kernel is the equivalence relation induced by identifying elements with the same image.
- From homomorphism to congruenceTake the kernel. Compatibility with the operations is inherited from α preserving them.
- From congruence to homomorphismTake the natural map A → A/θ sending a to a/θ. It is surjective and its kernel is θ.
- The two are mutually inverseUp to isomorphism of the codomain. This is the content of the first isomorphism theorem, and it is what makes Con A the right invariant to study.
05The symbol ≈ and why it is not =
The source distinguishes carefully between =, which asserts that two things are the same thing, and ≈, which is a formal symbol appearing inside equations. An equation p ≈ q is a syntactic object — a pair of terms — and it is the sort of thing that can be satisfied or fail in a given algebra. It is not a claim that p and q are equal.
The completeness theorem for equational logic states that an equation is derivable from a set of equations exactly when it holds in every algebra satisfying that set. That statement is only meaningful if equations are syntactic objects distinct from assertions of equality. Readers who silently read ≈ as = will find the entire chapter on equational logic trivial and will have misunderstood it.
The same distinction underlies the notation A ⊨ p ≈ q, read ‘A satisfies p ≈ q’, and its negation ⊭. The turnstile ⊢ is reserved for derivability within the formal system.
Frequently asked
Can I ignore the set/class distinction in practice?
Mostly, but not entirely. Day-to-day computation inside a fixed variety never touches it. It becomes unavoidable in three places: the statement of the HSP theorem, the construction of free algebras over an arbitrary class, and any argument that quantifies over all varieties of a given type. In those settings a proof that treats the class as a set is not merely informal, it is wrong.
Which composition convention should I use?
Whichever your source uses, consistently — but be aware that the two conventions give different answers for θ ∘ φ, and that results about permutability are stated symmetrically precisely so that the convention does not affect them. When reading across texts, check the definition before trusting a composite.
Is the kernel of a homomorphism ever not a congruence?
No — if α is a homomorphism, ker(α) is always a congruence. The converse direction is the one with content: every congruence arises as the kernel of some homomorphism, namely the natural map onto the quotient. The kernel of an arbitrary function is still an equivalence relation, just not generally compatible with the operations.
- 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.