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.
Engineering · Mathematics11 min readKV-MATH-0202
Learning objectives
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.
Key resultVarieties are proper classes
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.
Relational notation used throughout
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 ∘ θ ∘ φ ∘ … in general θ ∨ φ = θ ∘ φ when θ and φ permute
The general formula is the union over all finite alternating composites, which is why joins in a congruence lattice are hard to compute and meets are easy.
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.
ker(α) = {⟨a, b⟩ ∈ A × A : α(a) = α(b)}
Always an equivalence relation; a congruence exactly when α is a homomorphism.
From homomorphism to congruence
Take the kernel. Compatibility with the operations is inherited from α preserving them.
From congruence to homomorphism
Take the natural map A → A/θ sending a to a/θ. It is surjective and its kernel is θ.
The two are mutually inverse
Up 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.
CautionConflating ≈ with = collapses equational logic
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.
Sources and further reading
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.
Handbook application: from concept to controlled practice
Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Preliminaries: Sets, Classes, Relations and Notation. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.
The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.
Treat Preliminaries: Sets, Classes, Relations and Notation as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—classes, relations, sets, kernels, algebra—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.
For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.
A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.
Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.
Step-by-step operating method
Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.
Worked-example protocol
Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.
Stage
Record
Quality check
Input
Objects, domain, notation, assumptions
Every symbol is defined
Method
Permitted operation or cited result at each step
All hypotheses hold
Output
Exact result and representation
Correct type, domain and form
Verification
Substitution, invariant or alternative derivation
Independent agreement
Boundary test
Zero, identity, degenerate or failed hypothesis
Scope is understood
Common failure modes and recovery actions
1. Watch for
Using a theorem without checking every hypothesis.
Recovery: Return to the governing definition or requirement and restate the decision in one sentence.
2. Watch for
Treating a suggestive example as a proof of the general case.
Recovery: Separate evidence from assumption, assign an owner and set a date for validation.
3. Watch for
Changing notation or conventions part-way through an argument.
Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.
4. Watch for
Hiding a division-by-zero, convergence, finiteness or commutativity assumption.
Recovery: Record the consequence, decision and rationale, then update the controlled baseline.
5. Watch for
Reporting a computed result without a residual, substitution or structural check.
Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.
Review checklist
Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?
Questions for deeper application
What is the most important distinction a practitioner must preserve when applying Preliminaries: Sets, Classes, Relations and Notation?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which assumption about classes would change the result most if it proved false?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What evidence would allow an independent reviewer to reproduce or challenge the conclusion?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which boundary, exception or failure case has not yet been tested?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What must be handed over, monitored or reviewed after the immediate work is complete?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Authoritative references and use notes
The sources below were selected as institutional or primary guidance for the broader practice. They support the handbook method; they do not imply that every statement or clause in a source applies to every project. Confirm the current edition, jurisdiction, contract and application before treating any requirement as mandatory.
MIT OpenCourseWare — Algebra I — Massachusetts Institute of Technology. Used for groups, vector spaces, linear transformations and linear groups. Accessed 2026-08-13.
The Stacks Project — table of contents — The Stacks Project. Used for commutative algebra, homological algebra, modules and derived categories. Accessed 2026-08-13.
