Orientation

Set-Theoretic Preliminaries

The set-theoretic apparatus the subject actually uses: classes as well as sets, indexed families, direct products and powers, and the specific conventions that differ from ordinary practice.

Category Engineering / MathematicsSource PreliminariesPages 1-4Reading 2 minReviewed 2026-08-07

Learning objectives

Distinguish sets from classes and explain why the distinction is needed
Work with indexed families and their products
Apply the source's notational conventions for products and powers

Sets and classes

Universal algebra deals routinely with collections that are too large to be sets. The class of all groups is the standard example: it cannot be a set, because it contains groups on every set whatsoever. A variety is likewise a proper class.

A naive theory of sets and classes suffices for the subject. The working convention is that classes may be quantified over informally, but only sets may be members of anything. A class of sets is commonly called a family of sets.

Where it bites

Two places. First, the class operators H, S and P act on classes, not sets. Second, free algebras must be constructed rather than selected, because there is no set of all algebras to select from.

Indexed families and products

Definition — Indexed family

The notations A_i, i ∈ I, and (A_i)_i∈I both denote a family of sets indexed by a set I. The index set may be empty, finite or infinite.

The direct product ∏_i∈I A_i is the set of all functions a with domain I such that a(i) ∈ A_i for each i. When every A_i equals a fixed set A, the product is the direct power A^I, which is precisely the set of all functions from I to A.

A − B: set difference
|A|: cardinality of A
Su(A): the power set of A — all subsets
BA: the set of all functions from A to B
&langle;x1,…,xn&rangle;: ordered n-tuple
∏i∈I Ai: direct product of an indexed family
AI: direct power

A convention worth noting early

Su(A) versus Sub(A)

The source writes Su(A) for the power set of a plain set A, and Sub(A) for the set of subuniverses of an algebra A. The two are visually close and mean entirely different things. Bold face distinguishes an algebra A from its underlying set A throughout.

This bold/italic distinction is used consistently and carries real content: A = ⟨A, F⟩ says the algebra A has universe A and operations F. Losing the distinction makes several statements ambiguous.

The axiom of choice

Choice is used freely and often silently. The places where it is genuinely essential are worth knowing:

Existence of ultrafilters extending a given filter — used throughout Chapter IV §3 and Chapter V §2.
Zorn's lemma arguments for maximal congruences and maximal ideals.
The Boolean prime ideal theorem, which is strictly weaker than full choice but sufficient for most of the Boolean-algebraic material.
Birkhoff's subdirect representation theorem, whose standard proof selects meet-irreducible congruences.

Not a foundational text

The source does not track choice principles carefully, and nothing in it depends on doing so. Readers interested in reverse mathematics will find the Boolean prime ideal theorem is the pressure point.

Frequently asked questions

Why does the subject need proper classes at all?

Because varieties are the central object and every variety is a proper class. The class of all groups, all lattices, all Boolean algebras — each is too large to be a set. Restricting to sets would make the main theorems unstatable.

Is ZFC assumed?

Informally, yes, with classes handled naively rather than through a formal class theory like NBG. The source explicitly says a naive theory of sets and classes is sufficient for its purposes.

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 Preliminaries, book pages 1-4.

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.