Core Structure Theory
Irredundant Bases and the Irredundant Basis Theorem
Minimal generating sets that admit no proper generating subset, and the theorem constraining the possible sizes of such bases in an algebra with an algebraic closure operator.
Learning objectives
- Define irredundant basis and distinguish it from minimal generating set
- State the irredundant basis theorem
- Recognise when the theorem's conclusion is informative
Irredundant bases
A subset X of an algebra A is an irredundant basis if Sg(X) = A and no proper subset of X generates A.
Equivalently, X generates and for each x ∈ X, the element x does not lie in Sg(X − {x}).
An irredundant basis need not have the smallest possible size. An algebra can have irredundant bases of several different cardinalities — a phenomenon impossible for vector spaces, where all bases have the same size, and the reason the theorem is needed at all.
The theorem
Let C be an algebraic closure operator on a set A, and let IrB(A) denote the set of cardinalities of irredundant bases of A. If IrB(A) contains arbitrarily large finite numbers, then it contains all sufficiently large finite numbers — that is, the set of finite basis sizes has no arbitrarily large gaps.
Informally: irredundant basis sizes cannot be scattered arbitrarily. Once they become unbounded, they fill in a final segment of the natural numbers.
This section is one the source itself marks as omissible from the short course. It is not used in the main development; it is included because the question — how badly can generating sets behave? — is natural, and the answer is a genuine constraint rather than a triviality.
When bases are well behaved
| Setting | All irredundant bases the same size? |
|---|---|
| Vector space over a field | Yes — dimension is an invariant |
| Free module over a commutative ring | Yes — rank is an invariant |
| Free algebra in a variety | Yes — the free generating set is determined |
| Matroid / pregeometry | Yes — this is essentially the definition |
| General group | No |
| General algebra | No |
Bases have constant size exactly when the closure operator satisfies the exchange property: if a ∈ C(X ∪ {b}) − C(X), then b ∈ C(X ∪ {a}). Closure operators with this property are exactly the matroids, and linear span is the motivating instance.
A concrete failure
Groups with bases of different sizes
The symmetric group S3 is generated irredundantly by two transpositions — a basis of size 2. It is also generated by a transposition together with a 3-cycle, again size 2. But many finite simple groups admit irredundant generating sets of several distinct sizes, and the phenomenon is common rather than exceptional.
The exchange property fails for group generation: knowing that a is in the subgroup generated by X and b tells one nothing about whether b lies in the subgroup generated by X and a.
The irredundant basis theorem is the residual regularity that survives this failure. It does not restore constancy, but it rules out pathological gaps in the spectrum of basis sizes.
Frequently asked questions
Is this theorem used later in the book?
No. The source flags §4 as omissible from the introductory course, and nothing in the later chapters depends on it. It is included for completeness of the treatment of generation.
What is the relationship to matroid theory?
Direct. Closure operators satisfying the exchange property are matroids, where all bases have equal size. The irredundant basis theorem describes what can be salvaged when exchange fails, so it sits just outside matroid theory.
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.4, book pages 35-37.
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.