Boolean Constructions and Discriminator Varieties
The Spectrum of an Algebra
The spectrum as the Boolean space indexing an algebra's canonical decomposition, and the spectrum of a variety.
Learning objectives
- Define Spec A and Spec V
- Relate the spectrum to factor congruences
- Interpret spectra in familiar examples
The spectrum of an algebra
The Stone space of the Boolean algebra of factor congruences of A.
Points of Spec A are ultrafilters of factor congruences, and each determines a quotient of A. The algebra is represented as sections over its spectrum.
| Algebra | Factor congruences | Spec |
|---|---|---|
| Directly indecomposable | Only Δ and ∇ | A single point |
| A × B, both indecomposable | Four elements | Two points, discrete |
| An for indecomposable A | 2n elements | n points, discrete |
| A Boolean algebra B | All congruences are factor congruences | B* |
| A[B]* for simple A | ≅ B | B* |
A one-point spectrum means the algebra is directly indecomposable. A rich spectrum means many independent direct decompositions. The spectrum is the invariant that packages all of this information at once.
The spectrum of a variety
The set of cardinalities of finite algebras in a variety V — or, in the source's usage, the collection of spectra arising from members of V.
The spectrum problem for varieties asks which sets of natural numbers arise as the set of sizes of finite members of a variety. It is a question with a long history and connects to decidability.
| Variety | Sizes of finite members |
|---|---|
| Boolean algebras | Powers of 2 |
| Groups | All positive integers |
| K-vector spaces for finite K of size q | Powers of q |
| Distributive lattices | All positive integers |
| V(A) for primal A of size n | Powers of n |
When a variety is generated by a primal algebra of size n, every finite member is a finite Boolean power, hence of size nk. The spectrum being exactly the powers of n is a signature of this kind of rigid structure.
Spectra and representation
The spectrum is the index space in the Boolean product representation. Once it is identified, the representation follows.
The quality of the representation depends on what the quotients look like. If they are simple, the representation is as good as possible — and discriminator varieties are exactly the setting where this is guaranteed.
Connection to commutative algebra
The terminology is borrowed deliberately. For a commutative ring, the idempotents form a Boolean algebra and its Stone space is the space of connected components of the prime spectrum. The universal-algebraic spectrum generalises this.
| Commutative algebra | Universal algebra |
|---|---|
| Idempotent elements | Factor congruences |
| Boolean algebra of idempotents | Boolean algebra of factor congruences |
| Connected components of Spec | Points of Spec A |
| Ring is connected | Algebra is directly indecomposable |
| Structure sheaf | Boolean product representation |
The universal-algebraic spectrum captures only direct decomposition, whereas the prime spectrum of a ring carries much more information. The parallel is at the level of idempotents and connectedness, not the full scheme structure.
Frequently asked questions
Is Spec A always non-empty?
Yes, for a non-trivial algebra. The Boolean algebra of factor congruences is non-trivial, so it has at least one ultrafilter by BPI.
Does Spec A determine A?
No. It determines how A decomposes but not what the factors are. Two algebras with the same spectrum can have entirely different stalks.
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 IV.8, book pages 183-186.
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.