Boolean Constructions and Discriminator Varieties
Boolean Products: Definition and Motivation
A representation of an algebra as continuous sections over a Boolean space, replacing sheaf theory with a simpler and equally powerful formulation.
Learning objectives
- State the definition of a Boolean product
- Explain the two defining conditions
- Contrast Boolean products with subdirect and direct products
The definition
An algebra A is a Boolean product of a family (Ax)x∈X indexed by a Boolean space X if A is a subdirect product of the family and:
- Equalisers are clopen. For all a, b in A, the set [[a = b]] = {x : a(x) = b(x)} is clopen in X.
- Patchwork. If a, b ∈ A and N ⊆ X is clopen, then the function agreeing with a on N and with b off N also belongs to A.
The notation [[a = b]] for the equaliser is used throughout Chapter IV.
What the two conditions do
Clopen equalisers
Ensure the representation is compatible with the topology. Where two elements agree is a topologically well-behaved set, so agreement is a local property.
Patchwork
Ensures the representation is rich enough. Sections can be glued along clopen partitions, so the algebra contains all the elements the topology suggests it should.
The source's stated motivation is that general sheaf theory is cumbersome and that the Boolean product formulation captures what is needed with far less apparatus. The two conditions amount to saying the algebra is the algebra of global continuous sections of a sheaf of algebras over a Boolean space.
Comparison with the other product notions
| Construction | Index set | Conditions | Availability |
|---|---|---|---|
| Direct product | Arbitrary set | All tuples present | Rare as a decomposition |
| Boolean power | Boolean space | All factors equal | Restricted |
| Boolean product | Boolean space | Clopen equalisers, patchwork | Moderate |
| Weak Boolean product | Boolean space | Equalisers open, patchwork | Broader |
| Subdirect product | Arbitrary set | Projections onto | Always |
Boolean products occupy the useful middle: strong enough to support a representation theory, general enough to exist for interesting classes.
Where the Boolean space comes from
The index space is not chosen arbitrarily. For an algebra A, the factor congruences form a Boolean algebra, and its Stone space is the natural index space.
The observation in Chapter II §7 that factor congruences form a Boolean algebra is what makes this construction possible. That fact was recorded then as a curiosity; here it becomes the foundation of the representation theory.
Frequently asked questions
Is every algebra a Boolean product in some non-trivial way?
Every algebra is trivially a Boolean product over a one-point space, with itself as the single factor. Non-trivial representations exist when the algebra has non-trivial factor congruences, which requires it to be directly decomposable in some fashion.
Why not just use sheaves?
One could — the two formulations are equivalent. The source's judgement is that the sheaf formalism carries overhead disproportionate to the benefit for this application, and the two-condition definition is easier to verify in practice.
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 174-178.
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.