Boolean Constructions and Discriminator Varieties
Weak Boolean Products and Patchwork Properties
The relaxation of the Boolean product conditions that makes representations available more widely, and what is lost by the relaxation.
Learning objectives
- Define weak Boolean product
- Identify the practical difference from Boolean products
- Understand which representation theorems require which version
The relaxation
A subdirect representation over a Boolean space in which the equalisers [[a = b]] are required only to be open, rather than clopen, together with the patchwork condition.
Relaxing clopen to open is a genuine weakening: in a Boolean space every clopen set is open, but open sets need not be closed.
| Boolean product | Weak Boolean product | |
|---|---|---|
| Equalisers | Clopen | Open |
| Patchwork | Required | Required |
| Availability | Narrower | Broader |
| Structure recovered | Stronger | Weaker |
| Typical setting | Discriminator varieties | Congruence-distributive varieties generally |
What clopenness buys
When equalisers are clopen, the complement — the set where two elements differ — is also clopen, so both agreement and disagreement are topologically well behaved. This symmetry supports stronger conclusions.
In a Boolean product, if two elements agree everywhere they are equal; and by compactness, a covering of the space by clopen sets on which various pairs agree reduces to a finite subcovering. Finite reductions of this kind are the standard proof technique in Chapter IV §9–§11, and they require clopenness.
With merely open equalisers, the complement of an equaliser is closed but possibly not open, so the symmetric argument fails. Results that depend on reasoning about where elements differ do not transfer to the weak setting.
The patchwork condition in detail
Patchwork is retained in both versions because without it the representation carries too little information.
Patchwork stated concretely
Given a, b in the algebra and a clopen N, define c by c(x) = a(x) for x ∈ N and c(x) = b(x) otherwise. Patchwork requires c to belong to the algebra.
Iterating over a finite clopen partition allows arbitrary finite gluing, which is why elements of a Boolean product behave like locally constant selections.
Patchwork is exactly the condition making the clopen sets correspond to factor congruences. Given a clopen N, the pair of congruences “agree on N” and “agree off N” are complementary and permute, so they are factor congruences.
Which theorems need which
| Result | Version needed |
|---|---|
| Discriminator variety representation | Boolean product |
| Quasiprimal algebra structure | Boolean product |
| Primal algebra: Boolean power | Boolean product (indeed Boolean power) |
| General congruence-distributive representations | Weak Boolean product usually suffices |
| Semisimple variety analysis | Varies by hypothesis |
The pattern is that the strongest structural conclusions require the strongest product notion. Discriminator varieties sit at the top precisely because their members admit full Boolean product representations with simple factors.
Frequently asked questions
Is a weak Boolean product ever a Boolean product?
Yes, whenever the equalisers happen to be clopen. The weak notion is a generalisation, so every Boolean product is a weak one.
Why is the Boolean space required in both?
Because the topology is what makes 'open' and 'clopen' meaningful. Over an unstructured index set both conditions are vacuous and the notion collapses to an ordinary subdirect product.
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 178-183.
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.