Math 2.0

General algebraic systems Definition

Boolean product

Builds on Algebraic structureSubdirect productDirect product of algebrasStone spaceClopen set

Definition

Let XX be a Stone space, that is, a compact Hausdorff totally disconnected space. Let (𝐀x)x∈X(\mathbf{A}_x)_{x \in X} be a family of algebras of the same signature. A Boolean product of this family is a subdirect product 𝐀\mathbf{A} of the direct product ∏x∈X𝐀x\prod_{x \in X} \mathbf{A}_x with these two properties:

  1. for all a,b∈Aa, b \in A, the set {x∈X:a(x)=b(x)}\set{x \in X : a(x) = b(x)} is clopen in XX;
  2. for all a,b∈Aa, b \in A and every clopen set NβŠ†XN \subseteq X, the function cc with c(x)=a(x)c(x) = a(x) for x∈Nx \in N and c(x)=b(x)c(x) = b(x) for xβˆ‰Nx \notin N lies in AA.

Remarks

Clause 2 is called the patchwork property. An algebra is said to be representable as a Boolean product when it is isomorphic to a Boolean product of some family.

Example: let 𝐁\mathbf{B} be a Boolean algebra with Stone space XX, and let 𝟐\mathbf{2} be the two-element Boolean algebra. The continuous maps Xβ†’πŸX \to \mathbf{2}, with 𝟐\mathbf{2} discrete, form a Boolean product of copies of 𝟐\mathbf{2} isomorphic to 𝐁\mathbf{B}. Two continuous maps to a discrete space agree on a clopen set, and pasting two continuous maps along a clopen set gives a continuous map.

Non-example: let X={p,q}X = \set{p, q} be discrete, and let A={(0,0),(1,1)}A = \set{(0, 0), (1, 1)} inside πŸΓ—πŸ\mathbf{2} \times \mathbf{2}. This is a subdirect product and satisfies clause 1, but pasting (0,0)(0, 0) on {p}\set{p} with (1,1)(1, 1) on {q}\set{q} gives (0,1)βˆ‰A(0, 1) \notin A, so clause 2 fails.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Algebraic structure, concept, no evidence levelAlgebraic structureconceptClopen set, concept, no evidence levelClopen setconceptDirect product of algebras, concept, no evidence levelDirect product of alg…conceptStone space, concept, no evidence levelStone spaceconceptSubdirect product, concept, no evidence levelSubdirect productconceptBoolean product, definition, no evidence levelBoolean productdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Algebraic structure concept
  2. Clopen set concept
  3. Direct product of algebras concept
  4. Stone space concept
  5. Subdirect product concept
  6. Boolean product definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Boolean_algebras_canonically_defined
About this page
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Kinddefinition
WritingW0 stub
MSC08B25, 08B26, 06E15
DefinesBoolean product
UsesAlgebraic structure, Subdirect product, Direct product of algebras, Stone space, Clopen set
Changed2 November 2026

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