---
tag: CED4Q4TFCD
kind: concept
name: Boolean product
msc: [08B25, 08B26, "06E15"]
specializes: [88WN3AD2VQ]
sources:
  - url: https://en.wikipedia.org/wiki/Boolean_algebras_canonically_defined
---

## Summary

A Boolean product is a [[subdirect product|88WN3AD2VQ]] of [[algebras|QJGPT2W0FT]] indexed by the points of a [[Stone space|ZV38HA5PAQ]].
Its elements behave like continuous functions on that space.
The sets where two elements agree are clopen, and elements can be cut and pasted along clopen sets.
Boolean products give sheaf-like representations of algebras.
They are a central tool in the study of discriminator varieties.

## Explanations

### L1 human

Think of an element of a Boolean product as a function on a totally disconnected space, taking its value at each point in a factor algebra.
Two such functions agree on a set that is both open and closed.
Given two elements and a clopen set, you can take the first element on the clopen set and the second one elsewhere, and the result is again an element.
Every Boolean algebra is a Boolean product of copies of the two-element Boolean algebra, over its Stone space.