Boolean Constructions and Discriminator Varieties
Boolean Powers: Construction and Basic Properties
The Boolean power of an algebra by a Boolean algebra: a construction that transfers results about Boolean algebras to arbitrary varieties.
Learning objectives
- Construct the Boolean power A[B]*
- Describe its elements as locally constant functions
- State which properties transfer from A to the Boolean power
The construction
For an algebra A and a Boolean algebra B, the Boolean power A[B]* is the algebra of continuous functions from the Stone space B* to A with the discrete topology, with operations defined pointwise.
Continuity into a discrete space means locally constant: each function takes finitely many values, and the preimage of each value is clopen. So an element of the Boolean power is a finite clopen partition of B* together with a choice of element of A on each block.
- <strong>A</strong>[<strong>2</strong>]*
- ≅ A — the Stone space of 2 is a point
- <strong>A</strong>[Su(<em>n</em>)]*
- ≅ An — finitely many points
- <strong>A</strong>[<strong>B</strong>]* for atomless <strong>B</strong>
- A genuinely new algebra, not a direct power
Transfer properties
If A belongs to a variety V, so does A[B]*, since the operations are pointwise and identities are preserved by products and subalgebras.
| Property of <strong>A</strong> | Holds in <strong>A</strong>[<strong>B</strong>]*? |
|---|---|
| Satisfies an identity | Yes |
| Belongs to a variety V | Yes |
| Finite | Only if B is finite |
| Simple | No — the Boolean power acquires congruences from B |
| Congruence-distributive | Yes, as a property of the variety |
| Directly indecomposable | No, if B is non-trivial |
A Boolean power inherits the equational behaviour of A while acquiring the congruence structure of B. This lets facts about Boolean algebras — where everything is known — be exported into varieties where much less is known.
Congruences of a Boolean power
The congruence lattice of a Boolean power reflects both factors. When A is simple, the congruences of A[B]* correspond to the filters of B, so the congruence lattice of the Boolean power is that of B.
Boolean powers are the standard way to build algebras in a variety with prescribed congruence behaviour. Given a target lattice, find a Boolean algebra with that filter lattice and take the Boolean power of a simple member.
Relation to direct powers
The Boolean power generalises the direct power. Taking B = Su(I) for a set I gives Stone space the discrete space on I compactified, and the locally constant functions on it recover a subalgebra of AI.
For infinite I, the Boolean power by Su(I) consists only of functions taking finitely many values. The full direct power AI contains functions with infinitely many values and is strictly larger. The two coincide only when I is finite.
Frequently asked questions
Why locally constant rather than arbitrary functions?
Because continuity into a discrete space forces it, and because local constancy is what makes the construction respect the Boolean structure. Arbitrary functions would give the full direct power and lose the connection to B.
Is A[B]* determined by A and B up to isomorphism?
Yes. The construction depends only on the isomorphism types of A and B, since the Stone space is determined by B up to homeomorphism.
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.5, book pages 159-162.
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.