Boolean Constructions and Discriminator Varieties

Boolean Powers

A Boolean power is a direct power with continuity imposed. It transports the structure of a Boolean algebra into an arbitrary variety, and it is the first of the constructions that make Chapter IV work.

Engineering · Mathematics6 min readKV-MATH-0237

Learning objectives

Construct the Boolean power A[B]* in both the function and partition forms.
Show that Boolean powers preserve all identities of A.
Compute the Boolean power in degenerate cases.
Distinguish Boolean powers from arbitrary subalgebras of direct powers.
Define filtered Boolean powers and state why they were introduced.
Relate the construction to Stone duality.

01Two descriptions of the construction

Let A be an algebra and B a Boolean algebra with Stone space S(B). The Boolean power A[B]* is the algebra of continuous functions from S(B) to A, where A carries the discrete topology.

A[B]* = { f : S(B) → A f continuous, A discrete }

Continuity into a discrete space means locally constant. When A is finite this forces the image to be finite and each fibre to be clopen.

ProcedureThe equivalent partition description

in: A, B → out: the Boolean power A[B]*

input: algebra A, Boolean algebra B
an element of A[B]* is a map f : A → B with:
f(a) ∧ f(a′) = 0 for a ≠ a′ (the fibres are disjoint)
⋁ { f(a) : a ∈ A } = 1 (the fibres cover)
only finitely many f(a) are non-zero
read f(a) as 'the region of the Stone space where the value is a'
operations act coordinatewise: (g ⋆ h)(c) := ⋁ { g(a) ∧ h(b) : a ⋆ b = c }
output: the algebra A[B]* of type equal to that of A

The two descriptions agree by Stone duality: elements of B are clopen subsets of S(B), and a finite clopen partition labelled by elements of A is exactly a locally constant function. Caveat: the finiteness condition is essential — without it the operations need not be well defined for infinite A.

The partition form is the one to compute with; the function form is the one that explains what is going on. Each element of the Boolean power is a way of splitting the Stone space into finitely many clopen pieces and assigning a constant value from A to each.

02Boolean powers preserve identities

A[B]* is a subalgebra of the direct power A^S(B), so every identity holding in A holds in A[B]*.

Direct powers preserve identities
Identities are preserved by P, so A^I satisfies whatever A satisfies for any index set I.
Boolean powers are subalgebras of direct powers
The continuity condition cuts out a subuniverse: locally constant functions are closed under coordinatewise operations, since a finite meet of clopen partitions is a clopen partition.
Identities are preserved by S
So A[B]* satisfies every identity of A, and A[B]* lies in the variety generated by A.
Consequence
The construction never leaves the variety, which is what makes it usable for building new members of a prescribed variety.

Key resultBoolean powers stay inside V(A)

A[B]* ∈ SP({A}) ⊆ V(A) for every Boolean algebra B. So the construction generates an unbounded supply of members of V(A) parameterised by Boolean algebras, without ever escaping the variety.

03Degenerate and small cases

Boolean powers computed
B	S(B)	A[B]*	Note
2	one point	A	The trivial case: constants only.
2ⁿ	n discrete points	Aⁿ	Finite Boolean algebras give finite direct powers.
Power set of I	Stone–Čech of I	functions with finite range, locally constant	Not the full direct power.
Countable atomless	Cantor space	locally constant maps on Cantor space	The generic infinite case.
Trivial algebra	empty space	trivial algebra	Degenerate.

CautionA Boolean power is not the full direct power

For infinite Stone spaces, A[B]* is a proper subalgebra of A^S(B). Only locally constant functions qualify. Treating the two as the same object is a common error and destroys the results that depend on the continuity condition — including the decidability theorems, which fail outright for full direct powers.

04Why the construction is useful

Boolean powers do two distinct jobs, and it is worth keeping them apart.

Transfer tool

Boolean results into other varieties

Theorems about Boolean algebras become theorems about A[B]* for arbitrary A. This is how Boolean structure is exported.

Structure theory

Representing members of a variety

Certain varieties have every member representable as a Boolean power of a fixed algebra. Where this holds, the variety is understood completely.

The second use is the more ambitious. A variety in which every algebra is a Boolean power of a single finite algebra is as well behaved as a variety can be, and identifying such varieties is one of the goals of the chapter. Directly representable varieties, treated at the end of this stream, are the closest general answer.

05Filtered Boolean powers

The plain Boolean power is sometimes too rigid. Arens and Kaplansky introduced a refinement, later named the filtered Boolean power, which allows the values to be constrained by a filter.

ProcedureThe filtered Boolean power construction

in: A, B, filter data → out: filtered Boolean power

input: algebra A, Boolean algebra B, and a filter condition on the values
start from A[B]*, the locally constant functions S(B) → A
restrict to those functions whose behaviour respects a prescribed
family of subalgebras or congruences of A
the restriction is again a subuniverse of the direct power
output: a filtered Boolean power, still in V(A)

Introduced by Arens and Kaplansky in 1948 for finite fields, long before the general theory. Caveat: the exact filtering condition varies between sources; check the definition in use before comparing results.

The filtered version is what appears in the decidability theorems. Burris and Werner proved every finitely generated discriminator variety of finite type has a decidable first-order theory by taking a Boolean product representation, converting it to a filtered Boolean power, and applying a decidability result of Rabin on Boolean algebras with distinguished filters.

06Relation to Stone duality and to sheaves

The construction is Stone duality doing work. The Boolean algebra supplies a space; the algebra A supplies the values; continuity glues them.

Stone, 1936
The base duality
Boolean algebras become Boolean spaces. Elements become clopen sets.
Arens and Kaplansky, 1948
Filtered Boolean powers of finite fields
The first instance of the construction, arrived at independently of the general theory.
Foster and others
Boolean powers in general
The construction generalised to arbitrary algebras A.
Comer, 1970s
The sheaf perspective
Recognising Boolean powers as constant sheaves over a Boolean space, and the general case as arbitrary sheaves.
Burris and Werner, 1979
Boolean products
A formulation avoiding sheaf machinery entirely while retaining the content. Adopted by the source, and the subject of a later page in this stream.

A Boolean power is the constant-stalk case: every point of the Stone space carries the same algebra A. Allowing the stalk to vary from point to point gives the Boolean product, which is strictly more general and correspondingly more useful.

Frequently asked

Is every subalgebra of a direct power a Boolean power?

No, far from it. Boolean powers are a very restricted class of subalgebras of direct powers, cut out by local constancy. Subdirect products in general are much more abundant, and it is precisely the restriction that makes Boolean powers tractable.

What happens when A is infinite?

The finiteness condition in the partition description — only finitely many fibres non-zero — still applies, so elements of A[B]* have finite range even when A is infinite. This is what keeps the operations well defined. Some sources relax the condition, giving a different and larger construction, so check the definition.

Do Boolean powers preserve quasi-identities?

Not in general. Being a subalgebra of a direct power guarantees preservation of identities, and quasi-identities are preserved by S and P too, so in fact quasi-identities do transfer. What fails is preservation of arbitrary first-order sentences — that requires the Feferman–Vaught machinery and holds only for restricted classes of formulas.

Sources and further reading

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.

Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.