Boolean Constructions and Discriminator Varieties

Boolean Products

A Boolean product lets the stalk vary from point to point. It is sheaf theory with the machinery removed, and it is the representation that makes discriminator varieties tractable.

Engineering · Mathematics5 min readKV-MATH-0240

Learning objectives

Define a Boolean product and state the two conditions it must satisfy.
Contrast Boolean products with Boolean powers.
Explain the patching condition and why it is needed.
Relate Boolean products to sheaves over Boolean spaces.
State which first-order properties transfer to a Boolean product.
Identify the varieties whose members admit Boolean product representations.

01The definition

A Boolean product of a family of algebras indexed by a Boolean space X is a subdirect product satisfying two topological conditions.

The two conditions
Condition	Statement	Purpose
Equaliser condition	for f, g in the product, the set where they agree is clopen	makes agreement a topologically visible property
Patching condition	given f, g and a clopen N, the function agreeing with f on N and g off N is in the product	allows local data to be glued into global elements

A ≤ ∏_x∈X A_x with:
⟦f = g⟧ := { x ∈ X : f(x) = g(x) } is clopen
f|_N ∪ g|_X∖N ∈ A for clopen N

X is a Boolean space; the A_x are the stalks. Written A ≤ Γ^a(X, A_x) in the source's notation.

The double bracket notation ⟦f = g⟧ for the agreement set is standard and is used throughout the source. Reading it as 'the region where the statement holds' makes the sheaf intuition immediate.

02Boolean products versus Boolean powers

Boolean power A[B]*

Constant stalk

Every point of the Stone space carries the same algebra A. The construction is determined by A and B alone.

Boolean product

Varying stalks

Each point x carries its own algebra A_x. Strictly more general — a Boolean power is the special case where all stalks are equal and the space is S(B).

The generality matters. In a discriminator variety every algebra is a Boolean product of simple algebras, and the simple algebras appearing as stalks generally differ from one another. No Boolean power representation is available, but a Boolean product one is.

Key resultThe representation theorem to aim at

Every algebra in a discriminator variety is isomorphic to a Boolean product of simple algebras. This is the Bulman-Fleming, Keimel and Werner theorem, and it is the high point of the chapter.

03Why patching is needed

The equaliser condition alone makes the product topologically well behaved but does not let local information be assembled. Patching supplies that.

Local data
Suppose an element is specified on a clopen piece N by f and on the complement by g. Nothing so far guarantees the combined function lies in the algebra.
Patching guarantees it
The condition says exactly that the combined function is a member. So the algebra is closed under clopen case-splitting.
Consequence: local becomes global
Any property that can be established on each piece of a finite clopen partition holds globally, because the witnessing elements can be patched together.
Consequence: factor congruences
Each clopen set determines a factor congruence, so the factor congruences of a Boolean product contain a copy of the clopen algebra of X — which is a Boolean algebra.

The last point is the structural payoff. A Boolean product representation embeds a Boolean algebra into the factor congruences of the algebra, which is what makes the Boolean structure available for transfer arguments.

04The sheaf connection

A Boolean product is precisely a sheaf of algebras over a Boolean space, described without sheaf-theoretic vocabulary.

Comer, 1971–1976
Sheaves over Boolean spaces
Developed sheaf representations for cylindric algebras and then more broadly. Inspired the algebraic development.
Keimel and Werner, 1974
Boolean sheaf representations
Representation results for classes of algebras via sheaves over Boolean spaces.
Burris and Werner, 1979
The Boolean product formulation
Replaced the sheaf machinery with the two elementary conditions above, making the constructions accessible without sheaf theory.
Krauss and Clark, 1979
General sheaves algebraically
Showed the general sheaf construction admits a purely algebraic description, and posed a number of problems about which varieties admit such representations.

NoteWhy avoid sheaves

The source is explicit that the cumbersome formulation of general sheaf theory has been replaced by the considerably simpler definition of a Boolean product. Nothing is lost for the purposes at hand, and the two conditions can be checked directly. The sheaf language remains useful for connecting to topology and algebraic geometry.

05Transfer of first-order properties

Not every first-order property of the stalks transfers to a Boolean product, but a well-defined class does.

What transfers
Class of sentence	Transfers?	Reason
Identities	Yes	Preserved by S and P; a Boolean product is a subdirect product.
Quasi-identities	Yes	Also preserved by S and P.
Horn sentences	Yes	The classical preservation theorem for products.
Arbitrary first-order	No	Requires the Feferman–Vaught machinery.
Feferman–Vaught reducible	Yes	Comer formulated a version for Boolean products.

Comer's version of the Feferman–Vaught theorem for Boolean products is what makes the decidability results possible: it reduces the first-order theory of the product to the theories of the stalks together with the theory of the indexing Boolean algebra. Burris and Werner showed all known variations of Feferman–Vaught follow from Comer's version.

06Which algebras admit Boolean product representations

Discriminator varieties

Every member, over simple stalks

The Bulman-Fleming–Keimel–Werner theorem. The cleanest and most useful case, developed on the next page.

Arithmetical varieties

Members with enough factor congruences

Arithmeticity supplies the Boolean structure in the factor congruences needed for the representation.

Congruence-permutable

Factor congruences form a Boolean sublattice

The necessary starting point — without permutability the factor congruences are not well behaved enough.

General varieties

There is no representation theorem at this generality. Boolean products are a tool for well-behaved varieties, not a universal construction.

Krauss and Clark posed the general question of which varieties admit such representations, and the source lists it as Problem 12: for which varieties is every algebra a Boolean product of directly indecomposable algebras, of subdirectly irreducible algebras, or of simple algebras? All three variants were open in 1981.

Frequently asked

Is a Boolean product a subdirect product?

Yes, by definition — the two topological conditions are imposed on top of subdirectness. So every Boolean product is a subdirect product, but the converse fails badly: most subdirect products satisfy neither the equaliser nor the patching condition.

Can the indexing space be arbitrary?

It must be a Boolean space — compact, Hausdorff, totally disconnected. Total disconnectedness is what makes the clopen sets a basis, which both conditions rely on. Over a connected space the clopen sets are trivial and the conditions become vacuous.

What is the notation Γ^a(X, A_x)?

It denotes the algebra of global sections of the sheaf, restricted to those satisfying the two conditions — the source's notation for the largest Boolean product over the given data. An algebra is a Boolean product when it embeds appropriately into this, and notation varies between texts, so check before comparing.

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.