Boolean Products of Indecomposables

Stanley Burris

Recently Vaggione [2] proved that discriminator varieties V are characterized by the two properties: FHP and V = IΓ^{a}(V_{DI}). He asked if one could replace FHP by the weaker property BFC. In this note we point out that BFC + V = IΓ^{a}(V_{DI}) leads to every directly indecomposable being simple, and furthermore the class V_{S} of simples in V is defined by a set of universal sentences.

The proof that V_{DI} is axiomatized by universal sentences is similar to the proof in [1]. First suppose that B ≤ D ∈ V_{DI}, and we want to show that B ∈ V_{DI}. We assume that B is not trivial. Let A be the subalagebra of D[C]^{⋆} consisting of those f which satisfy f(x_{0}) ∈ B. Then A =_{a} PS(A) as A is a Boolean product with clopen equalizers, and the stalks on a dense set are directly indecomposable. From our assumptions we know A ≅ A′ ∈ Γ^{a}(V_{DI}), so we can assume Triv(A′) = Ø. Then A′ =_{a} PS(A′), so PS(A) =_{a} PS(A′). This guarantees that B ∈ V_{DI}, so V_{DI} is closed under subalgebras.

To show V_{DI} is closed under ultraproducts let D_{i} ∈ V_{DI}, for i ∈ I. Then we argue as in [1] that the stalks of PS(∏_{i∈I} D_{i}) are the ultraproducts ∏_{i∈I} D_{i}/U. By hypothesis there is a B such that

PS(∏_{i∈I} D_{i}) ≅ B ∈ Γ^{a}(V_{DI}).

We can assume Triv(B) = Ø. Thus PS(B) =_{a} B, so

PS(∏_{i∈I} D_{i}) =_{a} PS(B),

and thus each of the ultraproducts ∏_{i∈I} D_{i}/U is in V_{DI}.

Combining the above two results we see that V_{DI} is a class of algebras defined by universal sentences.

Now suppose α : D → B ∈ V_{S} where α is not 1-1 and B is not trivial. Let A be the modification of B[C]^{⋆} obtained by replacing the stalk B at x_{0} by D, and letting the elements of A be obtained by taking the elements f of B[C]^{⋆} by any g that agrees with f off x_{0}, and such that α(g(x_{0})) = f(x_{0}). Then A is in Γ(V_{DI}) \ Γ^{a}(V_{DI}). Choose A′ such that A ≅ A′ ∈ Γ^{a}(V_{DI}). We can assume that Triv(A′) = Ø.

Then A′ =_{a} PS(A′) =_{a} PS(A). For ϕ a factor congruence of A we have ϕ_{x} ∈ {Δ, ∇}. For x such that ϕ_{x} = ∇ we can choose f, g ∈ A such that f ≠ g on a neighborhood M of x. Then ϕ_{x} = ∇ on M, so ϕ_{x} = ∇ on an open subset of X. Likewise \overline{ϕ}_{x} = ∇ on an open subset of X, where ϕ, \overline{ϕ} is a pair of factor congruences of A. Consequently ϕ_{x} = Δ on a clopen set N, \overline{ϕ}_{x} = Δ on X \N, and this leads to ϕ being a transparent factor congruence. Thus A =_{a} PS(A), so A =_{a} A′. But this is a contradiction as one belongs to Γ^{a}(V_{DI}), but not the other.

From the last paragraph we see that every finitely generated A ∈ V_{DI} must be simple (otherwise it has a proper nontrivial simple quotient). And then every A ∈ V_{DI} is simple (as finitely generated subalgebras are in V_{DI}, and hence simple). Thus we have proved the following:

Theorem 1. Let V be a variety with BFC such that V = IΓ^{a}(V_{DI}). Then V_{DI} consists only of simple algebras, and it is a class defined by universal sentences.

References

[1] D. Bigelow and S. Burris, Boolean algebras of factor congruences. Acta Sci. Math. 54 (1990), 11–20.

[2] D. Vagggione, A characterization of discriminator varieties. Proc. Amer. Math. Soc. 129 (2001), 663–666.