USING WHALES TO COMPLETE A BOOLEAN ALGEBRA

Jason Holland

In this paper, 𝔹 will denote an arbitrary Boolean algebra. We use certain “large” subsets of 𝔹, known as whales, to construct an Archimedean Riesz space, also known as a vector lattice, which contains a natural, complete Boolean algebra. The latter Boolean algebra, known as the Boolean algebra of bands, will be used as the target for the completion of 𝔹. In our concluding remarks, we make a connection between the completion of 𝔹 and Stone’s Representation Theorem. It is known that every Boolean algebra has a completion. Moreover, a Boolean algebra is complete if and only if it is isomorphic to the regular open algebra of some topological space [2].

Whales were introduced in [1] and employed to give a short proof of the universal completion of an Archimedean Riesz space. We adapt the techniques in [1] to the setting of Boolean algebras and give a simple, constructive proof that any Boolean algebra can be represented as a dense subalgebra of the Boolean algebra of all bands of an Archimedean Riesz space. For an excellent text on elementary Riesz space theory, we refer the reader to [5].

In what follows, we denote the partial order of 𝔹 by ≤, the largest element by 1, and the smallest element by 0. If for every subset D of 𝔹, the least upper bound and the greatest lower bound of D exist, then 𝔹 is complete. A completion of 𝔹 is a complete Boolean algebra having 𝔹 as a dense subalgebra. If B ∈ 𝔹, then B′ will denote the complement of B. A subset A of a Boolean algebra 𝔹 is called a whale if

1. For every A ∈ A and every B ∈ 𝔹 with B ≤ A, we have B ∈ A, and

2. sup{A : A ∈ A} = 1.

It is easy to see that any union of whales is a whale. Also, if A and B are whales, then the set A ∧ B := {A ∧ B : A ∈ A, B ∈ B} is a whale.

1. Constructing a Riesz Space. Associated to each whale A, we define

ℝ_{A} := {f : A → ℝ such that f(0) = 0 and f(B) = f(A)

whenever B ≤ A, B ≠ 0}.

Under the usual pointwise addition and order, each ℝ_{A} is a Dedekind complete Riesz space. If f ∈ ℝ_{A}, we write dom f for A. Define a binary relation on ⋃{ℝ_{A} : A is a whale} by

f ∼ g := {f(A) = g(A) for all A ∈ dom f ∧ dom g}.

To see that ∼ is indeed an equivalence relation, suppose that f ∼ g and g ∼ h. Let A ∈ dom f ∧ dom h. Then for all B ∈ dom g,

f(A) = f(A ∧ B) = g(A ∧ B) = h(A ∧ B) = h(A).

That ∼ is reflexive and symmetric is obvious. We state the most important fact about ∼ in the next lemma. For a proof of Lemma 2, see [1].

\underline{Lemma 1}. Each equivalence class contains a unique element with maximal domain.

If f ∈ ∪{ℝ_{A} : A is a whale}, then we denote the equivalence class of f by [f] and the unique element of [f] with maximal domain by \overline{f}. We then denote the set of all elements with maximal domains by E(𝔹). If \overline{f}, \overline{g} ∈ E(𝔹), then one can find an \overline{h} for which \overline{h} = \overline{f} + \overline{g} as follows. Define an element h ∈ ℝ_{dom \overline{f}∧dom \overline{g}} by h(A) = \overline{f}(A) + \overline{g}(A) (A ∈ dom \overline{f} ∧ dom \overline{g}). By Lemma 2, \overline{h} exists and is unique. One routinely proves that E(𝔹) with this addition and the obvious scalar multiplication is a real vector space. We equip E(𝔹) with an order relation given by f ≤ g if f(A) ≤ g(A) for all A ∈ dom f ∧ dom g. This order relation enables us to prove the following theorem.

\underline{Theorem 2}. E(𝔹) is an Archimedean Riesz space.

\underline{Proof}. Lemma 1 guarantees that the order defined above is indeed a partial order. For example, if \overline{f} ≤ \overline{g} and \overline{g} ≤ \overline{f}, then \overline{f} ∼ \overline{g}. It follows then by Lemma 2 that \overline{f} = \overline{g}. We prove that the supremum of any two elements in E(𝔹) exists and leave the rest of the details to the reader. Let \overline{f}, \overline{g} ∈ E(𝔹). Define an element h ∈ ℝ_{dom \overline{f}∧dom \overline{g}} by

h(A) = max\{\overline{f}(A), \overline{g}(A) (A ∈ dom \overline{f} ∧ dom \overline{g})\}.

By Lemma 1, we obtain \overline{h} ∈ E(𝔹). Since \overline{h} ∼ h and dom h ⊂ dom \overline{h}, it follows that \overline{h} ≥ \overline{f} and \overline{h} ≥ \overline{g}. Suppose that \overline{y} ≥ \overline{f}, \overline{y} ≥ \overline{g}, and A ∈ dom \overline{y} ∧ dom \overline{h} such that A ≠ 0. Then

\overline{y}(A) = \overline{y}(A ∧ B) ≥ \overline{h}(A ∧ B) = \overline{h}(A).

Therefore, \overline{h} = \overline{f} ∨ \overline{g}.

2. Characteristic Functions. Every B ∈ 𝔹 gives rise to a whale A_{B} = {A ≤ B} ∪ {A ≤ B′}. Define a function χ_{B} ∈ ℝ_{A_{B}} by

χ_{B}(A) := {1 if A ≤ B, 0 if A ≤ B′}.

It is easy to see that χ_{B} is an element with maximal domain and thus a member of E(𝔹). We write B[E(𝔹)] for the Boolean algebra of all bands of E(𝔹). If f ∈ E(𝔹), then f′ is the band given by {g ∈ E(𝔹) : |g| ∧ |f| = 0}. The characteristic functions give us a natural way to define a map between the Boolean algebras 𝔹 and B[E(𝔹)]. We summarize the important properties of that map in Theorem 4. Before proving Theorem 4, we need the following lemma.

\underline{Lemma 3}. Let B and C be elements of 𝔹. Then (χ_{B})′′ ∩ (χ_{C})′′ = (χ_{B} ∧ χ_{C})′′.

\underline{Proof}. Since χ_{B∧C} = χ_{B} ∧ χ_{C} ∈ (χ_{B})′′ ∩ (χ_{C})′′, it follows that (χ_{B} ∧ χ_{C})′′ ⊂ (χ_{B})′′ ∩ (χ_{C})′′. For the other inclusion, let f ∈ (χ_{B})′′ ∩ (χ_{C})′′ and g ∈ (χ_{B} ∧ χ_{C})′. This implies that g ∈ (χ_{B})′ ∩(χ_{C})′. Thus, |f|∧|g| = 0 and f ∈ (χ_{B} ∧χ_{C})′′. Hence, (χ_{B} ∧ χ_{C})′′ = (χ_{B})′′ ∩ (χ_{C})′′.

\underline{Theorem 4}. Let φ: 𝔹 → B[E(𝔹)] be given by B ↦ (χ_{B})′′. Then φ is an injective Boolean homomorphism. Moreover, φ(𝔹) is a dense subalgebra of B[E(𝔹)].

\underline{Proof}. Applying the identity (χ_{B′})′ = (χ_{B})′′, together with Lemma 3, we get that φ is a Boolean homomorphism. If B ∈ 𝔹 and B ≠ 0, then χ_{B} is a nonzero element of (χ_{B})′′. Thus, φ is injective. For the density, let D ≠ {0} be an element of B[E(𝔹)] and let 0 < f ∈ D. There exists A ∈ dom f such that f(A) = k > 0.

If C ≤ A, then k · χ_{A}(C) = k = f(C). If C ≤ A′, then k · χ_{A}(C) = 0 ≤ f(C). It follows that k · χ_{A} ≤ f and

(k · χ_{A})′′ = (χ_{A})′′ ⊂ f′′ ⊂ D.

The Boolean algebra B[E(𝔹)] is complete [3]. The latter fact, in conjuction with Theorem 4.19 in [2] and Theorem 4, yields the next result.

\underline{Theorem 5}. The Boolean algebra B[E(𝔹)] is the completion of 𝔹.

Since the completion of a Boolean algebra can be represented as the bands of an Archimedean Riesz space, the following corollary is immediate.

\underline{Corollary 6}. Every Boolean algebra is isomorphic to a dense subalgebra of the Boolean algebra of all bands of an Archimedean Riesz space.

3. A Connection With Stone’s Theorem. Perhaps the most important result regarding the representation of a Boolean algebra is the topological version of Stone’s Representation Theorem. We state this as Theorem 7 below. For a proof see [2] or [4].

\underline{Theorem 7}. [Stone’s Representation Theorem]. Every Boolean algebra is isomorphic to the algebra of closed–open sets of a totally disconnnected, compact Hausdorff space.

The topological space mentioned in Theorem 7 is called the Stone space of the Boolean algebra. If a Boolean algebra is complete, then the corresponding Stone space is extremally disconnected [4]. It is straightforward, using the techniques in [1], to prove that E(𝔹) is a universally complete Riesz space. Applying the Ogasawara–Maeda representation theorem, E(𝔹) is of the form C^{∞}(X) for some compact, extremely disconnected Hausdorff space X. For a detailed account of this fact, see [3]. The Stone space of B[E(𝔹)] is precisely the topological space X. Thus, the Boolean algebra B[E(𝔹)] corresponds to the Boolean algebra of closed–open sets of X. The original Boolean algebra 𝔹 corresponds to a basis of closed–open sets of X.

\underline{References}

1. G. Buskes and A. van Rooij, “Whales and the Universal Completion,” Proceedings of the American Mathematical Society, 124 (1996), 423–427.

2. S. Koppelberg, Handbook of Boolean Algebras, Vol. I, North–Holland, Amsterdam - New York – Oxford – Tokyo, 1989.

3. W. A. Luxemburg and A. C. Zaanen, Riesz Spaces I, North Holland, Amsterdam, 1971.

4. R. Sikorski, Boolean Algebras, Springer–Verlag, New York, 1969.

5. A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer– Verlag, Berlin – Heidelberg – New York, 1997.

Jason Holland Department of Mathematics P. O. Box 11000 Oklahoma Christian University Oklahoma City, OK 73136-1100 email: jason.holland@oc.edu