# Complete Boolean algebra

A **complete Boolean algebra** is a [Boolean algebra](https://www.edgepedia.org/boolean-algebra) in which every subset has both a join (least upper bound) and a meet (greatest lower bound), extending the finite operations of an ordinary Boolean algebra to arbitrary, possibly infinite, collections of elements.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> Completeness is not expressible in the first-order language of Boolean algebras, and it is a strong condition: no countable Boolean algebra is complete.<sup>[2](https://math.stackexchange.com/questions/3419751/defining-completeness-in-the-theory-of-boolean-algebra)</sup><sup> • </sup><sup>[3](https://www.tau.ac.il/~landman/Online_Class_Notes_file/Boolean/5%20Completions%20and%20pseudocomplemented%20latttices.pdf)</sup> Complete Boolean algebras serve as the algebraic backbone of Boolean-valued models and forcing in set theory, and they are exactly the injective objects among Boolean algebras.<sup>[4](https://www.sciencedirect.com/science/article/pii/S0079816908611952)</sup><sup> • </sup><sup>[5](https://www.sciencedirect.com/topics/mathematics/complete-boolean-algebra)</sup>

| Key fact | Detail |
|---|---|
| Definition | Every subset C of the algebra has an arbitrary join ⋁C and an arbitrary meet ⋀C; by de Morgan's laws it suffices that arbitrary joins exist.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> |
| Basic example | The powerset P(S) of any set S, with union, intersection, and complement, is a complete Boolean algebra.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> |
| Infinite laws | Infinite distributivity x ∧ ⋁A = ⋁(x ∧ A) and infinite de Morgan laws (⋁A)* = ⋀A* hold.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> |
| Topological test | A Boolean algebra is complete if and only if its Stone space is extremally disconnected, meaning the closure of every open set is open.<sup>[6](https://encyclopediaofmath.org/wiki/Stone_space)</sup> |
| Size restriction | No countable Boolean algebra is complete; the only complete Boolean algebras generated by a chain are finite.<sup>[3](https://www.tau.ac.il/~landman/Online_Class_Notes_file/Boolean/5%20Completions%20and%20pseudocomplemented%20latttices.pdf)</sup> |
| Injectivity | By a theorem of Sikorski, the complete Boolean algebras are exactly the injectives in the category of Boolean algebras: any homomorphism from a subalgebra B₁ of B into a complete algebra A extends to all of B.<sup>[5](https://www.sciencedirect.com/topics/mathematics/complete-boolean-algebra)</sup> |
| Intermediate levels | A κ-complete Boolean algebra has joins of all subsets of size at most κ; countably complete means κ = ℵ₀, and a κ-complete algebra need not be complete.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> |

## Definition and first consequences

A Boolean algebra is a complemented distributive lattice, with operations join ∨, meet ∧, and complement. It is complete when every subset C has a join ⋁C and a meet ⋀C.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> Because complements reverse order, de Morgan's laws show that the two conditions are equivalent: if arbitrary joins exist, arbitrary meets exist, and conversely.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> The same equivalence holds in any lattice, not only Boolean ones: joins of arbitrary sets already give meets of arbitrary sets.<sup>[7](https://math.stackexchange.com/questions/4958845/any-complete-and-possibly-distributive-lattice-is-a-boolean-algebra)</sup>

Completeness is a lattice-theoretic condition, so it must be combined with the Boolean axioms; a complete distributive lattice is not automatically Boolean. The three-element linear order is a complete, even completely distributive, lattice but not a Boolean algebra, and the five-element lattice M₃ is complete without being Boolean.<sup>[7](https://math.stackexchange.com/questions/4958845/any-complete-and-possibly-distributive-lattice-is-a-boolean-algebra)</sup>

The canonical example is a powerset. For any set S, the collection P(S) of all subsets, ordered by inclusion with union, intersection, and complement, is complete: the join of any family of subsets is its union and the meet is its intersection.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> Another example comes from geometry: for any topological space, the regular open sets form a Boolean algebra, and the regular open algebra of a space is closed under the uncountably infinite versions of the operations, hence is complete.<sup>[8](https://arxiv.org/pdf/1703.02571)</sup><sup> • </sup><sup>[9](https://arxiv.org/abs/2112.06859)</sup>

In a complete Boolean algebra the finite distributive laws upgrade to infinite ones. For any element x and any family A,

\[ x \wedge \bigvee A = \bigvee (x \wedge A), \qquad x \vee \bigwedge A = \bigwedge (x \vee A), \]

and complements convert arbitrary joins into arbitrary meets:

\[ \left( \bigvee A \right)^{*} = \bigwedge A^{*}, \qquad \left( \bigwedge A \right)^{*} = \bigvee A^{*}, \]

where A* = {a* : a ∈ A}.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup>

## Homomorphisms and injectivity

In the category of complete Boolean algebras, a morphism is a Boolean algebra homomorphism that preserves arbitrary joins, equivalently arbitrary meets; such maps are called complete homomorphisms.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> Preserving arbitrary joins is strictly stronger than preserving finite ones, and the distinction matters in applications: a homomorphism f between complete Boolean algebras is complete when f(⋁{a : a ∈ C}) = ⋁{f(a) : a ∈ C} for every subset C.<sup>[10](https://math.stackexchange.com/questions/4859803/a-questions-on-complete-endomorphisms-of-2-mathbbn)</sup>

Complete Boolean algebras have a distinctive extension property. Sikorski's extension theorem states that if B₁ is a subalgebra of a Boolean algebra B, A is a complete Boolean algebra, and f : B₁ → A is a homomorphism, then f extends to a homomorphism g : B → A with g restricted to B₁ equal to f.<sup>[5](https://www.sciencedirect.com/topics/mathematics/complete-boolean-algebra)</sup> This says precisely that the complete Boolean algebras are the injective objects of the category of Boolean algebras. Under [Stone duality](https://www.edgepedia.org/stone-duality) the same fact appears topologically as [Gleason's theorem](https://www.edgepedia.org/gleasons-theorem), that extremally disconnected spaces are projective among Stone spaces.<sup>[6](https://encyclopediaofmath.org/wiki/Stone_space)</sup>

## The Stone space characterization

Stone's representation theorem identifies each Boolean algebra with the algebra of clopen (open-and-closed) sets of a compact, totally disconnected [Hausdorff space](https://www.edgepedia.org/hausdorff-space), its [Stone space](https://www.edgepedia.org/stone-space), built from the ultrafilters of the algebra.<sup>[11](https://encyclopediaofmath.org/wiki/Boolean_algebra)</sup><sup> • </sup><sup>[12](https://plato.stanford.edu/archives/spr2016/entries/boolalg-math/)</sup> Completeness has an exact topological translation: a Boolean algebra is complete if and only if its Stone space is extremally disconnected, meaning that the closure of every open set is again open.<sup>[6](https://encyclopediaofmath.org/wiki/Stone_space)</sup> Such spaces are called Stonean spaces.<sup>[13](https://mathoverflow.net/questions/378199/stone-%C4%8Cech-boundary-is-not-extremally-disconnected/477479)</sup>

This duality transfers algebraic questions to topological ones and back. The Stone space of a Boolean algebra is metrizable if and only if the algebra is countable, and since no countable Boolean algebra is complete, no metrizable Stone space of a Boolean algebra is extremally disconnected except in trivial finite cases.<sup>[6](https://encyclopediaofmath.org/wiki/Stone_space)</sup><sup> • </sup><sup>[3](https://www.tau.ac.il/~landman/Online_Class_Notes_file/Boolean/5%20Completions%20and%20pseudocomplemented%20latttices.pdf)</sup> The concept of extremally disconnected space originated with P. S. Urysohn; the completeness criterion was established in essence by Marshall Stone and formulated in modern terms by T. Ogasawara and, independently, B. Z. Vulikh.<sup>[14](https://math.nsc.ru/LBRT/g2/english/ssk/vladimirov.pdf)</sup>

## Completions

Every Boolean algebra, complete or not, can be embedded as a subalgebra of a complete one.<sup>[11](https://encyclopediaofmath.org/wiki/Boolean_algebra)</sup> The construction can be made canonical: every Boolean algebra A embeds in a complete Boolean algebra B in which every element of B is the least upper bound of a set of elements of A, and B is unique up to isomorphism fixing A; B is called the completion of A.<sup>[12](https://plato.stanford.edu/archives/spr2016/entries/boolalg-math/)</sup> One concrete realization identifies the completion with the algebra of normal ideals, or equivalently complete ideals, of A, and the completion preserves all joins and meets that already existed in A while adding only the missing ones.<sup>[3](https://www.tau.ac.il/~landman/Online_Class_Notes_file/Boolean/5%20Completions%20and%20pseudocomplemented%20latttices.pdf)</sup>

Completions are minimal in a precise sense: the completion of A embeds into any complete extension of A by a map that is the identity on A, and any two completions of A are isomorphic through a function fixing A.<sup>[3](https://www.tau.ac.il/~landman/Online_Class_Notes_file/Boolean/5%20Completions%20and%20pseudocomplemented%20latttices.pdf)</sup> The completion of the countable atomless Boolean algebra is a homogeneous, complete, atomless Boolean algebra of cardinality 2<sup>ℵ₀</sup>.<sup>[3](https://www.tau.ac.il/~landman/Online_Class_Notes_file/Boolean/5%20Completions%20and%20pseudocomplemented%20latttices.pdf)</sup>

## Intermediate completeness: κ-complete algebras

Between Boolean algebras and complete Boolean algebras lie infinitely many intermediate levels. For a cardinal κ, a Boolean algebra A is κ-complete when every subset C with |C| ≤ κ has a join ⋁C, and equivalently a meet; such an algebra is often called a κ-algebra, and the case κ = ℵ₀ gives the countably complete Boolean algebras.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> Every complete Boolean algebra is κ-complete for all κ, and κ-completeness implies λ-completeness for λ ≤ κ, but the hierarchy is strict: for a set S with κ < |S|, the collection of subsets T of S with either |T| ≤ κ or |S − T| ≤ κ is a κ-complete algebra that is not complete.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup> A homomorphism between κ-algebras is κ-complete when it preserves joins of subsets of size at most κ.<sup>[1](https://planetmath.org/completebooleanalgebra)</sup>

## Why completeness cannot be axiomatized in first-order logic

The operations of a complete Boolean algebra have infinite arity, so a complete Boolean algebra is not even a structure in the ordinary first-order language of Boolean algebras, which allows only finitary function symbols.<sup>[2](https://math.stackexchange.com/questions/3419751/defining-completeness-in-the-theory-of-boolean-algebra)</sup> The obstruction is deeper than syntax: there is no set Γ of first-order sentences in the language of Boolean algebras that holds in exactly the complete Boolean algebras. The downward [Löwenheim–Skolem theorem](https://www.edgepedia.org/lowenheim-skolem-theorem) gives the reason: the powerset of ℕ, viewed as a Boolean algebra, is complete, but no countable elementary subalgebra of it is complete, so any first-order theory true in it would also be true in an incomplete algebra.<sup>[2](https://math.stackexchange.com/questions/3419751/defining-completeness-in-the-theory-of-boolean-algebra)</sup> Completeness is thus an essentially infinitary property of Boolean algebras.

## Applications

**Set theory and forcing.** In the Boolean-valued approach to forcing, a partial order densely embeds into a complete Boolean algebra, and properties of the generic extension are determined by properties of the complete Boolean algebra used to construct it.<sup>[4](https://www.sciencedirect.com/science/article/pii/S0079816908611952)</sup><sup> • </sup><sup>[15](https://victoriagitman.github.io/talks/2018/05/01/boolean-valued-class-forcing.html)</sup> A filter in the complete Boolean algebra corresponds to a consistent theory in the forcing language, and this framework was used to show that Suslin's problem is independent of ZFC.<sup>[16](https://mathoverflow.net/questions/51187/what-is-the-generic-poset-used-in-forcing-really)</sup><sup> • </sup><sup>[4](https://www.sciencedirect.com/science/article/pii/S0079816908611952)</sup> For set forcing, Boolean completions are unique up to isomorphism, but for class forcing they need not be, and the forcing theorem itself is equivalent to the existence of a suitable Boolean completion.<sup>[17](https://arxiv.org/abs/1710.10820)</sup>

**Measure theory and probability.** A measure algebra is a pair (B, μ) where B is a Boolean σ-algebra and μ is a strictly positive measure; for μ(1) < ∞ the algebra B satisfies the countable antichain condition and, being σ-complete, is complete.<sup>[18](https://encyclopediaofmath.org/wiki/Measure_algebra_(measure_theory))</sup> More generally, a complete Boolean algebra is called normed when it carries a real-valued measure μ with μ(x) > 0 for x ≠ 0 and μ(sup E) equal to the sum of μ(x) over pairwise disjoint families E; in probability theory one takes μ(1) = 1 and reads μ(x) as the probability of the event x.<sup>[11](https://encyclopediaofmath.org/wiki/Boolean_algebra)</sup> Not every Boolean algebra can be normed, and normed Boolean algebras have been completely classified; for uniform normed algebras the only invariant is the weight.<sup>[11](https://encyclopediaofmath.org/wiki/Boolean_algebra)</sup>

**Other fields.** Complete Boolean algebras appear as the Lindenbaum–Tarski algebras underlying Boolean-valued models in logic, in the algebra of projections in functional analysis, and in classical mereology, whose models are complete Boolean algebras without a bottom element.<sup>[12](https://plato.stanford.edu/archives/spr2016/entries/boolalg-math/)</sup><sup> • </sup><sup>[19](https://academic.oup.com/book/41620/chapter-abstract/353448667)</sup> In constructive mathematics the classical notion has been reformulated as overlap algebras, which coincide with complete Boolean algebras classically but remain meaningful without excluded middle.<sup>[20](https://arxiv.org/abs/1904.13320)</sup>

## References

1. [Complete Boolean algebra, PlanetMath](https://planetmath.org/completebooleanalgebra)
2. [Defining Completeness in the Theory of Boolean Algebra, Mathematics Stack Exchange](https://math.stackexchange.com/questions/3419751/defining-completeness-in-the-theory-of-boolean-algebra)
3. [Completions and pseudocomplemented lattices, class notes, Tel Aviv University](https://www.tau.ac.il/~landman/Online_Class_Notes_file/Boolean/5%20Completions%20and%20pseudocomplemented%20latttices.pdf)
4. [Some Applications of Forcing, ScienceDirect](https://www.sciencedirect.com/science/article/pii/S0079816908611952)
5. [Complete Boolean Algebra – an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/complete-boolean-algebra)
6. [Stone space, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Stone_space)
7. [Any complete (and possibly distributive) lattice is a Boolean algebra, Mathematics Stack Exchange](https://math.stackexchange.com/questions/4958845/any-complete-and-possibly-distributive-lattice-is-a-boolean-algebra)
8. [Measure and integration on Boolean algebras of regular open sets, arXiv](https://arxiv.org/pdf/1703.02571)
9. [Choice-free Stone duality, arXiv](https://arxiv.org/abs/2112.06859)
10. [A question on complete endomorphisms of 2^ℕ, Mathematics Stack Exchange](https://math.stackexchange.com/questions/4859803/a-questions-on-complete-endomorphisms-of-2-mathbbn)
11. [Boolean algebra, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Boolean_algebra)
12. [The Mathematics of Boolean Algebra, Stanford Encyclopedia of Philosophy (Spring 2016)](https://plato.stanford.edu/archives/spr2016/entries/boolalg-math/)
13. [Stone–Čech boundary is not extremally disconnected, MathOverflow](https://mathoverflow.net/questions/378199/stone-%C4%8Cech-boundary-is-not-extremally-disconnected/477479)
14. [General Theory of Boolean Algebras, Vladimirov lecture notes](https://math.nsc.ru/LBRT/g2/english/ssk/vladimirov.pdf)
15. [Boolean-valued class forcing, Victoria Gitman](https://victoriagitman.github.io/talks/2018/05/01/boolean-valued-class-forcing.html)
16. [What is the generic poset used in forcing, really?, MathOverflow](https://mathoverflow.net/questions/51187/what-is-the-generic-poset-used-in-forcing-really)
17. [Class forcing, the forcing theorem and Boolean completions, arXiv](https://arxiv.org/abs/1710.10820)
18. [Measure algebra (measure theory), Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Measure_algebra_(measure_theory))
19. [Classical Mereology, in Mereology, Oxford Academic](https://academic.oup.com/book/41620/chapter-abstract/353448667)
20. [Overlap Algebras: a Constructive Look at Complete Boolean Algebras, arXiv](https://arxiv.org/abs/1904.13320)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Boolean and logic-related algebras › Complete and free Boolean algebras*

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