Complete Boolean algebra
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- This article is about a type of mathematical structure. For the notion from computer science, see complete Boolean algebra (computer science).
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum. Complete Boolean algebras are important in the theory of forcing. For every Boolean algebra A there is a smallest complete Boolean algebra of which A is a subalgebra. As a partially ordered set, this completion of A is the Dedekind completion.
[edit] Examples
Every finite Boolean algebra is complete.
The algebra of subsets of a given set is a complete Boolean algebra.
The regular open algebra corresponding to any topological space is a complete Boolean algebra. This example is of particular importance because every forcing poset can be considered as a topological space (a base for the topology consisting of sets that are the set of all elements less than or equal to a given element). The corresponding regular open algebra can be used to form Boolean-valued models which are then equivalent to generic extensions by the given forcing poset.
[edit] Nonexample
For an example of a Boolean algebra that is not complete, consider the collection of all sets of natural numbers, and ignore finite differences. The resulting object, denoted P(ω)/Fin, consists of all equivalence classes of sets of naturals, where the relevant equivalence relation is that two sets of naturals are equivalent if their symmetric difference is finite. The Boolean operations are defined analogously, for example, if A and B are two equivalence classes in P(ω)/Fin, we define 
to be the equivalence class of 
, where a and b are some (any) elements of A and B respectively.
Now let a0, a1,... be pairwise disjoint infinite sets of naturals, and let A0, A1,... be their corresponding equivalence classes in P(ω)/Fin . Then given any upper bound X of A0, A1,... in P(ω)/Fin, we can find a lesser upper bound, by removing from a representative for X one element of each an. Therefore the An have no supremum.
