# Boolean subalgebra

## Error message

• Notice: Trying to get property of non-object in question_unanswered_block() (line 404 of /home/jcorneli/beta/sites/all/modules/question/question.module).
• Notice: Trying to get property of non-object in question_unanswered_block() (line 404 of /home/jcorneli/beta/sites/all/modules/question/question.module).
• Notice: Trying to get property of non-object in question_unanswered_block() (line 404 of /home/jcorneli/beta/sites/all/modules/question/question.module).

## Primary tabs

Defines:
dense Boolean subalgebra
Synonym:
dense subalgebra
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

Let $A=P(N)$ be the algebra of subsets of the naturals. Let $B$ be the subalgebra of finite and cofinite sets, that is, $x\in A$ iff $|x|<\infty\ OR\ |x^{{\prime}}|<\infty$. Then $B$ is dense since every subset contains a finite subset. But it is not true that $(\forall x<=y)(\exists z\in B):\ x<=z<=y$. For example, take $y=2N,\ x=6N$ (i.e the set of naturals divisible by 2 and the set of naturals divisible by 6). Then $x<=y$ but no finite or cofinite subset can be between them because only infinite subsets with infinite compliment can be between these two sets.