# Thread: Intersection of collection of algebras

1. ## Intersection of collection of algebras

I want proof of the following statement
"Intersection of collection of algebras is also an algebra"...

2. ## Re: Intersection of collection of algebras

Do you mean algebras over a field or algebraic structures? Are you sure these topics belong in an undergraduate subforum?

3. ## Re: Intersection of collection of algebras

This discussion includes topics from graduate course of algebra

BTW I am talking about algebra of sets, which are closed under the operation of complementation and finite union

4. ## Re: Intersection of collection of algebras

Unless there are more restrictions on the definition of algebra, it is obvious that the intersection of any collection of algebras is again an algebra.

5. ## Re: Intersection of collection of algebras

This seems obvious. When trying to prove that the union of algebras is an algebra there is the following difficulty. Suppose $\displaystyle A,B\in\mathcal{F}_1\cup\mathcal{F}_2$. Then it may happen that $\displaystyle A\in\mathcal{F}_1$ and $\displaystyle B\in\mathcal{F}_2$. Even though $\displaystyle \mathcal{F}_1$ and $\displaystyle \mathcal{F}_2$ by themselves are closed under union, there is no reason to believe that $\displaystyle A\cup B\in\mathcal{F}_1$ or $\displaystyle A\cup B\in\mathcal{F}_2$. But with intersection of algebras, this can't happen.

Edit: Next time, post similar questions to the Advanced Algebra subforum since it is intended for university-level questions. The current subforum is for pre-university level questions.

ok thanks