Boolean Algebra

**dwsmith** · December 7th 2010, 02:44 PM

In order to show a Lattice is a Boolean Algebra, the diagram needs to be bounded, distributive, complemented, and $|L|=2^n \ n\geq 1 \ n\in\mathbb{Z}$ .

However, my book says, "A finite lattice is called a Boolean Algebra if it is isomorphic to $B_n$ for some nonnegative integer n."

What is $B_n$ ?

I know $D_n$ are the numbers that divide n but have no clue about this $B_n$ .

Thanks.

**Drexel28** · December 7th 2010, 02:58 PM

Originally Posted by dwsmith

In order to show a Lattice is a Boolean Algebra, the diagram needs to be bounded, distributive, complemented, and $|L|=2^n \ n\geq 1 \ n\in\mathbb{Z}$ .

However, my book says, "A finite lattice is called a Boolean Algebra if it is isomorphic to $B_n$ for some nonnegative integer n."

What is $B_n$ ?

I know $D_n$ are the numbers that divide n but have no clue about this $B_n$ .

Thanks.

I'm not quite sure what they mean, but a fundamental result in the study of Boolean algebras is that every finite Boolean algebra $B$ is isomorphic to $2^{[n]}$ (here $[n]=\{1,\cdots,n\}$ ) for some $n\in\mathbb{N}$ . This fact would make me guess that $B_n=2^{[n]}$ .

**dwsmith** · December 7th 2010, 03:01 PM

So if $\displaystyle |L|=8$ , what am I showing it is isomorphic too?

**Drexel28** · December 7th 2010, 03:26 PM

Originally Posted by dwsmith

So if $\displaystyle |L|=8$ , what am I showing it is isomorphic too?

$2^{[3]}$ .

**dwsmith** · December 7th 2010, 03:30 PM

Originally Posted by Drexel28

$2^{[3]}$ .

Unfortunately, that didn't help. I understand $8=2^3$ but how do I show it is isomorphic to $2^3$ ?

**Drexel28** · December 7th 2010, 03:43 PM

Originally Posted by dwsmith

Unfortunately, that didn't help. I understand $8=2^3$ but how do I show is isomorphic to $2^3$ ?

This isn't in general easy. The result takes a fair amount of background. Do you know what 'atomic' means?

**dwsmith** · December 7th 2010, 03:50 PM

Haven't a clue unless you are talking about the science sense or as in bomb.

**Drexel28** · December 7th 2010, 03:53 PM

Originally Posted by dwsmith

Haven't a clue unless you are talking about the science sense or as in bomb.

Hahaha. Nice. Then what do you have?

**dwsmith** · December 7th 2010, 04:13 PM

The book mentions making a Hasse diagram of $B_n$ . What would that look like?

Here is the Hasse diagram I need to show it is isomorphic to B_n.

Labeled a to h. a=O and h=I

$Boolean Algebra-hasse-diagram.jpg$

**Drexel28** · December 7th 2010, 04:21 PM

Originally Posted by dwsmith

The book mentions making a Hasse diagram of $B_n$ . What would that look like?

Here is the Hasse diagram I need to show it is isomorphic to B_n.

Labeled a to h. a=O and h=I

$Click image for larger version. Name: Hasse Diagram.jpg Views: 4 Size: 10.9 KB ID: 20015$

Here is the Hasse diagram for $2^{[3]}$

**dwsmith** · December 7th 2010, 04:23 PM

How is that the Hasse Diagram for $2^3\mbox{?}$ How do I come up with the Hasse diagram when I have $2^n$ is the better question?

**Drexel28** · December 7th 2010, 04:26 PM

Originally Posted by dwsmith

How is that the Hasse Diagram for $2^3\mbox{?}$ How do I come up with the Hasse diagram when I have $2^n$ is the better question?

Because it's the set of all subsets of a set with three elements. You do the same thing. Just draw out the diagram of the lattice for the power set of $\{1,2\}$ .

**dwsmith** · December 7th 2010, 04:30 PM

So 4 is the P(s)={1,2,3,4} for instance?

**Drexel28** · December 7th 2010, 06:12 PM

Originally Posted by dwsmith

So 4 is the P(s)={1,2,3,4} for instance?

What I mean, is that if you have a Boolean algebra $B$ with $\#(B)=2^n$ then $B\cong\mathcal{P}\left(\{1,\cdots,n\}\right)$ if that notation is more comfortable.

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