MHB Proving Boolean Lattice Complementarity in [a,b]

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The discussion focuses on proving that the interval [a,b] in a Boolean lattice L is also a Boolean lattice. The user initially demonstrated that [a,b] is a sublattice of L by showing that the join and meet operations are preserved within the interval. The main challenge was to prove that [a,b] is complemented, which was resolved by identifying the complement of an element x in [a,b] as z = (y∨a)∧b, where y is the complement of x in L. This construction successfully shows that every element in [a,b] has a complement, confirming that [a,b] is indeed a complemented lattice. The problem is now resolved, and the user suggests sharing the solution for others' benefit.
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The problem is this:

Let $L$ be a Boolean lattice. For all $a<b\in L$, prove that the interval $[a,b] = \uparrow a \cap \downarrow b$ is a Boolean lattice under the partial ordering inherited from $L$.

What I've managed to do so far:

I used the fact that $[a,b]$ was a poset and showed that, for any $x,y\in [a,b]$, $x\vee y\in [a,b]$ and $x\wedge y\in [a,b]$ which proved that $[a,b]$ was a sublattice of $L$. From this we get that $[a,b]$ is distributive for free, since if $L$ had a sublattice that wasn't distributive, then $L$ would not be distributive. All that remains is to show that $[a,b]$ is complemented. I cannot seem to figure this out. I know that, if we take some $x\in [a,b]$ then a complement exists in $L$ since $L$ is a Boolean lattice, but I don't know how to show that that complement is in $[a,b]$ also.

Any hints would be greatly appreciated!
 
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Nevermind, I figured it out. Close or delete if you like.
 
Aryth said:
Nevermind, I figured it out. Close or delete if you like.

If you want, and it is entirely up to you, you could post your solution for the benefit of others who might gain from your solution. :D
 
MarkFL said:
If you want, and it is entirely up to you, you could post your solution for the benefit of others who might gain from your solution. :D

Ah, OK then.

I'll limit the problem to proving that $[a,b]$ is complemented.

Let $x\in [a,b]$. Then, since $L$ is a Boolean lattice, $x$ has a complement in $L$, call it $y$. We claim that the complement of $x$ in $[a,b]$ is $z = (y\vee a)\wedge b$. To see this, observe that:
$x\wedge z = x\wedge (y\vee a)\wedge b = [(x\wedge y)\vee (x\wedge a)]\wedge b = (\bot\vee a)\wedge b = a$​
Similarly, $x\vee z = b$ and $z$ suffices as a complement for $x$ in $[a,b]$. Since $x$ was arbitrary, $[a,b]$ is complemented.
 
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