Boolean Lattice Problem Part 3

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In summary, a Boolean lattice $L$ is atomic if and only if the top element is the join of a set of atoms. The forward implication can be proved using Zorn's lemma, while the reverse implication can be proved by using the fact that Boolean lattices are join continuous and showing that every element in $L$ is the join of a set of atoms.
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Aryth1
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My problem for this thread is:

Let $L$ be a Boolean lattice. Prove that $L$ is atomic if and only if the top element is the join of a set of atoms.

For the forward implication, I am already done. I used Zorn's lemma to show that the set, $\mathcal{F}$, of the elements in $L$ which are the joins of some set of atoms has a maximal element, and that that element must be the top.

The reverse implication is what is tripping me up. Any help is greatly appreciated!
 
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  • #2
Well, I'm pretty sure I solved this a few days ago, but I forgot that posting here might help others or myself so here's the proof to the reverse implication. I used an earlier exercise that proved that Boolean lattices were join continuous. Turns out this problem also solved my next problem that needed to show that, in a complete atomic Boolean lattice, every element is the join of a set of atoms.

Let $L$ be a Boolean lattice where $\top$ is the join of a set of atoms and let $A$ be the set of all atoms from $L$. Then $\bigvee A = \top$. Let $y\in L$. Then $y\wedge \bigvee A = \bigvee\{y\wedge a: a\in A\} = y$ since $L$ is join continuous. If $y\wedge a = \bot$ for all $a\in A$, then it must be that $y = \bot$. So, if $y\neq \bot$, then there exists an $a\in A$ such that $y\wedge a = a$ and so $\downarrow y$ contains an atom from $L$. Since $y$ was arbitrary, $L$ is atomic.
 

1. What is the Boolean Lattice Problem Part 3?

The Boolean Lattice Problem Part 3 is a mathematical problem that involves finding the number of distinct Boolean lattices with a given number of elements. This problem is a continuation of the original Boolean Lattice Problem, which focused on finding the number of distinct Boolean lattices with a given number of elements and a certain set of operations.

2. What makes the Boolean Lattice Problem Part 3 significant?

The Boolean Lattice Problem Part 3 is significant because it helps us better understand the mathematical concept of lattices, which have many important applications in fields such as computer science, physics, and chemistry. It also has connections to other areas of mathematics, such as graph theory and abstract algebra.

3. What are some strategies for solving the Boolean Lattice Problem Part 3?

Some strategies for solving the Boolean Lattice Problem Part 3 include using generating functions, recurrence relations, and combinatorial techniques. Another approach is to use computer algorithms to calculate and verify the results. However, due to the complexity of the problem, finding an efficient and general solution remains an open research question.

4. How does the Boolean Lattice Problem Part 3 relate to the previous two parts?

The Boolean Lattice Problem Part 3 is a natural extension of the previous two parts, as it builds upon the same concept of finding the number of distinct Boolean lattices. However, Part 3 introduces a new element to the problem by considering different sets of operations, making it a more challenging and interesting problem to solve.

5. Are there any real-world applications of the Boolean Lattice Problem Part 3?

Yes, there are several real-world applications of the Boolean Lattice Problem Part 3, particularly in computer science. For example, lattices are used in data structures to efficiently store and retrieve information. They also have applications in cryptography, where they are used to create secure encryption algorithms. Additionally, lattices have been applied in the study of phase transitions in physics and in the analysis of chemical compounds in chemistry.

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