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The book "Introduction to Lattices and Order" says at Exercise 10.12 that an atomic Boolean lattice is weakly atomic.

Could you tell me why it holds?

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What is the definition of weakly atomic? Where to read about this (online)? –  porton Aug 7 '13 at 21:16
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1 Answer

up vote 2 down vote accepted

Let $L$ be our atomic Boolean lattice, and suppose $x, y \in L$ with $x < y$. We have to show that there exist $a,b \in L$ such that $x \leq a \prec b \leq y$.

Let $x'$ denote the complement of $x$, and consider the homomorphism \begin{align*} \downarrow x' &\longrightarrow \uparrow x, \\ c &\longmapsto c \vee x \end{align*} which is easily seen to be an isomorphism of lattices with inverse $d \mapsto d \wedge x'$. Now $\downarrow x'$ inherits atomicity from $L$, so $\uparrow x$ is atomic as well. Since $x < y$, there exists an $a \in \uparrow x$ such that $x \prec a \leq y$.

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I take it that $\prec$ is the covering (immediate successor) relation? –  Brian M. Scott Feb 6 '13 at 23:24
    
Yes. I use the notation of the book referred to in the question, so $\prec$ is the covering relation, $\downarrow x = \{ a \in L : a \leq x \}$ and similarly for $\uparrow x$. –  marlu Feb 6 '13 at 23:26
    
Thanks. I was familiar with the arrow notations, but I was inferring the other from context. (+1) –  Brian M. Scott Feb 6 '13 at 23:28
    
Thank you for your great answer! –  skymountain Feb 7 '13 at 8:53
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