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Every nontrivial (bounded) distributive lattice arises as a direct power of a certain number of nontrivial product-irreducible (bounded) distributive lattices. My question is how this number can be characterized by some properties of the lattice (of course, this number is $1$ in the case that the lattice is product-irreducible itself).

More precisely, let $\mathbf{D} := (D,0,1,\vee,\wedge)$ be a nontrivial bounded distributive lattice, and let $(\mathbf{A}_i)_{i \in I}$ be a family of nontrivial product-irreducible bounded distributive lattices. Note that this family is essentially unique (in particular, the cardinality of the set $I$ only depends on $\mathbf{D}$). How can we obtain the cardinality of $I$ without determining the exact product-decomposition?

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-1 for starting with "obviously" followed by a completely non-obvious statement. –  joriki May 23 '12 at 16:45
    
You are right. I have changed it. –  Cyrus May 24 '12 at 15:30
    
OK, I rescinded the downvote. –  joriki May 24 '12 at 17:47
    
Related: mathoverflow.net/questions/97844/… –  a3nm Feb 10 '13 at 22:29

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