**EDIT**

I now (strongly) believe that the following claim answers my question (see the text below). However, if it does, then I am sure that it is known. It is not difficult to prove and the question must have arisen before. Thus, can anybody tell me a reference where the following is shown or mentioned?

ClaimFor each $n \in \mathbb{N}$, a bounded distributive lattice $\mathbf{D}$ can be written as the direct product of $n$ product-irreducible bounded distributive lattices if the there exist $n$ (but not more) elements $a_1,\ldots,a_n \in D \setminus \{0\}$ such that $a_i \wedge a_j = 0$ for all $i \neq j$ and $\bigvee a_i = 1$.

I would like to use this fact for a paper, but since I assume its known, I would like to give a reference instead of a proof (in particular since it only appears as an example).

*I am sorry for having a question that is probably very basic for lattice theorists. However, I was not able to find the answer by looking into textbooks or by using the magic google machine.*

Let $\mathbf{D} = \langle D,0,1,\vee,\wedge \rangle$ be a bounded (not necessarily finite) distributive lattice and let $n$ be an integer such that $\mathbf{D}$ can be decomposed into the (direct) product of $n$ product-irreducible distributive lattices.

Is there any inner property of the lattice that characterizes the number $n$ in a different, preferably easy, way?

What I would like to have is a statement similar to the following one holding (I think) for Boolean algebras:

Proposition.For each $n \in \mathbb{N}$, a Boolean algebra $\mathbf{B}$ can be written as the direct product of $n$ product-irreducible Boolean algebras if and only if $\mathbf{B}$ has $2^n$ elements.

I believe the truth of this proposition can best be seen by using the duality between Boolean algebras and Stone spaces and the simple observation that any Stone space is the coproduct of $n$ coproduct-irreducible Stone spaces if and only if it contains exactly $n$ elements (a consequence of the fact that the coproduct in the category of Stone spaces is the disjoint union).

I think that the following statement is also true:

Proposition.For each $n \in \mathbb{N}$, afinitebounded distributive lattice $\mathbf{D}$ can be written as the direct product of $n$ product-irreducible bounded distributive lattices if the graph given by the partial order restricted to the nonzero join-irreducible elements among $\mathbf{D}$ has exactly $n$ connected components.

However, if the distributive lattice is infinite, then I am not so sure what could said about the number $n$ (if such a characterization is possible at all). Does anybody know?