Q:

Decomposition Theorem for Posets

There is in module theory the following (krull-shmidt) theorem:
If $(M_i)_{i\in I}$ and $(N_j)_{j\in J}$ are two families of simple module such that $\bigoplus\limits_{i\in I} M_i \simeq \bigoplus\limits_{j\in J} M_j$, then there exists a bijection $\sigma:I\rightarrow J$ such that $M_i \simeq N_{\sigma(i)}$.
Is there a similar kind of theorem for partially ordered sets? More precisely, is there a class of 'simple' partial orders such that if $(P_i)_{i\in I}$ and $(Q_j)_{j\in J}$ are 'simple' orders such that 
$\prod\limits_i P_i \simeq \prod\limits_j Q_j$, then there exists a bijection $\sigma:I \rightarrow J$ such that $P_i \simeq Q_{\sigma(i)}$.

A:

The question concerns identifying a class of partial orders whose products have unique factorizations. I will give positive and negative partial answers, both (co)authored by Hashimoto. 
The positive answer may be found in  

Hashimoto, Junji
On direct product decomposition of partially ordered sets.
Ann. of Math. (2) 54, (1951). 315-318. 

Here it is proved that 
any two direct product decompositions of a connected poset have a common refinement. Arbitrary products of connected orders are not always connected, but they are sometimes connected. Thus, one can almost take the word simple in the original question to mean connected and directly indecomposable. For example, you can take simple to mean connected and directly indecomposable if you are only concerned about finite products of posets.
The negative result (from an earlier paper of Hashimoto and Nakayama) concerns the nonconnected case. Hashimoto and Nakayama point out that $(1+x)(1+x^2+x^4)=(1+x^3)(1+x+x^2)$, and translate this fact into a statement about posets by letting $1={\bf 1}$ stand for the $1$-element chain, $x={\bf 2}$ stand for the $2$-element chain, multiplication stand for cartesian product, and sum stand for disjoint (parallel) sum.
In other words, there is an isomorphism $A\times B\cong C\times D$ where $A = {\bf 1}+{\bf 2}$ is the disjoint sum of a $1$-element chain and a $2$-element chain, $B={\bf 1}+{\bf 2}^2+{\bf 2}^4$, 
$C = {\bf 1}+{\bf 2}^3$, and $D = {\bf 1}+{\bf 2}+{\bf 2}^2$. Hashimoto and Nakayama argue that these factorizations have no common refinement.

