States on Orthomodular Posets of Decompositions

In Harding (Transactions of American Mathematical Society (1996) 348(5), 1839–1862), it was shown that the direct product decompositions of a set X naturally form an orthomodular poset Fact X. Here it is shown that Fact X has a state if and only if X is finite. An example is also given of a finite orthomodular poset that can be embedded into Fact X for X countable, but not for X finite.