Let P and P′ be
finite preordered sets, and let R be a ring for which the number of nonzero
summands in a direct decomposition of the regular module RR is bounded.
We show that if the incidence rings I(P,R) and I(P′,R) are isomorphic as
rings, then P and P′ are isomorphic as preordered sets. We give a stronger
version of this result in case P and P′ are partially ordered. We show that
various natural extensions of these results fail. Specifically, we show that if
{Pj∣j ∈ Ω} is any collection of (locally finite) preordered sets then there
exists a ring S such that the incidence rings {I(Pj,S)∣j ∈ Ω} are pairwise
isomorphic. Additionally, we verify that there exists a finite dimensional algebra R
and locally finite, nonisomorphic partially ordered sets P and P′ for which
I(P,R) ≃ I(P′,R).
