The isomorphism problem for incidence rings (Q5896381)

Let P be a locally finite pre-ordered set, hereafter shortened to pre- ordered set, and let R be a ring (with identity). The incidence ring I(P,R) consists of all functions \(f:P\times P\to R\) such that \(f(x,y)=0\) whenever \(x\nleq y\). If f,\(g\in I(P,R)\) and \(r\in R\) then \(f+g\), rf and fg are defined by the \(equations\) \((f+g)(x,y)=f(x,y)+g(x,y)\), \((rf)(x,y)=rf(x,y)\), \((fg)(x,y)=\sum_{x\leq z\leq y}f(x,z)g(z,y).\) Let C be a class of pre-ordered sets. We say that a ring R respects order in C if whenever P,\(Q\in C\) and \(I(P,R)\approx I(Q,R)\) then \(P\approx Q\). If C is the class of all pre-ordered (partially ordered) sets then we say that R respects preorder (partial order). Our main result is a solution of the isomorphism problem for incidence rings over products of indecomposable commutative rings. We will also study conditions under which commutative rings respect order in the class of connected pre- ordered sets and show that commutative rings respect order in the class of finite connected pre-ordered sets.