Isomorphic incidence algebras
Let P be a locally finite poset, R a commutative ring with 1 and I(P,R) the corresponding incidence algebra (i.e. the set of all functions f: \(P^ 2\to R\) such that f(x,y)\(\neq 0\) implies \(x\leq y\) with obviously defined addition and multiplication by elements of R and with multiplication defined by \((fg)(x,y)=\sum_{x\leq t\leq y}f(x,t)g(t,g)).\) The paper deals with the isomorphism problem, i.e. the problem whether the isomorphism of I(P,R) and I(Q,R) implies the isomorphism of P and Q. Main results are the following: (1) If P and Q are countable, then isomorphism of I(P,R) and I(Q,R) implies the isomorphism of P and Q. (2) P and Q are potentially isomorphic iff there exists a commutative ring R such that I(P,R) and I(Q,R) are isomorphic. A potential isomorphism of P and Q is a set \(\Phi\) of partial isomorphisms from P into Q such that for any \(\phi\in \Phi\) and \(x\in P\), \(y\in Q\) there exists a \(\psi\in \Phi\) which is an extension of \(\phi\) such that \(x\in dom \psi\), \(y\in ran \psi\).
scientific article; zbMATH DE number 4191753
scientific article