Affine isomorphism for partially ordered sets (Q1300339)

The strict zeta matrix of a poset \(P\) with the elements \(v_1, \dots, v_n\) is the matrix whose entry in the \(i\)th row and \(j\)th column is 1 for \(v_i<v_j\) and 0 otherwise. The authors consider matrices \(A,B\) which are either adjacency matrices of graphs \(G_1\), \(G_2\), or strict zeta matrices of posets \(P_1\), \(P_2\). To these matrices a certain affine space of matrices \(W_{A,B}\) is assigned. If there is a non-negative member of \(W_{A, B}\), then \(G_1\) and \(G_2\) resp. \(P_1\) and \(P_2\) are called fractionally isomorphic. This fractional isomorphism is not an equivalence. For posets \(P_1\), \(P_2\) the concept of affine isomorphism is introduced. They are affinely isomorphic if and only if \(W_{A,B}\) is non-empty. This is proved to be an equivalence and a certain characterization of affinely isomorphic pairs of posets is presented.