Affine Isomorphism for Partially Ordered Sets

Let A and B be the adjacency matrices of graphs G₁ and G₂ (or the strict zeta matrices of posets P₁ and P₂). Associated with A and B is a particular affine space of matrices, denoted _A,B, such that G₁ is isomorphic to G₂ (resp., P₁ is isomorphic to P₂) if and only if there is a 0-1 matrix in _A,B. Solving this integer programming problem is (notoriously) of unknown complexity, and researchers have considered its relaxation; if there is a nonnegative member of _A,B, then one says that G₁ is fractionally isomorphic to G₂ (resp., P₁ is fractionally isomorphic to P₂.) Several combinatorial characterizations of fractional isomorphism for graphs are known.In this paper we note that fractional isomorphism is not an equivalence relation for posets and introduce a further relaxation by defining P₁ to be affinely isomorphic to P₂ if _A,B is nonempty. (Asking whether _A,B is nonempty is a natural first question preceding the question of whether or not _A,B has a binary member, i.e., whether the graphs or posets are isomorphic.) We prove that affine isomorphism is indeed an equivalence relation on posets and that two posets are affinely ismorphic if and only if the f-vectors of their order complexes are the same. One consequence of this is a proof of an affine version of the Poset Reconstruction Conjecture.