played on partial isomorphisms

We study the poset 〈Iκ ,⊆〉 as a measure of how similar the structures A and B are to each other. A subset X of Iκ has the κ back-and-forth property if for all λ < κ ∀p ∈ X[∀a ∈ λA∃b ∈ B (p ∪ {〈a(i), b(i)〉 : i < λ} ∈ X) ∧ ∀b ∈ λB∃a ∈ A(p ∪ {〈a(i), b(i)〉 : i < λ} ∈ X)]. It is obvious that there is a largest κ-back-and-forth set which we denote by I∗ κ . The structures A and B are said to be partially isomorphic, A p B, if I∗ 2 = ∅. We get stronger criteria by demanding that I∗ κ is not just non-empty but “large”. This leads naturally to the condition: (σ )κ There is a set D ⊆ I∗ κ which has the κ-back-and-forth property and is σ -closed.