Models of PA: when two elements are necessarily order automorphic

We are interested in the question of how much the order of a non-standard model of PA can determine the model. In particular, for a model M , we want to characterize the complete types p(x, y) of non-standard elements (a, b) such that the linear orders {x : x < a} and {x : x < b} are necessarily isomorphic. It is proved that this set includes the complete types p(x, y) such that if the pair (a, b) realizes it (in M) then there is an element c such that for all standard n, c n < a, c n < b, a < bc and b < ac. We prove that this is optimal, because if ♦ ℵ 1 holds, then there is M of cardinality ℵ 1 for which we get equality. We also deal with how much the order in a model of PA may determine the addition.