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, cⁿ<a, cⁿ<b, a<bc, and b<ac. We prove that this is optimal, because if ⋄ℵ1 holds, then there is M of cardinality ℵ₁ for which we get equality. We also deal with how much the order in a model of PA may determine the addition.