Halmos defines the order continuation as follows:
We shall say that a well ordered set A is a continuation of well ordered set B if B is a subset of A, if, in fact, B is an intial segment of A and if, finally, the ordering of elements of B is the same as their ordering in A.Thus if X is well ordered and b and a are elements of X such that a
My questions are
a)If both A and B are subsets of a well ordered set is there any difference between inclusion and continuation.
b)Does "B is a subset of A" and "the ordering of elements of B is the same as their ordering in A" imply that "B is an intial segment of A".If not could someone illustrate that with a counter example.