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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.

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Consider the natural numbers with their usual order.

Continuation means that we only add new elements on tops. So $\{0,2,4,5\}$ is a continuation of $\{0,2,4\}$ but not a continuation of $\{0,2,5\}$.

(This shows both a difference between inclusion and continuation, as well a counterexample for the second question.)

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