Q: If both {1,2} x Z+ and Z+ x {1,2} are well ordered in the dictionary, do they have the same order type? Here, Z+ denotes positive integers. They cannot have the same order type. let A be the cartesian product such that  A={1,2} x Z+. and let B be the cartesian product such that B=Z+ x {1,2}.
Then the set A has this kind of ordering:
(1,1),(1,2),...,(2,1),(2,2),...
where all the elements in the first coordinate are either 1 or 2 and the second coordinate contains
On the other hand, the set B has this type of ordering:
(1,1),(1,2),(2,1),(2,2),...
where the first coordinate contains positive integers and the second coordinate contains either a 1 or a 2.
Now, there is no order-preserving bijection from A to B because if we define a map g: A->B,
and we let g(2,1)=(a,b) where a belongs to set of integers, and b belongs to the set {1,2}, then (2,1) contains infinitely many predecessors that no element in B contains. So there is no bijection between the two. Therefore, they do not have the same order type.
I am not sure if this is the right way to explain this. I know from my intuition they do not have the same order type. But why?

A: Your idea is right, only I think adding the extra notation with $g$ muddies things. The point is that on $A$, some elements (such as $(2,1)$) have infinitely many predecessors, whereas that is not true in $B$, where every element has finitely many predecessors.
