Both \(\{1,2\} \times \mathbb{Z}_{+}\) and \(\mathbb{Z}_{+} \times \{1,2\}\) are well-ordered in the dictionary order. Do they have the same order type?
Solution
In dictionary order, \[ \{1,2\}\times\mathbb Z_+ \] has all pairs \((1,n)\) first, followed by all pairs \((2,n)\). In particular, the element \((2,1)\) has infinitely many predecessors, namely all \((1,n)\).
On the other hand, \[ \mathbb Z_+\times\{1,2\} \] has the order \[ (1,1)<(1,2)<(2,1)<(2,2)<(3,1)<(3,2)<\cdots. \] Every element has only finitely many predecessors.
The property of an element having infinitely many predecessors is preserved by order isomorphism. Hence these two well-ordered sets do not have the same order type. Their order types are, respectively, \(\omega+\omega\) and \(\omega\).