Exercise 24.12 in Jech's Set Theory (3d Millenium Edition): The lexicographical ordering $\omega \times {\omega}_1$ does not have true cofinality.
True cofinality is defined (p.461) as the least cardinality of a cofinal chain (if it exists) in a given partial order (P,<), i.e. of a subset of P linearly ordered by the order relation.
But why not just take $\omega \times {\omega}_1$ itself as the desired subset (with cardinality $\aleph_1$ of course)? Surely it is a linearly ordered set, in the lexicographical order, which I take to mean $(a,b) < (c,d)$ if $a < c$, or $a = c$ and $b < d$. Or does Jech perhaps have something else in mind for the lexicographical ordering? (I have not managed to find his definition for it in the book.)