User blog comment:Wythagoras/Friendship function/@comment-1605058-20140717204945/@comment-5529393-20140718083229

Of course, we cannot make any correspondence between a non-well-ordered set a ordinals such that any nondecreasing sequence leads to a decreasing sequence of ordinals, since the existence of incomparable elements means we can order them either way. Nevertheless, for any well-partially-ordered set we can define the length to be the supremum of all order types of all linear extensions of the well partial order. This will give you an idea of the growth rate of your function. Note that, although a sequence does not have to obey a particular linearization of the WPO, it has to obey _some_ linearization, which cannot have an order type greater than the previously defined length.