A Theorem on Partial Well-ordering of Sets of Vectors

Now suppose that, inparticular, S is partially well-ordered (PWO) . This means ([3]) that whenever z o , . . ., 2 U, F, S there are indices x, ( such that a < P < w ; zx < zR . G . Higman proved ([2]) that, whenever S is P WO then 117s (< co) is PWO. On the other hand ([3]) there are PWO sets S such that 11's (w) is not PWO . Denote by V8(n) the set of all vectors of TVs(n) which have only a finite number of distinct components, and put Vs (< n_) = E(na < n) Vs(rn) . In [3] the conjecture was put forward that, for every n, Vs (< n) is PIVO whenever S is PWO, and this was proved for n = 0 . In the present note we prove the conjecture for every n < w''' . Our method is much simpler than the method used in [3] for the special case as = w3 . The same result, l's (< n) is PWO for every n < w", has been obtained by J . Kruskal (not published) who very kindly showed his manuscript to us and in this way stimulated the present investigation . His proof is considerably more complicated than ours . The following result is known ([3], Theorem 4) .