... matrix and G(M) is a forest then the strongcompletion of (Prof(M), ≤) is partially well-ordered.4 When profile classes are not partially well-orderedWe have half of Theorem 2.2 left to prove, and ... an mì n matrix and (i, j) [m]ì[n], we denoteby Mi,jthe entry of M in row i and column j.ForI ⊆ [m]andJ [n], we let MIìJstand for the submatrix (Mi,j)iI,jJ. We write Mt for the transpose ... is partially well-ordered if and only if p ∈{1, 12, 21, 132, 213, 231, 312}.We will rely heavily on the result of Higman [8] that the set of finite words over apartially well-ordered set is partially...