As shown in [1], the existence of a one-to-one order preserving m a p p i n g f from a partially ordered set P onto a partially ordered set Q and a one-to-one order preserving mapping g from Q onto P does not imply that P and Q are isomorphic. Below we show that, under these conditions, P 'and Q are isomorphic provided each is a partially well ordered set. Let us recall that a subset S of a partially ordered set is called diverse if and only if every two distinct elements of S are uncomparable. A partially ordered set is called partially well ordered [2] if and only if it has no infinite strictly decreasing sequence and no infinite diverse subset. Let (M, ~<) be a partially ordered set which has no infinite strictly decreasing sequence. Motivated by the canonical decomposition [2] of a partially well ordered set, we define a minimal decomposition of (M, ~< ) as follows. Exactly, as in [2], for every ordinal u we let: M u = {x ] x is a minimal element of M I._J Mi}. (1) i < u I f My = 0 for some ordinal v then