One sure sign of a successful generalization of a particular theory is an extension of language and technique to a larger domain permitting one to reconstruct considerable parts of previous theory on such a new basis. Another such sign is to provide new interpretations and insights captured by such an extended theory, especially involving results previously unobtainable or mysterious surprises hitherto possibly not even guessed at. The author demonstrates that the second condition has been met by providing some examples of applications in this paper in a somewhat abbreviated form for the sake of conservation of space including an analogue of MacMahon's calculation of the distribution of the major index for permutations. On the other hand, the first condition stated is met abundantly in that he is able to extend the notion of poset and, having done so, he is able to reconstruct a theory of \(P\)-partitions à la Stanley, to take the posets of order-ideal analogues through its paces and to consider \(EL\)-shellability properties of analogues of upper- semimodular lattices. The key to the generalization is to view partial orders for finite posets as subsets \(P\) of root systems: \(\Phi=\{e_ i-e_ j\mid 1\leq i\neq j\leq n\}\subset R^ n\), with the \(e_ i\) standard basis vectors and \(e_ i- e_ j\in P\) iff \(i