  Let $\Lambda^{\ast}$ be the free monoid of (finite) words over a not
necessarily finite alphabet $\Lambda$, which is equipped with some (partial)
order. This ordering lifts to $\Lambda^{\ast}$, where it extends the
divisibility ordering of words. The MacNeille completion of $\Lambda^{\ast}$
constitutes a complete lattice ordered monoid and is realized by the system of
"closed" lower sets in $\Lambda^*$ (ordered by inclusion) or its isomorphic
copy formed of the "closed" upper sets (ordered by reverse inclusion). Under
some additional hypothesis on $\Lambda$, one can easily identify the closed
lower sets as the finitely generated ones, whereas it is more complicated to
determine the closed upper sets. For a fairly large class of ordered sets
$\Lambda$ (including complete lattices as well as antichains) one can generate
the closure of any upper set of words by means of binary operations (
"syntactic rules") thus obtaining an efficient procedure to test closedness.
Closed upper set of words are involved in an embedding theorem for valuated
oriented graphs. In fact, generalized paths (so-called "zigzags") are encoded
by words over an alphabet $\Lambda$. Then the valuated oriented graphs which
are "isometrically" embeddable in a product of zigzags have the characteristic
property that the words corresponding to the zigzags between any pair of
vertices form a closed upper set in $\Lambda$.
