  The set of permutations on a finite set can be given the lattice structure
known as the weak Bruhat order. This lattice structure is generalized to the
set of words on a fixed alphabet $\Sigma$ = {x,y,z,...}, where each letter has
a fixed number of occurrences. These lattices are known as multinomial lattices
and, when card($\Sigma$) = 2, as lattices of lattice paths. By interpreting the
letters x, y, z, . . . as axes, these words can be interpreted as discrete
increasing paths on a grid of a d-dimensional cube, with d = card($\Sigma$).We
show how to extend this ordering to images of continuous monotone functions
from the unit interval to a d-dimensional cube and prove that this ordering is
a lattice, denoted by L(I^d). This construction relies on a few algebraic
properties of the quantale of join-continuous functions from the unit interval
of the reals to itself: it is cyclic $\star$-autonomous and it satisfies the
mix rule.We investigate structural properties of these lattices, which are
self-dual and not distributive. We characterize join-irreducible elements and
show that these lattices are generated under infinite joins from their
join-irreducible elements, they have no completely join-irreducible elements
nor compact elements. We study then embeddings of the d-dimensional multinomial
lattices into L(I^d). We show that these embeddings arise functorially from
subdivisions of the unit interval and observe that L(I^d) is the
Dedekind-MacNeille completion of the colimit of these embeddings. Yet, if we
restrict to embeddings that take rational values and if d > 2, then every
element of L(I^d) is only a join of meets of elements from the colimit of these
embeddings.
