Recently Edelman & Reiner} suggested two
poset structures S}1(n,d) and
S}2(n,d) on the set of all triangulations
of the cyclic d-polytope C(n,d) with n vertices.
Both posets are generalizations of the well-studied
Tamari lattice. While S}2(n,d) is bounded
by definition, the same is not obvious for
S}1(n,d). In the paper by Edelman
& Reiner} the bounds of S}2(n,d) were also
confirmed for S}1(n,d) whenever d \le 5,
leaving the general case as a conjecture. In this paper
their conjecture is answered in the affirmative for
all~d, using several new functorial constructions.
Moreover, a structure theorem is presented, stating
that the elements of S}1(n,d+1) are in
one-to-one correspondence to certain equivalence
classes of maximal chains in S}1(n,d). In
order to clarify the connection between
S}1(n,d) and the higher Bruhat order
B}(n-2,d-1) of Manin & Schechtman},
we define an order-preserving map from
B}(n-2,d-1) to S}1(n,d), thereby
concretizing a result by Kapranov & Voevodsky}
in the theory of ordered n-categories.