The Tamari lattice is a poset whose objects are ways of bracketing a sequence into pairs - for example (a(bc))d - with the covering relation being rightward application of the associative law, ie $((ab)c) \lessdot (a(bc))$.
(Equivalently, and what I'm actually interested in, the objects can be thought of as binary trees, and the covering relation as rightward rotation.)
Given two elements x,y in the lattice, is there a more efficient algorithm for finding out whether $x < y$ than just generating the up-set of x by iteration of the covering relation and then checking whether y is in it? For example, if you think of that as a graph search, is there a way to see early that a branch of the search can't possibly work, and prune it?
(I'm aware that there may be algorithms involving switching to the poset of permutations with the weak order, but if possible I'd prefer a more direct algorithm.)