  In this article we investigate the lattices of Dyck paths of type $A$ and $B$
under dominance order, and explicitly describe their Heyting algebra structure.
This means that each Dyck path of either type has a relative pseudocomplement
with respect to some other Dyck path of the same type. While the proof that
this lattice forms a Heyting algebra is quite straightforward, the explicit
computation of the relative pseudocomplements using the lattice-theoretic
definition is quite tedious. We give a combinatorial description of the Heyting
algebra operations join, meet, and relative pseudocomplement in terms of height
sequences, and we use these results to derive formulas for pseudocomplements
and to characterize the regular elements in these lattices.
