This paper is concerned with a study of inverse monoids presented by a set X X subject to relations of the form e i = f i {e_i} = {f_i} , i ∈ I i \in I , where e i {e_i} and f i {f_i} are Dyck words, i.e. idempotents of the free inverse monoid on X X . Some general results of Stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the Cayley graph of the free group on X X . In the finitely presented case the word problem is solved by using Rabin’s theorem on the second order monadic logic of the infinite binary tree. Some connections with the theory of rational subsets of the free group and the theory of context-free languages are explored.