Dyck Algebras, Interval Temporal Logic, and Posets of Intervals

We investigate a natural Heyting algebra structure on the set of Dyck paths of the same length. We provide a geometrical description of the pseudocomplement and relative pseudocomplement operations, as well as of regular elements. We also find a logic-theoretic interpretation of such Heyting algebras, which we call Dyck algebras, by showing that they are the algebraic counterpart of a certain fragment of a classical interval temporal logic (also known as Halpern–Shoham logic). Finally, we propose a generalization of our approach, suggesting a similar study of the Heyting algebra arising from the poset of intervals of a finite poset using Birkhoff duality. In order to illustrate this, we show how several combinatorial parameters of Dyck paths can be expressed in terms of the Heyting algebra structure of Dyck algebras, together with a certain total order on the set of atoms of each Dyck algebra.