Dyck algebras, interval temporal logic and posets of intervals

TL;DR

This paper explores the structure of Dyck algebras, a Heyting algebra on Dyck paths, providing geometric and logical interpretations, and extends the approach to posets of intervals, linking combinatorics, logic, and algebra.

Contribution

It introduces Dyck algebras as Heyting algebras with geometric and logical insights and generalizes the framework to posets of intervals using Birkhoff duality.

Findings

Dyck algebras form a Heyting algebra structure on Dyck paths.

A geometric description of pseudocomplement and regular elements is provided.

Several combinatorial parameters are expressed through the algebraic structure.

Abstract

We investigate a natural Heyting algebra structure on the set of Dyck paths of the same length. We provide a geometrical description of the operations of pseudocomplement and relative pseudocomplement, 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 Birkh\"off 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.