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 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.Comment: 17 pages, 3 figure

Generalizing Pauli Spin Matrices Using Cubic Lattices

In quantum mechanics, the connection between the operator algebraic realization and the logical models of measurement of state observables has long been an open question. In the approach that is presented here, we introduce a new application of the cubic lattice. We claim that the cubic lattice may be faithfully realized as a subset of the self-adjoint space of a von Neumann algebra. Furthermore, we obtain a unitary representation of the symmetry group of the cubic lattice. In so doing, we re-derive the classic quantum gates and gain a description of how they govern a system of qubits of arbitrary cardinality.Comment: 19 pages, 0 figure