NDLI: Dyck Algebras, Interval Temporal Logic, and Posets of Intervals

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Ferrari, Luca
Copyright Year	2016
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 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.
Sponsorship	Ministero dell'Istruzione, dell'Università e della Ricerca
Starting Page	1918
Ending Page	1937
Page Count	20
File Format	PDF
ISSN	08954801
DOI	10.1137/15M1016904
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	30
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-10-11
Access Restriction	Subscribed
Subject Keyword	poset of intervals Heyting algebra Combinatorics of partially ordered sets temporal logic Dyck path Heyting algebras Combinatorics Temporal logic
Content Type	Text
Resource Type	Article
Subject	Mathematics

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