8 research outputs found
Tamari Lattices and the symmetric Thompson monoid
We investigate the connection between Tamari lattices and the Thompson group
F, summarized in the fact that F is a group of fractions for a certain monoid
F+sym whose Cayley graph includes all Tamari lattices. Under this
correspondence, the Tamari lattice operations are the counterparts of the least
common multiple and greatest common divisor operations in F+sym. As an
application, we show that, for every n, there exists a length l chain in the
nth Tamari lattice whose endpoints are at distance at most 12l/n.Comment: 35page
KP line solitons and Tamari lattices
The KP-II equation possesses a class of line soliton solutions which can be
qualitatively described via a tropical approximation as a chain of rooted
binary trees, except at "critical" events where a transition to a different
rooted binary tree takes place. We prove that these correspond to maximal
chains in Tamari lattices (which are poset structures on associahedra). We
further derive results that allow to compute details of the evolution,
including the critical events. Moreover, we present some insights into the
structure of the more general line soliton solutions. All this yields a
characterization of possible evolutions of line soliton patterns on a shallow
fluid surface (provided that the KP-II approximation applies).Comment: 49 pages, 36 figures, second version: section 4 expande
On the rotation distance between binary trees
We develop combinatorial methods for computing the rotation distance between
binary trees, i.e., equivalently, the flip distance between triangulations of a
polygon. As an application, we prove that, for each n, there exist size n trees
at distance 2n - O(sqrt(n))
Polygraphs: From Rewriting to Higher Categories
Polygraphs are a higher-dimensional generalization of the notion of directed
graph. Based on those as unifying concept, this monograph on polygraphs
revisits the theory of rewriting in the context of strict higher categories,
adopting the abstract point of view offered by homotopical algebra. The first
half explores the theory of polygraphs in low dimensions and its applications
to the computation of the coherence of algebraic structures. It is meant to be
progressive, with little requirements on the background of the reader, apart
from basic category theory, and is illustrated with algorithmic computations on
algebraic structures. The second half introduces and studies the general notion
of n-polygraph, dealing with the homotopy theory of those. It constructs the
folk model structure on the category of strict higher categories and exhibits
polygraphs as cofibrant objects. This allows extending to higher dimensional
structures the coherence results developed in the first half