Associahedra, Tamari Lattices, and Related Structures
Tamari Memorial Festschrift
Progress in Mathematics, Vol. 299
Editors: Folkert Müller-Hoissen, Jean Pallo and Jim Stasheff
Tamari lattices originated from weakenings or reinterpretations of the familar
associativity law. This has been the subject of Dov Tamari's thesis at the Sorbonne in
Paris in 1951 and the central theme of his subsequent mathematical work. Tamari lattices
can be realized in terms of polytopes called associahedra, which in fact also appeared
first in Tamari's thesis. By now these beautiful structures have made their appearance in
many different areas of pure and applied mathematics, such as algebra, combinatorics,
computer science, category theory, geometry, topology, and also in physics. Their
interdisciplinary nature provides much fascination and value. On the occasion of Dov
Tamari's centennial birthday, this book provides an introduction to topical research
related to Tamari's work and ideas. Most of the articles collected in it are written in a
way accessible to a wide audience of students and researchers in mathematics and
mathematical physics and are accompanied by high quality illustrations.
- Folkert Müller-Hoissen and Hans-Otto Walther:
Dov Tamari (formerly Bernhard Teitler)
The life of Dov Tamari is described, including a brief introduction to his mathematical work.
- Carl Maxson:
On Being a Student of Dov Tamari
I reminisce about being a student of Dov Tamari at the State University
of New York at Buffalo in the late 1960's.
- Jim Stasheff:
How I 'met' Dov Tamari
Although I never met Dov Tamari, neither in person nor electronically,
our work had one important intersection - the associahedra.
This Festschrift has given me the opportunity to set the record straight:
the Stasheff polytope was in fact constructed by Tamari in 1951,
a full decade before my version. Here I will indulge in recollections of
some of the history of the associahedra, its generalizations and applications.
Others in this Festschrift will reveal still other aspects of Tamari's vision,
especially in more direct relation to the lattice/poset that bears his name.
- Jean-Louis Loday:
Dichotomy of the Addition of Natural Numbers
This is an elementary presentation of the arithmetic of trees. We show how it
is related to the Tamari poset. In the last part we investigate various ways of
realizing this poset as a polytope (associahedron), including one inferred from
- Susan H. Gensemer:
Partial Groupoid Embeddings in Semigroups
We examine a number of axiom systems guaranteeing the embedding of
a partial groupoid into a semigroup. These include the Tamari symmetric
partial groupoid and the Gensemer/Weinert equidivisible partial groupoid,
provided they satisfy an additional axiom, weak associativity. Both structures
share the one mountain property. More embedding results for partial groupoids
into other types of algebraic structures are presented as well.
- Satyan L. Devadoss, Benjamin Fehrman, Timothy Heath, and Aditi Vashist:
Moduli Spaces of Punctured Poincaré Disks
The Tamari lattice and the associahedron provide methods of measuring associativity
on a line. The real moduli space of marked curves captures the space of such
associativity. We consider a natural generalization by considering the moduli
space of marked particles on the Poincaré disk, extending Tamari's notion
of associativity based on nesting. A geometric and combinatorial construction
of this space is provided, which appears in Kontsevich's deformation quantization,
Voronov's swiss-cheese operad, and Kajiura and Stasheff's open-closed string theory.
- Cesar Ceballos and Günter M. Ziegler:
Realizing the Associahedron: Mysteries and Questions
There are many open problems and some mysteries connected to the realizations
of the associahedra as convex polytopes. In this note, we describe three
-- concerning special realizations with the vertices on a sphere,
the space of all possible realizations, and possible realizations of the
- Christophe Hohlweg:
Permutahedra and Associahedra
Permutahedra are a class of convex polytopes arising naturally from the study of
finite reflection groups, while generalized associahedra are a class of polytopes
indexed by finite reflection groups. We present the intimate links those two
classes of polytopes share.
- Victor M. Buchstaber and Vadim D. Volodin:
Combinatorial 2-truncated Cubes and Applications
We study a class of simple polytopes, called 2-truncated cubes.
These polytopes have remarkable properties and, in particular, satisfy Gal's
conjecture. Well-known polytopes (flag nestohedra, graph-associahedra and
graph-cubeahedra) are 2-truncated cubes.
- Stefan Forcey:
Extending the Tamari Lattice to Some Compositions of Species
An extension of the Tamari lattice to the multiplihedra is discussed,
along with projections to the composihedra and the Boolean lattice.
The multiplihedra and composihedra are sequences of polytopes that arose
in algebraic topology and category theory. Here we describe them in terms
of the composition of combinatorial species.
We define lattice structures on their vertices, indexed by painted trees,
which are extensions of the Tamari lattice and projections of the weak order on
the permutations. The projections from the weak order to the
Tamari lattice and the Boolean lattice are shown to be different
from the classical ones. We review how lattice structures often
interact with the Hopf algebra structures, following Aguiar and Sottile who
discovered the applications of M\"obius inversion on the Tamari lattice
to the Loday-Ronco Hopf algebra.
- Patrick Dehornoy:
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
whose Cayley graph includes all Tamari lattices. Under this correspondence,
the Tamari lattice meet and join are the counterparts of the least common
multiple and greatest common divisor operations in
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.
- Ross Street:
The Tamari lattice provides an example of a (non-symmetric) operad. We discuss
such operads and their associated monads and monoidal categories.
Freeness is an important aspect. The free structures are described in various
ways using well-formed words (in the spirit of some of Tamari's papers),
using string diagrams leading to forests, and in terms of rewrite rules.
- Frédéric Chapoton:
On the Categories of Modules over the Tamari Posets
One can attach an Abelian category to each Tamari poset, the
category of modules over its incidence algebra. This can also be
described as the category of modules over the Hasse diagram of the
poset, seen as a quiver with relations. The derived category of this
category seems to be a very interesting object, with nice properties
and many different descriptions. We recall known results and present
some conjectures on these derived categories.
- Hugh Thomas:
The Tamari Lattice as it Arises in Quiver Representations
In this chapter, we explain how the Tamari lattice arises in
the context of the representation theory of quivers, as the poset whose
elements are the torsion classes of a directed path quiver, with the order
relation given by inclusion.
- Nathan Reading:
From the Tamari lattice to Cambrian Lattices and Beyond
We trace the path from the Tamari lattice, via lattice congruences of the
weak order, to the definition of Cambrian lattices in the context of
finite Coxeter groups, and onward to the construction of Cambrian fans.
We then present sortable elements, the key combinatorial tool for studying
Cambrian lattices and fans.
The chapter concludes with a brief description of the applications of
Cambrian lattices and sortable elements to Coxeter-Catalan combinatorics
and to cluster algebras.
- Filippo Disanto, Luca Ferrari, Renzo Pinzani, Simone Rinaldi:
Catalan Lattices on Series Parallel Interval Orders
Using the notion of series parallel interval order, we
propose a unified setting to describe Dyck lattices and Tamari
lattices (two well-known lattice structures on Catalan objects) in
terms of basic notions of the theory of posets. As a consequence
of our approach, we find an extremely simple proof of the fact
that the Dyck order is a refinement of the Tamari one. Moreover,
we provide a description of both the weak and the strong Bruhat
order on 312-avoiding permutations, by recovering the proof of the
fact that they are isomorphic to the Tamari and the Dyck order,
respectively; our proof, which simplifies the existing ones,
relies on our results on series parallel interval orders.
- Maria Ronco:
Generalized Tamari Order
In their paper "Coxeter complexes and graph associahedra",
Topology and its Applications 153 (2006) 2155-2168,
M. Carr and S. Devadoss introduced the notion of tubing on
a finite simple graph G. When G is the linear graph
Ln, with n vertices, the polytope
KLn is the Stasheff polytope or associahedron.
Our goal is to describe a partial order on the set of tubings of a simple
graph, which generalizes the Tamari order on the set of vertices of
the associahedron. For certain families of graphs, this order induces
an associative product on the vector space spanned by the tubings of
all the graphs.
- Jörg Rambau and Victor Reiner:
A Survey of the Higher Stasheff-Tamari Orders
The Tamari lattice, thought as a poset on the set of
triangulations of a convex polygon with n vertices, generalizes
to the higher Stasheff-Tamari orders on the set of triangulations
of a cyclic d-dimensional polytope having n vertices.
This survey discusses what is known about these
orders, and what one would like to know about them.
- Aristophanes Dimakis and Folkert Müller-Hoissen:
KP Solitons, Higher Bruhat and Tamari Orders
In a tropical approximation, any tree-shaped line soliton solution,
a member of the simplest class of soliton solutions of the Kadomtsev-Petviashvili
(KP-II) equation, determines a chain of planar rooted binary trees, connected
by right rotation. More precisely, it determines a maximal chain of a Tamari
lattice. We show that an analysis of these solutions naturally involves
higher Bruhat and higher Tamari orders.
- Appendix: Dov Tamari's Publications