Related papers: Tamari lattices, forests and Thompson monoids

We prove that symmetric monoidal weak n-groupoids in the Tamsamani model provide a model for stable n-types. Moreover, we recover the classical statement that Picard categories model stable 1-types.

For any finite path $v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all the paths weakly above $v$ with the same number of north and east steps as $v$. For…

We study the existence of nontrivial and of representable (dual) weak complementations, along with the lattice congruences that preserve them, in different constructions of bounded lattices, then use this study to determine the finite…

Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared as ''cocycle equations'' in Street's category theory…

Let $T_{m,n}$ be toroidal square grid of size $m\times n$ and let both $m$ and $n$ be even. Let $P$ be a perfect matching of $T_{m,n}$ and let $D(P)$ be the cycle-rooted spanning forest of $P$ obtained by the generalized Temperley's…

A catalogue of simplicial hyperplane arrangements was first given by Gr\"unbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and the weak order through the poset of regions. For simplicial arrangements,…

We introduce new combinatorial objects, the interval- posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear operator that appears in the functional equation of Tamari intervals…

In earlier work we initiated a program to study relationships between finite groups and arithmetic geometric invariants of modular curves in a systematic way. In the present work we continue this program, with a focus on the two smallest…

Suppose $k$ is a nonarchimedean local field, $K$ is a maximally unramified extension of $k$, and $\mathbf{G}$ is a connected reductive $k$-group. In this paper we provide parameterizations via Bruhat-Tits theory of: the rational conjugacy…

A magma (or groupoid) is a set with a binary operation $(A,f)$. Roughly speaking, a magma is said to be lazy if compositions such as $f(x,f(f(y,z),u))$ depend on at most two variables. Recently, Kaprinai, Machida and Waldhauser described…

In a "naive" attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups T and F for any subfactor. The Thompson group elements are the "local scale transformations"…

The groups G_{k,1} of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call M_{k,1}, and to inverse monoids, called Inv_{k,1}; this is done by simply generalizing bijections to partial functions or…

It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on…

This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called…

In "Hopf algebra of the planar binary trees", Adv. Math. 139 (1998), no. 2, 293--309, we constructed by induction a graded associative product on the vector space generated by the planar binary trees (resp. the permutations). In the present…

This is a chapter in an upcoming Tamari Festscrift. 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…

We introduce a new combinatorial structure: the metasylvester lattice on decreasing trees. It appears in the context of the $m$-Tamari lattices and other related $m$-generalizations. The metasylvester congruence has been recently introduced…

Given a fixed integer $n\geq 2$, we construct two new finite lattices that we call the cyclic Tamari lattice and the affine Tamari lattice. The cyclic Tamari lattice is a sublattice and a quotient lattice of the cyclic Dyer lattice, which…

The set of permutations on a finite set can be given the lattice structure known as the weak Bruhat order. This lattice structure is generalized to the set of words on a fixed alphabet $\Sigma$ = {x,y,z,...}, where each letter has a fixed…

Let $\mathcal{F}$ be a Morse-Bott foliation on the solid torus $T=S^1\times D^2$ into $2$-tori parallel to the boundary and one singular central circle. Gluing two copies of $T$ by some diffeomorphism between their boundaries, one gets a…