Tamari lattices, forests and Thompson monoids

Tamari lattices, forests and Thompson monoids

A connection relating Tamari lattices on symmetric groups regarded as lattices under the weak Bruhat order to the positive monoid P of Thompson group F is presented. Tamari congruence classes correspond to classes of equivalent elements in P. The two well known normal forms in P correspond to endpoints of intervals in the weak Bruhat order that determine the Tamari classes. In the monoid P thes...

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 tha...

Counting smaller elements in the Tamari and m-Tamari lattices

An extension of Tamari lattices

For any finite path v on the square lattice consisting of north and east unit steps, we construct a poset Tam(v) that consists of all the paths lying weakly above v with the same endpoints as v. For particular choices of v, we recover the traditional Tamari lattice and the m-Tamari lattice. In particular this solves the problem of extending the m-Tamari lattice to any pair (a, b) of relatively ...

Tamari lattices and noncrossing partitions in type B and D

The usual, or type An, Tamari lattice is a partial order on T A n , the triangulations of an (n+3)-gon. We define a partial order on T B n , the set of centrally symmetric triangulations of a (2n + 2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the An Tamari lattice, and therefore that it deserves to be considered the Bn Tamari lattice. We define a bij...