Tamari Lattices and the symmetric Thompson monoid

TL;DR

This paper explores the relationship between Tamari lattices and the Thompson group F, revealing how lattice operations correspond to algebraic operations in a related monoid, with implications for lattice chain properties.

Contribution

It establishes a novel connection between Tamari lattices and the Thompson group F, showing how lattice operations relate to monoid operations and analyzing chain length properties.

Findings

F is a group of fractions for a monoid related to Tamari lattices

Tamari lattice operations correspond to least common multiple and gcd in the monoid

Existence of chains with bounded endpoint distances in Tamari lattices

Abstract

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.