Tamari lattices and the symmetric Thompson monoid

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 Fsym+ 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 Fsym+. 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.