  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.
