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 these correspond to the lexicographically largest and the lexicographically smallest form, while on the level of permutations they correspond to 132-avoiding and 231-avoiding permutations. s appear naturally in both contexts as they are used to model both permutations and elements of the Thompson monoid. nnection is then extended to Tamari orders on partitions of ( ( k − 1 ) n + 2 ) -gons into ( k + 1 ) -gons and Thompson monoids P k , k ≥ 2 .