Congruence lattices of finite diagram monoids

Abstract

We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I->M onto the minimal ideal, a congruence on M, and a normal subgroup of a maximal subgroup outside I.

V2: 49 pages, 13 figures, 4 tables - referee comments incorporated, to appear in Adv Math. V1: 47 pages, 12 figures, 3 tables