Double Catalan monoids

Abstract

In this paper, we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results of Barnabei et al. on admissible pairs of Dyck paths. We compute a presentation and determine the minimal dimension of an effective representation for the double Catalan monoid. We also determine the minimal dimension of an effective representation for the 0-Hecke monoid.

Effective dimension of finite semigroups

“…They used it (and a variation) to compute the effective dimension of the semigroups of Hall matrices and reflexive binary relations. The authors exploited this lemma in [27] to compute the effective dimension of the 0-Hecke monoid associated to a finite Coxeter group.…”

“…The effective dimension of groups (sometimes called the minimal faithful degree) is a classical topic, dating back to the origins of representation theory. There doesn't seem to be that much work in the literature on semigroups except for the paper [21] of Kim and Roush and previous work [27] of the authors. This could be due in part to the fact that the question is much trickier for semigroups because semigroup algebras are rarely semisimple.…”

“…Kim and Roush had already used this lemma (and a variant) to compute the effective dimension of the semigroups of Hall relations and reflexive relations. The authors used it in previous work to compute the effective dimension of 0-Hecke monoids associated to finite Coxeter groups, see [27]. In Section 6, we use it to compute the effective dimension of semigroups of transformations with a doubly transitive group of units and at least one singular transformation.…”

Identities in unitriangular and gossip monoids

“…By [11,Proposition 3] the set Conv n of all convex Boolean matrices is a submonoid of R n . Let C U n = Conv n ∩ U n denote the monoid of all convex upper unitriangular matrices, and for 1 ≤ i ≤ n − 1 let D(i) ∈ C U n be the matrix with 1's on the diagonal and a single off-diagonal 1 in position (i, i + 1).…”

NP-Completeness in the gossip monoid

“…The third set of relations, which are a kind of braid relation, can easily be verified by matrix multiplication. It follows immediately from the relations above that G n is the homomorphic image of an infinite 0-Hecke monoid (see [13]) corresponding to the Coxeter presentation with n 2 involutions c i,j satisfying relations analogous to (2) and (3) above. We also note that the double Catalan monoid DC n studied by Mazorchuk and Steinberg [13] is the submonoid of G n generated by the call matrices C[i, i + 1] for i = 1, .…”