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We introduce two new algebras that we call tied-boxed Hecke algebra and tied-boxed Temperley-Lieb algebra. The first one is a subalgebra of the algebra of braids and ties introduced by Aicardi and Juyumaya, and the second one is a tied-version of the well known Temperley-Lieb algebra. We study their representation theory and give cellular bases for them. Furthermore, we explore a strong connection between the tied-boxed Temperley-Lieb algebra and the so-called partition Temperley-Lieb algebra given by Juyumaya. Also, we show that both structures inherit diagrammatic interpretations from a new class of monoids that we call boxed ramified monoids. Additionally, we give presentations for the singular part of the ramified symmetric monoid and for the boxed ramified monoid associated to the Brauer monoid.

We introduce a structure on a Garside group that we call Dehornoy structure and we show that an iteration of such a structure leads to a left-order on the group. We define two conditions on a Garside group G and we show that, if G satisfies these two conditions, then G has a Dehornoy structure. Then we show that the Artin groups of type A and of type I 2 (m), m ≥ 4, satisfy these conditions, and therefore have Dehornoy structures. As indicated by the terminology, one of the orders obtained by this method on the Artin groups of type A coincides with the Dehornoy order. 20F36 * Supported by CONICYT Beca Doctorado "Becas Chile" 72130288.

We endow the space of rooted planar trees with an structure of Hopf algebra. We prove that variations of such a structure lead to Hopf algebras on the spaces of labelled trees, n-trees, increasing planar trees and sorted trees. These structures are used to construct Hopf algebras on different types of permutations. In particular, we obtain new characterizations of the Hopf algebras of Malvenuto-Reutenauer and Loday-Ronco via planar rooted trees.