# A simplicial complex spliting associativity

## TL;DR

This paper introduces a new family of simplicial objects called $ ext{Dy}^m$ operads, generalizing associative and dendriform algebras, with dimensions given by Fuss-Catalan numbers, and constructs $ ext{Dy}^m$ algebras from combinatorial posets.

## Contribution

It defines a novel simplicial object in non-symmetric operads that unifies associative and dendriform structures, extending to a family of $ ext{Dy}^m$ operads with Fuss-Catalan dimensions.

## Key findings

- $ ext{Dy}^m$ operads generalize associative and dendriform operads.
- Dimensions of $ ext{Dy}^m$ are given by Fuss-Catalan numbers.
- Construction of $ ext{Dy}^m$ algebras from combinatorial posets.

## Abstract

We introduce a simplicial object $(\{ \Dy^m\}_{m\geq 0}, {\mathbb F}_i, {\mathbb S}_j)$ in the category of non-symmetric algebraic operads, satisfying that $\Dy^0$ is the operad of associative algebras and $\Dy^1$ is J.-L. Loday\rq s operad of dendriform algebras. The dimensions of the operad $\Dy^m$ are given by the Fuss-Catalan numbers. Given a family of partially ordered sets ${\bold P}=\{P_n\}_{n\geq 1}$ we show that, under certain conditions, the vector space spanned by the set of $m$-simpleces of ${\bold P}$ is a $\Dy^m$ algebra. This construction, applied to certain combinatorial Hopf algebras, whose associative product comes from a dendriform structure, provides examples of $\Dy^m$ algebras.