Quotients of the Magmatic Operad: Lattice Structures and Convergent Rewrite Systems

Abstract

We study quotients of the magmatic operad that is the free nonsymmetric operad over one binary generator. In the linear setting, we show that the set of these quotients admits a lattice structure and we show an analog of the Grassmann formula for the dimensions of these operads. In the nonlinear setting, we define comb associative operads that are operads indexed by non-negative integers generalizing the associative operad. We show that the set of comb associative operads admits a lattice structure, isomorphic to the lattice of non-negative integers equipped with the division order. Driven by computer experimentations, we provide a finite convergent presentation for the comb associative operad in correspondence with 3. Finally, we study quotients of the magmatic operad by one cubic relation by expressing their Hilbert series and providing combinatorial realizations.

Keywords:

• operad
• tree
• lattice
• rewrite rule
• presentation
• enumeration

2010 Mathematics Subject Classification:

• 18D50
• 68Q42
• 05C05

Acknowledgments

The authors wish to thank Maxime Lucas for helpful discussions and Vladimir Dotsenko for his marks of interest and his bibliographic suggestions.

General notations and conventions

For any integers a and c, $[a,c]$ denotes the set $\{b\in \mathbb{N}:a ⩽ b⩽ c\}$ and $[n],$ the set $[1,n].$ The cardinality of a finite set S is denoted by $\#S.$ In all this work, $\mathbb{K}$ is a field of characteristic zero.