On the free commutative monoid over a positive operad
Dominique Manchon, Hedi Regeiba, Imen Rjaiba, Yannic Vargas
TL;DR
The paper addresses when a positive operad $\mathbf{q}$ can yield an operad-like structure on the free commutative monoid $\mathbf{E} \circ \mathbf{q}$, introducing μ-compatibility and a novel Nested Pre-Lie (NPL) operad notion to weaken associativity. It develops explicit constructions: a nonunital operad on $\mathbf{E} \circ \mathbf{q}$ from a μ-compatible operad on $\mathbf{q}$, and, more generally, NPL-operad structures on $\mathbf{E} \circ \mathbf{q}$ from (N)PL-structures on $\mathbf{q}$, with global compositions described by partition lattices. The work also exhibits NPL-operads on connected structures such as cycles and the permutation species, and defines algebras over NPL-operads via polynomial maps $\mathcal{E}(V)$, yielding broad, combinatorially flavored examples including partitions and linear partitions. The results extend operadic theory in the category of species, linking combinatorics of partitions to pre-Lie-like operadic behavior and suggesting potential geometric applications in hyperplane arrangements.
Abstract
We study algebraic structures on the free commutative twisted algebra generated by a positive operad $\mathbf q$, in the framework of vector species. Given a nonunital commutative twisted algebra structure $μ$ on $\mathbf q$, we introduce the notion of $μ$-compatible operad structure, leading to a nonunital operad structure on $\mathbf E \circ \mathbf q$, where $\mathbf E$ stands for the exponential species. Next, we define nested pre-Lie operads (NPL-operads), a weak form of the notion of operad, in which the nested associativity axiom is weakened down to a nested pre-Lie condition. This structure is new up to our knowledge. Several constructions of NPL-operads are presented. Finally, we define algebras over a NPL-operad, based on the notion of polynomial functions.