On the free commutative monoid over a positive operad
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.