2023/11/07 by Manchon, Dominique, Regeiba, Hedi, Rjaiba, Imen +1
#18M80 #47H60 #70G65 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2311.03808
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 ∘ \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.