YSYM

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Example 4.2 ySym is an operad in the category of vector spaces.
Our choice agrees with the product in ySym and CSym.
Theorem 4.8 For C [member of] {GSym, ySym, CSym}, the coalgebra compositions C o CSym and CSym o C are connections on CSym.
Note that CSym o ySym is a connection on both CSym and on ySym, which gives two distinct one-sided Hopf algebra structures.
We describe the key definitions of Section 3.1 and Section 4 for PSym := ySym o ySym.
The identity map on ySym makes PSym into a connection on ySym.
Definition 6 (action and coaction of ySym on MSym) Given b [member of] M., let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denote a p-splitting satisfying [absolute value of [b.sub.0] > 0.
Example 7 In the fundamental bases of MSym and ySym, the action looks like
Theorem 8 The maps * : MSym [cross product] YSym [right arrow] MSym and [rho] : MSym [right arrow] MSym [cross product] YSym are associative and coassociative, respectively.
We next introduce "monomial bases" for [??]Sym, MSym, and YSym. Given t [member of] [M.sub.n], define
Define the monomial bases of [??]Sym and YSym similarly (see (13) and (17) in [3]).
There, the right-grafting idea above is defined for pairs of planar binary trees and used to describe the coproduct structure of ySym in its monomial basis.