M Theory Lesson 101
Loday's later paper on stuffles (also called quasi-shuffles) discusses the MZV algebra in terms of trialgebras. It gives a trialgebra structure to $T(A)$, the tensor algebra for a (commutative) algebra $A$. For the stuffle product $\ast$ the operations left and right are given by
$x \ast y = x \dashv y + x \vdash y + x \perp y$
$ax \dashv by = a(x \ast by)$ with $1 \dashv x = 0$ and $x \dashv 1 = x$
$ax \vdash by = b(ax \ast y)$ with $1 \vdash x = x$ and $x \vdash 1 = 0$
so the left (right) operation has a right (left) unit. The $\perp$ operation is thought of as a deformation of shuffle product. The algebra determined by $\perp$ (with $x \perp y = y \perp x$) and $\dashv$ (with $x \vdash y = y \dashv x$) is a commutative example of trialgebra, leading to the idea that non-commutative zeta values may be studied in the context of trialgebras and their operads. Loday also shows that the functor from commutative algebras to commutative trialgebras of this type has a right adjoint which forgets the operation $\dashv$. Thus this type of algebra forms a natural triple with commutative and associative algebras.