Published online by Cambridge University Press: 28 February 2020
We discuss the free monoid and the cofree comonoid on a species (relative to a fixed hyperplane arrangement). In addition, we discuss the free bimonoid on a comonoid, and dually the cofree bimonoid on a monoid. More generally, for any scalar q, we have the free q-bimonoid on a comonoid and the cofree q-bimonoid on a monoid. An important special case is when the starting (co)monoid has trivial (co)product. We employ the terms concatenation and q-(quasi)shuffle for the products, and deconcatenation and q-de(quasi)shuffle for the coproducts. For q = 1, the q-(quasi)shuffle product is commutative, while the q-de(quasi)shuffle coproduct is cocommutative. The concatenation product and deconcatenation coproduct do not depend on q, and do not satisfy any commutativity property. In addition, we also discuss the free commutative monoid and the cofree cocommutative comonoid on a species and related constructions. These have signed analogues. We discuss the q-norm map between free and cofree q-bimonoids. It is an isomorphism when q is not a root of unity. Invertibility of the Varchenko matrix associated to the q-distance function plays a critical role here. We also discuss the (co)free graded (co)monoid on a graded species. Every species can be viewed as a graded species concentrated in degree 1. The free graded monoid on a species has a unique coproduct which turns it into a graded q-bimonoid. This is precisely the q-deshuffle coproduct. Dually, the q-shuffle product is the unique product which turns into a graded q-bimonoid.