Published online by Cambridge University Press: 28 February 2020
We discuss graded and filtered monoids, comonoids, bimonoids in species (relative to a fixed hyperplane arrangement). Every comonoid has a primitive part and more generally a primitive filtration which turns it into a filtered comonoid. Dually, every monoid has a decomposable part and more generally a decomposable filtration which turns it into a filtered monoid. The indecomposable part of a monoid is the quotient by its decomposable part. A map from a species to a comonoid is a coderivation if it maps into the primitive part of that comonoid. Dually, a map from a monoid to a species is a derivation if it factors through the indecomposable part of that monoid. A (co)derivation is the same as a (co)monoid morphism with the species viewed as a (co)monoid with the trivial (co)product. For a q-bimonoid, there is a canonical map from its primitive part to its indecomposable part. For a q-bimonoid for q not a root of unity, this map is bijective. For a bimonoid, this map is surjective iff the bimonoid is cocommutative, injective iff the bimonoid is commutative, and bijective iff the bimonoid is bicommutative. For a q-bimonoid, both the primitive and the decomposable filtrations turn it into a filtered q-bimonoid. Thus, for either filtration, we can consider the corresponding associated graded q-bimonoid. For q = 1, the associated graded bimonoid wrt the primitive filtration is commutative, and wrt the decomposable filtration is cocommutative. These are the Browder-Sweedler and Milnor-Moore (co)commutativity results.