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The definition of a bimonoid in species makes use of the Tits monoid of the hyperplane arrangement. The latter is a monoid structure on the set of faces.On the other hand, there is the bimonoid of faces, which is itself built out of faces.This double occurrence of faces acquires formal meaning now. Elements of the bimonoid of faces give rise to characteristic operations on any bimonoid.This yields a morphism from the bimonoid of faces to the biconvolution bimonoid associated to the given bimonoid. Further, when the given bimonoid is commutative or cocommutative, each face-component map of this morphism is an algebra antimap or an algebra map, with the Tits product on the former and composition product on the latter. The above story has a simpler commutative analogue. Bicommutative bimonoids can be formulated using the Birkhoff monoid. The latter is a monoid structure on the set of flats. On the other hand, there is the bimonoid of flats, which is itself built out of flats. Formally, elements of the bimonoid of flats give rise to commutative characteristic operations on any bicommutative bimonoid. This yields a morphism from the bimonoid of flats to the biconvolution bimonoid associated to the given bimonoid. Further, each flat-component of this morphism is an algebra map. There are more general operations one can consider on bimonoids by working with bifaces instead of faces. We call these the two-sided characteristic operations. The role of the Tits algebra is now played by the Janus algebra. More generally, one can also consider q-bimonoids whose face-components are acted upon by the q-Janus algebra.
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