  We introduce two new operations (compositional products and implication) on
Weihrauch degrees, and investigate the overall algebraic structure. The
validity of the various distributivity laws is studied and forms the basis for
a comparison with similar structures such as residuated lattices and concurrent
Kleene algebras. Introducing the notion of an ideal with respect to the
compositional product, we can consider suitable quotients of the Weihrauch
degrees. We also prove some specific characterizations using the implication.
In order to introduce and study compositional products and implications, we
introduce and study a function space of multi-valued continuous functions. This
space turns out to be particularly well-behaved for effectively traceable
spaces that are closely related to admissibly represented spaces.
