Dendriform algebras relative to a semigroup (Q784864)

Recently, numerous generalizations of Rota-Baxter algebras and dendriform algebras appear, where one or two parameters belonging to a semigroup are used: these are family algebras. This paper first shows that similar objects exist for associative, associative commutative, Lie, and many other types of algebras. It then gives a categorical interpretation of these objects: it is shown that they are in fact classical objects in a monoidal category of graded objects over the chosen monoid, satisfying a uniformity condition.