Dendriform Algebras Relative to a Semigroup

Loday's dendriform algebras and its siblings pre-Lie and zinbiel have received attention over the past two decades. In recent literature, there has been interest in a generalization of these types of algebra in which each individual operation is replaced by a family of operations indexed by a fixed semigroup $S$. The purpose of this note is twofold. First, we add to the existing work by showing that a similar extension is possible already for the most familiar types of algebra: commutative, associative, and Lie. Second, we show that these concepts arise naturally and in a unified manner from a categorical perspective. For this, one simply has to consider the standard types of algebra but in reference to the monoidal category of $S$-graded vector spaces.