Dendriform Algebras Relative to a Semigroup

Abstract

Loday's dendriform algebras and their 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 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 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.