In this paper, we first introduce a weighted derivation on algebras over an operad [Formula: see text], and prove that for the free [Formula: see text]-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential ([Formula: see text]-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential ([Formula: see text]-tri)dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts.
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Zhang et al. (2024) studied this question.
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