This paper aims to develop some results of M. Aguiar on dendriform algebras to anti-dendriform algebras connected to any algebras satisfying given set of multilinear relations. We introduce the notion of weak anti-Rota-Baxter operators and prove that such an operator gives rise to an anti- \(\mathrm {\mathscr {C}}\) -dendriform algebra. Additionally, we give the notion of anti-pre-Poisson algebras as an analogue of pre-Poisson algebras. Also we introduce the notion of anti-dendriform formal deformation of anti-dendriform algebras and prove that anti-pre-Poisson algebras are corresponding semi-classical limits. We study the concept of polarization to various kinds of algebras, and apply it to extend Aguiar’s results on deformations of anti-dendriform algebras. We introduce the notion of anti-NS-algebra and through using a bimodule property, we generalize this notion to arbitrary categories of algebras. Finally, we show that \(\textrm{H}\) -twisted anti-Rota-Baxter operator leads to anti- \(\mathrm {\mathscr {C}}\) -NS-algebra.
