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We introduce the notion of anti-dendriform algebras as a new approach ofsplitting the associativity. They are characterized as the algebras with twooperations whose sum is associative and the negative left and rightmultiplication operators compose the bimodules of the sum associative algebras,justifying the notion due to the comparison with the correspondingcharacterization of dendriform algebras. The notions of anti-$\mathcalO$-operators and anti-Rota-Baxter operators on associative algebras areintroduced to interpret anti-dendriform algebras. In particular, there arecompatible anti-dendriform algebra structures on associative algebras withnondegenerate commutative Connes cocycles. There is an important observationthat there are correspondences between certain subclasses of dendriform andanti-dendriform algebras in terms of $q$-algebras. As a direct consequence, wegive the notion of Novikov-type dendriform algebras as an analogue of Novikovalgebras for dendriform algebras, whose relationship with Novikov algebras isconsistent with the one between dendriform and pre-Lie algebras. Finally weextend to provide a general framework of introducing the notions of analoguesof anti-dendriform algebras, which interprets a new splitting of operations.

Anti-dendriform algebras, new splitting of operations and Novikov type algebras