  We introduce the notion of anti-dendriform algebras as a new approach of
splitting the associativity. They are characterized as the algebras with two
operations whose sum is associative and the negative left and right
multiplication operators compose the bimodules of the sum associative algebras,
justifying the notion due to the comparison with the corresponding
characterization of dendriform algebras. The notions of anti-$\mathcal
O$-operators and anti-Rota-Baxter operators on associative algebras are
introduced to interpret anti-dendriform algebras. In particular, there are
compatible anti-dendriform algebra structures on associative algebras with
nondegenerate commutative Connes cocycles. There is an important observation
that there are correspondences between certain subclasses of dendriform and
anti-dendriform algebras in terms of $q$-algebras. As a direct consequence, we
give the notion of Novikov-type dendriform algebras as an analogue of Novikov
algebras for dendriform algebras, whose relationship with Novikov algebras is
consistent with the one between dendriform and pre-Lie algebras. Finally we
extend to provide a general framework of introducing the notions of analogues
of anti-dendriform algebras, which interprets a new splitting of operations.
