  This article is devoted to the classification of anti-dendriform algebras
that are associated with associativity. They are characterized as algebras with
two operations whose sum is associative. In particular, the paper is devoted to
classifying anti-dendriform algebras associated with null-filiform associative
algebras and three-dimensional algebras. It is known that any
finite-dimensional associative algebra without a non-zero idempotent element is
nilpotent. If an associative algebra has a non-zero idempotent element, then
there does not exist a compatible anti-dendriform algebra structure associated
with associative algebras.
