  We introduce post-associative algebra structures and study their relationship
to post-Lie algebra structures, Rota--Baxter operators and decompositions of
associative algebras and Lie algebras. We show several results on the existence
of such structures. In particular we prove that there exists no post-Lie
algebra structure on a pair $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$
is a simple Lie algebra and $\mathfrak{g}$ is a reductive Lie algebra, which is
not isomorphic to $\mathfrak{n}$. We also show that there is no
post-associative algebra structure on a pair $(A,B)$ arising from a
Rota--Baxter operator of $B$, where $A$ is a semisimple associative algebra and
$B$ is not semisimple. The proofs use results on Rota--Baxter operators and
decompositions of algebras.
