  The cup product in the cohomology of algebras over quadratic operads has been
studied in the general setting of Koszul duality for operads. We study the cup
product on the cohomology of n-ary totally associative algebras with an
operation of even (homological) degree. This cup product endows the cohomology
with the structure of an n-ary partially associative algebra with an operation
of even or odd degree depending on the parity of n. In the cases n = 3 and n =
4, we provide an explicit definition of this cup product and prove its basis
properties.
