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The cup product in the cohomology of algebras over quadratic operads has beenstudied in the general setting of Koszul duality for operads. We study the cupproduct on the cohomology of n-ary totally associative algebras with anoperation of even (homological) degree. This cup product endows the cohomologywith the structure of an n-ary partially associative algebra with an operationof 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 basisproperties.

Operadic approach to cohomology of associative triple and N-tuple systems