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 basic properties.

中文翻译：

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关联三元组和 N 元组上同调的操作方法

已经在操作数的 Koszul 对偶的一般设置中研究了二次操作数上代数的上同调中的杯积。我们研究杯积的上同调n-ary 具有偶（同调）度运算的完全结合代数。这种杯子产品赋予上同调一个结构n-ary 部分结合代数，具有偶数或奇数运算，取决于奇偶性n. 在这种情况下n=3和n=4，我们提供了这个杯子产品的明确定义并证明了它的基本性质。