 Hochschild cup product

For $f\in C^p(A,A)$ and $g\in C^q(A,A)$, set $(f⌣g)(a_1,\dots ,a_{p+q})=f(a_1,\dots ,a_p)g(a_{p+1},\dots ,a_{p+q})$. This associative cochain multiplication descends to Hochschild cohomology, where it gives a graded commutative algebra. It is one operation in the Gerstenhaber algebra structure.