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The classical Hochschild cohomology theory of rings is extended to abelianheaps with distributing multiplication or trusses. This cohomology is thenemployed to give necessary and sufficient conditions for a Nijenhuis product ona truss (defined by the extension of the Nijenhuis product on an associativering introduced by Cari\~nena, Grabowski and Marmo in [Internat. J. ModernPhys. A 15 (2000), 4797-4810, arXiv:math-ph/0610011]) to be associative. Thedefinition of Nijenhuis product and operators on trusses is then linearised tothe case of affine spaces with compatible associative multiplications orassociative affgebras. It is shown that this construction leads to compatibleLie brackets on an affine space.

Affine Nijenhuis Operators and Hochschild Cohomology of Trusses