Abstract
The classical Hochschild cohomology theory of rings is extended to abelian heaps with distributing multiplication or trusses. This cohomology is then employed to give necessary and sufficient conditions for a Nijenhuis product on a truss (defined by the extension of the Nijenhuis product on an associative ring introduced by Carinena, Grabowski and Marmo in [Internat. J. Modern Phys. A 15 (2000), 4797-4810, arXiv:math-ph/0610011]) to be associative. The definition of Nijenhuis product and operators on trusses is then linearised to the case of affine spaces with compatible associative multiplications or associative affgebras. It is shown that this construction leads to compatible Lie brackets on an affine space.
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