The aim of this paper is to introduce the category of Hoch-algebras whose objects are
associative algebras equipped with an extra magmatic operation≻ verifying the following
relation motivated by the Hochschild two-cocycle identity: R2:(x≻ y)∗ z+(x∗ y)≻ z= x≻(y∗
z)+ x∗(y≻ z). Such algebras appear in mathematical physics with≻ associative under the
name of compatible products. Here, we relax the associativity condition. The free Hoch-
algebra over a K-vector space is then given in terms of planar rooted trees and the triple of …

Hochschild two-cocycles play an important role in the deformation\a la Gerstenhaber of
associative algebras. The aim of this paper is to introduce the category of Hoch-algebras
whose objects are associative algebras equipped with an extra magmatic operation\succ
verifying the Hochschild two-cocycle relation:(x\succ y)* z+ (x* y)\succ z= x\succ (y* z)+
x*(y\succ z). The free Hoch-algebra over a K-vector space is given in terms of planar rooted
trees and the triples of operads (As, Hoch, Mag^\infty) endowed with the infinitesimal …