2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78304We show that a graded commutative algebra A with any square zero odd differential operator is a natural generalization of a Batalin-Vilkovisky algebra. While such an operator of order 2 defines a Gerstenhaber (Lie) algebra structure on A, an operator of an order higher than 2 (Koszul-Akman definition) leads to the structure of a strongly homotopy Lie algebra (L$_\infty$-algebra) on A. This allows us to give a definition of a Batalin-Vilkovisky algebra up to homotopy. We also make a conjecture which is a generalization of the formality theorem of Kontsevich to the Batalin-Vilkovisky algebra level.9 pages, second version - minor grammatical changesQuantum AlgebraAlgebraic Topology17B70; 81T70; 58A50; 58D29text