We 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∞​-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.Comment: 9 pages, second version - minor grammatical change