Grobner bases for operads
It is now well-known that applications of the operad theory in general (and, in particular, to verifications of the Koszul property) are really difficult in particular computations. There was no known ``arithmetic'' of operations similar to the arithmetic of integers or polynomials (by an ``arithmetic'' we mean the usual notion of divisibility). The good analogue of multiplication for the operadic data is the composition of operations. But the action of the symmetric groups on the entries of operations contradicts with any possible functorial definition of divisibility. This paper contains a solution to this problem using the notion of Shuffle operads. The key idea is to forget about a certain part of the action of the symmetric group. In spite of being a very simple idea, it allowed us to introduce a theory of monomials, their divisibility and compatible orderings of monomials for operads. Summarizing these notions, we came up with the notion of Grobner bases for operads. Grobner bases is a remarkable technical tool initiated in the commutative algebra setting by Buchberger which allows one to solve systems of equations with many unknowns. The theory of Grobner bases for operads made it possible to provide a unified proof of the existing computational results in the field as well as to prove some new results. It is clear that there are many topics that can be successfully approached by these new methods.