Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A set of operations, each one having a fixed finite number of arguments and one output, which can be composed with others.
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Examples

Indeed, what we end up doing is forming the overlaps between leading terms by finding trees that parts of the trees from the leading terms from pairs of operad elements can embed into, and using the resulting procedure to build the same kind of bad cases we need to test for Buchberger's algorithm.

Furthermore, an operad O = {O (n)} has defined on it structure operations.

The vector space spanned by all such trees forms an operad where the composition operation works by taking a tree and attaching the root to the leaf numbered by whatever position we need to compose at.

I heard of this, and got my hands on the paper, when I first arrived in CIRM in Luminy, outside Marseille, for 2 weeks of operad theory with a master's course and a conference on the subject.

The resulting construction is the free operad on the generating set and takes the role that the polynomial ring had in the previous examples.

And what Dotsenko and Khoroshkin pointed out was that if we restrict the kinds of actions we allow from permutations on these trees somewhat, we end up with something that has exactly one representative that fits in the restricted context for each tree that occurs as a basis element of the free operad.

It turns out that GrÃ¶bner bases would be useful to prove operads to be Koszul  something I won't discuss here, but which is important in the operad theory context.
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