Notes for wClips-120321/0:22:35

"Addition" is never one of the operations in our structures - it is added later by allowing formal linear combinations of objects. I guess if you started with a structure that already has an addition operation - say, "connect sum" of knots - you'll have to rename it "original addition" so as to distinguish it from the "addition" we add later when we allow formal linear combinations.

When talking about powers of the augmentation ideal, we only use the original operations of our structure.

So in the example where the structure is a group $G$, its group-ring ${\mathbb Q}G$ is again a structure with just one binary operation (multiplication) plus an artificially-added auxiliary operation "addition" which does not participate in taking powers of the augmentation ideal.

An alternative to all that is to start with a structure whose ${\mathcal O}_\alpha$'s are linear spaces (or at least, ${\mathbb Z}$-modules) and all of whose operations are multi-linear. Here again "addition" will have a special role and will not participate in forming powers of the augmentation ideal. --Drorbn 15:46, 25 March 2012 (EDT)