# reasoning about almost-groups and almost-associativity?

I have a group-like structure with a unique property that makes it not quite a group. I've been thinking of it as "almost associativity" and I'm wondering what's been studied about it? To see what I mean, first consider an operation $*$ that is associative. Then:

$A*(B*C) = (A*B)*C$

But if it is an almost associative operation, all we can say is that this is "almost true" with some bounds. If we rearrange the terms and do some funkiness, we get:

$[A*(B*C)]*[(A*B)*C]^{-1} \le \epsilon$

This epsilon can either be a fixed quantity (e.g. 1), or it can be a function of the terms (e.g. 1%).

There is very simple example of "almost associativity" in computer science: operations on floating point numbers. Because we cannot represent fractions like (1/3) exactly, multiplication is not associative. For example:

$(3*\frac{1}{3})*4 \ne 3 * (\frac{1}{3}*4)$

But we can put an upper bound on what the error will be.

So my question is: are there names for structures like this, and where can I look up more references for reasoning about them?

• – Qiaochu Yuan Aug 13 '12 at 20:39