Knuth's non-associative "group" on ${\mathcal P}(\mathbb{N})$

TL;DR

It is shown that this operation is ``group-like'' in that it has a neutral element, inverses, but it is not associative, and that the power-set of $\mathbb{N}$ is the collection of infinite bit-strings.

Abstract

Donald Knuth introduced in The Art of Computer Programming (Vol 4a) a fast approximation to the addition of integers (given in binary) in terms of bit-wise operations by $a + b \; \approx \; a \oplus b \oplus ((a\land b) \ll 1).$ Generalizing this to infinite bit-strings we get a binary operation on ${\mathcal P}(\mathbb{N})$, the power-set of $\mathbb{N}$ (which we identify with the collection of infinite bit-strings). We show that this operation is ``group-like'' in that it has a neutral element, inverses, but it is not associative. There are a lot of questions left, which the author has not been able to answer.