> The given function is clearly only defined over naturals.
The post doesn't specify. Programming languages do define it over the negatives, however. I just didn't want anyone going into python and asserting that a & b >= min(a,b).
> there's no indication of how to encode negatives
Just do the standard thing: use 2s complement arithmetic [1]. More concretely: set b to infinity and take the limit under the 2-adic metric [2]. For example, -2 & -1 has partial sums 0, 10, 110, 1110, 11110, etc. It's limiting to ...1111110, which is the 2's complement representation of -2.
Excellent. I'm glad we agree. You'll notice that the stated identity f(x,x) = x fails to hold over x < 0. I hope this is sufficient to make it clear that the equation does not consider the negatives?
The post doesn't specify. Programming languages do define it over the negatives, however. I just didn't want anyone going into python and asserting that a & b >= min(a,b).
> there's no indication of how to encode negatives
Just do the standard thing: use 2s complement arithmetic [1]. More concretely: set b to infinity and take the limit under the 2-adic metric [2]. For example, -2 & -1 has partial sums 0, 10, 110, 1110, 11110, etc. It's limiting to ...1111110, which is the 2's complement representation of -2.
1: https://en.wikipedia.org/wiki/Two%27s_complement
2: https://en.wikipedia.org/wiki/P-adic_number