Almomorphism

This note is about "almost homomorphisms" -- Almomorphisms. We already use these: I'm assigning a name, and suggesting we might get some milage from formalizing our use. An almomorphism can be "almost" in a few ways. It might usually be true, or sometimes be true, or be true more often than chance; or it might be nearly correct, most of the time.

Example A: Residues modulo A and modulo B, with A < B < A+small. Then addition modulo A is frequently the same as addition modulo B. The answer c+d (mod A) is the same as c+d (mod B) when there's no wraparound, i.e. c+d < A.  This happens about half the time for randomly chosen c & d. Looking at multiplication, the no-wraparound condition becomes cd < A. This only happens with frequency about logA/A, which is somewhat more likely than would be expected by chance.

Both of those wraparound conditions are also apply to the almomorphism between the integers modulo A and rational integers. For example, any fixed-size integral type in typical computer languages where overflow is non-catastrophic. (Wrapping is typical, but saturation still satisfies the conditions.) Phil