**Lattice Rationals modulo zeroids and infinities**

Say that
x and y are “equal modulo zeroids”, a.k.a.
“x =

_{0}y”, if and only if:
x
+ 0/n = y +
0/n for some positive integer
n

Therefore
we have positive cancellation modulo zeroids:

(ac)/(bc) =

_{0}a/b if c>0
Equality
modulo zeroids is an equivalence relation; reflexive, symmetric, and
transitive. Therefore it defines equivalence classes, and operations on those
classes for well-defined on those classes. For instance:

If x
=

_{0}X and y =_{0}Y then;
x
+ y =

_{0}X + Y
x
- y =

_{0}X - Y
x
* y =

_{0}X * Y
But
reciprocal is not well-defined on the zeroids;

0/1 =

_{0}0/2 ; but 1/0 and 2/0 are not equal modulo zeroids.
Similarly,
reduction is not well-defined for the zeroids.

The
equivalence classes, with operations thus defined, equals the wheel numbers
plus the double ringlet of infinities. This is an arithmetic of the rationals,
plus alternator, plus all the infinities.

Reciprocally,
we could define equality modulo infinities:

“x =

_{1/0}y” if and only if :
x
1/+
n/0 = y 1/+
n/0 for some positive integer
n

This too
has positive cancellation:

(ac)/(bc) =

_{1/0}_{ }a/b if c>0
Reduction
and multiplication are well defined modulo infinities; but reciprocal and
addition are not well defined on the infinities. The equivalence classes modulo
infinities equals the wheel numbers plus
the double ringlet of zeroids.

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