Lattice Rationals, 10 of 10

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.