Lattice Rationals, 8 of 10

Double Ringlets of Zeroids  and Infinities

              

               Recall that in general:

                              (a/0) + (b/c)      =       (a/0)    if c does not equal zero;

                              (a/0) + (b/0)      =       (a+b)/0

               Infinities absorb finite quantities, but add like integers.

               They also multiply like integers, if we use the usual definitions:

                              (a/0)*(b/0)    =   (a*b) / 0

Now consider these laws for infinities:

                              (a/0) + (b/0)                       =             (a+b) / 0

                              (a/0)  1/+  (b/0)               =             lcm(a,b) / 0

                              (a/0) * (b/0)                       =             (a*b)/0

               Infinities  add by adding indices, reduce by lcm on indices, and multiply by multiplying indices. I call this the ‘double ringlet of infinities’; for in it multiplication double-distributes over addition in a ring and triple-distributes over reduction in a semi-ring.

               Zeroids have these laws:

                              (0/a) + (0/b)                       =             0 / lcm(a,b)

                              (0/a)  1/+  (0/b)               =             0 / (a+b)

                              (0/a) * (0/b)                       =             0 / (a*b)

               Zeroids reduce by adding indices, add by lcm on indices, and multiply by multiplying indices. I call this the ‘double ringlet of zeroids’; for in it multiplication double-distributes over reduction in a ring and triple-distributes over addition in a semi-ring.