## Wednesday, April 23, 2014

### 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.