**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.

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