User blog:Rgetar/Ordinal subtraction and integer extension of ordinals

Ordinal subtraction is operation opposite to ordinal addition.

Ordinal addition is non-commutative, so, there are two ordinal subtractions (as for exponentiation, which is also non-commutative and also has two opposite operations: root and logarithm) :

1. α-β: (α-β)+β = α

2. -β+α: β+(-β+α) = α

(Note: in Ordinals array function blog I denoted both subtractions as α-β, and later I started to denote second subtraction as -β+α).

As we can extend natural numbers and zero to integer numbers using subtraction, we can extend ordinals to "integer ordinals".

Opposite ordinals
Subtraction of β we may descibe as addition of opposite "integer ordinal" -β:

α-β = α+(-β)

-β+α = (-β)+α

-α is opposite to α:

α+(-α) = 0

(-α)+α=0

To get opposite to α we need write Cantor normal form of α backwards and change all its coefficients to opposite:

α = ωα0·n0 + ωα1·n1 + ωα2·n2 + ...

-α = ... + ωα2·(-n2) + ωα1·(-n1) + ω02·(-n0)

Examples:

α = ω+1

-α = -1-ω

α = ω8·4+ω3·2+1

-α = -1-ω3·2-ω8·4

Addition rules
Extension od addition rules on "integer ordinals" (see also Ordinal arithmetic blog, How to add two ordinals section):

1. ωα1·n1+ωα2·n2 = ωα1·(n1+n2) (if α1 = α2)

2. ωα1·n1+ωα2·n2 = ωα1·n1+ωα2·n2 (if (α1 > α2 and n2 > 0) or (α1 < α2 and n2 < 0))

3. ωα1·n1+ωα2·n2 = ωα2·n2 (if α1 < α2 and n2 > 0)

4. ωα1·n1+ωα2·n2 = ωα1·n1 (if α1 > α2 and n1 < 0)

So, ωαi·ni "eats" all terms leftward until it faces a term with larger αi, if ni > 0, and rightward, if ni < 0.

"Mixed integer ordinals"
There are "fully positive integer ordinals" with all coefficients of Cantor normal form positive (they are ordinals), such as

ω+1

And there are "fully negative integer ordinals" with all coefficients of Cantor normal form negative (they are opposite to ordinals), such as

-1-ω

There are also "mixed integer ordinals" with some coefficients of Cantor normal form positive, and some negative, such as

ω-1

1-ω

They are, however, also positive (> 0) or negative (< 0), but not "fully":

ω-1 is positive, but not fully positive

1-ω is negative, but not fully negative

fpp, fnp
An "integer ordinal" α can be uniquely represented as difference of two ordinals: fully positive part (fpp) of α and fully negative part (fnp) of α.

α = fpp(α)-fnp(α)

Example:

α = ω8 + ω5 + ω4 + ω3 + ω - 1 - ω - ω2 - ω6 - ω7

fpp(α) = ω8 + ω5 + ω4 + ω3 + ω

fnp(α) = ω7 + ω6 + ω2 + ω + 1

Properties:

if fpp(α) = 0, fnp(α) = 0 then α = 0 (it is also an ordinal)

if fpp(α) > 0, fnp(α) = 0 then α is a "fully positive ordinal" (or just an ordinal)

if fpp(α) = 0, fnp(α) > 0 then α is a "fully negative ordinal"

if fpp(α) > 0, fnp(α) > 0 then α is a "mixed ordinal"

if fpp(α) > fnp(α) then α is positive

if fnp(α) > fpp(α) then α is negative

-α = fnp(α)-fpp(α)

List of "integer ordinals"
List of all "integer ordinals" with Cantor normal form coefficients -1 or 1 and degrees of ω less than 3 in ascending order:

-1-ω-ω2

-ω-ω2

1-ω-ω2

-1-ω2

-ω2

1-ω2

ω-1-ω2

ω-ω2

ω+1-ω2

-1-ω

-ω

1-ω

-1

0

1

ω-1

ω

ω+1

ω2-1-ω

ω2-ω

ω2+1-ω

ω2-1

ω2

ω2+1

ω2+ω-1

ω2+ω

ω2+ω+1

All "integer ordinals" are successors:

0 is successor of -1

ω is successor of ω-1

-ω is successor of -1-ω