# SmallOrdinal¶

SmallOrdinal implements ordinal numbers up to `epsilon_0`. `+` and `*` are “natural” addition and multiplication of ordinals. Available separately are “ordered” operataions.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

<=: (%, %) -> Boolean

from PartialOrder

<: (%, %) -> Boolean

from PartialOrder

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean

from PartialOrder

>: (%, %) -> Boolean

from PartialOrder

^: (%, %) -> %

`o1^o2` returns `o1` to power `o2`, where power is inductively defined using successive natural multiplication from the left

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

antiCommutator: (%, %) -> %
coerce: % -> OutputForm
coerce: NonNegativeInteger -> %
hash: % -> SingleInteger

from Hashable

hashUpdate!: (HashState, %) -> HashState

from Hashable

integerPart: % -> NonNegativeInteger

`integerPart(o)` = `n` when `o` = `l` + `n` and `l` is a limit ordinal

latex: % -> String

from SetCategory

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

limitPart: % -> %

`limitPart(o)` = `l` when `o` = `l` + `n` and `l` is a limit ordinal and `n` is a nonnegative integer

max: (%, %) -> %

from OrderedSet

min: (%, %) -> %

from OrderedSet

omega: () -> %

`omega()` is the first infinite ordinal

omegapower: % -> %

`omegapower(p)` returns omega^p

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

ordinalAdd: (%, %) -> %

`ordinalAdd(o1, o2)` returns sum of `o1` and `o2` as ordered sets

ordinalMul: (%, %) -> %

`ordinalMul(o1, o2)` returns product of `o1` and `o2` as ordered sets

ordinalPower: (%, %) -> %

`ordinalPower(o1, o2)` returns `o1` to power `o2`, where power is inductively defined using successive ordinal multiplication from the left

recip: % -> Union(%, failed)

from MagmaWithUnit

retract: % -> NonNegativeInteger
retractIfCan: % -> Union(NonNegativeInteger, failed)
rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

smaller?: (%, %) -> Boolean

from Comparable

subtractIfCan: (%, %) -> Union(%, failed)
zero?: % -> Boolean

from AbelianMonoid

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

CancellationAbelianMonoid

Comparable

Hashable

Magma

MagmaWithUnit

Monoid

NonAssociativeSemiRing

NonAssociativeSemiRng

OrderedAbelianMonoid

OrderedAbelianSemiGroup

OrderedSet

PartialOrder

SemiGroup

SemiRing

SemiRng

SetCategory