SmallOrdinal

ordinal.spad line 1 [edit on github]

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

* : (%, %) -> %

from LeftModule(%)

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

< : (%, %) -> 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

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

coerce : NonNegativeInteger -> %

from CoercibleFrom(NonNegativeInteger)

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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

from RetractableTo(NonNegativeInteger)

retractIfCan : % -> Union(NonNegativeInteger, "failed")

from RetractableTo(NonNegativeInteger)

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

smaller? : (%, %) -> Boolean

from Comparable

subtractIfCan : (%, %) -> Union(%, "failed")

from CancellationAbelianMonoid

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

OrderedSet

RightModule(%)

Monoid

CancellationAbelianMonoid

OrderedAbelianSemiGroup

RetractableTo(NonNegativeInteger)

CoercibleFrom(NonNegativeInteger)

MagmaWithUnit

NonAssociativeSemiRng

CoercibleTo(OutputForm)

SemiRing

Hashable

AbelianSemiGroup

SetCategory

Comparable

LeftModule(%)

SemiGroup

AbelianMonoid

OrderedAbelianMonoid

BiModule(%, %)

PartialOrder

BasicType

NonAssociativeSemiRing

Magma

SemiRng