SmallOrdinalΒΆ

ordinal.spad line 1

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

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (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 succesive natural multiplication from the left
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: NonNegativeInteger -> %
from RetractableTo NonNegativeInteger
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
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 succesive 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

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

CancellationAbelianMonoid

CoercibleTo OutputForm

Comparable

LeftModule %

Magma

MagmaWithUnit

Monoid

NonAssociativeSemiRing

NonAssociativeSemiRng

OrderedAbelianMonoid

OrderedAbelianSemiGroup

OrderedSet

PartialOrder

RetractableTo NonNegativeInteger

RightModule %

SemiGroup

SemiRing

SemiRng

SetCategory