LeftOreRing

catdef.spad line 838 [edit on github]

This is category of left ore rings, that is noncommutaive rings without zero divisors where we can compute least left common multiple

* : (%, %) -> %

from Magma

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

associator : (%, %, %) -> %

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

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

from EntireRing

latex : % -> String

from SetCategory

lcmCoef : (%, %) -> Record(llcm_res : %, coeff1 : %, coeff2 : %)

lcmCoef(c1, c2) computes (llcm_res, coeff1, coeff2) such that llcm_res is least left common multiple of c1 and c2 and llcm_res = coeff1*c1 = coeff2*c2

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

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

from MagmaWithUnit

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

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

from MagmaWithUnit

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

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

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

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

from CancellationAbelianMonoid

unit? : % -> Boolean

from EntireRing

unitCanonical : % -> %

from EntireRing

unitNormal : % -> Record(unit : %, canonical : %, associate : %)

from EntireRing

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

RightModule(%)

EntireRing

Monoid

noZeroDivisors

Ring

SemiGroup

CancellationAbelianMonoid

LeftModule(%)

BasicType

unitsKnown

NonAssociativeRing

Rng

Magma

NonAssociativeSemiRng

SemiRing

AbelianGroup

NonAssociativeSemiRing

SetCategory

AbelianSemiGroup

AbelianMonoid

BiModule(%, %)

MagmaWithUnit

CoercibleTo(OutputForm)

SemiRng

NonAssociativeRng