FreeGroup(S)

free.spad line 383 [edit on github]

The free group on a set S is the group of finite products of the form reduce(*, [si ^ ni]) where the si's are in S, and the ni's are integers. The multiplication is not commutative.

* : (%, %) -> %
from Magma
* : (%, S) -> %

x * s returns the product of x by s on the right.

* : (S, %) -> %

s * x returns the product of x by s on the left.

/ : (%, %) -> %
from Group
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
^ : (%, Integer) -> %
from Group
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
^ : (S, Integer) -> %

s ^ n returns the product of s by itself n times.

coerce : S -> %
from CoercibleFrom(S)
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from Group
conjugate : (%, %) -> %
from Group
factors : % -> List(Record(gen : S, exp : Integer))

factors(a1^e1, ..., an^en) returns [[a1, e1], ..., [an, en]].

inv : % -> %
from Group
latex : % -> String
from SetCategory
leftPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower : (%, PositiveInteger) -> %
from Magma
leftRecip : % -> Union(%, "failed")
from MagmaWithUnit
mapExpon : (Mapping(Integer, Integer), %) -> %

mapExpon(f, a1^e1 ... an^en) returns a1^f(e1) ... an^f(en).

mapGen : (Mapping(S, S), %) -> %

mapGen(f, a1^e1 ... an^en) returns f(a1)^e1 ... f(an)^en.

nthExpon : (%, Integer) -> Integer

nthExpon(x, n) returns the exponent of the n^th monomial of x.

nthFactor : (%, Integer) -> S

nthFactor(x, n) returns the factor of the n^th monomial of x.

one? : % -> Boolean
from MagmaWithUnit
recip : % -> Union(%, "failed")
from MagmaWithUnit
retract : % -> S
from RetractableTo(S)
retractIfCan : % -> Union(S, "failed")
from RetractableTo(S)
rightPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower : (%, PositiveInteger) -> %
from Magma
rightRecip : % -> Union(%, "failed")
from MagmaWithUnit
sample : () -> %
from MagmaWithUnit
size : % -> NonNegativeInteger

size(x) returns the number of monomials in x.

smaller? : (%, %) -> Boolean if S has Comparable
from Comparable
~= : (%, %) -> Boolean
from BasicType

Monoid

SemiGroup

TwoSidedRecip

RetractableTo(S)

BasicType

unitsKnown

Magma

CoercibleTo(OutputForm)

Group

CoercibleFrom(S)

SetCategory

Comparable

MagmaWithUnit