PermutationCategory(S)

perm.spad line 1 [edit on github]

PermutationCategory provides a categorical environment for subgroups of bijections of a set (i.e. permutations)

* : (%, %) -> %
from Magma
/ : (%, %) -> %
from Group
1 : () -> %
from MagmaWithUnit
< : (%, %) -> Boolean

p < q is an order relation on permutations. Note: this order is only total if and only if S is totally ordered or S is finite.

<= : (%, %) -> Boolean if S has Finite or S has OrderedSet
from PartialOrder
= : (%, %) -> Boolean
from BasicType
> : (%, %) -> Boolean if S has Finite or S has OrderedSet
from PartialOrder
>= : (%, %) -> Boolean if S has Finite or S has OrderedSet
from PartialOrder
^ : (%, Integer) -> %
from Group
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from Group
conjugate : (%, %) -> %
from Group
cycle : List(S) -> %

cycle(ls) coerces a cycle ls, i.e. a list with not repetitions to a permutation, which maps ls.i to ls.i+1, indices modulo the length of the list. Error: if repetitions occur.

cycles : List(List(S)) -> %

cycles(lls) coerces a list list of cycles lls to a permutation, each cycle being a list with not repetitions, is coerced to the permutation, which maps ls.i to ls.i+1, indices modulo the length of the list, then these permutations are multiplied. Error: if repetitions occur in one cycle.

elt : (%, S) -> S

elt(p, el) returns the image of el under the permutation p.

eval : (%, S) -> S

eval(p, el) returns the image of el under the permutation p.

inv : % -> %
from Group
latex : % -> String
from SetCategory
leftPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower : (%, PositiveInteger) -> %
from Magma
leftRecip : % -> Union(%, "failed")
from MagmaWithUnit
max : (%, %) -> % if S has Finite or S has OrderedSet
from OrderedSet
min : (%, %) -> % if S has Finite or S has OrderedSet
from OrderedSet
one? : % -> Boolean
from MagmaWithUnit
orbit : (%, S) -> Set(S)

orbit(p, el) returns the orbit of el under the permutation p, i.e. the set which is given by applications of the powers of p to el.

recip : % -> Union(%, "failed")
from MagmaWithUnit
rightPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower : (%, PositiveInteger) -> %
from Magma
rightRecip : % -> Union(%, "failed")
from MagmaWithUnit
sample : () -> %
from MagmaWithUnit
smaller? : (%, %) -> Boolean if S has Finite or S has OrderedSet
from Comparable
~= : (%, %) -> Boolean
from BasicType

Monoid

SemiGroup

TwoSidedRecip

BasicType

unitsKnown

CoercibleTo(OutputForm)

OrderedSet

Group

Magma

SetCategory

Comparable

PartialOrder

MagmaWithUnit