PermutationCategory SΒΆ

perm.spad line 1

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

1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
/: (%, %) -> %
from Group
<: (%, %) -> 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
~=: (%, %) -> Boolean
from BasicType
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 mutiplied. 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.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
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

BasicType

CoercibleTo OutputForm

Comparable if S has Finite or S has OrderedSet

Group

Magma

MagmaWithUnit

Monoid

OrderedSet if S has Finite or S has OrderedSet

PartialOrder if S has Finite or S has OrderedSet

SemiGroup

SetCategory

unitsKnown