IndexedDirectProductObject(A, S)ΒΆ

indexedp.spad line 87

Indexed direct products of objects over a set A of generators indexed by an ordered set S. It currently provides the ground for, e.g. FreeModule which lies at the basis of polynomials of all sorts. All items have finite support. If A is a monoid, then only non-zero terms are stored. If A has additive structure, it is propagated coordinatewise to the product. Similarly, comparisons are propagated using lexicographic ordering.

0: % if A has AbelianMonoid
from AbelianMonoid
*: (Integer, %) -> % if A has AbelianGroup
from AbelianGroup
*: (NonNegativeInteger, %) -> % if A has AbelianMonoid
from AbelianMonoid
*: (PositiveInteger, %) -> % if A has AbelianMonoid
from AbelianSemiGroup
+: (%, %) -> % if A has AbelianMonoid
from AbelianSemiGroup
-: % -> % if A has AbelianGroup
from AbelianGroup
-: (%, %) -> % if A has AbelianGroup
from AbelianGroup
<: (%, %) -> Boolean if S has OrderedSet and A has OrderedAbelianMonoid or S has OrderedSet and A has OrderedAbelianMonoidSup
from PartialOrder
<=: (%, %) -> Boolean if S has OrderedSet and A has OrderedAbelianMonoid or S has OrderedSet and A has OrderedAbelianMonoidSup
from PartialOrder
=: (%, %) -> Boolean if A has Comparable or A has AbelianMonoid
from BasicType
>: (%, %) -> Boolean if S has OrderedSet and A has OrderedAbelianMonoid or S has OrderedSet and A has OrderedAbelianMonoidSup
from PartialOrder
>=: (%, %) -> Boolean if S has OrderedSet and A has OrderedAbelianMonoid or S has OrderedSet and A has OrderedAbelianMonoidSup
from PartialOrder
~=: (%, %) -> Boolean if A has Comparable or A has AbelianMonoid
from BasicType
coerce: % -> OutputForm if A has Comparable or A has AbelianMonoid
from CoercibleTo OutputForm
construct: List Record(k: S, c: A) -> %
from IndexedDirectProductCategory(A, S)
constructOrdered: List Record(k: S, c: A) -> %
from IndexedDirectProductCategory(A, S)
hash: % -> SingleInteger if A has Comparable or A has AbelianMonoid
from SetCategory
hashUpdate!: (HashState, %) -> HashState if A has Comparable or A has AbelianMonoid
from SetCategory
latex: % -> String if A has Comparable or A has AbelianMonoid
from SetCategory
leadingCoefficient: % -> A
from IndexedDirectProductCategory(A, S)
leadingMonomial: % -> %
from IndexedDirectProductCategory(A, S)
leadingSupport: % -> S
from IndexedDirectProductCategory(A, S)
leadingTerm: % -> Record(k: S, c: A)
from IndexedDirectProductCategory(A, S)
listOfTerms: % -> List Record(k: S, c: A)
from IndexedDirectProductCategory(A, S)
map: (A -> A, %) -> %
from IndexedDirectProductCategory(A, S)
max: (%, %) -> % if S has OrderedSet and A has OrderedAbelianMonoid or S has OrderedSet and A has OrderedAbelianMonoidSup
from OrderedSet
min: (%, %) -> % if S has OrderedSet and A has OrderedAbelianMonoid or S has OrderedSet and A has OrderedAbelianMonoidSup
from OrderedSet
monomial: (A, S) -> %
from IndexedDirectProductCategory(A, S)
monomial?: % -> Boolean
from IndexedDirectProductCategory(A, S)
numberOfMonomials: % -> NonNegativeInteger
from IndexedDirectProductCategory(A, S)
opposite?: (%, %) -> Boolean if A has AbelianMonoid
from AbelianMonoid
reductum: % -> %
from IndexedDirectProductCategory(A, S)
sample: % if A has AbelianMonoid
from AbelianMonoid
smaller?: (%, %) -> Boolean if A has Comparable
from Comparable
subtractIfCan: (%, %) -> Union(%, failed) if A has CancellationAbelianMonoid
from CancellationAbelianMonoid
sup: (%, %) -> % if S has OrderedSet and A has OrderedAbelianMonoidSup
from OrderedAbelianMonoidSup
zero?: % -> Boolean if A has AbelianMonoid
from AbelianMonoid

AbelianGroup if A has AbelianGroup

AbelianMonoid if A has AbelianMonoid

AbelianProductCategory A

AbelianSemiGroup if A has AbelianMonoid

BasicType if A has Comparable or A has AbelianMonoid

CancellationAbelianMonoid if A has CancellationAbelianMonoid

CoercibleTo OutputForm if A has Comparable or A has AbelianMonoid

Comparable if A has Comparable

IndexedDirectProductCategory(A, S)

OrderedAbelianMonoid if S has OrderedSet and A has OrderedAbelianMonoid or S has OrderedSet and A has OrderedAbelianMonoidSup

OrderedAbelianMonoidSup if S has OrderedSet and A has OrderedAbelianMonoidSup

OrderedAbelianSemiGroup if S has OrderedSet and A has OrderedAbelianMonoid or S has OrderedSet and A has OrderedAbelianMonoidSup

OrderedCancellationAbelianMonoid if S has OrderedSet and A has OrderedAbelianMonoidSup

OrderedSet if S has OrderedSet and A has OrderedAbelianMonoid or S has OrderedSet and A has OrderedAbelianMonoidSup

PartialOrder if S has OrderedSet and A has OrderedAbelianMonoid or S has OrderedSet and A has OrderedAbelianMonoidSup

SetCategory if A has Comparable or A has AbelianMonoid