# NormalizedTriangularSetCategory(R, E, V, P)ΒΆ

- R: GcdDomain
- E: OrderedAbelianMonoidSup
- V: OrderedSet
- P: RecursivePolynomialCategory(R, E, V)

The category of normalized triangular sets. A triangular set `ts`

is said normalized if for every algebraic variable `v`

of `ts`

the polynomial `select(ts, v)`

is normalized `w`

.`r`

.`t`

. every polynomial in `collectUnder(ts, v)`

. A polynomial `p`

is said normalized `w`

.`r`

.`t`

. a non-constant polynomial `q`

if `p`

is constant or `degree(p, mdeg(q)) = 0`

and `init(p)`

is normalized `w`

.`r`

.`t`

. `q`

. One of the important features of normalized triangular sets is that they are regular sets.

- #: % -> NonNegativeInteger
- from Aggregate
- =: (%, %) -> Boolean
- from BasicType
- ~=: (%, %) -> Boolean
- from BasicType
- algebraic?: (V, %) -> Boolean
- from TriangularSetCategory(R, E, V, P)
- algebraicCoefficients?: (P, %) -> Boolean
- from RegularTriangularSetCategory(R, E, V, P)
- algebraicVariables: % -> List V
- from TriangularSetCategory(R, E, V, P)
- any?: (P -> Boolean, %) -> Boolean
- from HomogeneousAggregate P
- augment: (List P, %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- augment: (List P, List %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- augment: (P, %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- augment: (P, List %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- autoReduced?: (%, (P, List P) -> Boolean) -> Boolean
- from TriangularSetCategory(R, E, V, P)
- basicSet: (List P, (P, P) -> Boolean) -> Union(Record(bas: %, top: List P), failed)
- from TriangularSetCategory(R, E, V, P)
- basicSet: (List P, P -> Boolean, (P, P) -> Boolean) -> Union(Record(bas: %, top: List P), failed)
- from TriangularSetCategory(R, E, V, P)
- coerce: % -> List P
- from CoercibleTo List P
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- coHeight: % -> NonNegativeInteger if V has Finite
- from TriangularSetCategory(R, E, V, P)
- collect: (%, V) -> %
- from PolynomialSetCategory(R, E, V, P)
- collectQuasiMonic: % -> %
- from TriangularSetCategory(R, E, V, P)
- collectUnder: (%, V) -> %
- from PolynomialSetCategory(R, E, V, P)
- collectUpper: (%, V) -> %
- from PolynomialSetCategory(R, E, V, P)
- construct: List P -> %
- from Collection P
- convert: % -> InputForm
- from ConvertibleTo InputForm
- copy: % -> %
- from Aggregate
- count: (P -> Boolean, %) -> NonNegativeInteger
- from HomogeneousAggregate P
- count: (P, %) -> NonNegativeInteger
- from HomogeneousAggregate P
- degree: % -> NonNegativeInteger
- from TriangularSetCategory(R, E, V, P)
- empty: () -> %
- from Aggregate
- empty?: % -> Boolean
- from Aggregate
- eq?: (%, %) -> Boolean
- from Aggregate
- eval: (%, Equation P) -> % if P has Evalable P
- from Evalable P
- eval: (%, List Equation P) -> % if P has Evalable P
- from Evalable P
- eval: (%, List P, List P) -> % if P has Evalable P
- from InnerEvalable(P, P)
- eval: (%, P, P) -> % if P has Evalable P
- from InnerEvalable(P, P)
- every?: (P -> Boolean, %) -> Boolean
- from HomogeneousAggregate P
- extend: (%, P) -> %
- from TriangularSetCategory(R, E, V, P)
- extend: (List P, %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- extend: (List P, List %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- extend: (P, %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- extend: (P, List %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- extendIfCan: (%, P) -> Union(%, failed)
- from TriangularSetCategory(R, E, V, P)
- find: (P -> Boolean, %) -> Union(P, failed)
- from Collection P
- first: % -> Union(P, failed)
- from TriangularSetCategory(R, E, V, P)
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- headReduce: (P, %) -> P
- from TriangularSetCategory(R, E, V, P)
- headReduced?: % -> Boolean
- from TriangularSetCategory(R, E, V, P)
- headReduced?: (P, %) -> Boolean
- from TriangularSetCategory(R, E, V, P)
- headRemainder: (P, %) -> Record(num: P, den: R)
- from PolynomialSetCategory(R, E, V, P)
- iexactQuo: (R, R) -> R
- from PolynomialSetCategory(R, E, V, P)
- infRittWu?: (%, %) -> Boolean
- from TriangularSetCategory(R, E, V, P)
- initiallyReduce: (P, %) -> P
- from TriangularSetCategory(R, E, V, P)
- initiallyReduced?: % -> Boolean
- from TriangularSetCategory(R, E, V, P)
- initiallyReduced?: (P, %) -> Boolean
- from TriangularSetCategory(R, E, V, P)
- initials: % -> List P
- from TriangularSetCategory(R, E, V, P)
- internalAugment: (List P, %) -> %
- from RegularTriangularSetCategory(R, E, V, P)
- internalAugment: (P, %) -> %
- from RegularTriangularSetCategory(R, E, V, P)
- intersect: (List P, %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- intersect: (List P, List %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- intersect: (P, %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- intersect: (P, List %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- invertible?: (P, %) -> Boolean
- from RegularTriangularSetCategory(R, E, V, P)
- invertible?: (P, %) -> List Record(val: Boolean, tower: %)
- from RegularTriangularSetCategory(R, E, V, P)
- invertibleElseSplit?: (P, %) -> Union(Boolean, List %)
- from RegularTriangularSetCategory(R, E, V, P)
- invertibleSet: (P, %) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- last: % -> Union(P, failed)
- from TriangularSetCategory(R, E, V, P)
- lastSubResultant: (P, P, %) -> List Record(val: P, tower: %)
- from RegularTriangularSetCategory(R, E, V, P)
- lastSubResultantElseSplit: (P, P, %) -> Union(P, List %)
- from RegularTriangularSetCategory(R, E, V, P)
- latex: % -> String
- from SetCategory
- less?: (%, NonNegativeInteger) -> Boolean
- from Aggregate
- mainVariable?: (V, %) -> Boolean
- from PolynomialSetCategory(R, E, V, P)
- mainVariables: % -> List V
- from PolynomialSetCategory(R, E, V, P)
- map!: (P -> P, %) -> %
- from HomogeneousAggregate P
- map: (P -> P, %) -> %
- from HomogeneousAggregate P
- member?: (P, %) -> Boolean
- from HomogeneousAggregate P
- members: % -> List P
- from HomogeneousAggregate P
- more?: (%, NonNegativeInteger) -> Boolean
- from Aggregate
- mvar: % -> V
- from PolynomialSetCategory(R, E, V, P)
- normalized?: % -> Boolean
- from TriangularSetCategory(R, E, V, P)
- normalized?: (P, %) -> Boolean
- from TriangularSetCategory(R, E, V, P)
- parts: % -> List P
- from HomogeneousAggregate P
- purelyAlgebraic?: % -> Boolean
- from RegularTriangularSetCategory(R, E, V, P)
- purelyAlgebraic?: (P, %) -> Boolean
- from RegularTriangularSetCategory(R, E, V, P)
- purelyAlgebraicLeadingMonomial?: (P, %) -> Boolean
- from RegularTriangularSetCategory(R, E, V, P)
- purelyTranscendental?: (P, %) -> Boolean
- from RegularTriangularSetCategory(R, E, V, P)
- quasiComponent: % -> Record(close: List P, open: List P)
- from TriangularSetCategory(R, E, V, P)
- reduce: ((P, P) -> P, %) -> P
- from Collection P
- reduce: ((P, P) -> P, %, P) -> P
- from Collection P
- reduce: ((P, P) -> P, %, P, P) -> P
- from Collection P
- reduce: (P, %, (P, P) -> P, (P, P) -> Boolean) -> P
- from TriangularSetCategory(R, E, V, P)
- reduceByQuasiMonic: (P, %) -> P
- from TriangularSetCategory(R, E, V, P)
- reduced?: (P, %, (P, P) -> Boolean) -> Boolean
- from TriangularSetCategory(R, E, V, P)
- remainder: (P, %) -> Record(rnum: R, polnum: P, den: R)
- from PolynomialSetCategory(R, E, V, P)
- remove: (P -> Boolean, %) -> %
- from Collection P
- remove: (P, %) -> %
- from Collection P
- removeDuplicates: % -> %
- from Collection P
- removeZero: (P, %) -> P
- from TriangularSetCategory(R, E, V, P)
- rest: % -> Union(%, failed)
- from TriangularSetCategory(R, E, V, P)
- retract: List P -> %
- from RetractableFrom List P
- retractIfCan: List P -> Union(%, failed)
- from RetractableFrom List P
- rewriteIdealWithHeadRemainder: (List P, %) -> List P
- from PolynomialSetCategory(R, E, V, P)
- rewriteIdealWithRemainder: (List P, %) -> List P
- from PolynomialSetCategory(R, E, V, P)
- rewriteSetWithReduction: (List P, %, (P, P) -> P, (P, P) -> Boolean) -> List P
- from TriangularSetCategory(R, E, V, P)
- roughBase?: % -> Boolean
- from PolynomialSetCategory(R, E, V, P)
- roughEqualIdeals?: (%, %) -> Boolean
- from PolynomialSetCategory(R, E, V, P)
- roughSubIdeal?: (%, %) -> Boolean
- from PolynomialSetCategory(R, E, V, P)
- roughUnitIdeal?: % -> Boolean
- from PolynomialSetCategory(R, E, V, P)
- sample: %
- from Aggregate
- select: (%, V) -> Union(P, failed)
- from TriangularSetCategory(R, E, V, P)
- select: (P -> Boolean, %) -> %
- from Collection P
- size?: (%, NonNegativeInteger) -> Boolean
- from Aggregate
- sort: (%, V) -> Record(under: %, floor: %, upper: %)
- from PolynomialSetCategory(R, E, V, P)
- squareFreePart: (P, %) -> List Record(val: P, tower: %)
- from RegularTriangularSetCategory(R, E, V, P)
- stronglyReduce: (P, %) -> P
- from TriangularSetCategory(R, E, V, P)
- stronglyReduced?: % -> Boolean
- from TriangularSetCategory(R, E, V, P)
- stronglyReduced?: (P, %) -> Boolean
- from TriangularSetCategory(R, E, V, P)
- triangular?: % -> Boolean
- from PolynomialSetCategory(R, E, V, P)
- trivialIdeal?: % -> Boolean
- from PolynomialSetCategory(R, E, V, P)
- variables: % -> List V
- from PolynomialSetCategory(R, E, V, P)
- zeroSetSplit: (List P, Boolean) -> List %
- from RegularTriangularSetCategory(R, E, V, P)
- zeroSetSplit: List P -> List %
- from TriangularSetCategory(R, E, V, P)
- zeroSetSplitIntoTriangularSystems: List P -> List Record(close: %, open: List P)
- from TriangularSetCategory(R, E, V, P)

Evalable P if P has Evalable P

InnerEvalable(P, P) if P has Evalable P

PolynomialSetCategory(R, E, V, P)

RegularTriangularSetCategory(R, E, V, P)

TriangularSetCategory(R, E, V, P)