Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Ind

### 12 Boolean algebra of constructible objects

#### 12.1 Constructors

##### 12.1-1 BooleanAlgebraOfConstructibleObjectsAsUnionOfMultipleDifferences
 ‣ BooleanAlgebraOfConstructibleObjectsAsUnionOfMultipleDifferences( D ) ( attribute )

Returns: a CAP category

Construct the Boolean algebra of constructible objects as union of objects in the poset D of multiple differences.

##### 12.1-2 UnionOfMultipleDifferences
 ‣ UnionOfMultipleDifferences( D1, D2, ... ) ( function )

Returns: an object in a CAP category

If D1=$$A1-B1$$, D2=$$A2-B2$$, ..., then the output is DirectProduct$$(A1,A2,...)$$ - Coproduct$$(B1,B2,...)$$.

##### 12.1-3 BooleanAlgebraOfConstructibleObjectsAsUnionOfDifferences
 ‣ BooleanAlgebraOfConstructibleObjectsAsUnionOfDifferences( D ) ( attribute )

Returns: a CAP category

Construct the Boolean algebra of constructible objects as union of objects in the poset D of single differences.

##### 12.1-4 UnionOfDifferences
 ‣ UnionOfDifferences( D1, D2, ... ) ( function )
 ‣ UnionOfDifferencesOfNormalizedObjects( D1, D2, ... ) ( function )

Returns: an object in a CAP category

If D1=$$A1-B1$$, D2=$$A2-B2$$, ..., then the output is DirectProduct$$(A1,A2,...)$$ - Coproduct$$(B1,B2,...)$$. AsFormalDifferenceOfNormalizedMorphisms assumes that the input is normalized.

#### 12.2 Properties

##### 12.2-1 IsOpen
 ‣ IsOpen( A ) ( property )

Returns: true or false

##### 12.2-2 IsClosed
 ‣ IsClosed( A ) ( operation )

##### 12.2-3 IsLocallyClosed
 ‣ IsLocallyClosed( A ) ( property )

Returns: true or false

#### 12.3 Attributes

##### 12.3-1 UnderlyingCategory
 ‣ UnderlyingCategory( C ) ( attribute )

Returns: a CAP category

The category underlying the Boolean algebra of constructible objects C.

##### 12.3-2 LocallyClosedPart
 ‣ LocallyClosedPart( A ) ( attribute )

##### 12.3-3 CanonicalObject
 ‣ CanonicalObject( A ) ( attribute )

##### 12.3-4 FactorizeObject
 ‣ FactorizeObject( A ) ( attribute )

##### 12.3-5 NormalizedObject
 ‣ NormalizedObject( A ) ( attribute )

##### 12.3-6 StandardizedObject
 ‣ StandardizedObject( A ) ( attribute )

##### 12.3-7 Closure
 ‣ Closure( A ) ( attribute )

Returns: an object in a co-Heyting algebra.

The closure of the constructible object A in the underlying co-Heyting algebra.

##### 12.3-8 ClosureAsConstructibleObject
 ‣ ClosureAsConstructibleObject( A ) ( attribute )

Returns: a constructible object.

The closure of the constructible object A as a constructible object.

##### 12.3-9 Dimension
 ‣ Dimension( A ) ( attribute )

#### 12.4 Operations

##### 12.4-1 NormalizeObject
 ‣ NormalizeObject( A ) ( operation )

##### 12.4-2 StandardizeObject
 ‣ StandardizeObject( A ) ( operation )

##### 12.4-3 Length
 ‣ Length( A ) ( attribute )

##### 12.4-4 
 ‣ ( A, i ) ( operation )

##### 12.4-5 Iterator
 ‣ Iterator( A ) ( operation )

##### 12.4-6 ForAllOp
 ‣ ForAllOp( A, f ) ( operation )

##### 12.4-7 ForAnyOp
 ‣ ForAnyOp( A, f ) ( operation )

##### 12.4-8 ListOfObjectsInMeetSemilatticeOfMultipleDifferences
 ‣ ListOfObjectsInMeetSemilatticeOfMultipleDifferences( A ) ( operation )

Returns: a list of CAP morphism

A list of morphisms in the underlying lattice representing the formal multiple difference A.

##### 12.4-9 ListOfObjectsInMeetSemilatticeOfDifferences
 ‣ ListOfObjectsInMeetSemilatticeOfDifferences( A ) ( operation )

Returns: a list of CAP morphism

A list of morphisms in the underlying lattice representing the formal multiple difference A.

#### 12.5 GAP categories

##### 12.5-1 IsBooleanAlgebraOfConstructibleObjects
 ‣ IsBooleanAlgebraOfConstructibleObjects( arg ) ( filter )

Returns: true or false

The GAP category of a Boolean algebra of constructible objects.

##### 12.5-2 IsConstructibleObject
 ‣ IsConstructibleObject( object ) ( filter )

Returns: true or false

The GAP category of objects in a Boolean algebra of constructible objects.

##### 12.5-3 IsMorphismBetweenConstructibleObjects
 ‣ IsMorphismBetweenConstructibleObjects( morphism ) ( filter )

Returns: true or false

The GAP category of morphisms in a Boolean algebra of constructible objects.

##### 12.5-4 IsBooleanAlgebraOfConstructibleObjectsAsUnionOfMultipleDifferences
 ‣ IsBooleanAlgebraOfConstructibleObjectsAsUnionOfMultipleDifferences( arg ) ( filter )

Returns: true or false

The GAP category of a Boolean algebra of constructible objects as union of multiple differences.

##### 12.5-5 IsConstructibleObjectAsUnionOfMultipleDifferences
 ‣ IsConstructibleObjectAsUnionOfMultipleDifferences( object ) ( filter )

Returns: true or false

The GAP category of objects in ...

##### 12.5-6 IsMorphismBetweenConstructibleObjectsAsUnionOfMultipleDifferences
 ‣ IsMorphismBetweenConstructibleObjectsAsUnionOfMultipleDifferences( morphism ) ( filter )

Returns: true or false

The GAP category of morphisms in ...

##### 12.5-7 IsBooleanAlgebraOfConstructibleObjectsAsUnionOfSingleDifferences
 ‣ IsBooleanAlgebraOfConstructibleObjectsAsUnionOfSingleDifferences( arg ) ( filter )

Returns: true or false

The GAP category of a Boolean algebra of constructible objects as union of single differences.

##### 12.5-8 IsConstructibleObjectAsUnionOfSingleDifferences
 ‣ IsConstructibleObjectAsUnionOfSingleDifferences( object ) ( filter )

Returns: true or false

The GAP category of objects in a Boolean algebra of constructible objects, the latter as unions of formal single differences.

##### 12.5-9 IsMorphismBetweenConstructibleObjectsAsUnionOfDifferences
 ‣ IsMorphismBetweenConstructibleObjectsAsUnionOfDifferences( morphism ) ( filter )

Returns: true or false

The GAP category of morphisms in a Boolean algebra of constructible objects, the latter as unions of formal single differences.

Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Ind

generated by GAPDoc2HTML