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### 9 Meet-semilattice of single/multiple differences

#### 9.1 Properties

##### 9.1-1 IsClosed
 ‣ IsClosed( A ) ( operation )

The embedding of the meet-semilattice of formal single/multiple differences into the underlying lattice has a right adjoint, at least in case the underying lattice is a co-Heyting algebra. A formal single difference is closed if its component of the unit of the above adjunction is an isomorphism.

##### 9.1-2 IsOpen
 ‣ IsOpen( A ) ( property )

Returns: true or false

Check if the complement (a constructible object) of A is closed.

##### 9.1-3 IsLocallyClosed
 ‣ IsLocallyClosed( A ) ( property )

Returns: true or false

#### 9.2 Attributes

##### 9.2-1 UnderlyingCategory
 ‣ UnderlyingCategory( D ) ( attribute )

Returns: a CAP category

The category underlying the meet-semilattice of single/multiple differences D.

##### 9.2-2 LocallyClosedPart
 ‣ LocallyClosedPart( A ) ( attribute )

##### 9.2-3 CanonicalObject
 ‣ CanonicalObject( A ) ( attribute )

##### 9.2-4 FactorizeObject
 ‣ FactorizeObject( A ) ( attribute )

##### 9.2-5 Closure
 ‣ Closure( A ) ( attribute )

Returns: an object in a co-Heyting algebra.

The closure of the formal single difference A in the underlying co-Heyting algebra.

##### 9.2-6 Dimension
 ‣ Dimension( A ) ( attribute )

#### 9.3 Operations

##### 9.3-1 NormalizeObject
 ‣ NormalizeObject( A ) ( operation )

##### 9.3-2 StandardizeObject
 ‣ StandardizeObject( A ) ( operation )

#### 9.4 GAP categories

##### 9.4-1 IsMeetSemilatticeOfDifferences
 ‣ IsMeetSemilatticeOfDifferences( object ) ( filter )

Returns: true or false

The GAP category of a meet-semilattice of single/multiple differences.

##### 9.4-2 IsObjectInMeetSemilatticeOfDifferences
 ‣ IsObjectInMeetSemilatticeOfDifferences( object ) ( filter )

Returns: true or false

The GAP category of objects in a meet-semilattice of single/multiple differences.

##### 9.4-3 IsMorphismInMeetSemilatticeOfDifferences
 ‣ IsMorphismInMeetSemilatticeOfDifferences( morphism ) ( filter )

Returns: true or false

The GAP category of morphisms in a meet-semilattice of single/multiple differences.

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