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### 10 Meet-semilattice of differences

#### 10.1 Constructors

##### 10.1-1 MeetSemilatticeOfSingleDifferences
 ‣ MeetSemilatticeOfSingleDifferences( L ) ( attribute )

Returns: a CAP category

Construct the meet-semilattice of differences from the lattice L.

##### 10.1-2 \-
 ‣ \-( A, B ) ( operation )
 ‣ FormalDifferenceOfNormalizedObjects( u ) ( operation )

Form the formal single difference object A - B. The expression A - 0 := A - InitialObject( CapCategory( A ) ). The expression - A := TerminalObject( CapCategory( A ) ) - A. FormalDifferenceOfNormalizedObjects assumes that A and B are normalized

#### 10.2 Operations

##### 10.2-1 MinuendAndSubtrahendInUnderlyingLattice
 ‣ MinuendAndSubtrahendInUnderlyingLattice( A ) ( operation )

Returns: a pair of objects in a thin category

A pair in the underlying lattice representing the formal single difference A.

##### 10.2-2 DistinguishedSubtrahend
 ‣ DistinguishedSubtrahend( A ) ( operation )

Returns: an object in a thin category

The output S should satisfy A.I - S = A. The standard method returns NormalizedDistinguishedSubtrahend( A ) if HasNormalizedDistinguishedSubtrahend( A ) = true or PreDistinguishedSubtrahend( A ) if HasPreDistinguishedSubtrahend( A ) = true. The remaining behavior is unspecified.

#### 10.3 GAP categories

##### 10.3-1 IsMeetSemilatticeOfSingleDifferences
 ‣ IsMeetSemilatticeOfSingleDifferences( arg ) ( filter )

Returns: true or false

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

##### 10.3-2 IsObjectInMeetSemilatticeOfSingleDifferences
 ‣ IsObjectInMeetSemilatticeOfSingleDifferences( object ) ( filter )

Returns: true or false

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

##### 10.3-3 IsMorphismInMeetSemilatticeOfSingleDifferences
 ‣ IsMorphismInMeetSemilatticeOfSingleDifferences( morphism ) ( filter )

Returns: true or false

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

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