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Locales

Locales, frames, coframes, meet semi-lattices of locally closed subsets, and Boolean algebras of constructible sets

2024.04-02

22 April 2024

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Markus Lange-Hegermann
Email: markus.lange-hegermann@th-owl.de
Homepage: https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/
Address:
Markus Lange-Hegermann
Hochschule Ostwestfalen-Lippe
Liebigstraße 87
32657 Lemgo
Germany

Tom Kuhmichel
Email: tom.kuhmichel@student.uni-siegen.de
Homepage: https://github.com/TKuh
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

2 Posets (partially ordered sets)

  2.1-1 IsPosetCategory
  2.1-2 IsTotalOrderCategory
  2.1-3 IsMonoidalPoset
  2.1-4 IsClosedMonoidalPoset
  2.1-5 IsCoclosedMonoidalPoset
  2.1-6 IsSymmetricMonoidalPoset
  2.1-7 IsSymmetricClosedMonoidalPoset
  2.1-8 IsSymmetricCoclosedMonoidalPoset

2.2-1 IsEqualForObjectsIfIsHomSetInhabited

2.3 Add-methods

2.3-1 AddIsEqualForObjectsIfIsHomSetInhabited

4 Heyting algebras

4.1-1 IsHeytingAlgebroid
4.1-2 IsHeytingAlgebra

  4.2-1 NegationOnObjects
  4.2-2 NegationOnMorphisms
  4.2-3 NegationOnMorphismsWithGivenNegations
  4.2-4 MorphismToDoubleNegation
  4.2-5 MorphismToDoubleNegationWithGivenDoubleNegation

4.3 Stable internal Hom

4.3-1 StableInternalHom

4.4 Add-methods

  4.4-1 AddMorphismToDoubleNegation
  4.4-2 AddMorphismToDoubleNegationWithGivenDoubleNegation
  4.4-3 AddNegationOnMorphisms
  4.4-4 AddNegationOnMorphismsWithGivenNegations
  4.4-5 AddNegationOnObjects

5 Co-Heyting algebras

5.1-1 IsCoHeytingAlgebroid
5.1-2 IsCoHeytingAlgebra

  5.2-1 ConegationOnObjects
  5.2-2 ConegationOnMorphisms
  5.2-3 ConegationOnMorphismsWithGivenConegations
  5.2-4 MorphismFromDoubleConegation
  5.2-5 MorphismFromDoubleConegationWithGivenDoubleConegation

5.3 Stable internal coHom

5.3-1 StableInternalCoHom

5.4 Add-methods

  5.4-1 AddConegationOnMorphisms
  5.4-2 AddConegationOnMorphismsWithGivenConegations
  5.4-3 AddConegationOnObjects
  5.4-4 AddMorphismFromDoubleConegation
  5.4-5 AddMorphismFromDoubleConegationWithGivenDoubleConegation

6 Boolean algebras

  6.1-1 IsBiHeytingAlgebroid
  6.1-2 IsBiHeytingAlgebra
  6.1-3 IsBooleanAlgebroid
  6.1-4 IsBooleanAlgebra

  6.2-1 MorphismFromDoubleNegation
  6.2-2 MorphismFromDoubleNegationWithGivenDoubleNegation
  6.2-3 MorphismToDoubleConegation
  6.2-4 MorphismToDoubleConegationWithGivenDoubleConegation

6.3 Add-methods

  6.3-1 AddMorphismFromDoubleNegation
  6.3-2 AddMorphismFromDoubleNegationWithGivenDoubleNegation
  6.3-3 AddMorphismToDoubleConegation
  6.3-4 AddMorphismToDoubleConegationWithGivenDoubleConegation

7 Preordered sets viewed as categories

7.1 Constructors

  7.1-1 ProsetAsCategory
  7.1-2 DiscreteSkeletalCategory
  7.1-3 TotalOrderAsCategory

  7.2-1 UnderlyingObjectFilter
  7.2-2 UnderlyingObjectMembershipFunction
  7.2-3 UnderlyingObjectEqualityFunction
  7.2-4 UnderlyingPreorderFunction
  7.2-5 UnderlyingObject

7.3 GAP categories

7.3-1 IsProsetAsCategory

8 The (stable) proset/poset of a category

8.1 Global variables

8.1-1 CAP_INTERNAL_METHOD_NAME_LIST_FOR_PREORDERED_SET_OF_CATEGORY

8.2 Constructors

  8.2-1 CreateProsetOrPosetOfCategory
  8.2-2 ProsetOfCategory
  8.2-3 StableProsetOfCategory
  8.2-4 PosetOfCategory
  8.2-5 StablePosetOfCategory
  8.2-6 AsCellOfProset
  8.2-7 AsCellOfStableProset
  8.2-8 AsCellOfPoset
  8.2-9 AsCellOfStablePoset
  8.2-10 /

8.3-1 AmbientCategory
8.3-2 UnderlyingCell

8.4-1 Ideal quotient

8.5 GAP categories

  8.5-1 Poset of category
  8.5-2 Stable poset of category
  8.5-3 IsProsetOrPosetOfCapCategory
  8.5-4 IsCellInProsetOrPosetOfCategory
  8.5-5 IsCellInStableProsetOrPosetOfCategory
  8.5-6 IsObjectInProsetOrPosetOfCategory
  8.5-7 IsProsetOfCapCategory
  8.5-8 IsCellInProsetOfCategory
  8.5-9 IsObjectInProsetOfCategory
  8.5-10 IsMorphismInProsetOfCategory
  8.5-11 IsPosetOfCapCategory
  8.5-12 IsCellInPosetOfCategory
  8.5-13 IsObjectInPosetOfCategory
  8.5-14 IsMorphismInPosetOfCategory

9 Meet-semilattice of single/multiple differences

  9.1-1 IsClosed
  9.1-2 IsOpen
  9.1-3 IsLocallyClosed

  9.2-1 UnderlyingCategory
  9.2-2 LocallyClosedPart
  9.2-3 CanonicalObject
  9.2-4 FactorizeObject
  9.2-5 Closure
  9.2-6 Dimension

9.3-1 NormalizeObject
9.3-2 StandardizeObject

9.4 GAP categories

  9.4-1 IsMeetSemilatticeOfDifferences
  9.4-2 IsObjectInMeetSemilatticeOfDifferences
  9.4-3 IsMorphismInMeetSemilatticeOfDifferences

10 Meet-semilattice of differences

10.1 Constructors

10.1-1 MeetSemilatticeOfSingleDifferences
10.1-2 \-

10.2 Operations

10.2-1 MinuendAndSubtrahendInUnderlyingLattice
10.2-2 DistinguishedSubtrahend

10.3 GAP categories

  10.3-1 IsMeetSemilatticeOfSingleDifferences
  10.3-2 IsObjectInMeetSemilatticeOfSingleDifferences
  10.3-3 IsMorphismInMeetSemilatticeOfSingleDifferences

11 Meet-semilattice of multiple differences

11.1 Constructors

11.1-1 MeetSemilatticeOfMultipleDifferences
11.1-2 AsMultipleDifference

11.2 Attributes

11.2-1 AsSingleDifference

11.3 Operations

11.3-1 ListOfSingleDifferences
11.3-2 \[\]

11.4 GAP categories

  11.4-1 IsMeetSemilatticeOfMultipleDifferences
  11.4-2 IsObjectInMeetSemilatticeOfMultipleDifferences
  11.4-3 IsMorphismInMeetSemilatticeOfMultipleDifferences

12 Boolean algebra of constructible objects

12.1 Constructors

  12.1-1 BooleanAlgebraOfConstructibleObjectsAsUnionOfMultipleDifferences
  12.1-2 UnionOfMultipleDifferences
  12.1-3 BooleanAlgebraOfConstructibleObjectsAsUnionOfDifferences
  12.1-4 UnionOfDifferences

12.2 Properties

  12.2-1 IsOpen
  12.2-2 IsClosed
  12.2-3 IsLocallyClosed

12.3 Attributes

  12.3-1 UnderlyingCategory
  12.3-2 LocallyClosedPart
  12.3-3 CanonicalObject
  12.3-4 FactorizeObject
  12.3-5 NormalizedObject
  12.3-6 StandardizedObject
  12.3-7 Closure
  12.3-8 ClosureAsConstructibleObject
  12.3-9 Dimension

12.4 Operations

  12.4-1 NormalizeObject
  12.4-2 StandardizeObject
  12.4-3 Length
  12.4-4 \[\]
  12.4-5 Iterator
  12.4-6 ForAllOp
  12.4-7 ForAnyOp
  12.4-8 ListOfObjectsInMeetSemilatticeOfMultipleDifferences
  12.4-9 ListOfObjectsInMeetSemilatticeOfDifferences

12.5 GAP categories

  12.5-1 IsBooleanAlgebraOfConstructibleObjects
  12.5-2 IsConstructibleObject
  12.5-3 IsMorphismBetweenConstructibleObjects
  12.5-4 IsBooleanAlgebraOfConstructibleObjectsAsUnionOfMultipleDifferences
  12.5-5 IsConstructibleObjectAsUnionOfMultipleDifferences
  12.5-6 IsMorphismBetweenConstructibleObjectsAsUnionOfMultipleDifferences
  12.5-7 IsBooleanAlgebraOfConstructibleObjectsAsUnionOfSingleDifferences
  12.5-8 IsConstructibleObjectAsUnionOfSingleDifferences
  12.5-9 IsMorphismBetweenConstructibleObjectsAsUnionOfDifferences

13 The interval category

13.1 Constructors

13.1-1 CreateIntervalCategory
13.1-2 IntervalCategory

13.2 Attributes

13.3 GAP categories

  13.3-1 IsIntervalCategory
  13.3-2 IsCellInIntervalCategory
  13.3-3 IsObjectInIntervalCategory
  13.3-4 IsMorphismInIntervalCategory

15.1 GAP categories

15.1-1 Precompilation

16 Locales automatic generated documentation

16.1 Locales automatic generated documentation of methods

16.1-1 DigraphOfPoset

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