:: A Characterization of Concept Lattices; Dual Concept Lattices
:: by Christoph Schwarzweller
::
:: Received August 17, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users

definition

let C be FormalContext;
let CP be strict FormalConcept of C;

func @ CP -> Element of (ConceptLattice C) equals :: CONLAT_2:def 1
CP;

coherence
CP is Element of (ConceptLattice C)

proof end;

end;

:: deftheorem defines @ CONLAT_2:def 1 :
for C being FormalContext
for CP being strict FormalConcept of C holds @ CP = CP;

registration

let C be FormalContext;

cluster ConceptLattice C -> bounded ;

coherence
ConceptLattice C is bounded

proof end;

end;

theorem Th1: :: CONLAT_2:1

for C being FormalContext holds
( Bottom (ConceptLattice C) = Concept-with-all-Attributes C & Top (ConceptLattice C) = Concept-with-all-Objects C )

proof end;

theorem Th2: :: CONLAT_2:2

for C being FormalContext
for D being non empty Subset-Family of the carrier of C holds (ObjectDerivation C) . (union D) = meet { ((ObjectDerivation C) . O) where O is Subset of the carrier of C : O in D }

proof end;

theorem Th3: :: CONLAT_2:3

for C being FormalContext
for D being non empty Subset-Family of the carrier' of C holds (AttributeDerivation C) . (union D) = meet { ((AttributeDerivation C) . A) where A is Subset of the carrier' of C : A in D }

proof end;

theorem Th4: :: CONLAT_2:4

for C being FormalContext
for D being Subset of (ConceptLattice C) holds
( "/\" (D,(ConceptLattice C)) is FormalConcept of C & "\/" (D,(ConceptLattice C)) is FormalConcept of C )

proof end;

definition

let C be FormalContext;
let D be Subset of (ConceptLattice C);

func "/\" (D,C) -> FormalConcept of C equals :: CONLAT_2:def 2
"/\" (D,(ConceptLattice C));

coherence
"/\" (D,(ConceptLattice C)) is FormalConcept of C by Th4;

func "\/" (D,C) -> FormalConcept of C equals :: CONLAT_2:def 3
"\/" (D,(ConceptLattice C));

coherence
"\/" (D,(ConceptLattice C)) is FormalConcept of C by Th4;

end;

:: deftheorem defines "/\" CONLAT_2:def 2 :
for C being FormalContext
for D being Subset of (ConceptLattice C) holds "/\" (D,C) = "/\" (D,(ConceptLattice C));

:: deftheorem defines "\/" CONLAT_2:def 3 :
for C being FormalContext
for D being Subset of (ConceptLattice C) holds "\/" (D,C) = "\/" (D,(ConceptLattice C));

theorem :: CONLAT_2:5

for C being FormalContext holds
( "\/" (({} (ConceptLattice C)),C) = Concept-with-all-Attributes C & "/\" (({} (ConceptLattice C)),C) = Concept-with-all-Objects C )

proof end;

theorem :: CONLAT_2:6

for C being FormalContext holds
( "\/" (([#] the carrier of (ConceptLattice C)),C) = Concept-with-all-Objects C & "/\" (([#] the carrier of (ConceptLattice C)),C) = Concept-with-all-Attributes C )

proof end;

theorem :: CONLAT_2:7

for C being FormalContext
for D being non empty Subset of (ConceptLattice C) holds
( the Extent of ("\/" (D,C)) = (AttributeDerivation C) . ((ObjectDerivation C) . (union { the Extent of ConceptStr(# E,I #) where E is Subset of the carrier of C, I is Subset of the carrier' of C : ConceptStr(# E,I #) in D } )) & the Intent of ("\/" (D,C)) = meet { the Intent of ConceptStr(# E,I #) where E is Subset of the carrier of C, I is Subset of the carrier' of C : ConceptStr(# E,I #) in D } )

proof end;

theorem :: CONLAT_2:8

for C being FormalContext
for D being non empty Subset of (ConceptLattice C) holds
( the Extent of ("/\" (D,C)) = meet { the Extent of ConceptStr(# E,I #) where E is Subset of the carrier of C, I is Subset of the carrier' of C : ConceptStr(# E,I #) in D } & the Intent of ("/\" (D,C)) = (ObjectDerivation C) . ((AttributeDerivation C) . (union { the Intent of ConceptStr(# E,I #) where E is Subset of the carrier of C, I is Subset of the carrier' of C : ConceptStr(# E,I #) in D } )) )

proof end;

theorem Th9: :: CONLAT_2:9

for C being FormalContext
for CP being strict FormalConcept of C holds "\/" ( { ConceptStr(# O,A #) where O is Subset of the carrier of C, A is Subset of the carrier' of C : ex o being Object of C st
( o in the Extent of CP & O = (AttributeDerivation C) . ((ObjectDerivation C) . {o}) & A = (ObjectDerivation C) . {o} ) } ,(ConceptLattice C)) = CP

proof end;

theorem Th10: :: CONLAT_2:10

for C being FormalContext
for CP being strict FormalConcept of C holds "/\" ( { ConceptStr(# O,A #) where O is Subset of the carrier of C, A is Subset of the carrier' of C : ex a being Attribute of C st
( a in the Intent of CP & O = (AttributeDerivation C) . {a} & A = (ObjectDerivation C) . ((AttributeDerivation C) . {a}) ) } ,(ConceptLattice C)) = CP

proof end;

definition

let C be FormalContext;

func gamma C -> Function of the carrier of C, the carrier of (ConceptLattice C) means :Def4: :: CONLAT_2:def 4
for o being Element of C ex O being Subset of the carrier of C ex A being Subset of the carrier' of C st
( it . o = ConceptStr(# O,A #) & O = (AttributeDerivation C) . ((ObjectDerivation C) . {o}) & A = (ObjectDerivation C) . {o} );

existence
ex b₁ being Function of the carrier of C, the carrier of (ConceptLattice C) st
for o being Element of C ex O being Subset of the carrier of C ex A being Subset of the carrier' of C st
( b₁ . o = ConceptStr(# O,A #) & O = (AttributeDerivation C) . ((ObjectDerivation C) . {o}) & A = (ObjectDerivation C) . {o} )

proof end;

uniqueness
for b₁, b₂ being Function of the carrier of C, the carrier of (ConceptLattice C) st ( for o being Element of C ex O being Subset of the carrier of C ex A being Subset of the carrier' of C st
( b₁ . o = ConceptStr(# O,A #) & O = (AttributeDerivation C) . ((ObjectDerivation C) . {o}) & A = (ObjectDerivation C) . {o} ) ) & ( for o being Element of C ex O being Subset of the carrier of C ex A being Subset of the carrier' of C st
( b₂ . o = ConceptStr(# O,A #) & O = (AttributeDerivation C) . ((ObjectDerivation C) . {o}) & A = (ObjectDerivation C) . {o} ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines gamma CONLAT_2:def 4 :
for C being FormalContext
for b₂ being Function of the carrier of C, the carrier of (ConceptLattice C) holds
( b₂ = gamma C iff for o being Element of C ex O being Subset of the carrier of C ex A being Subset of the carrier' of C st
( b₂ . o = ConceptStr(# O,A #) & O = (AttributeDerivation C) . ((ObjectDerivation C) . {o}) & A = (ObjectDerivation C) . {o} ) );

definition

let C be FormalContext;

func delta C -> Function of the carrier' of C, the carrier of (ConceptLattice C) means :Def5: :: CONLAT_2:def 5
for a being Element of the carrier' of C ex O being Subset of the carrier of C ex A being Subset of the carrier' of C st
( it . a = ConceptStr(# O,A #) & O = (AttributeDerivation C) . {a} & A = (ObjectDerivation C) . ((AttributeDerivation C) . {a}) );

existence
ex b₁ being Function of the carrier' of C, the carrier of (ConceptLattice C) st
for a being Element of the carrier' of C ex O being Subset of the carrier of C ex A being Subset of the carrier' of C st
( b₁ . a = ConceptStr(# O,A #) & O = (AttributeDerivation C) . {a} & A = (ObjectDerivation C) . ((AttributeDerivation C) . {a}) )

proof end;

uniqueness
for b₁, b₂ being Function of the carrier' of C, the carrier of (ConceptLattice C) st ( for a being Element of the carrier' of C ex O being Subset of the carrier of C ex A being Subset of the carrier' of C st
( b₁ . a = ConceptStr(# O,A #) & O = (AttributeDerivation C) . {a} & A = (ObjectDerivation C) . ((AttributeDerivation C) . {a}) ) ) & ( for a being Element of the carrier' of C ex O being Subset of the carrier of C ex A being Subset of the carrier' of C st
( b₂ . a = ConceptStr(# O,A #) & O = (AttributeDerivation C) . {a} & A = (ObjectDerivation C) . ((AttributeDerivation C) . {a}) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines delta CONLAT_2:def 5 :
for C being FormalContext
for b₂ being Function of the carrier' of C, the carrier of (ConceptLattice C) holds
( b₂ = delta C iff for a being Element of the carrier' of C ex O being Subset of the carrier of C ex A being Subset of the carrier' of C st
( b₂ . a = ConceptStr(# O,A #) & O = (AttributeDerivation C) . {a} & A = (ObjectDerivation C) . ((AttributeDerivation C) . {a}) ) );

theorem :: CONLAT_2:11

for C being FormalContext
for o being Object of C
for a being Attribute of C holds
( (gamma C) . o is FormalConcept of C & (delta C) . a is FormalConcept of C )

proof end;

theorem Th12: :: CONLAT_2:12

for C being FormalContext holds
( rng (gamma C) is supremum-dense & rng (delta C) is infimum-dense )

proof end;

theorem Th13: :: CONLAT_2:13

for C being FormalContext
for o being Object of C
for a being Attribute of C holds
( o is-connected-with a iff (gamma C) . o [= (delta C) . a )

proof end;

Lm1: for L1, L2 being Lattice
for f being Function of the carrier of L1, the carrier of L2 st ( for a, b being Element of L1 st f . a [= f . b holds
a [= b ) holds
f is one-to-one

proof end;

Lm2: for L1, L2 being complete Lattice
for f being Function of the carrier of L1, the carrier of L2 st f is one-to-one & rng f = the carrier of L2 & ( for a, b being Element of L1 holds
( a [= b iff f . a [= f . b ) ) holds
f is Homomorphism of L1,L2

proof end;

Lm3: for C being FormalContext
for CS being ConceptStr over C st (ObjectDerivation C) . the Extent of CS = the Intent of CS holds
not CS is empty

proof end;

theorem Th14: :: CONLAT_2:14

for L being complete Lattice
for C being FormalContext holds
( ConceptLattice C,L are_isomorphic iff ex g being Function of the carrier of C, the carrier of L ex d being Function of the carrier' of C, the carrier of L st
( rng g is supremum-dense & rng d is infimum-dense & ( for o being Object of C
for a being Attribute of C holds
( o is-connected-with a iff g . o [= d . a ) ) ) )

proof end;

definition

let L be Lattice;

func Context L -> non empty non void strict ContextStr equals :: CONLAT_2:def 6
ContextStr(# the carrier of L, the carrier of L,(LattRel L) #);

coherence
ContextStr(# the carrier of L, the carrier of L,(LattRel L) #) is non empty non void strict ContextStr ;

end;

:: deftheorem defines Context CONLAT_2:def 6 :
for L being Lattice holds Context L = ContextStr(# the carrier of L, the carrier of L,(LattRel L) #);

theorem Th15: :: CONLAT_2:15

for L being complete Lattice holds ConceptLattice (Context L),L are_isomorphic

proof end;

theorem :: CONLAT_2:16

for L being Lattice holds
( L is complete iff ex C being FormalContext st ConceptLattice C,L are_isomorphic )

proof end;

registration

let L be complete Lattice;

cluster L .: -> complete ;

coherence
L .: is complete

proof end;

end;

definition

let C be FormalContext;

func C .: -> non empty non void strict ContextStr equals :: CONLAT_2:def 7
ContextStr(# the carrier' of C, the carrier of C,( the Information of C ~) #);

coherence
ContextStr(# the carrier' of C, the carrier of C,( the Information of C ~) #) is non empty non void strict ContextStr ;

end;

:: deftheorem defines .: CONLAT_2:def 7 :
for C being FormalContext holds C .: = ContextStr(# the carrier' of C, the carrier of C,( the Information of C ~) #);

theorem :: CONLAT_2:17

for C being strict FormalContext holds (C .:) .: = C ;

theorem Th18: :: CONLAT_2:18

for C being FormalContext
for O being Subset of the carrier of C holds (ObjectDerivation C) . O = (AttributeDerivation (C .:)) . O

proof end;

theorem Th19: :: CONLAT_2:19

for C being FormalContext
for A being Subset of the carrier' of C holds (AttributeDerivation C) . A = (ObjectDerivation (C .:)) . A

proof end;

definition

let C be FormalContext;
let CP be ConceptStr over C;

func CP .: -> strict ConceptStr over C .: means :Def8: :: CONLAT_2:def 8
( the Extent of it = the Intent of CP & the Intent of it = the Extent of CP );

existence
ex b₁ being strict ConceptStr over C .: st
( the Extent of b₁ = the Intent of CP & the Intent of b₁ = the Extent of CP )

proof end;

uniqueness
for b₁, b₂ being strict ConceptStr over C .: st the Extent of b₁ = the Intent of CP & the Intent of b₁ = the Extent of CP & the Extent of b₂ = the Intent of CP & the Intent of b₂ = the Extent of CP holds
b₁ = b₂ ;

end;

:: deftheorem Def8 defines .: CONLAT_2:def 8 :
for C being FormalContext
for CP being ConceptStr over C
for b₃ being strict ConceptStr over C .: holds
( b₃ = CP .: iff ( the Extent of b₃ = the Intent of CP & the Intent of b₃ = the Extent of CP ) );

registration

let C be FormalContext;
let CP be FormalConcept of C;

cluster CP .: -> strict non empty concept-like ;

coherence
( not CP .: is empty & CP .: is concept-like )

proof end;

end;

theorem :: CONLAT_2:20

for C being strict FormalContext
for CP being strict FormalConcept of C holds (CP .:) .: = CP

proof end;

definition

let C be FormalContext;

func DualHomomorphism C -> Homomorphism of ((ConceptLattice C) .:),(ConceptLattice (C .:)) means :Def9: :: CONLAT_2:def 9
for CP being strict FormalConcept of C holds it . CP = CP .: ;

existence
ex b₁ being Homomorphism of ((ConceptLattice C) .:),(ConceptLattice (C .:)) st
for CP being strict FormalConcept of C holds b₁ . CP = CP .:

proof end;

uniqueness
for b₁, b₂ being Homomorphism of ((ConceptLattice C) .:),(ConceptLattice (C .:)) st ( for CP being strict FormalConcept of C holds b₁ . CP = CP .: ) & ( for CP being strict FormalConcept of C holds b₂ . CP = CP .: ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def9 defines DualHomomorphism CONLAT_2:def 9 :
for C being FormalContext
for b₂ being Homomorphism of ((ConceptLattice C) .:),(ConceptLattice (C .:)) holds
( b₂ = DualHomomorphism C iff for CP being strict FormalConcept of C holds b₂ . CP = CP .: );

theorem Th21: :: CONLAT_2:21

for C being FormalContext holds DualHomomorphism C is bijective

proof end;

theorem :: CONLAT_2:22

for C being FormalContext holds ConceptLattice (C .:),(ConceptLattice C) .: are_isomorphic

proof end;