:: Meet Continuous Lattices Revisited
:: by Artur Korni{\l}owicz
::
:: Received January 6, 2000
:: Copyright (c) 2000-2021 Association of Mizar Users

theorem Th1: :: WAYBEL30:1

for x being set
for D being non empty set holds x /\ (union D) = union { (x /\ d) where d is Element of D : verum }

proof end;

theorem Th2: :: WAYBEL30:2

for R being non empty reflexive transitive RelStr
for D being non empty directed Subset of (InclPoset (Ids R)) holds union D is Ideal of R

proof end;

Lm1: now :: thesis: for R being non empty reflexive transitive RelStr
for D being non empty directed Subset of (InclPoset (Ids R))
for UD being Element of (InclPoset (Ids R)) st UD = union D holds
D is_<=_than UD

let R be non empty reflexive transitive RelStr ; :: thesis: for D being non empty directed Subset of (InclPoset (Ids R))
for UD being Element of (InclPoset (Ids R)) st UD = union D holds
D is_<=_than UD
let D be non empty directed Subset of (InclPoset (Ids R)); :: thesis: for UD being Element of (InclPoset (Ids R)) st UD = union D holds
D is_<=_than UD
let UD be Element of (InclPoset (Ids R)); :: thesis: ( UD = union D implies D is_<=_than UD )
assume A1: UD = union D ; :: thesis: D is_<=_than UD
thus D is_<=_than UD :: thesis: verum

proof

let d be Element of (InclPoset (Ids R)); :: according to LATTICE3:def 9 :: thesis: ( not d in D or d <= UD )
assume A2: d in D ; :: thesis: d <= UD
d c= UD by A1, A2, TARSKI:def 4;
hence d <= UD by YELLOW_1:3; :: thesis: verum

end;

end;

Lm2: now :: thesis: for R being non empty reflexive transitive RelStr
for D being non empty directed Subset of (InclPoset (Ids R))
for UD being Element of (InclPoset (Ids R)) st UD = union D holds
for b being Element of (InclPoset (Ids R)) st b is_>=_than D holds
UD <= b

let R be non empty reflexive transitive RelStr ; :: thesis: for D being non empty directed Subset of (InclPoset (Ids R))
for UD being Element of (InclPoset (Ids R)) st UD = union D holds
for b being Element of (InclPoset (Ids R)) st b is_>=_than D holds
UD <= b
let D be non empty directed Subset of (InclPoset (Ids R)); :: thesis: for UD being Element of (InclPoset (Ids R)) st UD = union D holds
for b being Element of (InclPoset (Ids R)) st b is_>=_than D holds
UD <= b
let UD be Element of (InclPoset (Ids R)); :: thesis: ( UD = union D implies for b being Element of (InclPoset (Ids R)) st b is_>=_than D holds
UD <= b )
assume A1: UD = union D ; :: thesis: for b being Element of (InclPoset (Ids R)) st b is_>=_than D holds
UD <= b
thus for b being Element of (InclPoset (Ids R)) st b is_>=_than D holds
UD <= b :: thesis: verum

proof

let b be Element of (InclPoset (Ids R)); :: thesis: ( b is_>=_than D implies UD <= b )
assume A2: for a being Element of (InclPoset (Ids R)) st a in D holds
b >= a ; :: according to LATTICE3:def 9 :: thesis: UD <= b
UD c= b

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in UD or x in b )
assume x in UD ; :: thesis: x in b
then consider Z being set such that
A3: x in Z and
A4: Z in D by A1, TARSKI:def 4;
reconsider Z = Z as Element of (InclPoset (Ids R)) by A4;
b >= Z by A2, A4;
then Z c= b by YELLOW_1:3;
hence x in b by A3; :: thesis: verum

end;

hence UD <= b by YELLOW_1:3; :: thesis: verum

end;

end;

:: Exercise 2.16 (i), p. 16

registration

let R be non empty reflexive transitive RelStr ;

cluster InclPoset (Ids R) -> up-complete ;

coherence
InclPoset (Ids R) is up-complete

proof end;

end;

theorem Th3: :: WAYBEL30:3

for R being non empty reflexive transitive RelStr
for D being non empty directed Subset of (InclPoset (Ids R)) holds sup D = union D

proof end;

theorem Th4: :: WAYBEL30:4

for R being Semilattice
for D being non empty directed Subset of (InclPoset (Ids R))
for x being Element of (InclPoset (Ids R)) holds sup ({x} "/\" D) = union { (x /\ d) where d is Element of D : verum }

proof end;

:: Exercise 4.8 (i), p. 33

registration

let R be Semilattice;

cluster InclPoset (Ids R) -> satisfying_MC ;

coherence
InclPoset (Ids R) is satisfying_MC

proof end;

end;

registration

let R be 1 -element RelStr ;

cluster -> trivial for TopAugmentation of R;

coherence
for b₁ being TopAugmentation of R holds b₁ is trivial

proof end;

end;

theorem :: WAYBEL30:5

for S being complete Scott TopLattice
for T being complete LATTICE
for A being Scott TopAugmentation of T st RelStr(# the carrier of S, the InternalRel of S #) = RelStr(# the carrier of T, the InternalRel of T #) holds
TopRelStr(# the carrier of A, the InternalRel of A, the topology of A #) = TopRelStr(# the carrier of S, the InternalRel of S, the topology of S #)

proof end;

theorem :: WAYBEL30:6

for N being complete Lawson TopLattice
for T being complete LATTICE
for A being correct Lawson TopAugmentation of T st RelStr(# the carrier of N, the InternalRel of N #) = RelStr(# the carrier of T, the InternalRel of T #) holds
TopRelStr(# the carrier of A, the InternalRel of A, the topology of A #) = TopRelStr(# the carrier of N, the InternalRel of N, the topology of N #)

proof end;

theorem :: WAYBEL30:7

for N being complete Lawson TopLattice
for S being Scott TopAugmentation of N
for A being Subset of N
for J being Subset of S st A = J & J is closed holds
A is closed

proof end;

registration

let A be complete LATTICE;

cluster lambda A -> non empty ;

coherence
not lambda A is empty

proof end;

end;

registration

let S be complete Scott TopLattice;

cluster InclPoset (sigma S) -> non trivial complete ;

coherence
( InclPoset (sigma S) is complete & not InclPoset (sigma S) is trivial ) ;

end;

registration

let N be complete Lawson TopLattice;

cluster InclPoset (sigma N) -> non trivial complete ;

coherence
( InclPoset (sigma N) is complete & not InclPoset (sigma N) is trivial ) ;

cluster InclPoset (lambda N) -> non trivial complete ;

coherence
( InclPoset (lambda N) is complete & not InclPoset (lambda N) is trivial )

proof end;

end;

theorem Th8: :: WAYBEL30:8

for T being non empty reflexive RelStr holds sigma T c= { (W \ (uparrow F)) where W, F is Subset of T : ( W in sigma T & F is finite ) }

proof end;

theorem Th9: :: WAYBEL30:9

for N being complete Lawson TopLattice holds lambda N = the topology of N

proof end;

theorem Th10: :: WAYBEL30:10

for N being complete Lawson TopLattice holds sigma N c= lambda N

proof end;

theorem :: WAYBEL30:11

for M, N being complete LATTICE st RelStr(# the carrier of M, the InternalRel of M #) = RelStr(# the carrier of N, the InternalRel of N #) holds
lambda M = lambda N

proof end;

theorem Th12: :: WAYBEL30:12

for N being complete Lawson TopLattice
for X being Subset of N holds
( X in lambda N iff X is open ) by Th9;

registration

cluster 1 -element TopSpace-like reflexive -> 1 -element reflexive Scott for TopRelStr ;

coherence
for b₁ being 1 -element reflexive TopRelStr st b₁ is TopSpace-like holds
b₁ is Scott by TDLAT_3:15;

end;

registration

cluster trivial TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete -> complete Lawson for TopRelStr ;

coherence
for b₁ being complete TopLattice st b₁ is trivial holds
b₁ is Lawson

proof end;

end;

registration

cluster non empty TopSpace-like reflexive transitive antisymmetric lower-bounded upper-bounded V158() up-complete /\-complete with_suprema with_infima complete strict Scott continuous non void meet-continuous for TopRelStr ;

existence
ex b₁ being complete TopLattice st
( b₁ is strict & b₁ is continuous & b₁ is lower-bounded & b₁ is meet-continuous & b₁ is Scott )

proof end;

end;

registration

cluster non empty TopSpace-like Hausdorff reflexive transitive antisymmetric lower-bounded upper-bounded V158() up-complete /\-complete with_suprema with_infima complete strict Lawson continuous non void compact for TopRelStr ;

existence
ex b₁ being complete TopLattice st
( b₁ is strict & b₁ is continuous & b₁ is compact & b₁ is Hausdorff & b₁ is Lawson )

proof end;

end;

scheme :: WAYBEL30:sch 1
EmptySch{ F₁() -> Scott TopLattice, F₂() -> set , F₃( set ) -> set } :

{ F₃(w) where w is Element of F₁() : w in {} } = {}

proof end;

theorem Th13: :: WAYBEL30:13

for N being meet-continuous LATTICE
for A being Subset of N st A is property(S) holds
uparrow A is property(S)

proof end;

registration

let N be meet-continuous LATTICE;
let A be property(S) Subset of N;

cluster uparrow A -> property(S) ;

coherence
uparrow A is property(S) by Th13;

end;

:: Proposition 2.1 (i), p. 153

theorem Th14: :: WAYBEL30:14

for N being complete Lawson meet-continuous TopLattice
for S being Scott TopAugmentation of N
for A being Subset of N st A in lambda N holds
uparrow A in sigma S

proof end;

theorem Th15: :: WAYBEL30:15

for N being complete Lawson meet-continuous TopLattice
for S being Scott TopAugmentation of N
for A being Subset of N
for J being Subset of S st A = J & A is open holds
uparrow J is open

proof end;

theorem Th16: :: WAYBEL30:16

for N being complete Lawson meet-continuous TopLattice
for S being Scott TopAugmentation of N
for x being Point of S
for y being Point of N
for J being Basis of y st x = y holds
{ (uparrow A) where A is Subset of N : A in J } is Basis of x

proof end;

:: Proposition 2.1 (ii), p. 153

theorem Th17: :: WAYBEL30:17

for N being complete Lawson meet-continuous TopLattice
for S being Scott TopAugmentation of N
for X being upper Subset of N
for Y being Subset of S st X = Y holds
Int X = Int Y

proof end;

:: Proposition 2.1 (iii), p. 153

theorem :: WAYBEL30:18

for N being complete Lawson meet-continuous TopLattice
for S being Scott TopAugmentation of N
for X being lower Subset of N
for Y being Subset of S st X = Y holds
Cl X = Cl Y

proof end;

theorem Th19: :: WAYBEL30:19

for M, N being complete LATTICE
for LM being correct Lawson TopAugmentation of M
for LN being correct Lawson TopAugmentation of N st InclPoset (sigma N) is continuous holds
the topology of [:LM,LN:] = lambda [:M,N:]

proof end;

:: Proposition 2.2, p. 153

theorem Th20: :: WAYBEL30:20

for M, N being complete LATTICE
for P being correct Lawson TopAugmentation of [:M,N:]
for Q being correct Lawson TopAugmentation of M
for R being correct Lawson TopAugmentation of N st InclPoset (sigma N) is continuous holds
TopStruct(# the carrier of P, the topology of P #) = [:Q,R:]

proof end;

:: Theorem 2.3, p. 153, first conjunct

theorem Th21: :: WAYBEL30:21

for N being complete Lawson meet-continuous TopLattice
for x being Element of N holds x "/\" is continuous

proof end;

registration

let N be complete Lawson meet-continuous TopLattice;
let x be Element of N;

cluster x "/\" -> continuous ;

coherence
x "/\" is continuous by Th21;

end;

:: Theorem 2.3, p. 153, second conjunct

theorem Th22: :: WAYBEL30:22

for N being complete Lawson meet-continuous TopLattice st InclPoset (sigma N) is continuous holds
N is topological_semilattice

proof end;

Lm3: now :: thesis: for S, T, x1, x2 being set st x1 in S & x2 in T holds
<:(pr2 (S,T)),(pr1 (S,T)):> . [x1,x2] = [x2,x1]

let S, T, x1, x2 be set ; :: thesis: ( x1 in S & x2 in T implies <:(pr2 (S,T)),(pr1 (S,T)):> . [x1,x2] = [x2,x1] )
assume that
A1: x1 in S and
A2: x2 in T ; :: thesis: <:(pr2 (S,T)),(pr1 (S,T)):> . [x1,x2] = [x2,x1]
A3: dom <:(pr2 (S,T)),(pr1 (S,T)):> = (dom (pr2 (S,T))) /\ (dom (pr1 (S,T))) by FUNCT_3:def 7
.= (dom (pr2 (S,T))) /\ [:S,T:] by FUNCT_3:def 4
.= [:S,T:] /\ [:S,T:] by FUNCT_3:def 5
.= [:S,T:] ;
[x1,x2] in [:S,T:] by A1, A2, ZFMISC_1:87;
hence <:(pr2 (S,T)),(pr1 (S,T)):> . [x1,x2] = [((pr2 (S,T)) . (x1,x2)),((pr1 (S,T)) . (x1,x2))] by A3, FUNCT_3:def 7
.= [x2,((pr1 (S,T)) . (x1,x2))] by A1, A2, FUNCT_3:def 5
.= [x2,x1] by A1, A2, FUNCT_3:def 4 ;
:: thesis: verum

end;

:: Proposition 2.4, p. 153

theorem :: WAYBEL30:23

for N being complete Lawson meet-continuous TopLattice st InclPoset (sigma N) is continuous holds
( N is Hausdorff iff for X being Subset of [:N,N:] st X = the InternalRel of N holds
X is closed )

proof end;

:: Definition 2.5, p. 154

definition

let N be non empty reflexive RelStr ;
let X be Subset of N;

func X ^0 -> Subset of N equals :: WAYBEL30:def 1
{ u where u is Element of N : for D being non empty directed Subset of N st u <= sup D holds
X meets D } ;

coherence
{ u where u is Element of N : for D being non empty directed Subset of N st u <= sup D holds
X meets D } is Subset of N

proof end;

end;

:: deftheorem defines ^0 WAYBEL30:def 1 :
for N being non empty reflexive RelStr
for X being Subset of N holds X ^0 = { u where u is Element of N : for D being non empty directed Subset of N st u <= sup D holds
X meets D } ;

registration

let N be non empty reflexive antisymmetric RelStr ;
let X be empty Subset of N;

cluster X ^0 -> empty ;

coherence
X ^0 is empty

proof end;

end;

theorem :: WAYBEL30:24

for N being non empty reflexive RelStr
for A, J being Subset of N st A c= J holds
A ^0 c= J ^0

proof end;

:: Remark 2.6 (i), p. 154

theorem Th25: :: WAYBEL30:25

for N being non empty reflexive RelStr
for x being Element of N holds (uparrow x) ^0 = wayabove x

proof end;

:: Remark 2.6 (ii), p. 154

theorem Th26: :: WAYBEL30:26

for N being Scott TopLattice
for X being upper Subset of N holds Int X c= X ^0

proof end;

:: Lemma 2.7 (i), p. 154

theorem Th27: :: WAYBEL30:27

for N being non empty reflexive RelStr
for X, Y being Subset of N holds (X ^0) \/ (Y ^0) c= (X \/ Y) ^0

proof end;

:: Lemma 2.7 (ii), p. 154

theorem Th28: :: WAYBEL30:28

for N being meet-continuous LATTICE
for X, Y being upper Subset of N holds (X ^0) \/ (Y ^0) = (X \/ Y) ^0

proof end;

:: Lemma 2.8, p. 154

theorem Th29: :: WAYBEL30:29

for S being Scott meet-continuous TopLattice
for F being finite Subset of S holds Int (uparrow F) c= union { (wayabove x) where x is Element of S : x in F }

proof end;

:: Theorem 2.9, p. 154

theorem Th30: :: WAYBEL30:30

for N being complete Lawson TopLattice holds
( N is continuous iff ( N is meet-continuous & N is Hausdorff ) )

proof end;

registration

cluster TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson continuous -> Hausdorff complete for TopRelStr ;

coherence
for b₁ being complete TopLattice st b₁ is continuous & b₁ is Lawson holds
b₁ is Hausdorff ;

cluster TopSpace-like Hausdorff reflexive transitive antisymmetric with_suprema with_infima complete Lawson meet-continuous -> complete continuous for TopRelStr ;

coherence
for b₁ being complete TopLattice st b₁ is meet-continuous & b₁ is Lawson & b₁ is Hausdorff holds
b₁ is continuous by Th30;

end;

:: Definition 2.10, p. 155

definition

let N be non empty TopRelStr ;

attr N is with_small_semilattices means :: WAYBEL30:def 2
for x being Point of N ex J being basis of x st
for A being Subset of N st A in J holds
subrelstr A is meet-inheriting ;

attr N is with_compact_semilattices means :: WAYBEL30:def 3
for x being Point of N ex J being basis of x st
for A being Subset of N st A in J holds
( subrelstr A is meet-inheriting & A is compact );

attr N is with_open_semilattices means :Def4: :: WAYBEL30:def 4
for x being Point of N ex J being Basis of x st
for A being Subset of N st A in J holds
subrelstr A is meet-inheriting ;

end;

:: deftheorem defines with_small_semilattices WAYBEL30:def 2 :
for N being non empty TopRelStr holds
( N is with_small_semilattices iff for x being Point of N ex J being basis of x st
for A being Subset of N st A in J holds
subrelstr A is meet-inheriting );

:: deftheorem defines with_compact_semilattices WAYBEL30:def 3 :
for N being non empty TopRelStr holds
( N is with_compact_semilattices iff for x being Point of N ex J being basis of x st
for A being Subset of N st A in J holds
( subrelstr A is meet-inheriting & A is compact ) );

:: deftheorem Def4 defines with_open_semilattices WAYBEL30:def 4 :
for N being non empty TopRelStr holds
( N is with_open_semilattices iff for x being Point of N ex J being Basis of x st
for A being Subset of N st A in J holds
subrelstr A is meet-inheriting );

registration

cluster non empty TopSpace-like with_open_semilattices -> non empty TopSpace-like with_small_semilattices for TopRelStr ;

coherence
for b₁ being non empty TopSpace-like TopRelStr st b₁ is with_open_semilattices holds
b₁ is with_small_semilattices

proof end;

cluster non empty TopSpace-like with_compact_semilattices -> non empty TopSpace-like with_small_semilattices for TopRelStr ;

coherence
for b₁ being non empty TopSpace-like TopRelStr st b₁ is with_compact_semilattices holds
b₁ is with_small_semilattices

proof end;

cluster non empty anti-discrete -> non empty with_small_semilattices with_open_semilattices for TopRelStr ;

coherence
for b₁ being non empty TopRelStr st b₁ is anti-discrete holds
( b₁ is with_small_semilattices & b₁ is with_open_semilattices )

proof end;

cluster 1 -element TopSpace-like reflexive -> 1 -element with_compact_semilattices for TopRelStr ;

coherence
for b₁ being 1 -element TopRelStr st b₁ is reflexive & b₁ is TopSpace-like holds
b₁ is with_compact_semilattices

proof end;

end;

registration

cluster non empty trivial TopSpace-like reflexive transitive antisymmetric with_suprema with_infima strict lower non void for TopRelStr ;

existence
ex b₁ being TopLattice st
( b₁ is strict & b₁ is trivial & b₁ is lower )

proof end;

end;

theorem Th31: :: WAYBEL30:31

for N being with_infima topological_semilattice TopPoset
for C being Subset of N st subrelstr C is meet-inheriting holds
subrelstr (Cl C) is meet-inheriting

proof end;

:: Theorem 2.11, p. 155

theorem Th32: :: WAYBEL30:32

for N being complete Lawson meet-continuous TopLattice
for S being Scott TopAugmentation of N holds
( ( for x being Point of S ex J being Basis of x st
for W being Subset of S st W in J holds
W is Filter of S ) iff N is with_open_semilattices )

proof end;

:: Theorem 2.12, p. 155, r => l

theorem Th33: :: WAYBEL30:33

for N being complete Lawson TopLattice
for S being Scott TopAugmentation of N
for x being Element of N holds { (inf A) where A is Subset of S : ( x in A & A in sigma S ) } c= { (inf J) where J is Subset of N : ( x in J & J in lambda N ) }

proof end;

:: Theorem 2.12, p. 155

theorem Th34: :: WAYBEL30:34

for N being complete Lawson meet-continuous TopLattice
for S being Scott TopAugmentation of N
for x being Element of N holds { (inf A) where A is Subset of S : ( x in A & A in sigma S ) } = { (inf J) where J is Subset of N : ( x in J & J in lambda N ) }

proof end;

:: Theorem 2.13, p. 155, 1 <=> 2

theorem Th35: :: WAYBEL30:35

for N being complete Lawson meet-continuous TopLattice holds
( N is continuous iff ( N is with_open_semilattices & InclPoset (sigma N) is continuous ) )

proof end;

registration

cluster TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson continuous -> complete Lawson with_open_semilattices for TopRelStr ;

coherence
for b₁ being complete Lawson TopLattice st b₁ is continuous holds
b₁ is with_open_semilattices by Th35;

end;

registration

let N be complete Lawson continuous TopLattice;

cluster InclPoset (sigma N) -> continuous ;

coherence
InclPoset (sigma N) is continuous by Th35;

end;

:: Theorem 2.13, p. 155, 1 => 3

theorem :: WAYBEL30:36

for N being complete Lawson continuous TopLattice holds
( N is compact & N is Hausdorff & N is topological_semilattice & N is with_open_semilattices )

proof end;

:: Theorem 2.13, p. 155, 3 => 3'

theorem :: WAYBEL30:37

for N being Hausdorff complete Lawson topological_semilattice with_open_semilattices TopLattice holds N is with_compact_semilattices

proof end;

:: Theorem 2.13, p. 155, 3' => 4

theorem :: WAYBEL30:38

for N being Hausdorff complete Lawson meet-continuous TopLattice
for x being Element of N holds x = "\/" ( { (inf V) where V is Subset of N : ( x in V & V in lambda N ) } ,N)

proof end;

:: Theorem 2.13, p. 155, 1 <=> 4

theorem :: WAYBEL30:39

for N being complete Lawson meet-continuous TopLattice holds
( N is continuous iff for x being Element of N holds x = "\/" ( { (inf V) where V is Subset of N : ( x in V & V in lambda N ) } ,N) )

proof end;

:: Exercise 2.16, p. 156, 1 <=> 2

theorem Th40: :: WAYBEL30:40

for N being complete Lawson meet-continuous TopLattice holds
( N is algebraic iff ( N is with_open_semilattices & InclPoset (sigma N) is algebraic ) )

proof end;

registration

let N be complete algebraic Lawson meet-continuous TopLattice;

cluster InclPoset (sigma N) -> algebraic ;

coherence
InclPoset (sigma N) is algebraic by Th40;

end;