:: WAYBEL23 semantic presentation

for L being non empty Poset
for x being Element of L holds compactbelow x = (waybelow x) /\ the carrier of (CompactSublatt L)

proof end;

definition

let L be non empty reflexive transitive RelStr ;
let X be Subset of (InclPoset (Ids L));
:: original: union
redefine func union X -> Subset of L;
coherence

proof end;

end;

Lm1: for X being non empty set
for Y being Subset of (InclPoset X) st ex_sup_of Y, InclPoset X holds
union Y c= sup Y

proof end;

theorem Th2: :: WAYBEL23:2

for L being non empty RelStr
for X, Y being Subset of L st X c= Y holds
finsups X c= finsups Y

proof end;

theorem Th3: :: WAYBEL23:3

for L being non empty transitive RelStr
for S being non empty full sups-inheriting SubRelStr of L
for X being Subset of L
for Y being Subset of S st X = Y holds
finsups X c= finsups Y

proof end;

theorem :: WAYBEL23:4

for L being non empty transitive antisymmetric complete RelStr
for S being non empty full sups-inheriting SubRelStr of L
for X being Subset of L
for Y being Subset of S st X = Y holds
finsups X = finsups Y

proof end;

theorem Th5: :: WAYBEL23:5

for L being complete sup-Semilattice
for S being non empty full join-inheriting SubRelStr of L st Bottom L in the carrier of S holds
for X being Subset of L
for Y being Subset of S st X = Y holds
finsups Y c= finsups X

proof end;

theorem Th6: :: WAYBEL23:6

for L being lower-bounded sup-Semilattice
for X being Subset of (InclPoset (Ids L)) holds sup X = downarrow (finsups (union X))

proof end;

theorem Th7: :: WAYBEL23:7

for L being reflexive transitive RelStr
for X being Subset of L holds downarrow (downarrow X) = downarrow X

proof end;

theorem :: WAYBEL23:8

for L being reflexive transitive RelStr
for X being Subset of L holds uparrow (uparrow X) = uparrow X

proof end;

theorem :: WAYBEL23:9

for L being non empty reflexive transitive RelStr
for x being Element of L holds downarrow (downarrow x) = downarrow x

proof end;

theorem :: WAYBEL23:10

for L being non empty reflexive transitive RelStr
for x being Element of L holds uparrow (uparrow x) = uparrow x

proof end;

theorem Th11: :: WAYBEL23:11

for L being non empty RelStr
for S being non empty SubRelStr of L
for X being Subset of L
for Y being Subset of S st X = Y holds
downarrow Y c= downarrow X

proof end;

theorem Th12: :: WAYBEL23:12

for L being non empty RelStr
for S being non empty SubRelStr of L
for X being Subset of L
for Y being Subset of S st X = Y holds
uparrow Y c= uparrow X

proof end;

theorem :: WAYBEL23:13

for L being non empty RelStr
for S being non empty SubRelStr of L
for x being Element of L
for y being Element of S st x = y holds
downarrow y c= downarrow x

proof end;

theorem :: WAYBEL23:14

for L being non empty RelStr
for S being non empty SubRelStr of L
for x being Element of L
for y being Element of S st x = y holds
uparrow y c= uparrow x

proof end;

definition

let L be non empty RelStr ;
let S be Subset of L;
attr S is meet-closed means :Def1: :: WAYBEL23:def 1
subrelstr S is meet-inheriting;

end;

:: deftheorem Def1 defines meet-closed WAYBEL23:def 1 :

definition

let L be non empty RelStr ;
let S be Subset of L;
attr S is join-closed means :Def2: :: WAYBEL23:def 2
subrelstr S is join-inheriting;

end;

:: deftheorem Def2 defines join-closed WAYBEL23:def 2 :

definition

let L be non empty RelStr ;
let S be Subset of L;
attr S is infs-closed means :Def3: :: WAYBEL23:def 3
subrelstr S is infs-inheriting;

end;

:: deftheorem Def3 defines infs-closed WAYBEL23:def 3 :

definition

let L be non empty RelStr ;
let S be Subset of L;
attr S is sups-closed means :Def4: :: WAYBEL23:def 4
subrelstr S is sups-inheriting;

end;

:: deftheorem Def4 defines sups-closed WAYBEL23:def 4 :

registration

let L be non empty RelStr ;
cluster infs-closed -> meet-closed Element of K10(the carrier of L);
coherence

proof end;

cluster sups-closed -> join-closed Element of K10(the carrier of L);
coherence

proof end;

end;

registration

let L be non empty RelStr ;
cluster non empty meet-closed join-closed infs-closed sups-closed Element of K10(the carrier of L);
existence

proof end;

end;

theorem Th15: :: WAYBEL23:15

for L being non empty RelStr
for S being Subset of L holds
( S is meet-closed iff for x, y being Element of L st x in S & y in S & ex_inf_of {x,y},L holds
inf {x,y} in S )

proof end;

theorem Th16: :: WAYBEL23:16

for L being non empty RelStr
for S being Subset of L holds
( S is join-closed iff for x, y being Element of L st x in S & y in S & ex_sup_of {x,y},L holds
sup {x,y} in S )

proof end;

theorem :: WAYBEL23:17

for L being antisymmetric with_infima RelStr
for S being Subset of L holds
( S is meet-closed iff for x, y being Element of L st x in S & y in S holds
inf {x,y} in S )

proof end;

theorem Th18: :: WAYBEL23:18

for L being antisymmetric with_suprema RelStr
for S being Subset of L holds
( S is join-closed iff for x, y being Element of L st x in S & y in S holds
sup {x,y} in S )

proof end;

theorem :: WAYBEL23:19

for L being non empty RelStr
for S being Subset of L holds
( S is infs-closed iff for X being Subset of S st ex_inf_of X,L holds
"/\" X,L in S )

proof end;

theorem :: WAYBEL23:20

for L being non empty RelStr
for S being Subset of L holds
( S is sups-closed iff for X being Subset of S st ex_sup_of X,L holds
"\/" X,L in S )

proof end;

theorem Th21: :: WAYBEL23:21

for L being non empty transitive RelStr
for S being non empty infs-closed Subset of L
for X being Subset of S st ex_inf_of X,L holds
( ex_inf_of X, subrelstr S & "/\" X,(subrelstr S) = "/\" X,L )

proof end;

theorem Th22: :: WAYBEL23:22

for L being non empty transitive RelStr
for S being non empty sups-closed Subset of L
for X being Subset of S st ex_sup_of X,L holds
( ex_sup_of X, subrelstr S & "\/" X,(subrelstr S) = "\/" X,L )

proof end;

theorem :: WAYBEL23:23

for L being non empty transitive RelStr
for S being non empty meet-closed Subset of L
for x, y being Element of S st ex_inf_of {x,y},L holds
( ex_inf_of {x,y}, subrelstr S & "/\" {x,y},(subrelstr S) = "/\" {x,y},L )

proof end;

theorem :: WAYBEL23:24

for L being non empty transitive RelStr
for S being non empty join-closed Subset of L
for x, y being Element of S st ex_sup_of {x,y},L holds
( ex_sup_of {x,y}, subrelstr S & "\/" {x,y},(subrelstr S) = "\/" {x,y},L )

proof end;

theorem Th25: :: WAYBEL23:25

for L being transitive antisymmetric with_infima RelStr
for S being non empty meet-closed Subset of L holds subrelstr S is with_infima

proof end;

theorem Th26: :: WAYBEL23:26

for L being transitive antisymmetric with_suprema RelStr
for S being non empty join-closed Subset of L holds subrelstr S is with_suprema

proof end;

registration

let L be transitive antisymmetric with_infima RelStr ;
let S be non empty meet-closed Subset of L;
cluster subrelstr S -> with_infima ;
coherence by Th25;

end;

registration

let L be transitive antisymmetric with_suprema RelStr ;
let S be non empty join-closed Subset of L;
cluster subrelstr S -> with_suprema ;
coherence by Th26;

end;

theorem :: WAYBEL23:27

for L being non empty transitive antisymmetric complete RelStr
for S being non empty infs-closed Subset of L
for X being Subset of S holds "/\" X,(subrelstr S) = "/\" X,L

proof end;

theorem :: WAYBEL23:28

for L being non empty transitive antisymmetric complete RelStr
for S being non empty sups-closed Subset of L
for X being Subset of S holds "\/" X,(subrelstr S) = "\/" X,L

proof end;

theorem Th29: :: WAYBEL23:29

for L being Semilattice
for S being meet-closed Subset of L holds S is filtered

proof end;

theorem Th30: :: WAYBEL23:30

for L being sup-Semilattice
for S being join-closed Subset of L holds S is directed

proof end;

registration

let L be Semilattice;
cluster meet-closed -> filtered Element of K10(the carrier of L);
coherence by Th29;

end;

registration

let L be sup-Semilattice;
cluster join-closed -> directed Element of K10(the carrier of L);
coherence by Th30;

end;

theorem :: WAYBEL23:31

for L being Semilattice
for S being non empty upper Subset of L holds
( S is Filter of L iff S is meet-closed )

proof end;

theorem :: WAYBEL23:32

for L being sup-Semilattice
for S being non empty lower Subset of L holds
( S is Ideal of L iff S is join-closed )

proof end;

theorem Th33: :: WAYBEL23:33

for L being non empty RelStr
for S1, S2 being join-closed Subset of L holds S1 /\ S2 is join-closed

proof end;

theorem :: WAYBEL23:34

for L being non empty RelStr
for S1, S2 being meet-closed Subset of L holds S1 /\ S2 is meet-closed

proof end;

theorem Th35: :: WAYBEL23:35

for L being sup-Semilattice
for x being Element of L holds downarrow x is join-closed

proof end;

theorem Th36: :: WAYBEL23:36

for L being Semilattice
for x being Element of L holds downarrow x is meet-closed

proof end;

theorem Th37: :: WAYBEL23:37

for L being sup-Semilattice
for x being Element of L holds uparrow x is join-closed

proof end;

theorem Th38: :: WAYBEL23:38

for L being Semilattice
for x being Element of L holds uparrow x is meet-closed

proof end;

registration

let L be sup-Semilattice;
let x be Element of L;
cluster downarrow x -> join-closed ;
coherence by Th35;
cluster uparrow x -> directed join-closed ;
coherence by Th37;

end;

registration

let L be Semilattice;
let x be Element of L;
cluster downarrow x -> filtered meet-closed ;
coherence by Th36;
cluster uparrow x -> meet-closed ;
coherence by Th38;

end;

theorem Th39: :: WAYBEL23:39

for L being sup-Semilattice
for x being Element of L holds waybelow x is join-closed

proof end;

theorem Th40: :: WAYBEL23:40

for L being Semilattice
for x being Element of L holds waybelow x is meet-closed

proof end;

theorem Th41: :: WAYBEL23:41

for L being sup-Semilattice
for x being Element of L holds wayabove x is join-closed

proof end;

registration

let L be sup-Semilattice;
let x be Element of L;
cluster waybelow x -> join-closed ;
coherence by Th39;
cluster wayabove x -> directed join-closed ;
coherence by Th41;

end;

registration

let L be Semilattice;
let x be Element of L;
cluster waybelow x -> filtered meet-closed ;
coherence by Th40;

end;

definition

let T be TopStruct ;
func weight T -> Cardinal equals :: WAYBEL23:def 5
meet { (Card B) where B is Basis of T : verum } ;
coherence

proof end;

end;

:: deftheorem defines weight WAYBEL23:def 5 :

definition

let T be TopStruct ;
attr T is second-countable means :: WAYBEL23:def 6
weight T c= omega ;

end;

:: deftheorem defines second-countable WAYBEL23:def 6 :

definition

let L be continuous sup-Semilattice;
mode CLbasis of L -> Subset of L means :Def7: :: WAYBEL23:def 7
( it is join-closed & ( for x being Element of L holds x = sup ((waybelow x) /\ it) ) );
existence

proof end;

end;

:: deftheorem Def7 defines CLbasis WAYBEL23:def 7 :

definition

let L be non empty RelStr ;
let S be Subset of L;
attr S is with_bottom means :Def8: :: WAYBEL23:def 8
Bottom L in S;

end;

:: deftheorem Def8 defines with_bottom WAYBEL23:def 8 :

definition

let L be non empty RelStr ;
let S be Subset of L;
attr S is with_top means :Def9: :: WAYBEL23:def 9
Top L in S;

end;

:: deftheorem Def9 defines with_top WAYBEL23:def 9 :

registration

let L be non empty RelStr ;
cluster with_bottom -> non empty Element of K10(the carrier of L);
coherence by Def8;

end;

registration

let L be non empty RelStr ;
cluster with_top -> non empty Element of K10(the carrier of L);
coherence by Def9;

end;

registration

let L be non empty RelStr ;
cluster non empty with_bottom Element of K10(the carrier of L);
existence

proof end;

cluster non empty with_top Element of K10(the carrier of L);
existence

proof end;

end;

registration

let L be continuous sup-Semilattice;
cluster non empty with_bottom CLbasis of L;
existence

proof end;

cluster non empty with_top CLbasis of L;
existence

proof end;

end;

theorem Th42: :: WAYBEL23:42

for L being non empty antisymmetric lower-bounded RelStr
for S being with_bottom Subset of L holds subrelstr S is lower-bounded

proof end;

registration

let L be non empty antisymmetric lower-bounded RelStr ;
let S be with_bottom Subset of L;
cluster subrelstr S -> lower-bounded ;
coherence by Th42;

end;

registration

let L be continuous sup-Semilattice;
cluster -> directed join-closed CLbasis of L;
coherence by Def7;

end;

registration

cluster bounded continuous non trivial RelStr ;
existence

proof end;

end;

registration

let L be lower-bounded continuous non trivial sup-Semilattice;
cluster -> non empty directed join-closed CLbasis of L;
coherence

proof end;

end;

theorem Th43: :: WAYBEL23:43

for L being sup-Semilattice holds the carrier of (CompactSublatt L) is join-closed Subset of L

proof end;

theorem Th44: :: WAYBEL23:44

for L being lower-bounded algebraic LATTICE holds the carrier of (CompactSublatt L) is with_bottom CLbasis of L

proof end;

theorem :: WAYBEL23:45

for L being lower-bounded continuous sup-Semilattice st the carrier of (CompactSublatt L) is CLbasis of L holds
L is algebraic

proof end;

theorem Th46: :: WAYBEL23:46

for L being lower-bounded continuous LATTICE
for B being join-closed Subset of L holds
( B is CLbasis of L iff for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b << y ) )

proof end;

theorem Th47: :: WAYBEL23:47

for L being lower-bounded continuous LATTICE
for B being join-closed Subset of L st Bottom L in B holds
( B is CLbasis of L iff for x, y being Element of L st x << y holds
ex b being Element of L st
( b in B & x <= b & b << y ) )

proof end;

Lm2: for L being lower-bounded continuous LATTICE
for B being join-closed Subset of L st Bottom L in B & ( for x, y being Element of L st x << y holds
ex b being Element of L st
( b in B & x <= b & b << y ) ) holds
( the carrier of (CompactSublatt L) c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) )

proof end;

Lm3: for L being lower-bounded continuous LATTICE
for B being Subset of L st ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) holds
for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b << y )

proof end;

theorem Th48: :: WAYBEL23:48

for L being lower-bounded continuous LATTICE
for B being join-closed Subset of L st Bottom L in B holds
( B is CLbasis of L iff ( the carrier of (CompactSublatt L) c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) ) )

proof end;

theorem :: WAYBEL23:49

for L being lower-bounded continuous LATTICE
for B being join-closed Subset of L st Bottom L in B holds
( B is CLbasis of L iff for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) )

proof end;

theorem Th50: :: WAYBEL23:50

for L being lower-bounded sup-Semilattice
for S being non empty full SubRelStr of L st Bottom L in the carrier of S & the carrier of S is join-closed Subset of L holds
for x being Element of L holds (waybelow x) /\ the carrier of S is Ideal of S

proof end;

definition

let L be non empty reflexive transitive RelStr ;
let S be non empty full SubRelStr of L;
func supMap S -> Function of (InclPoset (Ids S)),L means :Def10: :: WAYBEL23:def 10
for I being Ideal of S holds it . I = "\/" I,L;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def10 defines supMap WAYBEL23:def 10 :

definition

let L be non empty reflexive transitive RelStr ;
let S be non empty full SubRelStr of L;
func idsMap S -> Function of (InclPoset (Ids S)),(InclPoset (Ids L)) means :Def11: :: WAYBEL23:def 11
for I being Ideal of S ex J being Subset of L st
( I = J & it . I = downarrow J );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def11 defines idsMap WAYBEL23:def 11 :

registration

let L be reflexive RelStr ;
let B be Subset of L;
cluster subrelstr B -> reflexive ;
coherence ;

end;

registration

let L be transitive RelStr ;
let B be Subset of L;
cluster subrelstr B -> transitive ;
coherence ;

end;

registration

let L be antisymmetric RelStr ;
let B be Subset of L;
cluster subrelstr B -> antisymmetric ;
coherence ;

end;

definition

let L be lower-bounded continuous sup-Semilattice;
let B be with_bottom CLbasis of L;
func baseMap B -> Function of L,(InclPoset (Ids (subrelstr B))) means :Def12: :: WAYBEL23:def 12
for x being Element of L holds it . x = (waybelow x) /\ B;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def12 defines baseMap WAYBEL23:def 12 :

theorem Th51: :: WAYBEL23:51

for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L holds
( dom (supMap S) = Ids S & rng (supMap S) is Subset of L )

proof end;

theorem Th52: :: WAYBEL23:52

for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L
for x being set holds
( x in dom (supMap S) iff x is Ideal of S )

proof end;

theorem Th53: :: WAYBEL23:53

for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L holds
( dom (idsMap S) = Ids S & rng (idsMap S) is Subset of (Ids L) )

proof end;

theorem Th54: :: WAYBEL23:54

for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L
for x being set holds
( x in dom (idsMap S) iff x is Ideal of S )

proof end;

theorem Th55: :: WAYBEL23:55

for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L
for x being set st x in rng (idsMap S) holds
x is Ideal of L

proof end;

theorem :: WAYBEL23:56

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds
( dom (baseMap B) = the carrier of L & rng (baseMap B) is Subset of (Ids (subrelstr B)) ) by FUNCT_2:def 1, YELLOW_1:1;

theorem :: WAYBEL23:57

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L
for x being set st x in rng (baseMap B) holds
x is Ideal of (subrelstr B)

proof end;

theorem Th58: :: WAYBEL23:58

for L being non empty up-complete Poset
for S being non empty full SubRelStr of L holds supMap S is monotone

proof end;

theorem Th59: :: WAYBEL23:59

for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L holds idsMap S is monotone

proof end;

theorem Th60: :: WAYBEL23:60

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds baseMap B is monotone

proof end;

registration

let L be non empty up-complete Poset;
let S be non empty full SubRelStr of L;
cluster supMap S -> monotone ;
coherence by Th58;

end;

registration

let L be non empty reflexive transitive RelStr ;
let S be non empty full SubRelStr of L;
cluster idsMap S -> monotone ;
coherence by Th59;

end;

registration

let L be lower-bounded continuous sup-Semilattice;
let B be with_bottom CLbasis of L;
cluster baseMap B -> monotone ;
coherence by Th60;

end;

theorem Th61: :: WAYBEL23:61

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds idsMap (subrelstr B) is sups-preserving

proof end;

theorem Th62: :: WAYBEL23:62

for L being non empty up-complete Poset
for S being non empty full SubRelStr of L holds supMap S = (SupMap L) * (idsMap S)

proof end;

theorem Th63: :: WAYBEL23:63

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds [(supMap (subrelstr B)),(baseMap B)] is Galois

proof end;

theorem Th64: :: WAYBEL23:64

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds
( supMap (subrelstr B) is upper_adjoint & baseMap B is lower_adjoint )

proof end;

theorem Th65: :: WAYBEL23:65

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds rng (supMap (subrelstr B)) = the carrier of L

proof end;

theorem Th66: :: WAYBEL23:66

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds
( supMap (subrelstr B) is infs-preserving & supMap (subrelstr B) is sups-preserving )

proof end;

theorem Th67: :: WAYBEL23:67

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds baseMap B is sups-preserving

proof end;

registration

let L be lower-bounded continuous sup-Semilattice;
let B be with_bottom CLbasis of L;
cluster supMap (subrelstr B) -> monotone infs-preserving sups-preserving ;
coherence by Th66;
cluster baseMap B -> monotone sups-preserving ;
coherence by Th67;

end;

theorem :: WAYBEL23:68

canceled;

theorem Th69: :: WAYBEL23:69

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds the carrier of (CompactSublatt (InclPoset (Ids (subrelstr B)))) = { (downarrow b) where b is Element of (subrelstr B) : verum }

proof end;

theorem :: WAYBEL23:70

for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds CompactSublatt (InclPoset (Ids (subrelstr B))), subrelstr B are_isomorphic

proof end;

Lm4: for L being lower-bounded continuous LATTICE st L is algebraic holds
( the carrier of (CompactSublatt L) is with_bottom CLbasis of L & ( for B being with_bottom CLbasis of L holds the carrier of (CompactSublatt L) c= B ) )

proof end;

theorem Th71: :: WAYBEL23:71

for L being lower-bounded continuous LATTICE
for B being with_bottom CLbasis of L st ( for B1 being with_bottom CLbasis of L holds B c= B1 ) holds
for J being Element of (InclPoset (Ids (subrelstr B))) holds J = (waybelow ("\/" J,L)) /\ B

proof end;

Lm5: for L being lower-bounded continuous LATTICE st ex B being with_bottom CLbasis of L st
for B1 being with_bottom CLbasis of L holds B c= B1 holds
L is algebraic

proof end;

theorem :: WAYBEL23:72

for L being lower-bounded continuous LATTICE holds
( L is algebraic iff ( the carrier of (CompactSublatt L) is with_bottom CLbasis of L & ( for B being with_bottom CLbasis of L holds the carrier of (CompactSublatt L) c= B ) ) ) by Lm4, Lm5;

theorem :: WAYBEL23:73

for L being lower-bounded continuous LATTICE holds
( L is algebraic iff ex B being with_bottom CLbasis of L st
for B1 being with_bottom CLbasis of L holds B c= B1 )

proof end;