:: Duality Based on Galois Connection. Part I
:: by Grzegorz Bancerek
::
:: Received August 8, 2001
:: Copyright (c) 2001-2021 Association of Mizar Users

::<section1>Infs-preserving and sups-preserving maps</section1>

registration

let S, T be complete LATTICE;

cluster Galois for Connection of S,T;

existence
ex b₁ being Connection of S,T st b₁ is Galois

proof end;

end;

theorem Th1: :: WAYBEL34:1

for S, T, S9, T9 being non empty RelStr st RelStr(# the carrier of S, the InternalRel of S #) = RelStr(# the carrier of S9, the InternalRel of S9 #) & RelStr(# the carrier of T, the InternalRel of T #) = RelStr(# the carrier of T9, the InternalRel of T9 #) holds
for c being Connection of S,T
for c9 being Connection of S9,T9 st c = c9 & c is Galois holds
c9 is Galois

proof end;

definition

let S, T be LATTICE;
let g be Function of S,T;
assume that
A1: S is complete and
A2: g is infs-preserving ;
A3: g is upper_adjoint by A1, A2, WAYBEL_1:16;

func LowerAdj g -> Function of T,S means :Def1: :: WAYBEL34:def 1
[g,it] is Galois ;

uniqueness
for b₁, b₂ being Function of T,S st [g,b₁] is Galois & [g,b₂] is Galois holds
b₁ = b₂

proof end;

existence
ex b₁ being Function of T,S st [g,b₁] is Galois by A3;

end;

:: deftheorem Def1 defines LowerAdj WAYBEL34:def 1 :
for S, T being LATTICE
for g being Function of S,T st S is complete & g is infs-preserving holds
for b₄ being Function of T,S holds
( b₄ = LowerAdj g iff [g,b₄] is Galois );

definition

let S, T be LATTICE;
let d be Function of T,S;
assume that
A1: T is complete and
A2: d is sups-preserving ;
A3: d is lower_adjoint by A1, A2, WAYBEL_1:17;

func UpperAdj d -> Function of S,T means :Def2: :: WAYBEL34:def 2
[it,d] is Galois ;

existence
ex b₁ being Function of S,T st [b₁,d] is Galois by A3;
correctness
uniqueness
for b₁, b₂ being Function of S,T st [b₁,d] is Galois & [b₂,d] is Galois holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def2 defines UpperAdj WAYBEL34:def 2 :
for S, T being LATTICE
for d being Function of T,S st T is complete & d is sups-preserving holds
for b₄ being Function of S,T holds
( b₄ = UpperAdj d iff [b₄,d] is Galois );

Lm1: now :: thesis: for S, T being LATTICE st S is complete & T is complete holds
for g being Function of S,T st g is infs-preserving holds
( LowerAdj g is monotone & LowerAdj g is lower_adjoint & LowerAdj g is sups-preserving )

let S, T be LATTICE; :: thesis: ( S is complete & T is complete implies for g being Function of S,T st g is infs-preserving holds
( LowerAdj g is monotone & LowerAdj g is lower_adjoint & LowerAdj g is sups-preserving ) )
assume that
A1: S is complete and
A2: T is complete ; :: thesis: for g being Function of S,T st g is infs-preserving holds
( LowerAdj g is monotone & LowerAdj g is lower_adjoint & LowerAdj g is sups-preserving )
reconsider S9 = S, T9 = T as complete LATTICE by A1, A2;
let g be Function of S,T; :: thesis: ( g is infs-preserving implies ( LowerAdj g is monotone & LowerAdj g is lower_adjoint & LowerAdj g is sups-preserving ) )
assume g is infs-preserving ; :: thesis: ( LowerAdj g is monotone & LowerAdj g is lower_adjoint & LowerAdj g is sups-preserving )
then reconsider g9 = g as infs-preserving Function of S9,T9 ;
[g9,(LowerAdj g9)] is Galois by Def1;
then ( LowerAdj g9 is lower_adjoint & LowerAdj g9 is monotone ) by WAYBEL_1:8;
hence ( LowerAdj g is monotone & LowerAdj g is lower_adjoint & LowerAdj g is sups-preserving ) ; :: thesis: verum

end;

Lm2: now :: thesis: for S, T being LATTICE st S is complete & T is complete holds
for g being Function of S,T st g is sups-preserving holds
( UpperAdj g is monotone & UpperAdj g is upper_adjoint & UpperAdj g is infs-preserving )

let S, T be LATTICE; :: thesis: ( S is complete & T is complete implies for g being Function of S,T st g is sups-preserving holds
( UpperAdj g is monotone & UpperAdj g is upper_adjoint & UpperAdj g is infs-preserving ) )
assume that
A1: S is complete and
A2: T is complete ; :: thesis: for g being Function of S,T st g is sups-preserving holds
( UpperAdj g is monotone & UpperAdj g is upper_adjoint & UpperAdj g is infs-preserving )
reconsider S9 = S, T9 = T as complete LATTICE by A1, A2;
let g be Function of S,T; :: thesis: ( g is sups-preserving implies ( UpperAdj g is monotone & UpperAdj g is upper_adjoint & UpperAdj g is infs-preserving ) )
assume g is sups-preserving ; :: thesis: ( UpperAdj g is monotone & UpperAdj g is upper_adjoint & UpperAdj g is infs-preserving )
then reconsider g9 = g as sups-preserving Function of S9,T9 ;
[(UpperAdj g9),g9] is Galois by Def2;
then ( UpperAdj g9 is upper_adjoint & UpperAdj g9 is monotone ) by WAYBEL_1:8;
hence ( UpperAdj g is monotone & UpperAdj g is upper_adjoint & UpperAdj g is infs-preserving ) ; :: thesis: verum

end;

registration

let S, T be complete LATTICE;
let g be infs-preserving Function of S,T;

cluster LowerAdj g -> lower_adjoint ;

coherence
LowerAdj g is lower_adjoint by Lm1;

end;

registration

let S, T be complete LATTICE;
let d be sups-preserving Function of T,S;

cluster UpperAdj d -> upper_adjoint ;

coherence
UpperAdj d is upper_adjoint by Lm2;

end;

theorem :: WAYBEL34:2

for S, T being complete LATTICE
for g being infs-preserving Function of S,T
for t being Element of T holds (LowerAdj g) . t = inf (g " (uparrow t))

proof end;

theorem :: WAYBEL34:3

for S, T being complete LATTICE
for d being sups-preserving Function of T,S
for s being Element of S holds (UpperAdj d) . s = sup (d " (downarrow s))

proof end;

definition

let S, T be RelStr ;
let f be Function of the carrier of S, the carrier of T;

func f opp -> Function of (S opp),(T opp) equals :: WAYBEL34:def 3
f;

coherence
f is Function of (S opp),(T opp) ;

end;

:: deftheorem defines opp WAYBEL34:def 3 :
for S, T being RelStr
for f being Function of the carrier of S, the carrier of T holds f opp = f;

registration

let S, T be complete LATTICE;
let g be infs-preserving Function of S,T;

cluster g opp -> lower_adjoint ;

coherence
g opp is lower_adjoint

proof end;

end;

registration

let S, T be complete LATTICE;
let d be sups-preserving Function of S,T;

cluster d opp -> upper_adjoint ;

coherence
d opp is upper_adjoint

proof end;

end;

theorem :: WAYBEL34:4

for S, T being complete LATTICE
for g being infs-preserving Function of S,T holds LowerAdj g = UpperAdj (g opp)

proof end;

theorem :: WAYBEL34:5

for S, T being complete LATTICE
for d being sups-preserving Function of S,T holds LowerAdj (d opp) = UpperAdj d

proof end;

theorem Th6: :: WAYBEL34:6

for L being non empty RelStr holds [(id L),(id L)] is Galois

proof end;

theorem Th7: :: WAYBEL34:7

for L being complete LATTICE holds
( LowerAdj (id L) = id L & UpperAdj (id L) = id L )

proof end;

theorem Th8: :: WAYBEL34:8

for L1, L2, L3 being complete LATTICE
for g1 being infs-preserving Function of L1,L2
for g2 being infs-preserving Function of L2,L3 holds LowerAdj (g2 * g1) = (LowerAdj g1) * (LowerAdj g2)

proof end;

theorem Th9: :: WAYBEL34:9

for L1, L2, L3 being complete LATTICE
for d1 being sups-preserving Function of L1,L2
for d2 being sups-preserving Function of L2,L3 holds UpperAdj (d2 * d1) = (UpperAdj d1) * (UpperAdj d2)

proof end;

theorem Th10: :: WAYBEL34:10

for S, T being complete LATTICE
for g being infs-preserving Function of S,T holds UpperAdj (LowerAdj g) = g

proof end;

theorem Th11: :: WAYBEL34:11

for S, T being complete LATTICE
for d being sups-preserving Function of S,T holds LowerAdj (UpperAdj d) = d

proof end;

theorem Th12: :: WAYBEL34:12

for C being non empty AltCatStr
for a, b, f being object st f in the Arrows of C . (a,b) holds
ex o1, o2 being Object of C st
( o1 = a & o2 = b & f in <^o1,o2^> & f is Morphism of o1,o2 )

proof end;

definition

let W be non empty set ;
defpred S₁[ LATTICE] means $1 is complete ;
defpred S₂[ LATTICE, LATTICE, Function of $1,$2] means $3 is infs-preserving ;
given w being Element of W such that A1: not w is empty ;

func W -INF_category -> strict lattice-wise category means :Def4: :: WAYBEL34:def 4
( ( for x being LATTICE holds
( x is Object of it iff ( x is strict & x is complete & the carrier of x in W ) ) ) & ( for a, b being Object of it
for f being monotone Function of (latt a),(latt b) holds
( f in <^a,b^> iff f is infs-preserving ) ) );

existence
ex b₁ being strict lattice-wise category st
( ( for x being LATTICE holds
( x is Object of b₁ iff ( x is strict & x is complete & the carrier of x in W ) ) ) & ( for a, b being Object of b₁
for f being monotone Function of (latt a),(latt b) holds
( f in <^a,b^> iff f is infs-preserving ) ) )

proof end;

uniqueness
for b₁, b₂ being strict lattice-wise category st ( for x being LATTICE holds
( x is Object of b₁ iff ( x is strict & x is complete & the carrier of x in W ) ) ) & ( for a, b being Object of b₁
for f being monotone Function of (latt a),(latt b) holds
( f in <^a,b^> iff f is infs-preserving ) ) & ( for x being LATTICE holds
( x is Object of b₂ iff ( x is strict & x is complete & the carrier of x in W ) ) ) & ( for a, b being Object of b₂
for f being monotone Function of (latt a),(latt b) holds
( f in <^a,b^> iff f is infs-preserving ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines -INF_category WAYBEL34:def 4 :
for W being non empty set st not for w being Element of W holds w is empty holds
for b₂ being strict lattice-wise category holds
( b₂ = W -INF_category iff ( ( for x being LATTICE holds
( x is Object of b₂ iff ( x is strict & x is complete & the carrier of x in W ) ) ) & ( for a, b being Object of b₂
for f being monotone Function of (latt a),(latt b) holds
( f in <^a,b^> iff f is infs-preserving ) ) ) );

Lm3: for W being with_non-empty_element set
for a, b being LATTICE
for f being Function of a,b holds
( f in the Arrows of (W -INF_category) . (a,b) iff ( a in the carrier of (W -INF_category) & b in the carrier of (W -INF_category) & a is complete & b is complete & f is infs-preserving ) )

proof end;

definition

let W be non empty set ;
defpred S₁[ LATTICE] means $1 is complete ;
defpred S₂[ LATTICE, LATTICE, Function of $1,$2] means $3 is sups-preserving ;
given w being Element of W such that A1: not w is empty ;

func W -SUP_category -> strict lattice-wise category means :Def5: :: WAYBEL34:def 5
( ( for x being LATTICE holds
( x is Object of it iff ( x is strict & x is complete & the carrier of x in W ) ) ) & ( for a, b being Object of it
for f being monotone Function of (latt a),(latt b) holds
( f in <^a,b^> iff f is sups-preserving ) ) );

existence
ex b₁ being strict lattice-wise category st
( ( for x being LATTICE holds
( x is Object of b₁ iff ( x is strict & x is complete & the carrier of x in W ) ) ) & ( for a, b being Object of b₁
for f being monotone Function of (latt a),(latt b) holds
( f in <^a,b^> iff f is sups-preserving ) ) )

proof end;

uniqueness
for b₁, b₂ being strict lattice-wise category st ( for x being LATTICE holds
( x is Object of b₁ iff ( x is strict & x is complete & the carrier of x in W ) ) ) & ( for a, b being Object of b₁
for f being monotone Function of (latt a),(latt b) holds
( f in <^a,b^> iff f is sups-preserving ) ) & ( for x being LATTICE holds
( x is Object of b₂ iff ( x is strict & x is complete & the carrier of x in W ) ) ) & ( for a, b being Object of b₂
for f being monotone Function of (latt a),(latt b) holds
( f in <^a,b^> iff f is sups-preserving ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines -SUP_category WAYBEL34:def 5 :
for W being non empty set st not for w being Element of W holds w is empty holds
for b₂ being strict lattice-wise category holds
( b₂ = W -SUP_category iff ( ( for x being LATTICE holds
( x is Object of b₂ iff ( x is strict & x is complete & the carrier of x in W ) ) ) & ( for a, b being Object of b₂
for f being monotone Function of (latt a),(latt b) holds
( f in <^a,b^> iff f is sups-preserving ) ) ) );

Lm4: for W being with_non-empty_element set
for a, b being LATTICE
for f being Function of a,b holds
( f in the Arrows of (W -SUP_category) . (a,b) iff ( a in the carrier of (W -SUP_category) & b in the carrier of (W -SUP_category) & a is complete & b is complete & f is sups-preserving ) )

proof end;

registration

let W be with_non-empty_element set ;

cluster W -INF_category -> strict lattice-wise with_complete_lattices ;

coherence
W -INF_category is with_complete_lattices

proof end;

cluster W -SUP_category -> strict lattice-wise with_complete_lattices ;

coherence
W -SUP_category is with_complete_lattices

proof end;

end;

theorem Th13: :: WAYBEL34:13

for W being with_non-empty_element set
for L being LATTICE holds
( L is Object of (W -INF_category) iff ( L is strict & L is complete & the carrier of L in W ) )

proof end;

theorem Th14: :: WAYBEL34:14

for W being with_non-empty_element set
for a, b being Object of (W -INF_category)
for f being set holds
( f in <^a,b^> iff f is infs-preserving Function of (latt a),(latt b) )

proof end;

theorem Th15: :: WAYBEL34:15

for W being with_non-empty_element set
for L being LATTICE holds
( L is Object of (W -SUP_category) iff ( L is strict & L is complete & the carrier of L in W ) )

proof end;

theorem Th16: :: WAYBEL34:16

for W being with_non-empty_element set
for a, b being Object of (W -SUP_category)
for f being set holds
( f in <^a,b^> iff f is sups-preserving Function of (latt a),(latt b) )

proof end;

theorem Th17: :: WAYBEL34:17

for W being with_non-empty_element set holds the carrier of (W -INF_category) = the carrier of (W -SUP_category)

proof end;

definition

let W be with_non-empty_element set ;
set A = W -INF_category ;
set B = W -SUP_category ;
deffunc H₁( LATTICE) -> LATTICE = $1;
deffunc H₂( LATTICE, LATTICE, Function of $1,$2) -> Function of $2,$1 = LowerAdj $3;
defpred S₁[ LATTICE, LATTICE, Function of $1,$2] means ( $1 is complete & $2 is complete & $3 is infs-preserving );
defpred S₂[ LATTICE, LATTICE, Function of $1,$2] means ( $1 is complete & $2 is complete & $3 is sups-preserving );
A1: for a, b being LATTICE
for f being Function of a,b holds
( f in the Arrows of (W -INF_category) . (a,b) iff ( a in the carrier of (W -INF_category) & b in the carrier of (W -INF_category) & S₁[a,b,f] ) ) by Lm3;
A2: for a, b being LATTICE
for f being Function of a,b holds
( f in the Arrows of (W -SUP_category) . (a,b) iff ( a in the carrier of (W -SUP_category) & b in the carrier of (W -SUP_category) & S₂[a,b,f] ) ) by Lm4;
A3: for a being LATTICE st a in the carrier of (W -INF_category) holds
H₁(a) in the carrier of (W -SUP_category) by Th17;
A4: for a, b being LATTICE
for f being Function of a,b st S₁[a,b,f] holds
( H₂(a,b,f) is Function of H₁(b),H₁(a) & S₂[H₁(b),H₁(a),H₂(a,b,f)] ) ;

A5: now :: thesis: for a being LATTICE st a in the carrier of (W -INF_category) holds
H₂(a,a, id a) = id H₁(a)

let a be LATTICE; :: thesis: ( a in the carrier of (W -INF_category) implies H₂(a,a, id a) = id H₁(a) )
assume a in the carrier of (W -INF_category) ; :: thesis: H₂(a,a, id a) = id H₁(a)
then a is complete by YELLOW21:def 5;
hence H₂(a,a, id a) = id H₁(a) by Th7; :: thesis: verum

end;

A6: for a, b, c being LATTICE
for f being Function of a,b
for g being Function of b,c st S₁[a,b,f] & S₁[b,c,g] holds
H₂(a,c,g * f) = H₂(a,b,f) * H₂(b,c,g) by Th8;

func W LowerAdj -> strict contravariant Functor of W -INF_category ,W -SUP_category means :Def6: :: WAYBEL34:def 6
( ( for a being Object of (W -INF_category) holds it . a = latt a ) & ( for a, b being Object of (W -INF_category) st <^a,b^> <> {} holds
for f being Morphism of a,b holds it . f = LowerAdj (@ f) ) );

existence
ex b₁ being strict contravariant Functor of W -INF_category ,W -SUP_category st
( ( for a being Object of (W -INF_category) holds b₁ . a = latt a ) & ( for a, b being Object of (W -INF_category) st <^a,b^> <> {} holds
for f being Morphism of a,b holds b₁ . f = LowerAdj (@ f) ) )

proof end;

uniqueness
for b₁, b₂ being strict contravariant Functor of W -INF_category ,W -SUP_category st ( for a being Object of (W -INF_category) holds b₁ . a = latt a ) & ( for a, b being Object of (W -INF_category) st <^a,b^> <> {} holds
for f being Morphism of a,b holds b₁ . f = LowerAdj (@ f) ) & ( for a being Object of (W -INF_category) holds b₂ . a = latt a ) & ( for a, b being Object of (W -INF_category) st <^a,b^> <> {} holds
for f being Morphism of a,b holds b₂ . f = LowerAdj (@ f) ) holds
b₁ = b₂

proof end;

deffunc H₃( LATTICE, LATTICE, Function of $1,$2) -> Function of $2,$1 = UpperAdj $3;
A13: for a being LATTICE st a in the carrier of (W -SUP_category) holds
H₁(a) in the carrier of (W -INF_category) by Th17;
A14: for a, b being LATTICE
for f being Function of a,b st S₂[a,b,f] holds
( H₃(a,b,f) is Function of H₁(b),H₁(a) & S₁[H₁(b),H₁(a),H₃(a,b,f)] ) ;

A15: now :: thesis: for a being LATTICE st a in the carrier of (W -SUP_category) holds
H₃(a,a, id a) = id H₁(a)

let a be LATTICE; :: thesis: ( a in the carrier of (W -SUP_category) implies H₃(a,a, id a) = id H₁(a) )
assume a in the carrier of (W -SUP_category) ; :: thesis: H₃(a,a, id a) = id H₁(a)
then a is complete by YELLOW21:def 5;
hence H₃(a,a, id a) = id H₁(a) by Th7; :: thesis: verum

end;

A16: for a, b, c being LATTICE
for f being Function of a,b
for g being Function of b,c st S₂[a,b,f] & S₂[b,c,g] holds
H₃(a,c,g * f) = H₃(a,b,f) * H₃(b,c,g) by Th9;

func W UpperAdj -> strict contravariant Functor of W -SUP_category ,W -INF_category means :Def7: :: WAYBEL34:def 7
( ( for a being Object of (W -SUP_category) holds it . a = latt a ) & ( for a, b being Object of (W -SUP_category) st <^a,b^> <> {} holds
for f being Morphism of a,b holds it . f = UpperAdj (@ f) ) );

existence
ex b₁ being strict contravariant Functor of W -SUP_category ,W -INF_category st
( ( for a being Object of (W -SUP_category) holds b₁ . a = latt a ) & ( for a, b being Object of (W -SUP_category) st <^a,b^> <> {} holds
for f being Morphism of a,b holds b₁ . f = UpperAdj (@ f) ) )

proof end;

uniqueness
for b₁, b₂ being strict contravariant Functor of W -SUP_category ,W -INF_category st ( for a being Object of (W -SUP_category) holds b₁ . a = latt a ) & ( for a, b being Object of (W -SUP_category) st <^a,b^> <> {} holds
for f being Morphism of a,b holds b₁ . f = UpperAdj (@ f) ) & ( for a being Object of (W -SUP_category) holds b₂ . a = latt a ) & ( for a, b being Object of (W -SUP_category) st <^a,b^> <> {} holds
for f being Morphism of a,b holds b₂ . f = UpperAdj (@ f) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines LowerAdj WAYBEL34:def 6 :
for W being with_non-empty_element set
for b₂ being strict contravariant Functor of W -INF_category ,W -SUP_category holds
( b₂ = W LowerAdj iff ( ( for a being Object of (W -INF_category) holds b₂ . a = latt a ) & ( for a, b being Object of (W -INF_category) st <^a,b^> <> {} holds
for f being Morphism of a,b holds b₂ . f = LowerAdj (@ f) ) ) );

:: deftheorem Def7 defines UpperAdj WAYBEL34:def 7 :
for W being with_non-empty_element set
for b₂ being strict contravariant Functor of W -SUP_category ,W -INF_category holds
( b₂ = W UpperAdj iff ( ( for a being Object of (W -SUP_category) holds b₂ . a = latt a ) & ( for a, b being Object of (W -SUP_category) st <^a,b^> <> {} holds
for f being Morphism of a,b holds b₂ . f = UpperAdj (@ f) ) ) );

registration

let W be with_non-empty_element set ;

cluster W LowerAdj -> strict contravariant bijective ;

coherence
W LowerAdj is bijective

proof end;

cluster W UpperAdj -> strict contravariant bijective ;

coherence
W UpperAdj is bijective

proof end;

end;

theorem Th18: :: WAYBEL34:18

for W being with_non-empty_element set holds
( (W LowerAdj) " = W UpperAdj & (W UpperAdj) " = W LowerAdj )

proof end;

theorem :: WAYBEL34:19

for W being with_non-empty_element set holds
( (W LowerAdj) * (W UpperAdj) = id (W -SUP_category) & (W UpperAdj) * (W LowerAdj) = id (W -INF_category) )

proof end;

theorem :: WAYBEL34:20

for W being with_non-empty_element set holds W -INF_category ,W -SUP_category are_anti-isomorphic

proof end;

::<section2>Scott continuous maps and continuous lattices</section2>

theorem Th21: :: WAYBEL34:21

for S, T being complete LATTICE
for g being infs-preserving Function of S,T holds
( g is directed-sups-preserving iff for X being Scott TopAugmentation of T
for Y being Scott TopAugmentation of S
for V being open Subset of X holds uparrow ((LowerAdj g) .: V) is open Subset of Y )

proof end;

definition

let S, T be non empty reflexive RelStr ;
let f be Function of S,T;

attr f is waybelow-preserving means :: WAYBEL34:def 8
for x, y being Element of S st x << y holds
f . x << f . y;

end;

:: deftheorem defines waybelow-preserving WAYBEL34:def 8 :
for S, T being non empty reflexive RelStr
for f being Function of S,T holds
( f is waybelow-preserving iff for x, y being Element of S st x << y holds
f . x << f . y );

theorem Th22: :: WAYBEL34:22

for S, T being complete LATTICE
for g being infs-preserving Function of S,T st g is directed-sups-preserving holds
LowerAdj g is waybelow-preserving

proof end;

theorem Th23: :: WAYBEL34:23

for S being complete LATTICE
for T being continuous complete LATTICE
for g being infs-preserving Function of S,T st LowerAdj g is waybelow-preserving holds
g is directed-sups-preserving

proof end;

definition

let S, T be TopSpace;
let f be Function of S,T;

attr f is relatively_open means :: WAYBEL34:def 9
for V being open Subset of S holds f .: V is open Subset of (T | (rng f));

end;

:: deftheorem defines relatively_open WAYBEL34:def 9 :
for S, T being TopSpace
for f being Function of S,T holds
( f is relatively_open iff for V being open Subset of S holds f .: V is open Subset of (T | (rng f)) );

theorem :: WAYBEL34:24

for X, Y being non empty TopSpace
for d being Function of X,Y holds
( d is relatively_open iff corestr d is open )

proof end;

theorem Th25: :: WAYBEL34:25

for S, T being complete LATTICE
for g being infs-preserving Function of S,T
for X being Scott TopAugmentation of T
for Y being Scott TopAugmentation of S
for V being open Subset of X holds (LowerAdj g) .: V = (rng (LowerAdj g)) /\ (uparrow ((LowerAdj g) .: V))

proof end;

theorem Th26: :: WAYBEL34:26

for S, T being complete LATTICE
for g being infs-preserving Function of S,T
for X being Scott TopAugmentation of T
for Y being Scott TopAugmentation of S st ( for V being open Subset of X holds uparrow ((LowerAdj g) .: V) is open Subset of Y ) holds
for d being Function of X,Y st d = LowerAdj g holds
d is relatively_open

proof end;

registration

let X, Y be complete LATTICE;
let f be sups-preserving Function of X,Y;

cluster Image f -> complete ;

coherence
Image f is complete by YELLOW_2:34;

end;

theorem :: WAYBEL34:27

for S, T being complete LATTICE
for g being infs-preserving Function of S,T
for X being Scott TopAugmentation of T
for Y being Scott TopAugmentation of S
for Z being Scott TopAugmentation of Image (LowerAdj g)
for d being Function of X,Y
for d9 being Function of X,Z st d = LowerAdj g & d9 = d & d is relatively_open holds
d9 is open

proof end;

theorem Th28: :: WAYBEL34:28

for S, T being complete LATTICE
for g being infs-preserving Function of S,T st g is V7() holds
for X being Scott TopAugmentation of T
for Y being Scott TopAugmentation of S
for d being Function of X,Y st d = LowerAdj g holds
( g is directed-sups-preserving iff d is open )

proof end;

registration

let X be complete LATTICE;
let f be projection Function of X,X;

cluster Image f -> complete ;

coherence
Image f is complete by WAYBEL_1:54;

end;

theorem Th29: :: WAYBEL34:29

for L being complete LATTICE
for k being kernel Function of L,L holds
( corestr k is infs-preserving & inclusion k is sups-preserving & LowerAdj (corestr k) = inclusion k & UpperAdj (inclusion k) = corestr k )

proof end;

theorem Th30: :: WAYBEL34:30

for L being complete LATTICE
for k being kernel Function of L,L holds
( k is directed-sups-preserving iff corestr k is directed-sups-preserving )

proof end;

theorem :: WAYBEL34:31

for L being complete LATTICE
for k being kernel Function of L,L holds
( k is directed-sups-preserving iff for X being Scott TopAugmentation of Image k
for Y being Scott TopAugmentation of L
for V being Subset of L st V is open Subset of X holds
uparrow V is open Subset of Y )

proof end;

theorem Th32: :: WAYBEL34:32

for L being complete LATTICE
for S being non empty full sups-inheriting SubRelStr of L
for x, y being Element of L
for a, b being Element of S st a = x & b = y & x << y holds
a << b

proof end;

theorem :: WAYBEL34:33

for L being complete LATTICE
for k being kernel Function of L,L st k is directed-sups-preserving holds
for x, y being Element of L
for a, b being Element of (Image k) st a = x & b = y holds
( x << y iff a << b )

proof end;

theorem :: WAYBEL34:34

for L being complete LATTICE
for k being kernel Function of L,L st Image k is continuous & ( for x, y being Element of L
for a, b being Element of (Image k) st a = x & b = y holds
( x << y iff a << b ) ) holds
k is directed-sups-preserving

proof end;

theorem Th35: :: WAYBEL34:35

for L being complete LATTICE
for c being closure Function of L,L holds
( corestr c is sups-preserving & inclusion c is infs-preserving & UpperAdj (corestr c) = inclusion c & LowerAdj (inclusion c) = corestr c )

proof end;

theorem Th36: :: WAYBEL34:36

for L being complete LATTICE
for c being closure Function of L,L holds
( Image c is directed-sups-inheriting iff inclusion c is directed-sups-preserving )

proof end;

theorem :: WAYBEL34:37

for L being complete LATTICE
for c being closure Function of L,L holds
( Image c is directed-sups-inheriting iff for X being Scott TopAugmentation of Image c
for Y being Scott TopAugmentation of L
for f being Function of Y,X st f = c holds
f is open )

proof end;

theorem :: WAYBEL34:38

for L being complete LATTICE
for c being closure Function of L,L st Image c is directed-sups-inheriting holds
corestr c is waybelow-preserving

proof end;

theorem :: WAYBEL34:39

for L being continuous complete LATTICE
for c being closure Function of L,L st corestr c is waybelow-preserving holds
Image c is directed-sups-inheriting

proof end;

::<section3>Duality of subcategories of {\it INF} and {\it SUP}</section3>

definition

let W be non empty set ;
set A = W -INF_category ;
defpred S₁[ set ] means verum;
defpred S₂[ Object of (W -INF_category), Object of (W -INF_category), Morphism of $1,$2] means @ $3 is directed-sups-preserving ;
A1: ex a being Object of (W -INF_category) st S₁[a] ;
A2: for a, b, c being Object of (W -INF_category) st S₁[a] & S₁[b] & S₁[c] & <^a,b^> <> {} & <^b,c^> <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c st S₂[a,b,f] & S₂[b,c,g] holds
S₂[a,c,g * f]

proof end;

A8: for a being Object of (W -INF_category) st S₁[a] holds
S₂[a,a, idm a]

proof end;

func W -INF(SC)_category -> non empty strict subcategory of W -INF_category means :Def10: :: WAYBEL34:def 10
( ( for a being Object of (W -INF_category) holds a is Object of it ) & ( for a, b being Object of (W -INF_category)
for a9, b9 being Object of it st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff @ f is directed-sups-preserving ) ) );

existence
ex b₁ being non empty strict subcategory of W -INF_category st
( ( for a being Object of (W -INF_category) holds a is Object of b₁ ) & ( for a, b being Object of (W -INF_category)
for a9, b9 being Object of b₁ st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff @ f is directed-sups-preserving ) ) )

proof end;

correctness
uniqueness
for b₁, b₂ being non empty strict subcategory of W -INF_category st ( for a being Object of (W -INF_category) holds a is Object of b₁ ) & ( for a, b being Object of (W -INF_category)
for a9, b9 being Object of b₁ st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff @ f is directed-sups-preserving ) ) & ( for a being Object of (W -INF_category) holds a is Object of b₂ ) & ( for a, b being Object of (W -INF_category)
for a9, b9 being Object of b₂ st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff @ f is directed-sups-preserving ) ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def10 defines -INF(SC)_category WAYBEL34:def 10 :
for W being non empty set
for b₂ being non empty strict subcategory of W -INF_category holds
( b₂ = W -INF(SC)_category iff ( ( for a being Object of (W -INF_category) holds a is Object of b₂ ) & ( for a, b being Object of (W -INF_category)
for a9, b9 being Object of b₂ st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff @ f is directed-sups-preserving ) ) ) );

definition

let W be with_non-empty_element set ;
A1: ex w being non empty set st w in W by SETFAM_1:def 10;
set A = W -SUP_category ;
defpred S₁[ set ] means verum;
defpred S₂[ Object of (W -SUP_category), Object of (W -SUP_category), Morphism of $1,$2] means UpperAdj (@ $3) is directed-sups-preserving ;
A2: ex a being Object of (W -SUP_category) st S₁[a] ;
A3: for a, b, c being Object of (W -SUP_category) st S₁[a] & S₁[b] & S₁[c] & <^a,b^> <> {} & <^b,c^> <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c st S₂[a,b,f] & S₂[b,c,g] holds
S₂[a,c,g * f]

proof end;

A12: for a being Object of (W -SUP_category) st S₁[a] holds
S₂[a,a, idm a]

proof end;

func W -SUP(SO)_category -> non empty strict subcategory of W -SUP_category means :Def11: :: WAYBEL34:def 11
( ( for a being Object of (W -SUP_category) holds a is Object of it ) & ( for a, b being Object of (W -SUP_category)
for a9, b9 being Object of it st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff UpperAdj (@ f) is directed-sups-preserving ) ) );

existence
ex b₁ being non empty strict subcategory of W -SUP_category st
( ( for a being Object of (W -SUP_category) holds a is Object of b₁ ) & ( for a, b being Object of (W -SUP_category)
for a9, b9 being Object of b₁ st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff UpperAdj (@ f) is directed-sups-preserving ) ) )

proof end;

correctness
uniqueness
for b₁, b₂ being non empty strict subcategory of W -SUP_category st ( for a being Object of (W -SUP_category) holds a is Object of b₁ ) & ( for a, b being Object of (W -SUP_category)
for a9, b9 being Object of b₁ st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff UpperAdj (@ f) is directed-sups-preserving ) ) & ( for a being Object of (W -SUP_category) holds a is Object of b₂ ) & ( for a, b being Object of (W -SUP_category)
for a9, b9 being Object of b₂ st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff UpperAdj (@ f) is directed-sups-preserving ) ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def11 defines -SUP(SO)_category WAYBEL34:def 11 :
for W being with_non-empty_element set
for b₂ being non empty strict subcategory of W -SUP_category holds
( b₂ = W -SUP(SO)_category iff ( ( for a being Object of (W -SUP_category) holds a is Object of b₂ ) & ( for a, b being Object of (W -SUP_category)
for a9, b9 being Object of b₂ st a9 = a & b9 = b & <^a,b^> <> {} holds
for f being Morphism of a,b holds
( f in <^a9,b9^> iff UpperAdj (@ f) is directed-sups-preserving ) ) ) );

theorem Th40: :: WAYBEL34:40

for S being non empty RelStr
for T being non empty reflexive antisymmetric RelStr
for t being Element of T
for X being non empty Subset of S holds
( S --> t preserves_sup_of X & S --> t preserves_inf_of X )

proof end;

theorem Th41: :: WAYBEL34:41

for S being non empty RelStr
for T being non empty reflexive antisymmetric lower-bounded RelStr holds S --> (Bottom T) is sups-preserving

proof end;

theorem Th42: :: WAYBEL34:42

for S being non empty RelStr
for T being non empty reflexive antisymmetric upper-bounded RelStr holds S --> (Top T) is infs-preserving

proof end;

registration

let S be non empty RelStr ;
let T be non empty reflexive antisymmetric upper-bounded RelStr ;

cluster S --> (Top T) -> infs-preserving directed-sups-preserving ;

coherence
( S --> (Top T) is directed-sups-preserving & S --> (Top T) is infs-preserving ) by Th40, Th42;

end;

registration

let S be non empty RelStr ;
let T be non empty reflexive antisymmetric lower-bounded RelStr ;

cluster S --> (Bottom T) -> sups-preserving filtered-infs-preserving ;

coherence
( S --> (Bottom T) is filtered-infs-preserving & S --> (Bottom T) is sups-preserving ) by Th40, Th41;

end;

registration

let S be non empty RelStr ;
let T be non empty reflexive antisymmetric upper-bounded RelStr ;

cluster Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving directed-sups-preserving for Element of bool [: the carrier of S, the carrier of T:];

existence
ex b₁ being Function of S,T st
( b₁ is directed-sups-preserving & b₁ is infs-preserving )

proof end;

end;

registration

let S be non empty RelStr ;
let T be non empty reflexive antisymmetric lower-bounded RelStr ;

cluster Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving filtered-infs-preserving for Element of bool [: the carrier of S, the carrier of T:];

existence
ex b₁ being Function of S,T st
( b₁ is filtered-infs-preserving & b₁ is sups-preserving )

proof end;

end;

theorem Th43: :: WAYBEL34:43

for W being with_non-empty_element set
for L being LATTICE holds
( L is Object of (W -INF(SC)_category) iff ( L is strict & L is complete & the carrier of L in W ) )

proof end;

theorem Th44: :: WAYBEL34:44

for W being with_non-empty_element set
for a, b being Object of (W -INF(SC)_category)
for f being set holds
( f in <^a,b^> iff f is infs-preserving directed-sups-preserving Function of (latt a),(latt b) )

proof end;

theorem :: WAYBEL34:45

for W being with_non-empty_element set
for L being LATTICE holds
( L is Object of (W -SUP(SO)_category) iff ( L is strict & L is complete & the carrier of L in W ) )

proof end;

theorem Th46: :: WAYBEL34:46

for W being with_non-empty_element set
for a, b being Object of (W -SUP(SO)_category)
for f being set holds
( f in <^a,b^> iff ex g being sups-preserving Function of (latt a),(latt b) st
( g = f & UpperAdj g is directed-sups-preserving ) )

proof end;

theorem :: WAYBEL34:47

for W being with_non-empty_element set holds W -INF(SC)_category = Intersect ((W -INF_category),(W -UPS_category))

proof end;

definition

let W be with_non-empty_element set ;
defpred S₁[ Object of (W -INF(SC)_category)] means latt $1 is continuous ;
consider a being non empty set such that
A1: a in W by SETFAM_1:def 10;
set r = the well-ordering upper-bounded Order of a;
set b = RelStr(# a, the well-ordering upper-bounded Order of a #);

func W -CL_category -> non empty strict full subcategory of W -INF(SC)_category means :Def12: :: WAYBEL34:def 12
for a being Object of (W -INF(SC)_category) holds
( a is Object of it iff latt a is continuous );

existence
ex b₁ being non empty strict full subcategory of W -INF(SC)_category st
for a being Object of (W -INF(SC)_category) holds
( a is Object of b₁ iff latt a is continuous )

proof end;

correctness
uniqueness
for b₁, b₂ being non empty strict full subcategory of W -INF(SC)_category st ( for a being Object of (W -INF(SC)_category) holds
( a is Object of b₁ iff latt a is continuous ) ) & ( for a being Object of (W -INF(SC)_category) holds
( a is Object of b₂ iff latt a is continuous ) ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def12 defines -CL_category WAYBEL34:def 12 :
for W being with_non-empty_element set
for b₂ being non empty strict full subcategory of W -INF(SC)_category holds
( b₂ = W -CL_category iff for a being Object of (W -INF(SC)_category) holds
( a is Object of b₂ iff latt a is continuous ) );

registration

let W be with_non-empty_element set ;

cluster W -CL_category -> non empty strict full with_complete_lattices ;

coherence
W -CL_category is with_complete_lattices ;

end;

theorem Th48: :: WAYBEL34:48

for W being with_non-empty_element set
for L being LATTICE st the carrier of L in W holds
( L is Object of (W -CL_category) iff ( L is strict & L is complete & L is continuous ) )

proof end;

theorem Th49: :: WAYBEL34:49

for W being with_non-empty_element set
for a, b being Object of (W -CL_category)
for f being set holds
( f in <^a,b^> iff f is infs-preserving directed-sups-preserving Function of (latt a),(latt b) )

proof end;

definition

let W be with_non-empty_element set ;
defpred S₁[ Object of (W -SUP(SO)_category)] means latt $1 is continuous ;
consider a being non empty set such that
A1: a in W by SETFAM_1:def 10;
set r = the well-ordering upper-bounded Order of a;
set b = RelStr(# a, the well-ordering upper-bounded Order of a #);

func W -CL-opp_category -> non empty strict full subcategory of W -SUP(SO)_category means :Def13: :: WAYBEL34:def 13
for a being Object of (W -SUP(SO)_category) holds
( a is Object of it iff latt a is continuous );

existence
ex b₁ being non empty strict full subcategory of W -SUP(SO)_category st
for a being Object of (W -SUP(SO)_category) holds
( a is Object of b₁ iff latt a is continuous )

proof end;

correctness
uniqueness
for b₁, b₂ being non empty strict full subcategory of W -SUP(SO)_category st ( for a being Object of (W -SUP(SO)_category) holds
( a is Object of b₁ iff latt a is continuous ) ) & ( for a being Object of (W -SUP(SO)_category) holds
( a is Object of b₂ iff latt a is continuous ) ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def13 defines -CL-opp_category WAYBEL34:def 13 :
for W being with_non-empty_element set
for b₂ being non empty strict full subcategory of W -SUP(SO)_category holds
( b₂ = W -CL-opp_category iff for a being Object of (W -SUP(SO)_category) holds
( a is Object of b₂ iff latt a is continuous ) );

theorem Th50: :: WAYBEL34:50

for W being with_non-empty_element set
for L being LATTICE st the carrier of L in W holds
( L is Object of (W -CL-opp_category) iff ( L is strict & L is complete & L is continuous ) )

proof end;

theorem Th51: :: WAYBEL34:51

for W being with_non-empty_element set
for a, b being Object of (W -CL-opp_category)
for f being set holds
( f in <^a,b^> iff ex g being sups-preserving Function of (latt a),(latt b) st
( g = f & UpperAdj g is directed-sups-preserving ) )

proof end;

:: 1.10. THEOREM, p. 182

theorem Th52: :: WAYBEL34:52

for W being with_non-empty_element set holds W -INF(SC)_category ,W -SUP(SO)_category are_anti-isomorphic_under W LowerAdj

proof end;

theorem :: WAYBEL34:53

for W being with_non-empty_element set holds W -SUP(SO)_category ,W -INF(SC)_category are_anti-isomorphic_under W UpperAdj

proof end;

theorem Th54: :: WAYBEL34:54

for W being with_non-empty_element set holds W -CL_category ,W -CL-opp_category are_anti-isomorphic_under W LowerAdj

proof end;

theorem :: WAYBEL34:55

for W being with_non-empty_element set holds W -CL-opp_category ,W -CL_category are_anti-isomorphic_under W UpperAdj

proof end;

::<section4>Compact preserving maps and sup-semilattices morphisms</section4>

definition

let S, T be non empty reflexive RelStr ;
let f be Function of S,T;

attr f is compact-preserving means :: WAYBEL34:def 14
for s being Element of S st s is compact holds
f . s is compact ;

end;

:: deftheorem defines compact-preserving WAYBEL34:def 14 :
for S, T being non empty reflexive RelStr
for f being Function of S,T holds
( f is compact-preserving iff for s being Element of S st s is compact holds
f . s is compact );

theorem Th56: :: WAYBEL34:56

for S, T being complete LATTICE
for d being sups-preserving Function of T,S st d is waybelow-preserving holds
d is compact-preserving

proof end;

theorem Th57: :: WAYBEL34:57

for S, T being complete LATTICE
for d being sups-preserving Function of T,S st T is algebraic & d is compact-preserving holds
d is waybelow-preserving

proof end;

theorem Th58: :: WAYBEL34:58

for R, S, T being non empty RelStr
for X being Subset of R
for f being Function of R,S
for g being Function of S,T st f preserves_sup_of X & g preserves_sup_of f .: X holds
g * f preserves_sup_of X

proof end;

definition

let S, T be non empty RelStr ;
let f be Function of S,T;

attr f is finite-sups-preserving means :: WAYBEL34:def 15
for X being finite Subset of S holds f preserves_sup_of X;

attr f is bottom-preserving means :: WAYBEL34:def 16
f preserves_sup_of {} S;

end;

:: deftheorem defines finite-sups-preserving WAYBEL34:def 15 :
for S, T being non empty RelStr
for f being Function of S,T holds
( f is finite-sups-preserving iff for X being finite Subset of S holds f preserves_sup_of X );

:: deftheorem defines bottom-preserving WAYBEL34:def 16 :
for S, T being non empty RelStr
for f being Function of S,T holds
( f is bottom-preserving iff f preserves_sup_of {} S );

theorem :: WAYBEL34:59

for R, S, T being non empty RelStr
for f being Function of R,S
for g being Function of S,T st f is finite-sups-preserving & g is finite-sups-preserving holds
g * f is finite-sups-preserving

proof end;

definition

let S, T be non empty antisymmetric lower-bounded RelStr ;
let f be Function of S,T;

redefine attr f is bottom-preserving means :Def17: :: WAYBEL34:def 17
f . (Bottom S) = Bottom T;

compatibility
( f is bottom-preserving iff f . (Bottom S) = Bottom T )

proof end;

end;

:: deftheorem Def17 defines bottom-preserving WAYBEL34:def 17 :
for S, T being non empty antisymmetric lower-bounded RelStr
for f being Function of S,T holds
( f is bottom-preserving iff f . (Bottom S) = Bottom T );

definition

let L be non empty RelStr ;
let S be SubRelStr of L;

attr S is finite-sups-inheriting means :: WAYBEL34:def 18
for X being finite Subset of S st ex_sup_of X,L holds
"\/" (X,L) in the carrier of S;

attr S is bottom-inheriting means :Def19: :: WAYBEL34:def 19
Bottom L in the carrier of S;

end;

:: deftheorem defines finite-sups-inheriting WAYBEL34:def 18 :
for L being non empty RelStr
for S being SubRelStr of L holds
( S is finite-sups-inheriting iff for X being finite Subset of S st ex_sup_of X,L holds
"\/" (X,L) in the carrier of S );

:: deftheorem Def19 defines bottom-inheriting WAYBEL34:def 19 :
for L being non empty RelStr
for S being SubRelStr of L holds
( S is bottom-inheriting iff Bottom L in the carrier of S );

registration

let S, T be non empty RelStr ;

cluster Function-like quasi_total sups-preserving -> bottom-preserving for Element of bool [: the carrier of S, the carrier of T:];

coherence
for b₁ being Function of S,T st b₁ is sups-preserving holds
b₁ is bottom-preserving ;

end;

registration

let L be non empty antisymmetric lower-bounded RelStr ;

cluster finite-sups-inheriting -> join-inheriting bottom-inheriting for SubRelStr of L;

coherence
for b₁ being SubRelStr of L st b₁ is finite-sups-inheriting holds
( b₁ is bottom-inheriting & b₁ is join-inheriting )

proof end;

end;

registration

let L be non empty RelStr ;

cluster sups-inheriting -> finite-sups-inheriting for SubRelStr of L;

coherence
for b₁ being SubRelStr of L st b₁ is sups-inheriting holds
b₁ is finite-sups-inheriting ;

end;

registration

let S, T be non empty lower-bounded Poset;

cluster Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving for Element of bool [: the carrier of S, the carrier of T:];

existence
ex b₁ being Function of S,T st b₁ is sups-preserving

proof end;

end;

registration

let L be non empty antisymmetric lower-bounded RelStr ;

cluster full bottom-inheriting -> non empty lower-bounded full for SubRelStr of L;

coherence
for b₁ being full SubRelStr of L st b₁ is bottom-inheriting holds
( not b₁ is empty & b₁ is lower-bounded )

proof end;

end;

registration

let L be non empty antisymmetric lower-bounded RelStr ;

cluster non empty full sups-inheriting finite-sups-inheriting bottom-inheriting for SubRelStr of L;

existence
ex b₁ being SubRelStr of L st
( not b₁ is empty & b₁ is sups-inheriting & b₁ is finite-sups-inheriting & b₁ is bottom-inheriting & b₁ is full )

proof end;

end;

theorem Th60: :: WAYBEL34:60

for L being non empty antisymmetric lower-bounded RelStr
for S being non empty full bottom-inheriting SubRelStr of L holds Bottom S = Bottom L

proof end;

registration

let L be non empty lower-bounded with_suprema Poset;

cluster full join-inheriting bottom-inheriting -> full finite-sups-inheriting for SubRelStr of L;

coherence
for b₁ being full SubRelStr of L st b₁ is bottom-inheriting & b₁ is join-inheriting holds
b₁ is finite-sups-inheriting

proof end;

end;

theorem :: WAYBEL34:61

for S, T being non empty RelStr
for f being Function of S,T st f is finite-sups-preserving holds
( f is join-preserving & f is bottom-preserving ) ;

theorem Th62: :: WAYBEL34:62

for S, T being lower-bounded with_suprema Poset
for f being Function of S,T st f is join-preserving & f is bottom-preserving holds
f is finite-sups-preserving

proof end;

registration

let S, T be non empty RelStr ;

cluster Function-like quasi_total sups-preserving -> finite-sups-preserving for Element of bool [: the carrier of S, the carrier of T:];

coherence
for b₁ being Function of S,T st b₁ is sups-preserving holds
b₁ is finite-sups-preserving ;

cluster Function-like quasi_total finite-sups-preserving -> join-preserving bottom-preserving for Element of bool [: the carrier of S, the carrier of T:];

coherence
for b₁ being Function of S,T st b₁ is finite-sups-preserving holds
( b₁ is join-preserving & b₁ is bottom-preserving ) ;

end;

registration

let S be non empty RelStr ;
let T be non empty reflexive antisymmetric lower-bounded RelStr ;

cluster Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving finite-sups-preserving for Element of bool [: the carrier of S, the carrier of T:];

existence
ex b₁ being Function of S,T st
( b₁ is sups-preserving & b₁ is finite-sups-preserving )

proof end;

end;

registration

let L be non empty lower-bounded Poset;

cluster CompactSublatt L -> lower-bounded ;

coherence
CompactSublatt L is lower-bounded

proof end;

end;

theorem Th63: :: WAYBEL34:63

for S being RelStr
for T being non empty RelStr
for f being Function of S,T
for S9 being SubRelStr of S
for T9 being SubRelStr of T st f .: the carrier of S9 c= the carrier of T9 holds
f | the carrier of S9 is Function of S9,T9

proof end;

theorem Th64: :: WAYBEL34:64

for S, T being LATTICE
for f being join-preserving Function of S,T
for S9 being non empty full join-inheriting SubRelStr of S
for T9 being non empty full join-inheriting SubRelStr of T
for g being Function of S9,T9 st g = f | the carrier of S9 holds
g is join-preserving

proof end;

theorem Th65: :: WAYBEL34:65

for S, T being lower-bounded LATTICE
for f being finite-sups-preserving Function of S,T
for S9 being non empty full finite-sups-inheriting SubRelStr of S
for T9 being non empty full finite-sups-inheriting SubRelStr of T
for g being Function of S9,T9 st g = f | the carrier of S9 holds
g is finite-sups-preserving

proof end;

registration

let L be complete LATTICE;

cluster CompactSublatt L -> finite-sups-inheriting ;

coherence
CompactSublatt L is finite-sups-inheriting

proof end;

end;

theorem Th66: :: WAYBEL34:66

for S, T being complete LATTICE
for d being sups-preserving Function of T,S holds
( d is compact-preserving iff d | the carrier of (CompactSublatt T) is finite-sups-preserving Function of (CompactSublatt T),(CompactSublatt S) )

proof end;

theorem :: WAYBEL34:67

for S, T being complete LATTICE st T is algebraic holds
for g being infs-preserving Function of S,T holds
( g is directed-sups-preserving iff (LowerAdj g) | the carrier of (CompactSublatt T) is finite-sups-preserving Function of (CompactSublatt T),(CompactSublatt S) )

proof end;