:: Duality Based on Galois Connection. Part I
:: by Grzegorz Bancerek
::
:: 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 ;
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 ;
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
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

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

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:
then reconsider g9 = g as infs-preserving Function of S9,T9 ;
[g9,(LowerAdj g9)] is Galois by Def1;
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
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

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

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:
then reconsider g9 = g as sups-preserving Function of S9,T9 ;
[(UpperAdj g9),g9] is Galois by Def2;
end;

registration
let S, T be complete LATTICE;
let g be infs-preserving Function of S,T;
coherence by Lm1;
end;

registration
let S, T be complete LATTICE;
let d be sups-preserving Function of T,S;
coherence 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 () . t = inf (g " ())
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 () . s = sup (d " ())
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;
coherence
proof end;
end;

registration
let S, T be complete LATTICE;
let d be sups-preserving Function of S,T;
coherence
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 () = g
proof end;

theorem Th11: :: WAYBEL34:11
for S, T being complete LATTICE
for d being sups-preserving Function of S,T holds LowerAdj () = 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 () . (a,b) iff ( a in the carrier of () & b in the carrier of () & 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 () . (a,b) iff ( a in the carrier of () & b in the carrier of () & a is complete & b is complete & f is sups-preserving ) )

proof end;

registration
let W be with_non-empty_element set ;
coherence
proof end;
coherence
proof end;
end;

theorem Th13: :: WAYBEL34:13
for W being with_non-empty_element set
for L being LATTICE holds
( L is Object of () 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 ()
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 () 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 ()
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 () = the carrier of ()
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 () . (a,b) iff ( a in the carrier of () & b in the carrier of () & 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 () . (a,b) iff ( a in the carrier of () & b in the carrier of () & S2[a,b,f] ) ) by Lm4;
A3: for a being LATTICE st a in the carrier of () holds
H1(a) in the carrier of () 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 () holds
H2(a,a, id a) = id H1(a)
let a be LATTICE; :: thesis: ( a in the carrier of () implies H2(a,a, id a) = id H1(a) )
assume a in the carrier of () ; :: 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 () holds it . a = latt a ) & ( for a, b being Object of () 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 () holds b1 . a = latt a ) & ( for a, b being Object of () 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 () holds b1 . a = latt a ) & ( for a, b being Object of () st <^a,b^> <> {} holds
for f being Morphism of a,b holds b1 . f = LowerAdj (@ f) ) & ( for a being Object of () holds b2 . a = latt a ) & ( for a, b being Object of () 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 () holds H1(a) in the carrier of () 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 () holds H3(a,a, id a) = id H1(a) let a be LATTICE; :: thesis: ( a in the carrier of () implies H3(a,a, id a) = id H1(a) ) assume a in the carrier of () ; :: 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 () holds it . a = latt a ) & ( for a, b being Object of () 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 () holds b1 . a = latt a ) & ( for a, b being Object of () 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 () holds b1 . a = latt a ) & ( for a, b being Object of () st <^a,b^> <> {} holds for f being Morphism of a,b holds b1 . f = UpperAdj (@ f) ) & ( for a being Object of () holds b2 . a = latt a ) & ( for a, b being Object of () 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 () holds b2 . a = latt a ) & ( for a, b being Object of () 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 () holds b2 . a = latt a ) & ( for a, b being Object of () 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 ; coherence proof end; coherence proof end; end; theorem Th18: :: WAYBEL34:18 for W being with_non-empty_element set holds ( () " = W UpperAdj & () " = W LowerAdj ) proof end; theorem :: WAYBEL34:19 for W being with_non-empty_element set holds ( () * () = id () & () * () = id () ) proof end; theorem :: WAYBEL34:20 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 (() .: 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 () .: V = (rng ()) /\ (uparrow (() .: 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 (() .: 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; coherence 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 () 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; coherence 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 () = inclusion k & UpperAdj () = 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 () 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 () 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 () = inclusion c & LowerAdj () = 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 (), Object of (), Morphism of$1,$2] means @$3 is directed-sups-preserving ;
A1: ex a being Object of () st S1[a] ;
A2: for a, b, c being Object of () 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 () 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 () holds a is Object of it ) & ( for a, b being Object of ()
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 () holds a is Object of b1 ) & ( for a, b being Object of ()
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 () holds a is Object of b1 ) & ( for a, b being Object of ()
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 () holds a is Object of b2 ) & ( for a, b being Object of ()
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 () holds a is Object of b2 ) & ( for a, b being Object of ()
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 (), Object of (), Morphism of $1,$2] means UpperAdj (@ $3) is directed-sups-preserving ; A2: ex a being Object of () st S1[a] ; A3: for a, b, c being Object of () 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 () 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 () holds a is Object of it ) & ( for a, b being Object of () 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 () holds a is Object of b1 ) & ( for a, b being Object of () 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 () holds a is Object of b1 ) & ( for a, b being Object of () 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 () holds a is Object of b2 ) & ( for a, b being Object of () 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 () holds a is Object of b2 ) & ( for a, b being Object of () 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 --> () 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 ; coherence ( S --> (Top T) is directed-sups-preserving & S --> (Top T) is infs-preserving ) by ; end; registration let S be non empty RelStr ; let T be non empty reflexive antisymmetric lower-bounded RelStr ; coherence ( S --> () is filtered-infs-preserving & S --> () is sups-preserving ) by ; 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 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 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 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 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 ((),()) proof end; definition let W be with_non-empty_element set ; defpred S1[ Object of ] 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 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 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 holds
( a is Object of b1 iff latt a is continuous ) ) & ( for a being Object of 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 holds
( a is Object of b2 iff latt a is continuous ) );

registration
let W be with_non-empty_element set ;
coherence ;
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 () 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 ()
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 ] 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 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 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 holds
( a is Object of b1 iff latt a is continuous ) ) & ( for a being Object of 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 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 () 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 ()
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
proof end;

theorem :: WAYBEL34:53
proof end;

theorem Th54: :: WAYBEL34:54
proof end;

theorem :: WAYBEL34:55
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;
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 T;
compatibility
( f is bottom-preserving iff f . () = 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 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 ;
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 ;
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 ;
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 ;
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;
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;
coherence
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;
coherence
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 () is finite-sups-preserving Function of (),() )
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 () | the carrier of () is finite-sups-preserving Function of (),() )
proof end;