:: WAYBEL34 semantic presentation
theorem Th1: :: WAYBEL34:1
definition
let c
1, c
2 be
LATTICE;
let c
3 be
Function of c
1,c
2;
assume
( c
1 is
complete & c
2 is
complete & c
3 is
infs-preserving )
;
then E2:
c
3 has_a_lower_adjoint
by WAYBEL_1:17;
func LowerAdj c
3 -> Function of a
2,a
1 means :
Def1:
:: WAYBEL34:def 1
[a3,a4] is
Galois;
uniqueness
for b1, b2 being Function of c2,c1 holds
( [c3,b1] is Galois & [c3,b2] is Galois implies b1 = b2 )
existence
ex b1 being Function of c2,c1 st [c3,b1] is Galois
by E2, WAYBEL_1:def 11;
end;
:: deftheorem Def1 defines LowerAdj WAYBEL34:def 1 :
definition
let c
1, c
2 be
LATTICE;
let c
3 be
Function of c
2,c
1;
assume
( c
1 is
complete & c
2 is
complete & c
3 is
sups-preserving )
;
then E3:
c
3 has_an_upper_adjoint
by WAYBEL_1:18;
func UpperAdj c
3 -> Function of a
1,a
2 means :
Def2:
:: WAYBEL34:def 2
[a4,a3] is
Galois;
existence
ex b1 being Function of c1,c2 st [b1,c3] is Galois
by E3, WAYBEL_1:def 12;
correctness
uniqueness
for b1, b2 being Function of c1,c2 holds
( [b1,c3] is Galois & [b2,c3] is Galois implies b1 = b2 );
end;
:: deftheorem Def2 defines UpperAdj WAYBEL34:def 2 :
theorem Th2: :: WAYBEL34:2
theorem Th3: :: WAYBEL34:3
:: deftheorem Def3 defines opp WAYBEL34:def 3 :
theorem Th4: :: WAYBEL34:4
theorem Th5: :: WAYBEL34:5
theorem Th6: :: WAYBEL34:6
theorem Th7: :: WAYBEL34:7
theorem Th8: :: WAYBEL34:8
theorem Th9: :: WAYBEL34:9
theorem Th10: :: WAYBEL34:10
theorem Th11: :: WAYBEL34:11
theorem Th12: :: WAYBEL34:12
definition
let c
1 be non
empty set ;
defpred S
1[
LATTICE] means a
1 is
complete;
defpred S
2[
LATTICE,
LATTICE,
Function of a
1,a
2] means a
3 is
infs-preserving;
given c
2 being
Element of c
1 such that E13:
not c
2 is
empty
;
func c
1 -INF_category -> strict lattice-wise category means :
Def4:
:: WAYBEL34:def 4
( ( for b
1 being
LATTICE holds
( b
1 is
object of a
2 iff ( b
1 is
strict & b
1 is
complete & the
carrier of b
1 in a
1 ) ) ) & ( for b
1, b
2 being
object of a
2for b
3 being
monotone Function of
(latt b1),
(latt b2) holds
( b
3 in <^b1,b2^> iff b
3 is
infs-preserving ) ) );
existence
ex b1 being strict lattice-wise category st
( ( for b2 being LATTICE holds
( b2 is object of b1 iff ( b2 is strict & b2 is complete & the carrier of b2 in c1 ) ) ) & ( for b2, b3 being object of b1
for b4 being monotone Function of (latt b2),(latt b3) holds
( b4 in <^b2,b3^> iff b4 is infs-preserving ) ) )
uniqueness
for b1, b2 being strict lattice-wise category holds
( ( for b3 being LATTICE holds
( b3 is object of b1 iff ( b3 is strict & b3 is complete & the carrier of b3 in c1 ) ) ) & ( for b3, b4 being object of b1
for b5 being monotone Function of (latt b3),(latt b4) holds
( b5 in <^b3,b4^> iff b5 is infs-preserving ) ) & ( for b3 being LATTICE holds
( b3 is object of b2 iff ( b3 is strict & b3 is complete & the carrier of b3 in c1 ) ) ) & ( for b3, b4 being object of b2
for b5 being monotone Function of (latt b3),(latt b4) holds
( b5 in <^b3,b4^> iff b5 is infs-preserving ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines -INF_category WAYBEL34:def 4 :
Lemma14:
for b1 being with_non-empty_element set
for b2, b3 being LATTICE
for b4 being Function of b2,b3 holds
( b4 in the Arrows of (b1 -INF_category ) . b2,b3 iff ( b2 in the carrier of (b1 -INF_category ) & b3 in the carrier of (b1 -INF_category ) & b2 is complete & b3 is complete & b4 is infs-preserving ) )
definition
let c
1 be non
empty set ;
defpred S
1[
LATTICE] means a
1 is
complete;
defpred S
2[
LATTICE,
LATTICE,
Function of a
1,a
2] means a
3 is
sups-preserving;
given c
2 being
Element of c
1 such that E15:
not c
2 is
empty
;
func c
1 -SUP_category -> strict lattice-wise category means :
Def5:
:: WAYBEL34:def 5
( ( for b
1 being
LATTICE holds
( b
1 is
object of a
2 iff ( b
1 is
strict & b
1 is
complete & the
carrier of b
1 in a
1 ) ) ) & ( for b
1, b
2 being
object of a
2for b
3 being
monotone Function of
(latt b1),
(latt b2) holds
( b
3 in <^b1,b2^> iff b
3 is
sups-preserving ) ) );
existence
ex b1 being strict lattice-wise category st
( ( for b2 being LATTICE holds
( b2 is object of b1 iff ( b2 is strict & b2 is complete & the carrier of b2 in c1 ) ) ) & ( for b2, b3 being object of b1
for b4 being monotone Function of (latt b2),(latt b3) holds
( b4 in <^b2,b3^> iff b4 is sups-preserving ) ) )
uniqueness
for b1, b2 being strict lattice-wise category holds
( ( for b3 being LATTICE holds
( b3 is object of b1 iff ( b3 is strict & b3 is complete & the carrier of b3 in c1 ) ) ) & ( for b3, b4 being object of b1
for b5 being monotone Function of (latt b3),(latt b4) holds
( b5 in <^b3,b4^> iff b5 is sups-preserving ) ) & ( for b3 being LATTICE holds
( b3 is object of b2 iff ( b3 is strict & b3 is complete & the carrier of b3 in c1 ) ) ) & ( for b3, b4 being object of b2
for b5 being monotone Function of (latt b3),(latt b4) holds
( b5 in <^b3,b4^> iff b5 is sups-preserving ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines -SUP_category WAYBEL34:def 5 :
Lemma16:
for b1 being with_non-empty_element set
for b2, b3 being LATTICE
for b4 being Function of b2,b3 holds
( b4 in the Arrows of (b1 -SUP_category ) . b2,b3 iff ( b2 in the carrier of (b1 -SUP_category ) & b3 in the carrier of (b1 -SUP_category ) & b2 is complete & b3 is complete & b4 is sups-preserving ) )
theorem Th13: :: WAYBEL34:13
theorem Th14: :: WAYBEL34:14
theorem Th15: :: WAYBEL34:15
theorem Th16: :: WAYBEL34:16
theorem Th17: :: WAYBEL34:17
definition
let c
1 be
with_non-empty_element set ;
set c
2 = c
1 -INF_category ;
set c
3 = c
1 -SUP_category ;
deffunc H
1(
LATTICE)
-> LATTICE = a
1;
deffunc H
2(
LATTICE,
LATTICE,
Function of a
1,a
2)
-> Function of a
2,a
1 =
LowerAdj a
3;
defpred S
1[
LATTICE,
LATTICE,
Function of a
1,a
2] means ( a
1 is
complete & a
2 is
complete & a
3 is
infs-preserving );
defpred S
2[
LATTICE,
LATTICE,
Function of a
1,a
2] means ( a
1 is
complete & a
2 is
complete & a
3 is
sups-preserving );
E22:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( b
3 in the
Arrows of
(c1 -INF_category ) . b
1,b
2 iff ( b
1 in the
carrier of
(c1 -INF_category ) & b
2 in the
carrier of
(c1 -INF_category ) & S
1[b
1,b
2,b
3] ) )
by Lemma14;
E23:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( b
3 in the
Arrows of
(c1 -SUP_category ) . b
1,b
2 iff ( b
1 in the
carrier of
(c1 -SUP_category ) & b
2 in the
carrier of
(c1 -SUP_category ) & S
2[b
1,b
2,b
3] ) )
by Lemma16;
E24:
for b
1 being
LATTICE holds
( b
1 in the
carrier of
(c1 -INF_category ) implies H
1(b
1)
in the
carrier of
(c1 -SUP_category ) )
by Th17;
E25:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( S
1[b
1,b
2,b
3] implies ( H
2(b
1,b
2,b
3) is
Function of H
1(b
2),H
1(b
1) & S
2[H
1(b
2),H
1(b
1),H
2(b
1,b
2,b
3)] ) )
by Lemma4;
E27:
for b
1, b
2, b
3 being
LATTICEfor b
4 being
Function of b
1,b
2for b
5 being
Function of b
2,b
3 holds
( S
1[b
1,b
2,b
4] & S
1[b
2,b
3,b
5] implies H
2(b
1,b
3,b
5 * b
4)
= H
2(b
1,b
2,b
4)
* H
2(b
2,b
3,b
5) )
by Th8;
func c
1 LowerAdj -> strict contravariant Functor of a
1 -INF_category ,a
1 -SUP_category means :
Def6:
:: WAYBEL34:def 6
( ( for b
1 being
object of
(a1 -INF_category ) holds a
2 . b
1 = latt b
1 ) & ( for b
1, b
2 being
object of
(a1 -INF_category ) holds
(
<^b1,b2^> <> {} implies for b
3 being
Morphism of b
1,b
2 holds a
2 . b
3 = LowerAdj (@ b3) ) ) );
existence
ex b1 being strict contravariant Functor of c1 -INF_category ,c1 -SUP_category st
( ( for b2 being object of (c1 -INF_category ) holds b1 . b2 = latt b2 ) & ( for b2, b3 being object of (c1 -INF_category ) holds
( <^b2,b3^> <> {} implies for b4 being Morphism of b2,b3 holds b1 . b4 = LowerAdj (@ b4) ) ) )
uniqueness
for b1, b2 being strict contravariant Functor of c1 -INF_category ,c1 -SUP_category holds
( ( for b3 being object of (c1 -INF_category ) holds b1 . b3 = latt b3 ) & ( for b3, b4 being object of (c1 -INF_category ) holds
( <^b3,b4^> <> {} implies for b5 being Morphism of b3,b4 holds b1 . b5 = LowerAdj (@ b5) ) ) & ( for b3 being object of (c1 -INF_category ) holds b2 . b3 = latt b3 ) & ( for b3, b4 being object of (c1 -INF_category ) holds
( <^b3,b4^> <> {} implies for b5 being Morphism of b3,b4 holds b2 . b5 = LowerAdj (@ b5) ) ) implies b1 = b2 )
deffunc H
3(
LATTICE,
LATTICE,
Function of a
1,a
2)
-> Function of a
2,a
1 =
UpperAdj a
3;
E28:
for b
1 being
LATTICE holds
( b
1 in the
carrier of
(c1 -SUP_category ) implies H
1(b
1)
in the
carrier of
(c1 -INF_category ) )
by Th17;
E29:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( S
2[b
1,b
2,b
3] implies ( H
3(b
1,b
2,b
3) is
Function of H
1(b
2),H
1(b
1) & S
1[H
1(b
2),H
1(b
1),H
3(b
1,b
2,b
3)] ) )
by Lemma5;
E31:
for b
1, b
2, b
3 being
LATTICEfor b
4 being
Function of b
1,b
2for b
5 being
Function of b
2,b
3 holds
( S
2[b
1,b
2,b
4] & S
2[b
2,b
3,b
5] implies H
3(b
1,b
3,b
5 * b
4)
= H
3(b
1,b
2,b
4)
* H
3(b
2,b
3,b
5) )
by Th9;
func c
1 UpperAdj -> strict contravariant Functor of a
1 -SUP_category ,a
1 -INF_category means :
Def7:
:: WAYBEL34:def 7
( ( for b
1 being
object of
(a1 -SUP_category ) holds a
2 . b
1 = latt b
1 ) & ( for b
1, b
2 being
object of
(a1 -SUP_category ) holds
(
<^b1,b2^> <> {} implies for b
3 being
Morphism of b
1,b
2 holds a
2 . b
3 = UpperAdj (@ b3) ) ) );
existence
ex b1 being strict contravariant Functor of c1 -SUP_category ,c1 -INF_category st
( ( for b2 being object of (c1 -SUP_category ) holds b1 . b2 = latt b2 ) & ( for b2, b3 being object of (c1 -SUP_category ) holds
( <^b2,b3^> <> {} implies for b4 being Morphism of b2,b3 holds b1 . b4 = UpperAdj (@ b4) ) ) )
uniqueness
for b1, b2 being strict contravariant Functor of c1 -SUP_category ,c1 -INF_category holds
( ( for b3 being object of (c1 -SUP_category ) holds b1 . b3 = latt b3 ) & ( for b3, b4 being object of (c1 -SUP_category ) holds
( <^b3,b4^> <> {} implies for b5 being Morphism of b3,b4 holds b1 . b5 = UpperAdj (@ b5) ) ) & ( for b3 being object of (c1 -SUP_category ) holds b2 . b3 = latt b3 ) & ( for b3, b4 being object of (c1 -SUP_category ) holds
( <^b3,b4^> <> {} implies for b5 being Morphism of b3,b4 holds b2 . b5 = UpperAdj (@ b5) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines LowerAdj WAYBEL34:def 6 :
:: deftheorem Def7 defines UpperAdj WAYBEL34:def 7 :
theorem Th18: :: WAYBEL34:18
theorem Th19: :: WAYBEL34:19
theorem Th20: :: WAYBEL34:20
theorem Th21: :: WAYBEL34:21
canceled;
theorem Th22: :: WAYBEL34:22
canceled;
theorem Th23: :: WAYBEL34:23
:: deftheorem Def8 defines waybelow-preserving WAYBEL34:def 8 :
theorem Th24: :: WAYBEL34:24
theorem Th25: :: WAYBEL34:25
:: deftheorem Def9 defines relatively_open WAYBEL34:def 9 :
theorem Th26: :: WAYBEL34:26
theorem Th27: :: WAYBEL34:27
theorem Th28: :: WAYBEL34:28
theorem Th29: :: WAYBEL34:29
theorem Th30: :: WAYBEL34:30
theorem Th31: :: WAYBEL34:31
theorem Th32: :: WAYBEL34:32
theorem Th33: :: WAYBEL34:33
theorem Th34: :: WAYBEL34:34
theorem Th35: :: WAYBEL34:35
theorem Th36: :: WAYBEL34:36
theorem Th37: :: WAYBEL34:37
theorem Th38: :: WAYBEL34:38
theorem Th39: :: WAYBEL34:39
theorem Th40: :: WAYBEL34:40
theorem Th41: :: WAYBEL34:41
theorem Th42: :: WAYBEL34:42
theorem Th43: :: WAYBEL34:43
definition
let c
1 be non
empty set ;
set c
2 = c
1 -INF_category ;
defpred S
1[
set ] means verum;
defpred S
2[
object of
(c1 -INF_category ),
object of
(c1 -INF_category ),
Morphism of a
1,a
2] means
@ a
3 is
directed-sups-preserving;
E39:
ex b
1 being
object of
(c1 -INF_category ) st
S
1[b
1]
;
E40:
for b
1, b
2, b
3 being
object of
(c1 -INF_category ) holds
( S
1[b
1] & S
1[b
2] & S
1[b
3] &
<^b1,b2^> <> {} &
<^b2,b3^> <> {} implies for b
4 being
Morphism of b
1,b
2for b
5 being
Morphism of b
2,b
3 holds
( S
2[b
1,b
2,b
4] & S
2[b
2,b
3,b
5] implies S
2[b
1,b
3,b
5 * b
4] ) )
E41:
for b
1 being
object of
(c1 -INF_category ) holds
( S
1[b
1] implies S
2[b
1,b
1,
idm b
1] )
func c
1 -INF(SC)_category -> non
empty strict subcategory of a
1 -INF_category means :
Def10:
:: WAYBEL34:def 10
( ( for b
1 being
object of
(a1 -INF_category ) holds
b
1 is
object of a
2 ) & ( for b
1, b
2 being
object of
(a1 -INF_category )for b
3, b
4 being
object of a
2 holds
( b
3 = b
1 & b
4 = b
2 &
<^b1,b2^> <> {} implies for b
5 being
Morphism of b
1,b
2 holds
( b
5 in <^b3,b4^> iff
@ b
5 is
directed-sups-preserving ) ) ) );
existence
ex b1 being non empty strict subcategory of c1 -INF_category st
( ( for b2 being object of (c1 -INF_category ) holds
b2 is object of b1 ) & ( for b2, b3 being object of (c1 -INF_category )
for b4, b5 being object of b1 holds
( b4 = b2 & b5 = b3 & <^b2,b3^> <> {} implies for b6 being Morphism of b2,b3 holds
( b6 in <^b4,b5^> iff @ b6 is directed-sups-preserving ) ) ) )
correctness
uniqueness
for b1, b2 being non empty strict subcategory of c1 -INF_category holds
( ( for b3 being object of (c1 -INF_category ) holds
b3 is object of b1 ) & ( for b3, b4 being object of (c1 -INF_category )
for b5, b6 being object of b1 holds
( b5 = b3 & b6 = b4 & <^b3,b4^> <> {} implies for b7 being Morphism of b3,b4 holds
( b7 in <^b5,b6^> iff @ b7 is directed-sups-preserving ) ) ) & ( for b3 being object of (c1 -INF_category ) holds
b3 is object of b2 ) & ( for b3, b4 being object of (c1 -INF_category )
for b5, b6 being object of b2 holds
( b5 = b3 & b6 = b4 & <^b3,b4^> <> {} implies for b7 being Morphism of b3,b4 holds
( b7 in <^b5,b6^> iff @ b7 is directed-sups-preserving ) ) ) implies b1 = b2 );
end;
:: deftheorem Def10 defines -INF(SC)_category WAYBEL34:def 10 :
definition
let c
1 be
with_non-empty_element set ;
E40:
ex b
1 being non
empty set st b
1 in c
1
by SETFAM_1:def 11;
set c
2 = c
1 -SUP_category ;
defpred S
1[
set ] means verum;
defpred S
2[
object of
(c1 -SUP_category ),
object of
(c1 -SUP_category ),
Morphism of a
1,a
2] means
UpperAdj (@ a3) is
directed-sups-preserving;
E41:
ex b
1 being
object of
(c1 -SUP_category ) st
S
1[b
1]
;
E42:
for b
1, b
2, b
3 being
object of
(c1 -SUP_category ) holds
( S
1[b
1] & S
1[b
2] & S
1[b
3] &
<^b1,b2^> <> {} &
<^b2,b3^> <> {} implies for b
4 being
Morphism of b
1,b
2for b
5 being
Morphism of b
2,b
3 holds
( S
2[b
1,b
2,b
4] & S
2[b
2,b
3,b
5] implies S
2[b
1,b
3,b
5 * b
4] ) )
E43:
for b
1 being
object of
(c1 -SUP_category ) holds
( S
1[b
1] implies S
2[b
1,b
1,
idm b
1] )
func c
1 -SUP(SO)_category -> non
empty strict subcategory of a
1 -SUP_category means :
Def11:
:: WAYBEL34:def 11
( ( for b
1 being
object of
(a1 -SUP_category ) holds
b
1 is
object of a
2 ) & ( for b
1, b
2 being
object of
(a1 -SUP_category )for b
3, b
4 being
object of a
2 holds
( b
3 = b
1 & b
4 = b
2 &
<^b1,b2^> <> {} implies for b
5 being
Morphism of b
1,b
2 holds
( b
5 in <^b3,b4^> iff
UpperAdj (@ b5) is
directed-sups-preserving ) ) ) );
existence
ex b1 being non empty strict subcategory of c1 -SUP_category st
( ( for b2 being object of (c1 -SUP_category ) holds
b2 is object of b1 ) & ( for b2, b3 being object of (c1 -SUP_category )
for b4, b5 being object of b1 holds
( b4 = b2 & b5 = b3 & <^b2,b3^> <> {} implies for b6 being Morphism of b2,b3 holds
( b6 in <^b4,b5^> iff UpperAdj (@ b6) is directed-sups-preserving ) ) ) )
correctness
uniqueness
for b1, b2 being non empty strict subcategory of c1 -SUP_category holds
( ( for b3 being object of (c1 -SUP_category ) holds
b3 is object of b1 ) & ( for b3, b4 being object of (c1 -SUP_category )
for b5, b6 being object of b1 holds
( b5 = b3 & b6 = b4 & <^b3,b4^> <> {} implies for b7 being Morphism of b3,b4 holds
( b7 in <^b5,b6^> iff UpperAdj (@ b7) is directed-sups-preserving ) ) ) & ( for b3 being object of (c1 -SUP_category ) holds
b3 is object of b2 ) & ( for b3, b4 being object of (c1 -SUP_category )
for b5, b6 being object of b2 holds
( b5 = b3 & b6 = b4 & <^b3,b4^> <> {} implies for b7 being Morphism of b3,b4 holds
( b7 in <^b5,b6^> iff UpperAdj (@ b7) is directed-sups-preserving ) ) ) implies b1 = b2 );
end;
:: deftheorem Def11 defines -SUP(SO)_category WAYBEL34:def 11 :
theorem Th44: :: WAYBEL34:44
theorem Th45: :: WAYBEL34:45
theorem Th46: :: WAYBEL34:46
theorem Th47: :: WAYBEL34:47
theorem Th48: :: WAYBEL34:48
theorem Th49: :: WAYBEL34:49
theorem Th50: :: WAYBEL34:50
theorem Th51: :: WAYBEL34:51
:: deftheorem Def12 defines -CL_category WAYBEL34:def 12 :
theorem Th52: :: WAYBEL34:52
theorem Th53: :: WAYBEL34:53
:: deftheorem Def13 defines -CL-opp_category WAYBEL34:def 13 :
theorem Th54: :: WAYBEL34:54
theorem Th55: :: WAYBEL34:55
theorem Th56: :: WAYBEL34:56
theorem Th57: :: WAYBEL34:57
theorem Th58: :: WAYBEL34:58
theorem Th59: :: WAYBEL34:59
:: deftheorem Def14 defines compact-preserving WAYBEL34:def 14 :
theorem Th60: :: WAYBEL34:60
theorem Th61: :: WAYBEL34:61
theorem Th62: :: WAYBEL34:62
:: deftheorem Def15 defines finite-sups-preserving WAYBEL34:def 15 :
:: deftheorem Def16 defines bottom-preserving WAYBEL34:def 16 :
theorem Th63: :: WAYBEL34:63
:: deftheorem Def17 defines bottom-preserving WAYBEL34:def 17 :
:: deftheorem Def18 defines finite-sups-inheriting WAYBEL34:def 18 :
:: deftheorem Def19 defines bottom-inheriting WAYBEL34:def 19 :
theorem Th64: :: WAYBEL34:64
theorem Th65: :: WAYBEL34:65
theorem Th66: :: WAYBEL34:66
theorem Th67: :: WAYBEL34:67
theorem Th68: :: WAYBEL34:68
theorem Th69: :: WAYBEL34:69
theorem Th70: :: WAYBEL34:70
theorem Th71: :: WAYBEL34:71