:: WAYBEL15 semantic presentation

theorem Th1: :: WAYBEL15:1
for b1 being RelStr
for b2 being full SubRelStr of b1
for b3 being full SubRelStr of b2 holds
b3 is full SubRelStr of b1
proof end;

theorem Th2: :: WAYBEL15:2
for b1 being 1-sorted
for b2, b3 being non empty 1-sorted
for b4 being Function of b1,b2
for b5 being Function of b2,b3 holds
( b4 is onto & b5 is onto implies b5 * b4 is onto )
proof end;

theorem Th3: :: WAYBEL15:3
for b1 being 1-sorted
for b2 being Subset of b1 holds (id b1) .: b2 = b2 by FUNCT_1:162;

theorem Th4: :: WAYBEL15:4
for b1 being set
for b2 being Element of (BoolePoset b1) holds uparrow b2 = { b3 where B is Subset of b1 : b2 c= b3 }
proof end;

theorem Th5: :: WAYBEL15:5
for b1 being non empty antisymmetric upper-bounded RelStr
for b2 being Element of b1 holds
( Top b1 <= b2 implies b2 = Top b1 )
proof end;

theorem Th6: :: WAYBEL15:6
for b1, b2 being non empty Poset
for b3 being Function of b1,b2
for b4 being Function of b2,b1 holds
( b3 is onto & [b3,b4] is Galois implies b2, Image b4 are_isomorphic )
proof end;

theorem Th7: :: WAYBEL15:7
for b1, b2, b3 being non empty Poset
for b4 being Function of b1,b2
for b5 being Function of b2,b3
for b6 being Function of b2,b1
for b7 being Function of b3,b2 holds
( [b4,b6] is Galois & [b5,b7] is Galois implies [(b5 * b4),(b6 * b7)] is Galois )
proof end;

theorem Th8: :: WAYBEL15:8
for b1, b2 being non empty Poset
for b3 being Function of b1,b2
for b4 being Function of b2,b1 holds
( b4 = b3 " & b3 is isomorphic implies ( [b3,b4] is Galois & [b4,b3] is Galois ) )
proof end;

theorem Th9: :: WAYBEL15:9
for b1 being set holds BoolePoset b1 is arithmetic
proof end;

registration
let c1 be set ;
cluster BoolePoset a1 -> arithmetic ;
coherence
BoolePoset c1 is arithmetic
by Th9;
end;

Lemma8: for b1, b2 being non empty RelStr
for b3 being Function of b1,b2 holds
( b3 is sups-preserving implies b3 is directed-sups-preserving )
proof end;

theorem Th10: :: WAYBEL15:10
for b1, b2 being non empty up-complete Poset
for b3 being Function of b1,b2 holds
( b3 is isomorphic implies for b4 being Element of b1 holds b3 .: (waybelow b4) = waybelow (b3 . b4) )
proof end;

theorem Th11: :: WAYBEL15:11
for b1, b2 being non empty Poset holds
( b1,b2 are_isomorphic & b1 is continuous implies b2 is continuous )
proof end;

theorem Th12: :: WAYBEL15:12
for b1, b2 being LATTICE holds
( b1,b2 are_isomorphic & b1 is lower-bounded & b1 is arithmetic implies b2 is arithmetic )
proof end;

Lemma12: for b1 being lower-bounded LATTICE holds
not ( b1 is continuous & ( for b2 being lower-bounded arithmetic LATTICE
for b3 being Function of b2,b1 holds
not ( b3 is onto & b3 is infs-preserving & b3 is directed-sups-preserving ) ) )
proof end;

theorem Th13: :: WAYBEL15:13
for b1, b2, b3 being non empty Poset
for b4 being Function of b1,b2
for b5 being Function of b2,b3 holds
( b4 is directed-sups-preserving & b5 is directed-sups-preserving implies b5 * b4 is directed-sups-preserving )
proof end;

theorem Th14: :: WAYBEL15:14
for b1, b2 being non empty RelStr
for b3 being Function of b1,b2
for b4 being Subset of (Image b3) holds (inclusion b3) .: b4 = b4 by Th3;

theorem Th15: :: WAYBEL15:15
for b1 being set
for b2 being Function of (BoolePoset b1),(BoolePoset b1) holds
( b2 is idempotent & b2 is directed-sups-preserving implies inclusion b2 is directed-sups-preserving )
proof end;

Lemma16: for b1 being lower-bounded LATTICE holds
not ( ex b2 being lower-bounded algebraic LATTICEex b3 being Function of b2,b1 st
( b3 is onto & b3 is infs-preserving & b3 is directed-sups-preserving ) & ( for b2 being non empty set
for b3 being projection Function of (BoolePoset b2),(BoolePoset b2) holds
not ( b3 is directed-sups-preserving & b1, Image b3 are_isomorphic ) ) )
proof end;

theorem Th16: :: WAYBEL15:16
for b1 being complete continuous LATTICE
for b2 being kernel Function of b1,b1 holds
( b2 is directed-sups-preserving implies Image b2 is continuous LATTICE )
proof end;

theorem Th17: :: WAYBEL15:17
for b1 being complete continuous LATTICE
for b2 being projection Function of b1,b1 holds
( b2 is directed-sups-preserving implies Image b2 is continuous LATTICE )
proof end;

Lemma19: for b1 being LATTICE holds
( ex b2 being set ex b3 being projection Function of (BoolePoset b2),(BoolePoset b2) st
( b3 is directed-sups-preserving & b1, Image b3 are_isomorphic ) implies b1 is continuous )
proof end;

theorem Th18: :: WAYBEL15:18
for b1 being lower-bounded LATTICE holds
( b1 is continuous iff ex b2 being lower-bounded arithmetic LATTICEex b3 being Function of b2,b1 st
( b3 is onto & b3 is infs-preserving & b3 is directed-sups-preserving ) )
proof end;

theorem Th19: :: WAYBEL15:19
for b1 being lower-bounded LATTICE holds
( b1 is continuous iff ex b2 being lower-bounded algebraic LATTICEex b3 being Function of b2,b1 st
( b3 is onto & b3 is infs-preserving & b3 is directed-sups-preserving ) )
proof end;

theorem Th20: :: WAYBEL15:20
for b1 being lower-bounded LATTICE holds
( not ( b1 is continuous & ( for b2 being non empty set
for b3 being projection Function of (BoolePoset b2),(BoolePoset b2) holds
not ( b3 is directed-sups-preserving & b1, Image b3 are_isomorphic ) ) ) & ( ex b2 being set ex b3 being projection Function of (BoolePoset b2),(BoolePoset b2) st
( b3 is directed-sups-preserving & b1, Image b3 are_isomorphic ) implies b1 is continuous ) )
proof end;

theorem Th21: :: WAYBEL15:21
for b1 being non empty RelStr
for b2 being Element of b1 holds
( b2 in PRIME (b1 opp ) iff b2 is co-prime )
proof end;

definition
let c1 be non empty RelStr ;
let c2 be Element of c1;
attr a2 is atom means :Def1: :: WAYBEL15:def 1
( Bottom a1 < a2 & ( for b1 being Element of a1 holds
( Bottom a1 < b1 & b1 <= a2 implies b1 = a2 ) ) );
end;

:: deftheorem Def1 defines atom WAYBEL15:def 1 :
for b1 being non empty RelStr
for b2 being Element of b1 holds
( b2 is atom iff ( Bottom b1 < b2 & ( for b3 being Element of b1 holds
( Bottom b1 < b3 & b3 <= b2 implies b3 = b2 ) ) ) );

definition
let c1 be non empty RelStr ;
func ATOM c1 -> Subset of a1 means :Def2: :: WAYBEL15:def 2
for b1 being Element of a1 holds
( b1 in a2 iff b1 is atom );
existence
ex b1 being Subset of c1 st
for b2 being Element of c1 holds
( b2 in b1 iff b2 is atom )
proof end;
uniqueness
for b1, b2 being Subset of c1 holds
( ( for b3 being Element of c1 holds
( b3 in b1 iff b3 is atom ) ) & ( for b3 being Element of c1 holds
( b3 in b2 iff b3 is atom ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def2 defines ATOM WAYBEL15:def 2 :
for b1 being non empty RelStr
for b2 being Subset of b1 holds
( b2 = ATOM b1 iff for b3 being Element of b1 holds
( b3 in b2 iff b3 is atom ) );

theorem Th22: :: WAYBEL15:22
canceled;

theorem Th23: :: WAYBEL15:23
for b1 being Boolean LATTICE
for b2 being Element of b1 holds
( b2 is atom iff ( b2 is co-prime & b2 <> Bottom b1 ) )
proof end;

registration
let c1 be Boolean LATTICE;
cluster atom -> co-prime Element of the carrier of a1;
coherence
for b1 being Element of c1 holds
( b1 is atom implies b1 is co-prime )
by Th23;
end;

theorem Th24: :: WAYBEL15:24
for b1 being Boolean LATTICE holds ATOM b1 = (PRIME (b1 opp )) \ {(Bottom b1)}
proof end;

theorem Th25: :: WAYBEL15:25
for b1 being Boolean LATTICE
for b2, b3 being Element of b1 holds
( b3 is atom implies ( b3 <= b2 iff not b3 <= 'not' b2 ) )
proof end;

theorem Th26: :: WAYBEL15:26
for b1 being complete Boolean LATTICE
for b2 being Subset of b1
for b3 being Element of b1 holds b3 "/\" (sup b2) = "\/" { (b3 "/\" b4) where B is Element of b1 : b4 in b2 } ,b1
proof end;

theorem Th27: :: WAYBEL15:27
for b1 being non empty antisymmetric with_infima lower-bounded RelStr
for b2, b3 being Element of b1 holds
( b2 is atom & b3 is atom & b2 <> b3 implies b2 "/\" b3 = Bottom b1 )
proof end;

theorem Th28: :: WAYBEL15:28
for b1 being complete Boolean LATTICE
for b2 being Element of b1
for b3 being Subset of b1 holds
( b3 c= ATOM b1 implies ( b2 in b3 iff ( b2 is atom & b2 <= sup b3 ) ) )
proof end;

theorem Th29: :: WAYBEL15:29
for b1 being complete Boolean LATTICE
for b2, b3 being Subset of b1 holds
( b2 c= ATOM b1 & b3 c= ATOM b1 implies ( b2 c= b3 iff sup b2 <= sup b3 ) )
proof end;

Lemma27: for b1 being Boolean LATTICE holds
( ex b2 being set st b1, BoolePoset b2 are_isomorphic implies b1 is arithmetic )
proof end;

Lemma28: for b1 being Boolean LATTICE holds
( b1 is continuous implies b1 opp is continuous )
proof end;

Lemma29: for b1 being Boolean LATTICE holds
( ( b1 is continuous & b1 opp is continuous ) iff b1 is completely-distributive )
proof end;

Lemma30: for b1 being Boolean LATTICE holds
( b1 is completely-distributive implies ( b1 is complete & ( for b2 being Element of b1 holds
ex b3 being Subset of b1 st
( b3 c= ATOM b1 & b2 = sup b3 ) ) ) )
proof end;

Lemma31: for b1 being Boolean LATTICE holds
not ( b1 is complete & ( for b2 being Element of b1 holds
ex b3 being Subset of b1 st
( b3 c= ATOM b1 & b2 = sup b3 ) ) & ( for b2 being set holds
not b1, BoolePoset b2 are_isomorphic ) )
proof end;

theorem Th30: :: WAYBEL15:30
for b1 being Boolean LATTICE holds
( b1 is arithmetic iff ex b2 being set st b1, BoolePoset b2 are_isomorphic )
proof end;

theorem Th31: :: WAYBEL15:31
for b1 being Boolean LATTICE holds
( b1 is arithmetic iff b1 is algebraic )
proof end;

theorem Th32: :: WAYBEL15:32
for b1 being Boolean LATTICE holds
( b1 is arithmetic iff b1 is continuous )
proof end;

theorem Th33: :: WAYBEL15:33
for b1 being Boolean LATTICE holds
( b1 is arithmetic iff ( b1 is continuous & b1 opp is continuous ) )
proof end;

theorem Th34: :: WAYBEL15:34
for b1 being Boolean LATTICE holds
( b1 is arithmetic iff b1 is completely-distributive )
proof end;

theorem Th35: :: WAYBEL15:35
for b1 being Boolean LATTICE holds
( b1 is arithmetic iff ( b1 is complete & ( for b2 being Element of b1 holds
ex b3 being Subset of b1 st
( b3 c= ATOM b1 & b2 = sup b3 ) ) ) )
proof end;