:: Algebraic and Arithmetic Lattices
:: by Robert Milewski
::
:: Copyright (c) 1997-2019 Association of Mizar Users

theorem Th1: :: WAYBEL13:1
for L being non empty reflexive transitive RelStr
for x, y being Element of L st x <= y holds
compactbelow x c= compactbelow y
proof end;

theorem Th2: :: WAYBEL13:2
for L being non empty reflexive RelStr
for x being Element of L holds compactbelow x is Subset of ()
proof end;

theorem Th3: :: WAYBEL13:3
for L being RelStr
for S being SubRelStr of L
for X being Subset of S holds X is Subset of L
proof end;

theorem Th4: :: WAYBEL13:4
for L being non empty reflexive transitive with_suprema RelStr holds the carrier of L is Ideal of L
proof end;

theorem Th5: :: WAYBEL13:5
for L1 being non empty reflexive antisymmetric lower-bounded RelStr
for L2 being non empty reflexive antisymmetric RelStr st RelStr(# the carrier of L1, the InternalRel of L1 #) = RelStr(# the carrier of L2, the InternalRel of L2 #) & L1 is up-complete holds
the carrier of () = the carrier of ()
proof end;

theorem Th6: :: WAYBEL13:6
for L being lower-bounded algebraic LATTICE
for S being CLSubFrame of L holds S is algebraic
proof end;

theorem Th7: :: WAYBEL13:7
for X, E being set
for L being CLSubFrame of BoolePoset X holds
( E in the carrier of () iff ex F being Element of () st
( F is finite & E = meet { Y where Y is Element of L : F c= Y } & F c= E ) )
proof end;

theorem Th8: :: WAYBEL13:8
for L being lower-bounded sup-Semilattice holds InclPoset (Ids L) is CLSubFrame of BoolePoset the carrier of L
proof end;

registration
let L be non empty reflexive transitive RelStr ;
cluster non empty directed lower principal for Element of K32( the carrier of L);
existence
ex b1 being Ideal of L st b1 is principal
proof end;
end;

theorem Th9: :: WAYBEL13:9
for L being lower-bounded sup-Semilattice
for X being non empty directed Subset of (InclPoset (Ids L)) holds sup X = union X
proof end;

theorem Th10: :: WAYBEL13:10
for S being lower-bounded sup-Semilattice holds InclPoset (Ids S) is algebraic
proof end;

theorem Th11: :: WAYBEL13:11
for S being lower-bounded sup-Semilattice
for x being Element of (InclPoset (Ids S)) holds
( x is compact iff x is principal Ideal of S )
proof end;

theorem Th12: :: WAYBEL13:12
for S being lower-bounded sup-Semilattice
for x being Element of (InclPoset (Ids S)) holds
( x is compact iff ex a being Element of S st x = downarrow a )
proof end;

theorem :: WAYBEL13:13
for L being lower-bounded sup-Semilattice
for f being Function of L,(CompactSublatt (InclPoset (Ids L))) st ( for x being Element of L holds f . x = downarrow x ) holds
f is isomorphic
proof end;

theorem :: WAYBEL13:14
for S being lower-bounded LATTICE holds InclPoset (Ids S) is arithmetic
proof end;

theorem Th15: :: WAYBEL13:15
proof end;

theorem Th16: :: WAYBEL13:16
for L being lower-bounded algebraic sup-Semilattice
for f being Function of L,(InclPoset ()) st ( for x being Element of L holds f . x = compactbelow x ) holds
f is isomorphic
proof end;

theorem :: WAYBEL13:17
for L being lower-bounded algebraic sup-Semilattice
for x being Element of L holds
( compactbelow x is principal Ideal of () iff x is compact )
proof end;

theorem Th18: :: WAYBEL13:18
for L1, L2 being non empty RelStr
for X being Subset of L1
for x being Element of L1
for f being Function of L1,L2 st f is isomorphic holds
( x is_<=_than X iff f . x is_<=_than f .: X )
proof end;

theorem Th19: :: WAYBEL13:19
for L1, L2 being non empty RelStr
for X being Subset of L1
for x being Element of L1
for f being Function of L1,L2 st f is isomorphic holds
( x is_>=_than X iff f . x is_>=_than f .: X )
proof end;

theorem Th20: :: WAYBEL13:20
for L1, L2 being non empty antisymmetric RelStr
for f being Function of L1,L2 st f is isomorphic holds
( f is infs-preserving & f is sups-preserving )
proof end;

registration
let L1, L2 be non empty antisymmetric RelStr ;
cluster Function-like V21( the carrier of L1, the carrier of L2) isomorphic -> infs-preserving sups-preserving for Element of K32(K33( the carrier of L1, the carrier of L2));
coherence
for b1 being Function of L1,L2 st b1 is isomorphic holds
( b1 is infs-preserving & b1 is sups-preserving )
by Th20;
end;

theorem Th21: :: WAYBEL13:21
for L1, L2, L3 being non empty transitive antisymmetric RelStr
for f being Function of L1,L2 st f is infs-preserving & L2 is full infs-inheriting SubRelStr of L3 & L3 is complete holds
ex g being Function of L1,L3 st
( f = g & g is infs-preserving )
proof end;

theorem Th22: :: WAYBEL13:22
for L1, L2, L3 being non empty transitive antisymmetric RelStr
for f being Function of L1,L2 st f is monotone & f is directed-sups-preserving & L2 is full directed-sups-inheriting SubRelStr of L3 & L3 is complete holds
ex g being Function of L1,L3 st
( f = g & g is directed-sups-preserving )
proof end;

theorem Th23: :: WAYBEL13:23
for L being lower-bounded sup-Semilattice holds InclPoset () is CLSubFrame of BoolePoset the carrier of ()
proof end;

theorem Th24: :: WAYBEL13:24
for L being lower-bounded algebraic LATTICE ex g being Function of L,(BoolePoset the carrier of ()) st
( g is infs-preserving & g is directed-sups-preserving & g is V13() & ( for x being Element of L holds g . x = compactbelow x ) )
proof end;

theorem :: WAYBEL13:25
for I being non empty set
for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st ( for i being Element of I holds J . i is lower-bounded algebraic LATTICE ) holds
product J is lower-bounded algebraic LATTICE
proof end;

Lm1: for L being lower-bounded LATTICE st L is algebraic holds
ex X being non empty set ex S being full SubRelStr of BoolePoset X st
( S is infs-inheriting & S is directed-sups-inheriting & L,S are_isomorphic )

proof end;

theorem Th26: :: WAYBEL13:26
for L1, L2 being non empty RelStr st RelStr(# the carrier of L1, the InternalRel of L1 #) = RelStr(# the carrier of L2, the InternalRel of L2 #) holds
L1,L2 are_isomorphic
proof end;

Lm2: for L being LATTICE st ex X being non empty set ex S being full SubRelStr of BoolePoset X st
( S is infs-inheriting & S is directed-sups-inheriting & L,S are_isomorphic ) holds
ex X being non empty set ex c being closure Function of (),() st
( c is directed-sups-preserving & L, Image c are_isomorphic )

proof end;

Lm3: for L being LATTICE st ex X being set ex S being full SubRelStr of BoolePoset X st
( S is infs-inheriting & S is directed-sups-inheriting & L,S are_isomorphic ) holds
ex X being set ex c being closure Function of (),() st
( c is directed-sups-preserving & L, Image c are_isomorphic )

proof end;

Lm4: for L1, L2 being non empty up-complete Poset
for f being Function of L1,L2 st f is isomorphic holds
for x, y being Element of L1 st x << y holds
f . x << f . y

proof end;

theorem Th27: :: WAYBEL13:27
for L1, L2 being non empty up-complete Poset
for f being Function of L1,L2 st f is isomorphic holds
for x, y being Element of L1 holds
( x << y iff f . x << f . y )
proof end;

theorem Th28: :: WAYBEL13:28
for L1, L2 being non empty up-complete Poset
for f being Function of L1,L2 st f is isomorphic holds
for x being Element of L1 holds
( x is compact iff f . x is compact )
proof end;

theorem Th29: :: WAYBEL13:29
for L1, L2 being non empty up-complete Poset
for f being Function of L1,L2 st f is isomorphic holds
for x being Element of L1 holds f .: () = compactbelow (f . x)
proof end;

theorem Th30: :: WAYBEL13:30
for L1, L2 being non empty Poset st L1,L2 are_isomorphic & L1 is up-complete holds
L2 is up-complete
proof end;

theorem Th31: :: WAYBEL13:31
for L1, L2 being non empty Poset st L1,L2 are_isomorphic & L1 is complete & L1 is satisfying_axiom_K holds
L2 is satisfying_axiom_K
proof end;

theorem Th32: :: WAYBEL13:32
for L1, L2 being sup-Semilattice st L1,L2 are_isomorphic & L1 is lower-bounded & L1 is algebraic holds
L2 is algebraic
proof end;

Lm5: for L being LATTICE st ex X being set ex c being closure Function of (),() st
( c is directed-sups-preserving & L, Image c are_isomorphic ) holds
L is algebraic

proof end;

theorem :: WAYBEL13:33
for L being lower-bounded continuous sup-Semilattice holds
( SupMap L is infs-preserving & SupMap L is sups-preserving ) by ;

:: THEOREM 4.15. (1) iff (2)
theorem :: WAYBEL13:34
for L being lower-bounded LATTICE holds
( ( L is algebraic implies ex X being non empty set ex S being full SubRelStr of BoolePoset X st
( S is infs-inheriting & S is directed-sups-inheriting & L,S are_isomorphic ) ) & ( ex X being set ex S being full SubRelStr of BoolePoset X st
( S is infs-inheriting & S is directed-sups-inheriting & L,S are_isomorphic ) implies L is algebraic ) )
proof end;

:: THEOREM 4.15. (1) iff (3)
theorem :: WAYBEL13:35
for L being lower-bounded LATTICE holds
( ( L is algebraic implies ex X being non empty set ex c being closure Function of (),() st
( c is directed-sups-preserving & L, Image c are_isomorphic ) ) & ( ex X being set ex c being closure Function of (),() st
( c is directed-sups-preserving & L, Image c are_isomorphic ) implies L is algebraic ) )
proof end;