:: Prime Ideals and Filters
:: by Grzegorz Bancerek
::
:: Copyright (c) 1996-2021 Association of Mizar Users

theorem Th1: :: WAYBEL_7:1
for L being complete LATTICE
for X, Y being set st X c= Y holds
( "\/" (X,L) <= "\/" (Y,L) & "/\" (X,L) >= "/\" (Y,L) )
proof end;

theorem Th2: :: WAYBEL_7:2
for X being set holds the carrier of () = bool X
proof end;

theorem :: WAYBEL_7:3
for L being non empty antisymmetric bounded RelStr holds
( L is trivial iff Top L = Bottom L )
proof end;

registration
let X be set ;
coherence ;
end;

registration
let X be non empty set ;
cluster BoolePoset X -> non trivial ;
coherence
not BoolePoset X is trivial
proof end;
end;

theorem Th4: :: WAYBEL_7:4
for L being non empty lower-bounded Poset
for F being Filter of L holds
( F is proper iff not Bottom L in F )
proof end;

registration
existence
ex b1 being LATTICE st
( not b1 is trivial & b1 is Boolean & b1 is strict )
proof end;
end;

registration
let L be non trivial upper-bounded Poset;
cluster non empty proper filtered upper for Element of bool the carrier of L;
existence
ex b1 being Filter of L st b1 is proper
proof end;
end;

theorem Th5: :: WAYBEL_7:5
for X being set
for a being Element of () holds 'not' a = X \ a
proof end;

theorem :: WAYBEL_7:6
for X being set
for Y being Subset of () holds
( Y is lower iff for x, y being set st x c= y & y in Y holds
x in Y )
proof end;

theorem Th7: :: WAYBEL_7:7
for X being set
for Y being Subset of () holds
( Y is upper iff for x, y being set st x c= y & y c= X & x in Y holds
y in Y )
proof end;

theorem :: WAYBEL_7:8
for X being set
for Y being lower Subset of () holds
( Y is directed iff for x, y being set st x in Y & y in Y holds
x \/ y in Y )
proof end;

theorem Th9: :: WAYBEL_7:9
for X being set
for Y being upper Subset of () holds
( Y is filtered iff for x, y being set st x in Y & y in Y holds
x /\ y in Y )
proof end;

theorem :: WAYBEL_7:10
for X being set
for Y being non empty lower Subset of () holds
( Y is directed iff for Z being finite Subset-Family of X st Z c= Y holds
union Z in Y )
proof end;

theorem Th11: :: WAYBEL_7:11
for X being set
for Y being non empty upper Subset of () holds
( Y is filtered iff for Z being finite Subset-Family of X st Z c= Y holds
Intersect Z in Y )
proof end;

definition
let L be with_infima Poset;
let I be Ideal of L;
attr I is prime means :Def1: :: WAYBEL_7:def 1
for x, y being Element of L holds
( not x "/\" y in I or x in I or y in I );
end;

:: deftheorem Def1 defines prime WAYBEL_7:def 1 :
for L being with_infima Poset
for I being Ideal of L holds
( I is prime iff for x, y being Element of L holds
( not x "/\" y in I or x in I or y in I ) );

theorem Th12: :: WAYBEL_7:12
for L being with_infima Poset
for I being Ideal of L holds
( I is prime iff for A being non empty finite Subset of L st inf A in I holds
ex a being Element of L st
( a in A & a in I ) )
proof end;

registration
let L be LATTICE;
cluster non empty directed lower prime for Element of bool the carrier of L;
existence
ex b1 being Ideal of L st b1 is prime
proof end;
end;

theorem :: WAYBEL_7:13
for L1, L2 being LATTICE st RelStr(# the carrier of L1, the InternalRel of L1 #) = RelStr(# the carrier of L2, the InternalRel of L2 #) holds
for x being set st x is prime Ideal of L1 holds
x is prime Ideal of L2
proof end;

definition
let L be with_suprema Poset;
let F be Filter of L;
attr F is prime means :Def2: :: WAYBEL_7:def 2
for x, y being Element of L holds
( not x "\/" y in F or x in F or y in F );
end;

:: deftheorem Def2 defines prime WAYBEL_7:def 2 :
for L being with_suprema Poset
for F being Filter of L holds
( F is prime iff for x, y being Element of L holds
( not x "\/" y in F or x in F or y in F ) );

theorem :: WAYBEL_7:14
for L being with_suprema Poset
for F being Filter of L holds
( F is prime iff for A being non empty finite Subset of L st sup A in F holds
ex a being Element of L st
( a in A & a in F ) )
proof end;

registration
let L be LATTICE;
cluster non empty filtered upper prime for Element of bool the carrier of L;
existence
ex b1 being Filter of L st b1 is prime
proof end;
end;

theorem Th15: :: WAYBEL_7:15
for L1, L2 being LATTICE st RelStr(# the carrier of L1, the InternalRel of L1 #) = RelStr(# the carrier of L2, the InternalRel of L2 #) holds
for x being set st x is prime Filter of L1 holds
x is prime Filter of L2
proof end;

theorem Th16: :: WAYBEL_7:16
for L being LATTICE
for x being set holds
( x is prime Ideal of L iff x is prime Filter of (L opp) )
proof end;

theorem Th17: :: WAYBEL_7:17
for L being LATTICE
for x being set holds
( x is prime Filter of L iff x is prime Ideal of (L opp) )
proof end;

:: Remark 3.16: (3) iff (2)
theorem :: WAYBEL_7:18
for L being with_infima Poset
for I being Ideal of L holds
( I is prime iff ( I  is Filter of L or I  = {} ) )
proof end;

:: Remark 3.16: (3) iff (1)
theorem :: WAYBEL_7:19
for L being LATTICE
for I being Ideal of L holds
( I is prime iff I in PRIME (InclPoset (Ids L)) )
proof end;

theorem Th20: :: WAYBEL_7:20
for L being Boolean LATTICE
for F being Filter of L holds
( F is prime iff for a being Element of L holds
( a in F or 'not' a in F ) )
proof end;

:: Remark 3.18: (1) iff (2)
theorem Th21: :: WAYBEL_7:21
for X being set
for F being Filter of () holds
( F is prime iff for A being Subset of X holds
( A in F or X \ A in F ) )
proof end;

definition
let L be non empty Poset;
let F be Filter of L;
attr F is ultra means :: WAYBEL_7:def 3
( F is proper & ( for G being Filter of L holds
( not F c= G or F = G or G = the carrier of L ) ) );
end;

:: deftheorem defines ultra WAYBEL_7:def 3 :
for L being non empty Poset
for F being Filter of L holds
( F is ultra iff ( F is proper & ( for G being Filter of L holds
( not F c= G or F = G or G = the carrier of L ) ) ) );

registration
let L be non empty Poset;
cluster non empty filtered upper ultra -> proper for Element of bool the carrier of L;
coherence
for b1 being Filter of L st b1 is ultra holds
b1 is proper
;
end;

Lm1: for L being with_infima Poset
for F being Filter of L
for X being non empty finite Subset of L
for x being Element of L st x in uparrow (fininfs (F \/ X)) holds
ex a being Element of L st
( a in F & x >= a "/\" (inf X) )

proof end;

:: Remark 3.18: (1)+proper iff (3)
theorem Th22: :: WAYBEL_7:22
for L being Boolean LATTICE
for F being Filter of L holds
( ( F is proper & F is prime ) iff F is ultra )
proof end;

Lm2: for L being with_suprema Poset
for I being Ideal of L
for X being non empty finite Subset of L
for x being Element of L st x in downarrow (finsups (I \/ X)) holds
ex i being Element of L st
( i in I & x <= i "\/" (sup X) )

proof end;

:: Lemma 3.19
theorem Th23: :: WAYBEL_7:23
for L being distributive LATTICE
for I being Ideal of L
for F being Filter of L st I misses F holds
ex P being Ideal of L st
( P is prime & I c= P & P misses F )
proof end;

theorem Th24: :: WAYBEL_7:24
for L being distributive LATTICE
for I being Ideal of L
for x being Element of L st not x in I holds
ex P being Ideal of L st
( P is prime & I c= P & not x in P )
proof end;

theorem Th25: :: WAYBEL_7:25
for L being distributive LATTICE
for I being Ideal of L
for F being Filter of L st I misses F holds
ex P being Filter of L st
( P is prime & F c= P & I misses P )
proof end;

theorem Th26: :: WAYBEL_7:26
for L being non trivial Boolean LATTICE
for F being proper Filter of L ex G being Filter of L st
( F c= G & G is ultra )
proof end;

definition
let T be TopSpace;
let F be set ;
let x be object ;
pred x is_a_cluster_point_of F,T means :: WAYBEL_7:def 4
for A being Subset of T st A is open & x in A holds
for B being set st B in F holds
A meets B;
pred x is_a_convergence_point_of F,T means :: WAYBEL_7:def 5
for A being Subset of T st A is open & x in A holds
A in F;
end;

:: deftheorem defines is_a_cluster_point_of WAYBEL_7:def 4 :
for T being TopSpace
for F being set
for x being object holds
( x is_a_cluster_point_of F,T iff for A being Subset of T st A is open & x in A holds
for B being set st B in F holds
A meets B );

:: deftheorem defines is_a_convergence_point_of WAYBEL_7:def 5 :
for T being TopSpace
for F being set
for x being object holds
( x is_a_convergence_point_of F,T iff for A being Subset of T st A is open & x in A holds
A in F );

registration
let X be non empty set ;
cluster non empty filtered upper ultra for Element of bool the carrier of ();
existence
ex b1 being Filter of () st b1 is ultra
proof end;
end;

theorem Th27: :: WAYBEL_7:27
for T being non empty TopSpace
for F being ultra Filter of (BoolePoset the carrier of T)
for p being set holds
( p is_a_cluster_point_of F,T iff p is_a_convergence_point_of F,T )
proof end;

:: Proposition 3.20 (1) => (2)
theorem Th28: :: WAYBEL_7:28
for T being non empty TopSpace
for x, y being Element of (InclPoset the topology of T) st x << y holds
for F being proper Filter of (BoolePoset the carrier of T) st x in F holds
ex p being Element of T st
( p in y & p is_a_cluster_point_of F,T )
proof end;

:: Proposition 3.20: (1) => (3)
theorem :: WAYBEL_7:29
for T being non empty TopSpace
for x, y being Element of (InclPoset the topology of T) st x << y holds
for F being ultra Filter of (BoolePoset the carrier of T) st x in F holds
ex p being Element of T st
( p in y & p is_a_convergence_point_of F,T )
proof end;

:: Proposition 3.20: (3) => (1)
theorem Th30: :: WAYBEL_7:30
for T being non empty TopSpace
for x, y being Element of (InclPoset the topology of T) st x c= y & ( for F being ultra Filter of (BoolePoset the carrier of T) st x in F holds
ex p being Element of T st
( p in y & p is_a_convergence_point_of F,T ) ) holds
x << y
proof end;

:: Proposition 3.21
:: Alexander's Lemma
theorem :: WAYBEL_7:31
for T being non empty TopSpace
for B being prebasis of T
for x, y being Element of (InclPoset the topology of T) st x c= y holds
( x << y iff for F being Subset of B st y c= union F holds
ex G being finite Subset of F st x c= union G )
proof end;

:: Proposition 3.22
theorem :: WAYBEL_7:32
for L being distributive complete LATTICE
for x, y being Element of L holds
( x << y iff for P being prime Ideal of L st y <= sup P holds
x in P )
proof end;

theorem Th33: :: WAYBEL_7:33
for L being LATTICE
for p being Element of L st p is prime holds
downarrow p is prime
proof end;

definition
let L be LATTICE;
let p be Element of L;
attr p is pseudoprime means :: WAYBEL_7:def 6
ex P being prime Ideal of L st p = sup P;
end;

:: deftheorem defines pseudoprime WAYBEL_7:def 6 :
for L being LATTICE
for p being Element of L holds
( p is pseudoprime iff ex P being prime Ideal of L st p = sup P );

theorem Th34: :: WAYBEL_7:34
for L being LATTICE
for p being Element of L st p is prime holds
p is pseudoprime
proof end;

:: Proposition 3.24: (1) => (2)
theorem Th35: :: WAYBEL_7:35
for L being continuous LATTICE
for p being Element of L st p is pseudoprime holds
for A being non empty finite Subset of L st inf A << p holds
ex a being Element of L st
( a in A & a <= p )
proof end;

:: Proposition 3.24: (2)+? => (3)
theorem :: WAYBEL_7:36
for L being continuous LATTICE
for p being Element of L st ( p <> Top L or not Top L is compact ) & ( for A being non empty finite Subset of L st inf A << p holds
ex a being Element of L st
( a in A & a <= p ) ) holds
uparrow (fininfs (() )) misses waybelow p
proof end;

:: It is not true that: Proposition 3.24: (2) => (3)
theorem :: WAYBEL_7:37
for L being continuous LATTICE st Top L is compact holds
( ( for A being non empty finite Subset of L st inf A << Top L holds
ex a being Element of L st
( a in A & a <= Top L ) ) & uparrow (fininfs ((downarrow (Top L)) )) meets waybelow (Top L) )
proof end;

:: Proposition 3.24: (2) <= (3)
theorem :: WAYBEL_7:38
for L being continuous LATTICE
for p being Element of L st uparrow (fininfs (() )) misses waybelow p holds
for A being non empty finite Subset of L st inf A << p holds
ex a being Element of L st
( a in A & a <= p )
proof end;

:: Proposition 3.24: (3) => (1)
theorem :: WAYBEL_7:39
for L being distributive continuous LATTICE
for p being Element of L st uparrow (fininfs (() )) misses waybelow p holds
p is pseudoprime
proof end;

definition
let L be non empty RelStr ;
let R be Relation of the carrier of L;
attr R is multiplicative means :: WAYBEL_7:def 7
for a, x, y being Element of L st [a,x] in R & [a,y] in R holds
[a,(x "/\" y)] in R;
end;

:: deftheorem defines multiplicative WAYBEL_7:def 7 :
for L being non empty RelStr
for R being Relation of the carrier of L holds
( R is multiplicative iff for a, x, y being Element of L st [a,x] in R & [a,y] in R holds
[a,(x "/\" y)] in R );

registration
let L be lower-bounded sup-Semilattice;
let R be auxiliary Relation of L;
let x be Element of L;
cluster R -above x -> upper ;
coherence
R -above x is upper
proof end;
end;

:: Lemma 3.25: (1) iff (2)-[non empty]
theorem :: WAYBEL_7:40
for L being lower-bounded LATTICE
for R being auxiliary Relation of L holds
( R is multiplicative iff for x being Element of L holds R -above x is filtered )
proof end;

:: Lemma 3.25: (1) iff (3)
theorem Th41: :: WAYBEL_7:41
for L being lower-bounded LATTICE
for R being auxiliary Relation of L holds
( R is multiplicative iff for a, b, x, y being Element of L st [a,x] in R & [b,y] in R holds
[(a "/\" b),(x "/\" y)] in R )
proof end;

:: Lemma 3.25: (1) iff (4)
theorem :: WAYBEL_7:42
for L being lower-bounded LATTICE
for R being auxiliary Relation of L holds
( R is multiplicative iff for S being full SubRelStr of [:L,L:] st the carrier of S = R holds
S is meet-inheriting )
proof end;

:: Lemma 3.25: (1) iff (5)
theorem :: WAYBEL_7:43
for L being lower-bounded LATTICE
for R being auxiliary Relation of L holds
( R is multiplicative iff R -below is meet-preserving )
proof end;

Lm3: now :: thesis: for L being lower-bounded continuous LATTICE
for p being Element of L st L -waybelow is multiplicative & ( for a, b being Element of L holds
( not a "/\" b << p or a <= p or b <= p ) ) holds
p is prime
let L be lower-bounded continuous LATTICE; :: thesis: for p being Element of L st L -waybelow is multiplicative & ( for a, b being Element of L holds
( not a "/\" b << p or a <= p or b <= p ) ) holds
p is prime

let p be Element of L; :: thesis: ( L -waybelow is multiplicative & ( for a, b being Element of L holds
( not a "/\" b << p or a <= p or b <= p ) ) implies p is prime )

assume that
A1: L -waybelow is multiplicative and
A2: for a, b being Element of L holds
( not a "/\" b << p or a <= p or b <= p ) and
A3: not p is prime ; :: thesis: contradiction
consider x, y being Element of L such that
A4: x "/\" y <= p and
A5: not x <= p and
A6: not y <= p by A3;
A7: for a being Element of L holds
( not waybelow a is empty & waybelow a is directed ) ;
then consider u being Element of L such that
A8: u << x and
A9: not u <= p by ;
consider v being Element of L such that
A10: v << y and
A11: not v <= p by ;
A12: [v,y] in L -waybelow by ;
[u,x] in L -waybelow by ;
then [(u "/\" v),(x "/\" y)] in L -waybelow by ;
then u "/\" v << x "/\" y by WAYBEL_4:def 1;
then u "/\" v << p by ;
hence contradiction by A2, A9, A11; :: thesis: verum
end;

theorem Th44: :: WAYBEL_7:44
for L being lower-bounded continuous LATTICE st L -waybelow is multiplicative holds
for p being Element of L holds
( p is pseudoprime iff for a, b being Element of L holds
( not a "/\" b << p or a <= p or b <= p ) )
proof end;

theorem :: WAYBEL_7:45
for L being lower-bounded continuous LATTICE st L -waybelow is multiplicative holds
for p being Element of L st p is pseudoprime holds
p is prime
proof end;

theorem :: WAYBEL_7:46
for L being lower-bounded distributive continuous LATTICE st ( for p being Element of L st p is pseudoprime holds
p is prime ) holds
L -waybelow is multiplicative
proof end;