:: Algebraic Lattices
:: by Robert Milewski
::
:: Received December 1, 1996
:: Copyright (c) 1996-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies XBOOLE_0, RELAT_2, ORDERS_2, STRUCT_0, CAT_1, YELLOW_0, SUBSET_1,
RCOMP_1, WELLORD1, TARSKI, LATTICES, REWRITE1, WAYBEL_0, XXREAL_0,
WAYBEL_3, WAYBEL_6, CARD_FIL, LATTICE3, EQREL_1, ORDINAL2, FINSET_1,
ZFMISC_1, WAYBEL_4, MSSUBFAM, WAYBEL_7, INT_2, WAYBEL_1, FUNCT_1,
GROUP_6, RELAT_1, BINOP_1, SEQM_3, YELLOW_1, FILTER_1, SETFAM_1,
WAYBEL_8, CARD_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, CARD_1, FINSET_1,
RELAT_1, TOLER_1, FUNCT_1, RELSET_1, FUNCT_2, DOMAIN_1, RELAT_2,
STRUCT_0, ORDERS_2, LATTICE3, QUANTAL1, YELLOW_0, YELLOW_1, YELLOW_2,
WAYBEL_0, WAYBEL_1, WAYBEL_3, WAYBEL_4, WAYBEL_6, WAYBEL_7;
constructors DOMAIN_1, TOLER_1, QUANTAL1, ORDERS_3, WAYBEL_1, YELLOW_3,
WAYBEL_3, WAYBEL_4, WAYBEL_6, WAYBEL_7, RELSET_1, PRALG_1;
registrations XBOOLE_0, SUBSET_1, RELSET_1, FINSET_1, STRUCT_0, LATTICE3,
YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_2, WAYBEL_1, WAYBEL_2, WAYBEL_3,
WAYBEL_4, WAYBEL_6;
requirements SUBSET, BOOLE, NUMERALS;
definitions TARSKI, RELAT_2, ORDERS_2;
equalities YELLOW_2, STRUCT_0;
expansions TARSKI, RELAT_2, ORDERS_2;
theorems TARSKI, STRUCT_0, FINSET_1, FUNCT_1, FUNCT_2, ORDERS_2, LATTICE3,
YELLOW_0, YELLOW_1, YELLOW_2, YELLOW_3, YELLOW_5, WAYBEL_0, WAYBEL_1,
WAYBEL_3, WAYBEL_4, WAYBEL_6, WAYBEL_7, RELAT_1, YELLOW_7, XBOOLE_0,
XBOOLE_1;
schemes SUBSET_1, FUNCT_1;
begin :: The Subset of All Compact Elements
definition :: DEFINITION 4.1
let L be non empty reflexive RelStr;
func CompactSublatt L -> strict full SubRelStr of L means
:Def1:
for x be Element of L holds x in the carrier of it iff x is compact;
existence
proof
defpred P[set] means ex y be Element of L st y = $1 & y is compact;
consider X being Subset of L such that
A1: for x be set holds x in X iff x in the carrier of L & P[x] from
SUBSET_1:sch 1;
reconsider S = RelStr(#X, (the InternalRel of L)|_2 X#) as strict full
SubRelStr of L by YELLOW_0:56;
take S;
let x be Element of L;
thus x in the carrier of S implies x is compact
proof
assume x in the carrier of S;
then ex y be Element of L st y = x & y is compact by A1;
hence thesis;
end;
thus thesis by A1;
end;
uniqueness
proof
let K1,K2 be strict full SubRelStr of L such that
A2: for x be Element of L holds x in the carrier of K1 iff x is compact and
A3: for x be Element of L holds x in the carrier of K2 iff x is compact;
now
let x be object;
thus x in the carrier of K1 implies x in the carrier of K2
proof
assume
A4: x in the carrier of K1;
the carrier of K1 c= the carrier of L by YELLOW_0:def 13;
then reconsider x9 = x as Element of L by A4;
x9 is compact by A2,A4;
hence thesis by A3;
end;
thus x in the carrier of K2 implies x in the carrier of K1
proof
assume
A5: x in the carrier of K2;
the carrier of K2 c= the carrier of L by YELLOW_0:def 13;
then reconsider x9 = x as Element of L by A5;
x9 is compact by A3,A5;
hence thesis by A2;
end;
end;
then the carrier of K1 = the carrier of K2 by TARSKI:2;
hence thesis by YELLOW_0:57;
end;
end;
registration
let L be lower-bounded non empty reflexive antisymmetric RelStr;
cluster CompactSublatt L -> non empty;
coherence
proof
Bottom L is compact by WAYBEL_3:15;
hence thesis by Def1;
end;
end;
theorem
for L be complete LATTICE for x,y,k be Element of L holds x <= k & k
<= y & k in the carrier of CompactSublatt L implies x << y
proof
let L be complete LATTICE;
let x,y,k be Element of L;
assume that
A1: x <= k & k <= y and
A2: k in the carrier of CompactSublatt L;
k is compact by A2,Def1;
then k << k by WAYBEL_3:def 2;
hence thesis by A1,WAYBEL_3:2;
end;
theorem :: REMARK 4.2
for L be complete LATTICE for x be Element of L holds uparrow x is
Open Filter of L iff x is compact
proof
let L be complete LATTICE;
let x be Element of L;
thus uparrow x is Open Filter of L implies x is compact
proof
x <= x;
then
A1: x in uparrow x by WAYBEL_0:18;
assume uparrow x is Open Filter of L;
then consider y be Element of L such that
A2: y in uparrow x and
A3: y << x by A1,WAYBEL_6:def 1;
x <= y by A2,WAYBEL_0:18;
then x << x by A3,WAYBEL_3:2;
hence thesis by WAYBEL_3:def 2;
end;
assume
A4: x is compact;
now
let u be Element of L;
assume u in uparrow x;
then
A5: x <= u by WAYBEL_0:18;
take x2 = x;
x <= x2;
hence x2 in uparrow x by WAYBEL_0:18;
x << x by A4,WAYBEL_3:def 2;
hence x2 << u by A5,WAYBEL_3:2;
end;
hence thesis by WAYBEL_6:def 1;
end;
theorem :: REMARK 4.3
for L be lower-bounded with_suprema non empty Poset holds
CompactSublatt L is join-inheriting & Bottom L in the carrier of CompactSublatt
L
proof
let L be lower-bounded with_suprema non empty Poset;
now
let x,y be Element of L;
assume that
A1: x in the carrier of CompactSublatt L and
A2: y in the carrier of CompactSublatt L and
A3: ex_sup_of {x,y},L;
y is compact by A2,Def1;
then
A4: y << y by WAYBEL_3:def 2;
x is compact by A1,Def1;
then
A5: x << x by WAYBEL_3:def 2;
y <= x "\/" y by A3,YELLOW_0:18;
then
A6: y << x "\/" y by A4,WAYBEL_3:2;
x <= x "\/" y by A3,YELLOW_0:18;
then x << x "\/" y by A5,WAYBEL_3:2;
then x "\/" y << x "\/" y by A6,WAYBEL_3:3;
then x "\/" y is compact by WAYBEL_3:def 2;
then sup {x,y} is compact by YELLOW_0:41;
hence sup {x,y} in the carrier of CompactSublatt L by Def1;
end;
hence CompactSublatt L is join-inheriting by YELLOW_0:def 17;
Bottom L is compact by WAYBEL_3:15;
hence thesis by Def1;
end;
definition
let L be non empty reflexive RelStr;
let x be Element of L;
func compactbelow x -> Subset of L equals
{y where y is Element of L: x >= y
& y is compact};
coherence
proof
set Z = {y where y is Element of L: x >= y & y is compact};
defpred P[Element of L] means x >= $1 & $1 is compact;
consider X being Subset of L such that
A1: for y being Element of L holds y in X iff P[y] from SUBSET_1:sch 3;
now
let z be object;
thus z in X implies z in Z
proof
assume
A2: z in X;
then reconsider z1 = z as Element of L;
x >= z1 & z1 is compact by A1,A2;
hence thesis;
end;
thus z in Z implies z in X
proof
assume z in Z;
then ex v be Element of L st v = z & x >= v & v is compact;
hence thesis by A1;
end;
end;
hence thesis by TARSKI:2;
end;
end;
theorem Th4:
for L be non empty reflexive RelStr for x,y be Element of L holds
y in compactbelow x iff x >= y & y is compact
proof
let L be non empty reflexive RelStr;
let x,y be Element of L;
thus y in compactbelow x implies x >= y & y is compact
proof
assume y in compactbelow x;
then ex z be Element of L st z = y & x >= z & z is compact;
hence thesis;
end;
assume x >= y & y is compact;
hence thesis;
end;
theorem Th5:
for L be non empty reflexive RelStr for x be Element of L holds
compactbelow x = downarrow x /\ the carrier of CompactSublatt L
proof
let L be non empty reflexive RelStr;
let x be Element of L;
now
let y be object;
assume
A1: y in downarrow x /\ the carrier of CompactSublatt L;
then reconsider y9 = y as Element of L;
y in downarrow x by A1,XBOOLE_0:def 4;
then
A2: y9 <= x by WAYBEL_0:17;
y in the carrier of CompactSublatt L by A1,XBOOLE_0:def 4;
then y9 is compact by Def1;
hence y in compactbelow x by A2;
end;
then
A3: downarrow x /\ the carrier of CompactSublatt L c= compactbelow x;
now
let y be object;
assume y in compactbelow x;
then consider y9 be Element of L such that
A4: y9 = y and
A5: y9 <= x & y9 is compact;
y9 in downarrow x & y9 in the carrier of CompactSublatt L by A5,Def1,
WAYBEL_0:17;
hence y in downarrow x /\ the carrier of CompactSublatt L by A4,
XBOOLE_0:def 4;
end;
then compactbelow x c= downarrow x /\ the carrier of CompactSublatt L;
hence thesis by A3,XBOOLE_0:def 10;
end;
theorem Th6:
for L be non empty reflexive transitive RelStr for x be Element
of L holds compactbelow x c= waybelow x
proof
let L be non empty reflexive transitive RelStr;
let x be Element of L;
now
let z be object;
assume z in compactbelow x;
then consider z9 be Element of L such that
A1: z9 = z and
A2: x >= z9 and
A3: z9 is compact;
z9 << z9 by A3,WAYBEL_3:def 2;
then z9 << x by A2,WAYBEL_3:2;
hence z in waybelow x by A1,WAYBEL_3:7;
end;
hence thesis;
end;
registration
let L be non empty lower-bounded reflexive antisymmetric RelStr;
let x be Element of L;
cluster compactbelow x -> non empty;
coherence
proof
x >= Bottom L & Bottom L is compact by WAYBEL_3:15,YELLOW_0:44;
then Bottom L in {y where y is Element of L: x >= y & y is compact};
hence thesis;
end;
end;
begin :: Algebraic Lattices
definition
let L be non empty reflexive RelStr;
attr L is satisfying_axiom_K means
:Def3:
for x be Element of L holds x = sup compactbelow x;
end;
definition :: DEFINITION 4.4
let L be non empty reflexive RelStr;
attr L is algebraic means
:Def4:
(for x being Element of L holds
compactbelow x is non empty directed) & L is up-complete satisfying_axiom_K;
end;
theorem Th7: :: PROPOSITION 4.5
for L be LATTICE holds L is algebraic iff ( L is continuous & for
x,y be Element of L st x << y ex k be Element of L st k in the carrier of
CompactSublatt L & x <= k & k <= y )
proof
let L be LATTICE;
thus L is algebraic implies ( L is continuous & for x,y be Element of L st x
<< y ex k be Element of L st k in the carrier of CompactSublatt L & x <= k & k
<= y )
proof
assume
A1: L is algebraic;
then
A2: L is up-complete satisfying_axiom_K;
now
let x be Element of L;
A3: compactbelow x is non empty directed by A1;
then
A4: ex s be object st s in compactbelow x by XBOOLE_0:def 1;
compactbelow x c= waybelow x by Th6;
then
A5: ex_sup_of waybelow x,L by A2,A4,WAYBEL_0:75;
ex_sup_of compactbelow x,L by A2,A3,WAYBEL_0:75;
then sup compactbelow x <= sup waybelow x by A5,Th6,YELLOW_0:34;
then
A6: x <= sup waybelow x by A2;
waybelow x is_<=_than x by WAYBEL_3:9;
then sup waybelow x <= x by A5,YELLOW_0:def 9;
hence x = sup waybelow x by A6,ORDERS_2:2;
end;
then
A7: L is satisfying_axiom_of_approximation by WAYBEL_3:def 5;
for x be Element of L holds waybelow x is non empty directed
proof
let x be Element of L;
compactbelow x c= waybelow x by Th6;
hence thesis by A1;
end;
hence L is continuous by A2,A7,WAYBEL_3:def 6;
let x,y be Element of L;
reconsider D = compactbelow y as non empty directed Subset of L by A1;
assume
A8: x << y;
y = sup D by A2;
then consider d being Element of L such that
A9: d in D and
A10: x <= d by A8,WAYBEL_3:def 1;
take d;
d is compact by A9,Th4;
hence d in the carrier of CompactSublatt L by Def1;
thus thesis by A9,A10,Th4;
end;
assume that
A11: L is continuous and
A12: for x,y be Element of L st x << y ex k be Element of L st k in the
carrier of CompactSublatt L & x <= k & k <= y;
now
let x be Element of L;
A13: now
let z be Element of L;
thus z is_>=_than waybelow x implies z is_>=_than compactbelow x
proof
assume
A14: z is_>=_than waybelow x;
now
let d be Element of L;
assume
A15: d in compactbelow x;
then d is compact by Th4;
then
A16: d << d by WAYBEL_3:def 2;
d <= x by A15,Th4;
then d << x by A16,WAYBEL_3:2;
then d in waybelow x by WAYBEL_3:7;
hence d <= z by A14,LATTICE3:def 9;
end;
hence thesis by LATTICE3:def 9;
end;
thus z is_>=_than compactbelow x implies z is_>=_than waybelow x
proof
assume
A17: z is_>=_than compactbelow x;
now
let d be Element of L;
assume d in waybelow x;
then d << x by WAYBEL_3:7;
then consider k be Element of L such that
A18: k in the carrier of CompactSublatt L and
A19: d <= k and
A20: k <= x by A12;
k is compact by A18,Def1;
then k in compactbelow x by A20;
then k <= z by A17,LATTICE3:def 9;
hence d <= z by A19,ORDERS_2:3;
end;
hence thesis by LATTICE3:def 9;
end;
end;
x = sup waybelow x & ex_sup_of waybelow x,L by A11,WAYBEL_0:75
,WAYBEL_3:def 5;
hence x = sup compactbelow x by A13,YELLOW_0:47;
end;
then
A21: L is satisfying_axiom_K;
for x be Element of L holds compactbelow x is non empty directed
proof
let x be Element of L;
now
let Y be finite Subset of compactbelow x;
compactbelow x c= waybelow x by Th6;
then Y is finite Subset of waybelow x by XBOOLE_1:1;
then consider b be Element of L such that
A22: b in waybelow x and
A23: b is_>=_than Y by A11,WAYBEL_0:1;
b << x by A22,WAYBEL_3:7;
then consider c be Element of L such that
A24: c in the carrier of CompactSublatt L and
A25: b <= c and
A26: c <= x by A12;
take c;
c is compact by A24,Def1;
hence c in compactbelow x by A26;
thus c is_>=_than Y by A23,A25,YELLOW_0:4;
end;
hence thesis by WAYBEL_0:1;
end;
hence thesis by A11,A21;
end;
registration
cluster algebraic -> continuous for LATTICE;
coherence by Th7;
end;
registration
cluster algebraic -> up-complete satisfying_axiom_K for non empty reflexive
RelStr;
coherence;
end;
registration
let L be non empty with_suprema Poset;
cluster CompactSublatt L -> join-inheriting;
coherence
proof
now
let x,y be Element of L;
assume that
A1: x in the carrier of CompactSublatt L and
A2: y in the carrier of CompactSublatt L and
A3: ex_sup_of {x,y},L;
y is compact by A2,Def1;
then
A4: y << y by WAYBEL_3:def 2;
x is compact by A1,Def1;
then
A5: x << x by WAYBEL_3:def 2;
y <= x "\/" y by A3,YELLOW_0:18;
then
A6: y << x "\/" y by A4,WAYBEL_3:2;
x <= x "\/" y by A3,YELLOW_0:18;
then x << x "\/" y by A5,WAYBEL_3:2;
then x "\/" y << x "\/" y by A6,WAYBEL_3:3;
then x "\/" y is compact by WAYBEL_3:def 2;
then sup {x,y} is compact by YELLOW_0:41;
hence sup {x,y} in the carrier of CompactSublatt L by Def1;
end;
hence thesis by YELLOW_0:def 17;
end;
end;
definition
let L be non empty reflexive RelStr;
attr L is arithmetic means
L is algebraic & CompactSublatt L is meet-inheriting;
end;
begin :: Arithmetic Lattices
registration
cluster arithmetic -> algebraic for LATTICE;
coherence;
end;
registration
cluster trivial -> arithmetic for LATTICE;
coherence
proof
let L be LATTICE;
assume
A1: L is trivial;
reconsider L9 = L as 1-element LATTICE by A1;
A2: for x,y be Element of L9 st x << y ex k be Element of L9 st k in the
carrier of CompactSublatt L9 & x <= k & k <= y
proof
let x,y be Element of L9;
assume
A3: x << y;
take k = x;
x = y by STRUCT_0:def 10;
then k is compact by A3,WAYBEL_3:def 2;
hence k in the carrier of CompactSublatt L9 by Def1;
thus thesis by STRUCT_0:def 10;
end;
A4: L9 is algebraic by Th7,A2;
for z,v be Element of L9 st z in the carrier of CompactSublatt L9 & v
in the carrier of CompactSublatt L9 & ex_inf_of {z,v},L9 holds inf {z,v} in the
carrier of CompactSublatt L9 by STRUCT_0:def 10;
then CompactSublatt L9 is meet-inheriting by YELLOW_0:def 16;
hence thesis by A4;
end;
end;
registration
cluster 1-element strict for LATTICE;
existence
proof
set B =the strict 1-element reflexive RelStr;
take B;
thus thesis;
end;
end;
theorem Th8:
for L1,L2 be non empty reflexive antisymmetric RelStr st the
RelStr of L1 = the RelStr of L2 & L1 is up-complete for x1,y1 be Element of L1
for x2,y2 be Element of L2 st x1 = x2 & y1 = y2 & x1 << y1 holds x2 << y2
proof
let L1,L2 be non empty reflexive antisymmetric RelStr;
assume that
A1: the RelStr of L1 = the RelStr of L2 and
A2: L1 is up-complete;
let x1,y1 be Element of L1;
let x2,y2 be Element of L2;
assume that
A3: x1 = x2 and
A4: y1 = y2 and
A5: x1 << y1;
now
let D2 be non empty directed Subset of L2;
reconsider D1 = D2 as Subset of L1 by A1;
reconsider D1 as non empty directed Subset of L1 by A1,WAYBEL_0:3;
ex_sup_of D1,L1 by A2,WAYBEL_0:75;
then
A6: sup D1 = sup D2 by A1,YELLOW_0:26;
assume y2 <= sup D2;
then y1 <= sup D1 by A1,A4,A6;
then consider d1 be Element of L1 such that
A7: d1 in D1 and
A8: x1 <= d1 by A5,WAYBEL_3:def 1;
reconsider d2 = d1 as Element of L2 by A1;
take d2;
thus d2 in D2 by A7;
thus x2 <= d2 by A1,A3,A8;
end;
hence thesis by WAYBEL_3:def 1;
end;
theorem Th9:
for L1,L2 be non empty reflexive antisymmetric RelStr st the
RelStr of L1 = the RelStr of L2 & L1 is up-complete for x be Element of L1 for
y be Element of L2 st x = y & x is compact holds y is compact
proof
let L1,L2 be non empty reflexive antisymmetric RelStr;
assume
A1: the RelStr of L1 = the RelStr of L2 & L1 is up-complete;
let x be Element of L1;
let y be Element of L2;
assume that
A2: x = y and
A3: x is compact;
x << x by A3,WAYBEL_3:def 2;
then y << y by A1,A2,Th8;
hence thesis by WAYBEL_3:def 2;
end;
theorem Th10:
for L1,L2 be up-complete non empty reflexive antisymmetric
RelStr st the RelStr of L1 = the RelStr of L2 for x be Element of L1 for y be
Element of L2 st x = y holds compactbelow x = compactbelow y
proof
let L1,L2 be up-complete non empty reflexive antisymmetric RelStr;
assume
A1: the RelStr of L1 = the RelStr of L2;
let x be Element of L1;
let y be Element of L2;
assume
A2: x = y;
now
let z be object;
assume
A3: z in compactbelow y;
then reconsider z2 = z as Element of L2;
reconsider z1 = z2 as Element of L1 by A1;
z2 is compact by A3,Th4;
then
A4: z1 is compact by A1,Th9;
z2 <= y by A3,Th4;
then z1 <= x by A1,A2;
hence z in compactbelow x by A4;
end;
then
A5: compactbelow y c= compactbelow x;
now
let z be object;
assume
A6: z in compactbelow x;
then reconsider z1 = z as Element of L1;
reconsider z2 = z1 as Element of L2 by A1;
z1 is compact by A6,Th4;
then
A7: z2 is compact by A1,Th9;
z1 <= x by A6,Th4;
then z2 <= y by A1,A2;
hence z in compactbelow y by A7;
end;
then compactbelow x c= compactbelow y;
hence thesis by A5,XBOOLE_0:def 10;
end;
theorem
for L1,L2 be RelStr st the RelStr of L1 = the RelStr of L2 & L1 is non
empty holds L2 is non empty;
theorem Th12:
for L1,L2 be RelStr st the RelStr of L1 = the RelStr of L2 & L1
is reflexive holds L2 is reflexive;
theorem Th13:
for L1,L2 be RelStr st the RelStr of L1 = the RelStr of L2 & L1
is transitive holds L2 is transitive;
theorem Th14:
for L1,L2 be RelStr st the RelStr of L1 = the RelStr of L2 & L1
is antisymmetric holds L2 is antisymmetric;
theorem Th15:
for L1,L2 be non empty reflexive RelStr st the RelStr of L1 =
the RelStr of L2 & L1 is up-complete holds L2 is up-complete
proof
let L1,L2 be non empty reflexive RelStr;
assume that
A1: the RelStr of L1 = the RelStr of L2 and
A2: L1 is up-complete;
now
let X be non empty directed Subset of L2;
reconsider X9 = X as Subset of L1 by A1;
reconsider X9 as non empty directed Subset of L1 by A1,WAYBEL_0:3;
consider x9 be Element of L1 such that
A3: x9 is_>=_than X9 and
A4: for y9 be Element of L1 st y9 is_>=_than X9 holds x9 <= y9 by A2,
WAYBEL_0:def 39;
reconsider x = x9 as Element of L2 by A1;
take x;
thus x is_>=_than X by A1,A3,YELLOW_0:2;
let y be Element of L2 such that
A5: y is_>=_than X;
reconsider y9 = y as Element of L1 by A1;
x9 <= y9 by A1,A4,A5,YELLOW_0:2;
hence x <= y by A1;
end;
hence thesis by WAYBEL_0:def 39;
end;
theorem Th16:
for L1,L2 be up-complete non empty reflexive antisymmetric
RelStr st the RelStr of L1 = the RelStr of L2 & L1 is satisfying_axiom_K & for
x be Element of L1 holds compactbelow x is non empty directed holds L2 is
satisfying_axiom_K
proof
let L1,L2 be up-complete non empty reflexive antisymmetric RelStr;
assume that
A1: the RelStr of L1 = the RelStr of L2 and
A2: L1 is satisfying_axiom_K and
A3: for x be Element of L1 holds compactbelow x is non empty directed;
now
let x be Element of L2;
reconsider x9 = x as Element of L1 by A1;
compactbelow x9 is non empty directed by A3;
then
A4: ex_sup_of compactbelow x9,L1 by WAYBEL_0:75;
x9 = sup compactbelow x9 & compactbelow x = compactbelow x9 by A1,A2,Th10;
hence x = sup compactbelow x by A1,A4,YELLOW_0:26;
end;
hence thesis;
end;
theorem Th17:
for L1,L2 be non empty reflexive antisymmetric RelStr st the
RelStr of L1 = the RelStr of L2 & L1 is algebraic holds L2 is algebraic
proof
let L1,L2 be non empty reflexive antisymmetric RelStr;
assume that
A1: the RelStr of L1 = the RelStr of L2 and
A2: L1 is algebraic;
A3: L2 is up-complete by A1,A2,Th15;
A4: for x be Element of L2 holds compactbelow x is non empty directed
proof
let x be Element of L2;
reconsider x9 = x as Element of L1 by A1;
compactbelow x9 is non empty directed by A2;
hence thesis by A1,A2,A3,Th10,WAYBEL_0:3;
end;
L2 is satisfying_axiom_K by A1,A2,A3,Th16;
hence thesis by A3,A4;
end;
theorem Th18:
for L1,L2 be LATTICE st the RelStr of L1 = the RelStr of L2 & L1
is arithmetic holds L2 is arithmetic
proof
let L1,L2 be LATTICE;
assume that
A1: the RelStr of L1 = the RelStr of L2 and
A2: L1 is arithmetic;
A3: L2 is algebraic by A1,A2,Th17;
A4: CompactSublatt L1 is meet-inheriting by A2;
now
let x2,y2 be Element of L2;
reconsider x1 = x2, y1 = y2 as Element of L1 by A1;
assume that
A5: x2 in the carrier of CompactSublatt L2 and
A6: y2 in the carrier of CompactSublatt L2 and
A7: ex_inf_of {x2,y2},L2;
x2 is compact by A5,Def1;
then x1 is compact by A1,A3,Th9;
then
A8: x1 in the carrier of CompactSublatt L1 by Def1;
y2 is compact by A6,Def1;
then y1 is compact by A1,A3,Th9;
then
A9: y1 in the carrier of CompactSublatt L1 by Def1;
ex_inf_of {x1,y1},L1 by A1,A7,YELLOW_0:14;
then inf {x1,y1} in the carrier of CompactSublatt L1 by A4,A8,A9,
YELLOW_0:def 16;
then
A10: inf {x1,y1} is compact by Def1;
inf {x1,y1} = inf {x2,y2} by A1,A7,YELLOW_0:27;
then inf {x2,y2} is compact by A1,A2,A10,Th9;
hence inf {x2,y2} in the carrier of CompactSublatt L2 by Def1;
end;
then CompactSublatt L2 is meet-inheriting by YELLOW_0:def 16;
hence thesis by A3;
end;
registration
let L be non empty RelStr;
cluster the RelStr of L -> non empty;
coherence;
end;
registration
let L be non empty reflexive RelStr;
cluster the RelStr of L -> reflexive;
coherence by Th12;
end;
registration
let L be transitive RelStr;
cluster the RelStr of L -> transitive;
coherence by Th13;
end;
registration
let L be antisymmetric RelStr;
cluster the RelStr of L -> antisymmetric;
coherence by Th14;
end;
registration
let L be with_infima RelStr;
cluster the RelStr of L -> with_infima;
coherence by YELLOW_7:14;
end;
registration
let L be with_suprema RelStr;
cluster the RelStr of L -> with_suprema;
coherence by YELLOW_7:15;
end;
registration
let L be up-complete non empty reflexive RelStr;
cluster the RelStr of L -> up-complete;
coherence by Th15;
end;
registration
let L be algebraic non empty reflexive antisymmetric RelStr;
cluster the RelStr of L -> algebraic;
coherence by Th17;
end;
registration
let L be arithmetic LATTICE;
cluster the RelStr of L -> arithmetic;
coherence by Th18;
end;
theorem :: PROPOSITION 4.7 a)
for L be algebraic LATTICE holds L is arithmetic iff CompactSublatt L
is LATTICE
proof
let L be algebraic LATTICE;
thus L is arithmetic implies CompactSublatt L is LATTICE
proof
set x = the Element of L;
assume
A1: L is arithmetic;
compactbelow x is non empty by Def4;
then consider z be object such that
A2: z in compactbelow x by XBOOLE_0:def 1;
ex z9 be Element of L st z9 = z & x >= z9 & z9 is compact by A2;
then CompactSublatt L is non empty join-inheriting meet-inheriting full
SubRelStr of L by A1,Def1;
hence thesis;
end;
assume
A3: CompactSublatt L is LATTICE;
now
let x,y be Element of L;
assume that
A4: x in the carrier of CompactSublatt L and
A5: y in the carrier of CompactSublatt L and
ex_inf_of {x,y},L;
reconsider L9 = CompactSublatt L as non empty full SubRelStr of L by A4;
reconsider x9 = x, y9 = y as Element of L9 by A4,A5;
set X = compactbelow inf {x,y};
reconsider c = "/\"({x,y},L9) as Element of L by YELLOW_0:58;
A6: inf {x,y} = sup X by Def3;
X is non empty directed by Def4;
then
A7: ex_sup_of X,L by WAYBEL_0:75;
A8: ex_inf_of {x9,y9},L9 by A3,YELLOW_0:21;
then
A9: "/\"({x,y},L9) is_<=_than {x,y} by YELLOW_0:31;
now
let z be object;
assume z in X;
then consider z1 be Element of L such that
A10: z = z1 and
A11: inf {x,y} >= z1 and
A12: z1 is compact;
A13: z1 <= x "/\" y by A11,YELLOW_0:40;
x "/\" y <= y by YELLOW_0:23;
then z1 <= y by A13,ORDERS_2:3;
then
A14: z in compactbelow y by A10,A12;
x "/\" y <= x by YELLOW_0:23;
then z1 <= x by A13,ORDERS_2:3;
then z in compactbelow x by A10,A12;
hence z in compactbelow x /\ compactbelow y by A14,XBOOLE_0:def 4;
end;
then
A15: X c= compactbelow x /\ compactbelow y;
now
let b9 be Element of L9;
reconsider b = b9 as Element of L by YELLOW_0:58;
assume
A16: b9 in X;
then b9 in compactbelow y by A15,XBOOLE_0:def 4;
then b <= y by Th4;
then
A17: b9 <= y9 by YELLOW_0:60;
b9 in compactbelow x by A15,A16,XBOOLE_0:def 4;
then b <= x by Th4;
then b9 <= x9 by YELLOW_0:60;
then b9 <= x9 "/\" y9 by A8,A17,YELLOW_0:19;
hence b9 <= "/\"({x,y},L9) by A3,YELLOW_0:40;
end;
then
A18: X is_<=_than "/\"({x,y},L9) by LATTICE3:def 9;
now
let d be object;
assume d in X;
then
ex d9 be Element of L st d9 = d & inf {x,y} >= d9 & d9 is compact;
hence d in the carrier of L9 by Def1;
end;
then X c= the carrier of L9;
then X is_<=_than c by A18,YELLOW_0:62;
then
A19: sup X <= c by A7,YELLOW_0:30;
y9 in {x,y} by TARSKI:def 2;
then "/\"({x,y},L9) <= y9 by A9,LATTICE3:def 8;
then
A20: c <= y by YELLOW_0:59;
x9 in {x,y} by TARSKI:def 2;
then "/\"({x,y},L9) <= x9 by A9,LATTICE3:def 8;
then c <= x by YELLOW_0:59;
then c <= x "/\" y by A20,YELLOW_0:23;
then c <= sup X by A6,YELLOW_0:40;
then c = sup X by A19,ORDERS_2:2;
hence inf {x,y} in the carrier of CompactSublatt L by A6;
end;
then CompactSublatt L is meet-inheriting by YELLOW_0:def 16;
hence thesis;
end;
theorem Th20: :: PROPOSITION 4.7 b)
for L be algebraic lower-bounded LATTICE holds L is arithmetic
iff L-waybelow is multiplicative
proof
let L be algebraic lower-bounded LATTICE;
thus L is arithmetic implies L-waybelow is multiplicative
proof
assume
A1: L is arithmetic;
now
reconsider C = CompactSublatt L as meet-inheriting non empty full
SubRelStr of L by A1;
let a,x,y be Element of L;
assume that
A2: [a,x] in L-waybelow and
A3: [a,y] in L-waybelow;
a << x by A2,WAYBEL_4:def 1;
then consider c be Element of L such that
A4: c in the carrier of CompactSublatt L and
A5: a <= c and
A6: c <= x by Th7;
a << y by A3,WAYBEL_4:def 1;
then consider k be Element of L such that
A7: k in the carrier of CompactSublatt L and
A8: a <= k and
A9: k <= y by Th7;
A10: c"/\"k <= x"/\"y by A6,A9,YELLOW_3:2;
reconsider c9=c, k9=k as Element of C by A4,A7;
c9"/\"k9 in the carrier of CompactSublatt L;
then c"/\"k in the carrier of CompactSublatt L by YELLOW_0:69;
then c"/\"k is compact by Def1;
then
A11: c"/\"k << c"/\"k by WAYBEL_3:def 2;
a"/\"a = a by YELLOW_5:2;
then a <= c"/\"k by A5,A8,YELLOW_3:2;
then a << x"/\"y by A10,A11,WAYBEL_3:2;
hence [a,x"/\"y] in L-waybelow by WAYBEL_4:def 1;
end;
hence thesis by WAYBEL_7:def 7;
end;
assume
A12: L-waybelow is multiplicative;
now
let x,y be Element of L;
assume that
A13: x in the carrier of CompactSublatt L and
A14: y in the carrier of CompactSublatt L and
ex_inf_of {x,y},L;
y is compact by A14,Def1;
then y << y by WAYBEL_3:def 2;
then
A15: [y,y] in L-waybelow by WAYBEL_4:def 1;
x is compact by A13,Def1;
then x << x by WAYBEL_3:def 2;
then [x,x] in L-waybelow by WAYBEL_4:def 1;
then [x "/\" y,x "/\" y] in L-waybelow by A12,A15,WAYBEL_7:41;
then x "/\" y << x "/\" y by WAYBEL_4:def 1;
then x "/\" y is compact by WAYBEL_3:def 2;
then x "/\" y in the carrier of CompactSublatt L by Def1;
hence inf {x,y} in the carrier of CompactSublatt L by YELLOW_0:40;
end;
then CompactSublatt L is meet-inheriting by YELLOW_0:def 16;
hence thesis;
end;
theorem :: COROLLARY 4.8.a)
for L be arithmetic lower-bounded LATTICE, p be Element of L holds p
is pseudoprime implies p is prime
proof
let L be arithmetic lower-bounded LATTICE;
let p be Element of L;
L-waybelow is multiplicative by Th20;
hence thesis by WAYBEL_7:45;
end;
theorem :: COROLLARY 4.8.b)
for L be algebraic distributive lower-bounded LATTICE st for p being
Element of L st p is pseudoprime holds p is prime holds L is arithmetic
proof
let L be algebraic distributive lower-bounded LATTICE;
assume for p be Element of L st p is pseudoprime holds p is prime;
then L-waybelow is multiplicative by WAYBEL_7:46;
hence thesis by Th20;
end;
registration
let L be algebraic LATTICE;
let c be closure Function of L,L;
cluster non empty directed for Subset of Image c;
existence
proof
set x = the Element of Image c;
take downarrow x;
thus thesis;
end;
end;
theorem Th23:
for L be algebraic LATTICE for c be closure Function of L,L st c
is directed-sups-preserving holds c.:([#]CompactSublatt L) c= [#]CompactSublatt
Image c
proof
let L be algebraic LATTICE;
let c be closure Function of L,L;
assume
A1: c is directed-sups-preserving;
let x be object;
A2: c is idempotent monotone by WAYBEL_1:def 13;
assume x in c.:([#]CompactSublatt L);
then consider y be object such that
A3: y in dom c and
A4: y in [#]CompactSublatt L and
A5: x = c.y by FUNCT_1:def 6;
A6: x in rng c by A3,A5,FUNCT_1:def 3;
then reconsider x9 = x as Element of Image c by YELLOW_0:def 15;
reconsider x1 = x9 as Element of L by A6;
reconsider y9 = y as Element of L by A3;
y9 is compact by A4,Def1;
then
A7: y9 << y9 by WAYBEL_3:def 2;
now
id(L) <= c by WAYBEL_1:def 14;
then id(L).y9 <= c.y9 by YELLOW_2:9;
then
A8: y9 <= x1 by A5,FUNCT_1:18;
let D be non empty directed Subset of Image c;
assume
A9: x9 <= sup D;
D c= the carrier of Image c;
then
A10: D c= rng c by YELLOW_0:def 15;
then reconsider D9 = D as non empty (Subset of L) by XBOOLE_1:1;
reconsider D9 as non empty directed (Subset of L) by YELLOW_2:7;
A11: ex_sup_of D9,L by WAYBEL_0:75;
c preserves_sup_of D9 by A1,WAYBEL_0:def 37;
then
A12: c.sup D9 = sup (c.:D9) by A11,WAYBEL_0:def 31
.= sup D9 by A2,A10,YELLOW_2:20;
c.sup D9 = sup D by A11,WAYBEL_1:55;
then x1 <= sup D9 by A9,A12,YELLOW_0:59;
then y9 <= sup D9 by A8,ORDERS_2:3;
then consider d9 be Element of L such that
A13: d9 in D9 and
A14: y9 <= d9 by A7,WAYBEL_3:def 1;
reconsider d = d9 as Element of Image c by A13;
take d;
thus d in D by A13;
d in the carrier of Image c;
then d in rng c by YELLOW_0:def 15;
then d9 in {z where z is Element of L: z = c.z} by A2,YELLOW_2:19;
then
A15: ex z9 be Element of L st d9 = z9 & z9 = c.z9;
c.y9 <= c.d9 by A2,A14,WAYBEL_1:def 2;
hence x9 <= d by A5,A15,YELLOW_0:60;
end;
then x9 << x9 by WAYBEL_3:def 1;
then x9 is compact by WAYBEL_3:def 2;
hence thesis by Def1;
end;
theorem :: PROPOSITION 4.9.(i)
for L be algebraic lower-bounded LATTICE for c be closure Function of
L,L st c is directed-sups-preserving holds Image c is algebraic LATTICE
proof
let L be algebraic lower-bounded LATTICE;
let c be closure Function of L,L;
assume
A1: c is directed-sups-preserving;
c is idempotent by WAYBEL_1:def 13;
then reconsider Imc = Image c as complete LATTICE by A1,YELLOW_2:35;
A2: now
let x be Element of Imc;
now
let y,z be Element of Imc;
assume that
A3: y in compactbelow x and
A4: z in compactbelow x;
y is compact by A3,Th4;
then
A5: y << y by WAYBEL_3:def 2;
z is compact by A4,Th4;
then
A6: z << z by WAYBEL_3:def 2;
take v = y "\/" z;
z <= v by YELLOW_0:22;
then
A7: z << v by A6,WAYBEL_3:2;
y <= v by YELLOW_0:22;
then y << v by A5,WAYBEL_3:2;
then v << v by A7,WAYBEL_3:3;
then
A8: v is compact by WAYBEL_3:def 2;
y <= x & z <= x by A3,A4,Th4;
then v <= x by YELLOW_0:22;
hence v in compactbelow x & y <= v & z <= v by A8,YELLOW_0:22;
end;
hence compactbelow x is non empty directed by WAYBEL_0:def 1;
end;
now
let x be Element of Imc;
consider y be Element of L such that
A9: c.y = x by YELLOW_2:10;
sup compactbelow y in the carrier of L;
then
A10: sup compactbelow y in dom c by FUNCT_2:def 1;
compactbelow y is non empty directed by Def4;
then
A11: c preserves_sup_of compactbelow y & ex_sup_of compactbelow y,L by A1,
WAYBEL_0:75,def 37;
then c.sup compactbelow y = sup (c.:(compactbelow y)) by WAYBEL_0:def 31;
then sup (c.:(compactbelow y)) in rng c by A10,FUNCT_1:def 3;
then
A12: ex_sup_of (c.:(compactbelow y)),L & sup (c.:(compactbelow y)) in the
carrier of Image c by YELLOW_0:17,def 15;
for b be Element of Imc st b in compactbelow x holds b <= x by Th4;
then x is_>=_than compactbelow x by LATTICE3:def 9;
then
A13: sup compactbelow x <= x by YELLOW_0:32;
A14: c.:([#]CompactSublatt L) c= [#]CompactSublatt Image c by A1,Th23;
A15: c is monotone by A1,YELLOW_2:16;
compactbelow y = downarrow y /\ [#]CompactSublatt L by Th5;
then
A16: c.:(compactbelow y) c= c.:(downarrow y) /\ c.: ([#]CompactSublatt L)
by RELAT_1:121;
now
let z be object;
assume
A17: z in c.:(compactbelow y);
then consider v be object such that
A18: v in dom c and
A19: v in compactbelow y and
A20: z = c.v by FUNCT_1:def 6;
z in rng c by A18,A20,FUNCT_1:def 3;
then reconsider z1 = z as Element of Imc by YELLOW_0:def 15;
reconsider v as Element of L by A18;
v <= y by A19,Th4;
then c.v <= c.y by A15,WAYBEL_1:def 2;
then
A21: z1 <= x by A9,A20,YELLOW_0:60;
z in c.:([#]CompactSublatt L) by A16,A17,XBOOLE_0:def 4;
then z1 is compact by A14,Def1;
hence z in compactbelow x by A21;
end;
then
A22: c.:(compactbelow y) c= compactbelow x;
compactbelow x is directed by A2;
then
A23: ex_sup_of compactbelow x,Imc by WAYBEL_0:75;
c.:(compactbelow y) c= rng c by RELAT_1:111;
then c.:(compactbelow y) is Subset of Imc by YELLOW_0:def 15;
then
A24: ex_sup_of c.:(compactbelow y),Imc & sup (c.:(compactbelow y)) = "\/"(
c.:( compactbelow y),Imc) by A12,YELLOW_0:64;
c.y = c.sup compactbelow y by Def3
.= sup (c.:(compactbelow y)) by A11,WAYBEL_0:def 31;
then x <= sup compactbelow x by A9,A23,A24,A22,YELLOW_0:34;
hence x = sup compactbelow x by A13,ORDERS_2:2;
end;
then Imc is satisfying_axiom_K;
hence thesis by A2,Def4;
end;
theorem :: PROPOSITION 4.9.(ii)
for L be algebraic lower-bounded LATTICE, c be closure Function of L,L
st c is directed-sups-preserving holds c.:([#]CompactSublatt L) = [#]
CompactSublatt Image c
proof
let L be algebraic lower-bounded LATTICE;
let c be closure Function of L,L;
assume
A1: c is directed-sups-preserving;
now
c is idempotent by WAYBEL_1:def 13;
then
A2: rng c = {x where x is Element of L: x = c.x} by YELLOW_2:19;
c is idempotent by WAYBEL_1:def 13;
then reconsider Imc = Image c as complete LATTICE by A1,YELLOW_2:35;
let a9 be object;
assume
A3: a9 in [#]CompactSublatt Image c;
A4: the carrier of CompactSublatt Imc c= the carrier of Imc by YELLOW_0:def 13;
then reconsider a = a9 as Element of Imc by A3;
a is compact by A3,Def1;
then
A5: a << a by WAYBEL_3:def 2;
a9 in the carrier of Imc by A3,A4;
then a in rng c by YELLOW_0:def 15;
then consider a1 be Element of L such that
A6: a = a1 and
A7: a1 = c.a1 by A2;
compactbelow a1 is non empty directed Subset of L by Def4;
then
A8: c preserves_sup_of compactbelow a1 & ex_sup_of compactbelow a1,L by A1,
WAYBEL_0:75,def 37;
A9: c is monotone by A1,YELLOW_2:16;
now
let z be object;
assume z in c.:(compactbelow a1);
then consider v be object such that
A10: v in dom c and
A11: v in compactbelow a1 and
A12: z = c.v by FUNCT_1:def 6;
reconsider v9 = v as Element of L by A11;
A13: v in downarrow a1 /\ the carrier of CompactSublatt L by A11,Th5;
then v in downarrow a1 by XBOOLE_0:def 4;
then v9 <= a1 by WAYBEL_0:17;
then c.v9 <= a1 by A7,A9,WAYBEL_1:def 2;
then
A14: z in (downarrow a1) by A12,WAYBEL_0:17;
v in [#]CompactSublatt L by A13,XBOOLE_0:def 4;
then z in c.:([#]CompactSublatt L) by A10,A12,FUNCT_1:def 6;
hence z in (downarrow a1) /\ (c.:([#]CompactSublatt L)) by A14,
XBOOLE_0:def 4;
end;
then
A15: c.:(compactbelow a1) c= (downarrow a1) /\ (c.:([#] CompactSublatt L));
a = sup compactbelow a1 by A6,Def3;
then
A16: a = sup (c.:(compactbelow a1)) by A6,A7,A8,WAYBEL_0:def 31;
c.:(compactbelow a1) c= rng c by RELAT_1:111;
then
A17: c.:(compactbelow a1) c= the carrier of Imc by YELLOW_0:def 15;
A18: downarrow a /\ c.:([#]CompactSublatt L) is non empty directed Subset
of Imc
proof
(c.:[#]CompactSublatt L) /\ downarrow a is Subset of Imc;
then reconsider
D = downarrow a /\ c.:[#]CompactSublatt L as Subset of Imc;
A19: Bottom Imc <= a by YELLOW_0:44;
A20: now
let x,y be Element of Imc;
assume that
A21: x in D and
A22: y in D;
x in c.:([#]CompactSublatt L) by A21,XBOOLE_0:def 4;
then consider d be object such that
A23: d in dom c and
A24: d in [#]CompactSublatt L and
A25: x = c.d by FUNCT_1:def 6;
y in c.: ([#]CompactSublatt L) by A22,XBOOLE_0:def 4;
then consider e be object such that
A26: e in dom c and
A27: e in [#]CompactSublatt L and
A28: y = c.e by FUNCT_1:def 6;
reconsider e as Element of L by A26;
y in downarrow a by A22,XBOOLE_0:def 4;
then y <= a by WAYBEL_0:17;
then
A29: c.e <= a1 by A6,A28,YELLOW_0:59;
reconsider d as Element of L by A23;
A30: d <= d "\/" e by YELLOW_0:22;
x in downarrow a by A21,XBOOLE_0:def 4;
then x <= a by WAYBEL_0:17;
then c.d <= a1 by A6,A25,YELLOW_0:59;
then
A31: c.d "\/" c.e <= a1 by A29,YELLOW_0:22;
d "\/" e in the carrier of L;
then d "\/" e in dom c by FUNCT_2:def 1;
then c.(d "\/" e) in rng c by FUNCT_1:def 3;
then reconsider z = c.(d "\/" e) as Element of Imc by YELLOW_0:def 15;
take z;
A32: id(L) <= c by WAYBEL_1:def 14;
then id(L).e <= c.e by YELLOW_2:9;
then
A33: e <= c.e by FUNCT_1:18;
id(L).d <= c.d by A32,YELLOW_2:9;
then d <= c.d by FUNCT_1:18;
then d "\/" e <= c.d "\/" c.e by A33,YELLOW_3:3;
then d "\/" e <= a1 by A31,ORDERS_2:3;
then c.(d "\/" e) <= a1 by A7,A9,WAYBEL_1:def 2;
then z <= a by A6,YELLOW_0:60;
then
A34: z in downarrow a by WAYBEL_0:17;
A35: e <= d "\/" e by YELLOW_0:22;
then
A36: c.e <= c.(d "\/" e) by A9,WAYBEL_1:def 2;
e is compact by A27,Def1;
then e << e by WAYBEL_3:def 2;
then
A37: e << d "\/" e by A35,WAYBEL_3:2;
d is compact by A24,Def1;
then d << d by WAYBEL_3:def 2;
then d << d "\/" e by A30,WAYBEL_3:2;
then d "\/" e << d "\/" e by A37,WAYBEL_3:3;
then d "\/" e is compact by WAYBEL_3:def 2;
then
A38: d "\/" e in [#]CompactSublatt L by Def1;
d "\/" e in the carrier of L;
then d "\/" e in dom c by FUNCT_2:def 1;
then z in c.:([#]CompactSublatt L) by A38,FUNCT_1:def 6;
hence z in D by A34,XBOOLE_0:def 4;
c.d <= c.(d "\/" e) by A9,A30,WAYBEL_1:def 2;
hence x <= z & y <= z by A25,A28,A36,YELLOW_0:60;
end;
Bottom L is compact by WAYBEL_3:15;
then dom c = the carrier of L & Bottom L in [#]CompactSublatt L by Def1,
FUNCT_2:def 1;
then
A39: c.Bottom L in c.:([#]CompactSublatt L) by FUNCT_1:def 6;
A40: ex_sup_of {},L & {} c= the carrier of L by YELLOW_0:42;
A41: {} c= the carrier of Imc;
c.Bottom L = c."\/"({},L) by YELLOW_0:def 11
.= "\/"({},Imc) by A40,A41,WAYBEL_1:55
.= Bottom Imc by YELLOW_0:def 11;
then c.Bottom L in downarrow a by A19,WAYBEL_0:17;
hence thesis by A39,A20,WAYBEL_0:def 1,XBOOLE_0:def 4;
end;
A42: c.:([#]CompactSublatt L) c= rng c by RELAT_1:111;
now
let z be object;
assume
A43: z in (downarrow a1) /\ (c.:([#]CompactSublatt L));
then reconsider z1 = z as Element of L;
A44: z in c.:([#] CompactSublatt L) by A43,XBOOLE_0:def 4;
then reconsider z2 = z1 as Element of Imc by A42,YELLOW_0:def 15;
z in downarrow a1 by A43,XBOOLE_0:def 4;
then z1 <= a1 by WAYBEL_0:17;
then z2 <= a by A6,YELLOW_0:60;
then z in (downarrow a) by WAYBEL_0:17;
hence z in (downarrow a) /\ (c.: ([#]CompactSublatt L)) by A44,
XBOOLE_0:def 4;
end;
then
A45: (downarrow a1) /\ (c.:([#]CompactSublatt L)) c= (downarrow a) /\ (c
.: ([#]CompactSublatt L));
ex_sup_of c.:(compactbelow a1),L by A8,WAYBEL_0:def 31;
then
A46: a = "\/"(c.:(compactbelow a1),Imc) by A6,A7,A17,A16,WAYBEL_1:55;
ex_sup_of c.:(compactbelow a1),Imc & ex_sup_of (downarrow a) /\ (c.:(
[#] CompactSublatt L)),Imc by YELLOW_0:17;
then a <= "\/"((downarrow a) /\ (c.:([#]CompactSublatt L)),Imc) by A46,A15
,A45,XBOOLE_1:1,YELLOW_0:34;
then consider k be Element of Imc such that
A47: k in ((downarrow a) /\ (c.:([#]CompactSublatt L))) and
A48: a <= k by A5,A18,WAYBEL_3:def 1;
k in downarrow a by A47,XBOOLE_0:def 4;
then
A49: k <= a by WAYBEL_0:17;
k in c.:([#]CompactSublatt L) by A47,XBOOLE_0:def 4;
hence a9 in c.:([#]CompactSublatt L) by A48,A49,ORDERS_2:2;
end;
then
A50: [#]CompactSublatt Image c c= c.:([#]CompactSublatt L);
c.:([#]CompactSublatt L) c= [#]CompactSublatt Image c by A1,Th23;
hence thesis by A50,XBOOLE_0:def 10;
end;
begin :: Boolean Posets as Algebraic Lattices
theorem Th26:
for X,x be set holds x is Element of BoolePoset X iff x c= X
proof
let X,x be set;
LattPOSet BooleLatt X = RelStr(#the carrier of BooleLatt X, LattRel (
BooleLatt X)#) by LATTICE3:def 2;
then
A1: the carrier of BoolePoset X = the carrier of BooleLatt X by YELLOW_1:def 2
.= bool X by LATTICE3:def 1;
hence x is Element of BoolePoset X implies x c= X;
thus thesis by A1;
end;
theorem Th27:
for X be set for x,y be Element of BoolePoset X holds x << y iff
for Y be Subset-Family of X st y c= union Y ex Z be finite Subset of Y st x c=
union Z
proof
let X be set;
let x,y be Element of BoolePoset X;
LattPOSet BooleLatt X = RelStr(#the carrier of BooleLatt X, LattRel (
BooleLatt X)#) by LATTICE3:def 2;
then
A1: the carrier of BoolePoset X = the carrier of BooleLatt X by YELLOW_1:def 2
.= bool X by LATTICE3:def 1;
thus x << y implies for Y be Subset-Family of X st y c= union Y ex Z be
finite Subset of Y st x c= union Z
proof
assume
A2: x << y;
let Y be Subset-Family of X;
reconsider Y9 = Y as Subset of BoolePoset X by A1;
assume y c= union Y;
then y c= sup Y9 by YELLOW_1:21;
then y <= sup Y9 by YELLOW_1:2;
then consider Z be finite Subset of BoolePoset X such that
A3: Z c= Y and
A4: x <= sup Z by A2,WAYBEL_3:18;
reconsider Z9 = Z as finite Subset of Y by A3;
take Z9;
x c= sup Z by A4,YELLOW_1:2;
hence thesis by YELLOW_1:21;
end;
thus (for Y be Subset-Family of X st y c= union Y ex Z be finite Subset of Y
st x c= union Z) implies x << y
proof
assume
A5: for Y be Subset-Family of X st y c= union Y ex Z be finite Subset
of Y st x c= union Z;
now
let Y be Subset of BoolePoset X;
reconsider Y9 = Y as Subset-Family of X by A1;
assume y <= sup Y;
then y c= sup Y by YELLOW_1:2;
then y c= union Y9 by YELLOW_1:21;
then consider Z9 be finite Subset of Y9 such that
A6: x c= union Z9 by A5;
reconsider Z = Z9 as finite Subset of BoolePoset X by XBOOLE_1:1;
take Z;
thus Z c= Y;
x c= sup Z by A6,YELLOW_1:21;
hence x <= sup Z by YELLOW_1:2;
end;
hence thesis by WAYBEL_3:19;
end;
end;
theorem Th28:
for X be set for x be Element of BoolePoset X holds x is finite
iff x is compact
proof
let X be set;
let x be Element of BoolePoset X;
per cases;
suppose
A1: x is empty;
then x = Bottom BoolePoset X by YELLOW_1:18;
hence thesis by A1,WAYBEL_3:15;
end;
suppose
A2: x is non empty;
thus x is finite implies x is compact
proof
assume
A3: x is finite;
now
defpred P[object,object] means ex A being set st A = $2 & $1 in A;
let Y be Subset-Family of X;
assume
A4: x c= union Y;
A5: for e be object st e in x ex u be object st u in Y & P[e,u]
proof
let e be object;
assume e in x;
then ex u be set st e in u & u in Y by A4,TARSKI:def 4;
hence thesis;
end;
consider f be Function such that
A6: dom f = x and
A7: rng f c= Y and
A8: for e be object st e in x holds P[e,f.e] from FUNCT_1:sch 6(A5);
reconsider Z = rng f as Subset of Y by A7;
reconsider Z as finite Subset of Y by A3,A6,FINSET_1:8;
take Z;
now
let z be object;
assume
A9: z in x;
then P[z,f.z] by A8;
then z in f.z & f.z in Z by A6,FUNCT_1:def 3,A9;
hence z in union Z by TARSKI:def 4;
end;
hence x c= union Z;
end;
then x << x by Th27;
hence thesis by WAYBEL_3:def 2;
end;
thus x is compact implies x is finite
proof
reconsider x9 = x as non empty set by A2;
set Y = the set of all {y} where y is Element of x9 ;
Y c= bool X
proof
let t be object;
reconsider tt=t as set by TARSKI:1;
assume t in Y;
then
A10: ex y9 be Element of x9 st t = {y9};
for k being object st k in tt holds k in X by A10,Th26,TARSKI:def 3;
then tt c= X;
hence thesis;
end;
then reconsider Y as Subset-Family of X;
now
let y be object;
assume y in x;
then
A11: {y} in Y;
y in {y} by TARSKI:def 1;
hence y in union Y by A11,TARSKI:def 4;
end;
then
A12: x c= union Y;
assume x is compact;
then x << x by WAYBEL_3:def 2;
then consider Z be finite Subset of Y such that
A13: x c= union Z by A12,Th27;
now
let k be set;
assume k in Z;
then k in Y;
then ex y9 be Element of x9 st k = {y9};
hence k is finite;
end;
then union Z is finite by FINSET_1:7;
hence thesis by A13;
end;
end;
end;
theorem Th29:
for X be set for x be Element of BoolePoset X holds compactbelow
x = the set of all y where y is finite Subset of x
proof
let X be set;
let x be Element of BoolePoset X;
now
let z be object;
assume z in the set of all y where y is finite Subset of x ;
then consider z9 be finite Subset of x such that
A1: z9 = z;
x c= X by Th26;
then z9 c= X;
then reconsider z1 = z9 as Element of BoolePoset X by Th26;
z1 is compact & z1 <= x by Th28,YELLOW_1:2;
hence z in compactbelow x by A1;
end;
then
A2: the set of all y where y is finite Subset of x c= compactbelow x;
now
let z be object;
assume z in compactbelow x;
then consider z9 be Element of BoolePoset X such that
A3: z9 = z and
A4: x >= z9 & z9 is compact;
z9 is finite & z9 c= x by A4,Th28,YELLOW_1:2;
hence z in the set of all y where y is finite Subset of x by A3;
end;
then
compactbelow x c= the set of all y where y is finite Subset of x ;
hence thesis by A2,XBOOLE_0:def 10;
end;
theorem
for X be set for F be Subset of X holds F in the carrier of
CompactSublatt BoolePoset X iff F is finite
proof
let X be set;
let F be Subset of X;
reconsider F9 = F as Element of BoolePoset X by Th26;
A1: F c= F;
thus F in the carrier of CompactSublatt BoolePoset X implies F is finite
proof
assume
A2: F in the carrier of CompactSublatt BoolePoset X;
the carrier of CompactSublatt BoolePoset X c= the carrier of
BoolePoset X by YELLOW_0:def 13;
then reconsider F9 = F as Element of BoolePoset X by A2;
F9 <= F9 & F9 is compact by A2,Def1;
then F9 in compactbelow F9;
then F9 in the set of all y where y is finite Subset of F9 by Th29;
then ex F1 be finite Subset of F9 st F1 = F9;
hence thesis;
end;
assume F is finite;
then F in the set of all y where y is finite Subset of F9 by A1;
then F9 in compactbelow F9 by Th29;
then F9 is compact by Th4;
hence thesis by Def1;
end;
registration
let X be set;
let x be Element of BoolePoset X;
cluster compactbelow x -> lower directed;
coherence
proof
now
let a,b be set;
assume that
A1: a c= b and
A2: b in compactbelow x;
b in the set of all y where y is finite Subset of x by A2,Th29;
then ex b1 be finite Subset of x st b = b1;
then a is finite Subset of x by A1,XBOOLE_1:1;
then a in the set of all y where y is finite Subset of x ;
hence a in compactbelow x by Th29;
end;
hence
A3: compactbelow x is lower by WAYBEL_7:6;
now
let a,b be set;
assume that
A4: a in compactbelow x and
A5: b in compactbelow x;
b in the set of all y where y is finite Subset of x by A5,Th29;
then
A6: ex b1 be finite Subset of x st b = b1;
a in the set of all y where y is finite Subset of x by A4,Th29;
then ex a1 be finite Subset of x st a = a1;
then a \/ b is finite Subset of x by A6,XBOOLE_1:8;
then a \/ b in the set of all y where y is finite Subset of x ;
hence a \/ b in compactbelow x by Th29;
end;
hence thesis by A3,WAYBEL_7:8;
end;
end;
theorem Th31: :: EXAMPLES 4.11.(1b)
for X be set holds BoolePoset X is algebraic
proof
let X be set;
now
let x be Element of BoolePoset X;
A1: now
let a be Element of BoolePoset X;
assume
A2: a is_>=_than compactbelow x;
now
let t be object;
assume
A3: t in x;
A4: x c= X by Th26;
now
let k be object;
assume k in {t};
then k = t by TARSKI:def 1;
hence k in X by A3,A4;
end;
then {t} c= X;
then reconsider t1 = {t} as Element of BoolePoset X by Th26;
for k be object st k in {t} holds k in x by A3,TARSKI:def 1;
then {t} is finite Subset of x by TARSKI:def 3;
then {t} in the set of all y where y is finite Subset of x ;
then {t} in compactbelow x by Th29;
then t1 <= a by A2,LATTICE3:def 9;
then t in {t} & {t} c= a by TARSKI:def 1,YELLOW_1:2;
hence t in a;
end;
then x c= a;
hence x <= a by YELLOW_1:2;
end;
for b be Element of BoolePoset X st b in compactbelow x holds b <= x
by Th4;
then x is_>=_than compactbelow x by LATTICE3:def 9;
hence x = sup compactbelow x by A1,YELLOW_0:32;
end;
then ( for x be Element of BoolePoset X holds compactbelow x is non empty
directed)& BoolePoset X is satisfying_axiom_K;
hence thesis;
end;
registration
let X be set;
cluster BoolePoset X -> algebraic;
coherence by Th31;
end;