:: Stone Lattices
:: by Adam Grabowski
::
:: Received October 22, 2015
:: Copyright (c) 2015-2017 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 SUBSET_1, LATTICES, LATSTONE, XBOOLE_0, TARSKI, EQREL_1, PBOOLE,
ORDINAL4, XXREAL_2, STRUCT_0, ZFMISC_1, LATTICE5, FILTER_0, LATTICE4,
TOPS_1, REWRITE1, MOEBIUS2, NAT_1, FINSET_1, XCMPLX_0, MOEBIUS1, RELAT_1,
BINOP_1, REALSET1, LATTICEA, XXREAL_0, CARD_1, NUMBERS, NEWTON, ARYTM_3;
notations TARSKI, ZFMISC_1, XBOOLE_0, RELAT_1, ORDINAL1, NAT_1, FINSET_1,
ARYTM_3, BINOP_1, REALSET1, SUBSET_1, NAT_D, ENUMSET1, XCMPLX_0,
XXREAL_0, CARD_1, NUMBERS, NEWTON, INT_2, FUNCT_1, FUNCT_2, STRUCT_0,
LATTICES, LATTICE3, FILTER_0, FILTER_1, FILTER_2, LATTICE4, MOEBIUS1,
MOEBIUS2, LATTICEA;
constructors DOMAIN_1, XBOOLE_0, FILTER_1, NAT_1, ORDINAL1, INT_2, NEWTON,
ARYTM_3, NAT_D, SQUARE_1, ENUMSET1, ZFMISC_1, FILTER_0, FINSET_1,
FILTER_2, LATTICE4, MOEBIUS1, XREAL_0, XCMPLX_0, XXREAL_0, BINOP_1,
REALSET2, LATTICE3, PARTFUN1, MOEBIUS2, SUBSET_1, BOOLEALG, REALSET1,
LATTICEA;
registrations STRUCT_0, LATTICES, PARTFUN1, FUNCT_2, REALSET1, INT_1,
FILTER_1, LATTICE3, MOEBIUS1, MOEBIUS2, XCMPLX_0, CARD_1, INT_2, INT_3,
NEWTON, NAT_1, SUBSET_1, ORDINAL1, NUMBERS, XXREAL_0, XREAL_0, RELSET_1,
PBOOLE, LATTICE6;
requirements SUBSET, BOOLE, REAL, NUMERALS, ARITHM;
definitions TARSKI, XBOOLE_0, LATTICES;
equalities FILTER_1;
theorems LATTICES, FILTER_0, FILTER_2, MSUALG_7, MOEBIUS2, MOEBIUS1, NEWTON,
ENUMSET1, INT_2, NAT_5, TARSKI, BOOLEALG, FILTER_1, ZFMISC_1, XTUPLE_0,
LATTICEA, LATTICE4;
begin :: Preliminaries
:: The main aim of this article is to provide a formal counterpart
:: for the paper of Jouni J\"arvinen "Lattice theory for rough sets",
:: in Transactions of Rough Sets, VI, LNCS vol. 4374, pp. 400--498, 2007.
:: The corresponding items are marked by the acronym LTRS. We deal
:: with Section 4.3, page 423--426.
theorem ThA:
for L being distributive Lattice, S being Sublattice of L holds
S is distributive
proof
let L be distributive Lattice, S be Sublattice of L;
let a,b,c be Element of S;
reconsider aa = a, bb = b, cc = c as Element of L by FILTER_2:68;
A2: b "\/" c = bb "\/" cc by MSUALG_7:11;
reconsider ab = aa "/\" bb, ac = aa "/\" cc as Element of L;
A4: ab = a "/\" b & ac = a "/\" c by MSUALG_7:11;
a "/\" (b "\/" c) = aa "/\" (bb "\/" cc) by MSUALG_7:11,A2
.= (aa "/\" bb) "\/" (aa "/\" cc) by LATTICES:def 11
.= (a "/\" b) "\/" (a "/\" c) by MSUALG_7:11,A4;
hence thesis;
end;
registration let L be distributive Lattice;
cluster -> distributive for Sublattice of L;
coherence by ThA;
end;
registration let L1, L2 be bounded Lattice;
cluster [:L1,L2:] -> bounded;
coherence by FILTER_1:41;
end;
reserve L for Lattice;
reserve I,P for non empty ClosedSubset of L;
theorem ThB:
L is lower-bounded & Bottom L in I implies latt (L,I) is lower-bounded
& Bottom latt (L,I) = Bottom L
proof
set b9 = the Element of latt (L,I);
reconsider b = b9 as Element of L by FILTER_2:68;
assume
A0: L is lower-bounded & Bottom L in I;
set c = Bottom L;
reconsider c9 = c as Element of latt (L,I) by FILTER_2:72,A0;
ex c9 being Element of latt (L,I) st
for a9 being Element of latt (L,I) holds c9 "/\" a9 = c9 & a9 "/\" c9 = c9
proof
take c9;
let a9 be Element of latt (L,I);
reconsider a = a9 as Element of L by FILTER_2:68;
thus c9 "/\" a9 = c "/\" a by FILTER_2:73 .= c9 by A0;
hence thesis;
end;
hence
W1: latt (L,I) is lower-bounded by LATTICES:def 13;
for a9 being Element of latt (L,I) holds c9 "/\" a9 = c9 & a9 "/\" c9 = c9
proof
let a9 be Element of latt (L,I);
reconsider a = a9 as Element of L by FILTER_2:68;
thus c9 "/\" a9 = c "/\" a by FILTER_2:73 .= c9 by A0;
hence thesis;
end;
hence thesis by LATTICES:def 16,W1;
end;
theorem ThC:
L is upper-bounded & Top L in I implies latt (L,I) is upper-bounded
& Top latt (L,I) = Top L
proof
set b9 = the Element of latt (L,I);
reconsider b = b9 as Element of L by FILTER_2:68;
assume
A0: L is upper-bounded & Top L in I;
set c = Top L;
reconsider c9 = c as Element of latt (L,I) by FILTER_2:72,A0;
ex c9 being Element of latt (L,I) st
for a9 being Element of latt (L,I) holds c9 "\/" a9 = c9 & a9 "\/" c9 = c9
proof
take c9;
let a9 be Element of latt (L,I);
reconsider a = a9 as Element of L by FILTER_2:68;
thus c9 "\/" a9 = c "\/" a by FILTER_2:73 .= c9 by A0;
hence thesis;
end;
hence
W1: latt (L,I) is upper-bounded by LATTICES:def 14;
for a9 being Element of latt (L,I) holds c9 "\/" a9 = c9 & a9 "\/" c9 = c9
proof
let a9 be Element of latt (L,I);
reconsider a = a9 as Element of L by FILTER_2:68;
thus c9 "\/" a9 = c "\/" a by FILTER_2:73 .= c9 by A0;
hence thesis;
end;
hence thesis by LATTICES:def 17,W1;
end;
begin :: Pseudocomplements in Lattices
::$N Pseudocomplement
definition let L be non empty LattStr;
let a,b be Element of L;
pred a is_a_pseudocomplement_of b means
a "/\" b = Bottom L &
for x being Element of L st b "/\" x = Bottom L holds x [= a;
end;
:: p. 122 6.1 Definition from Gratzer
definition let L be non empty LattStr;
attr L is pseudocomplemented means :def2:
for x being Element of L
ex y being Element of L st y is_a_pseudocomplement_of x;
end;
:: LTRS: Example 43(a)
theorem Th1:
for L being Boolean Lattice holds L is pseudocomplemented
proof
let L be Boolean Lattice;
L is pseudocomplemented
proof
let x be Element of L;
take y = x`;
for z being Element of L st x "/\" z = Bottom L holds z [= y
proof
let z be Element of L;
assume E1: x "/\" z = Bottom L;
z = z "/\" Top L .= z "/\" (x "\/" y) by LATTICES:21
.= (z "/\" x) "\/" (z "/\" y) by LATTICES:def 11 .= z "/\" y by E1;
hence z [= y by LATTICES:4;
end;
hence thesis by LATTICES:20;
end;
hence thesis;
end;
registration
cluster Boolean -> pseudocomplemented for Lattice;
coherence by Th1;
end;
registration
cluster Boolean pseudocomplemented bounded for Lattice;
existence
proof
set L = the Boolean Lattice;
take L;
thus thesis;
end;
end;
:: Pseudocomplements are unique
theorem Th2:
for L being pseudocomplemented lower-bounded Lattice,
a, b, x being Element of L st a is_a_pseudocomplement_of x &
b is_a_pseudocomplement_of x holds a = b
proof
let L be pseudocomplemented lower-bounded Lattice;
let a, b, x be Element of L;
assume that
a1: a is_a_pseudocomplement_of x and
b1: b is_a_pseudocomplement_of x;
a2: a "/\" x = Bottom L by a1;
b2: b "/\" x = Bottom L by b1;
ab1:b [= a by a1,b2;
a [= b by b1,a2;
hence a = b by ab1,LATTICES:8;
end;
:: The unique pseudocomplement
definition let L be non empty LattStr, x be Element of L;
assume A1:L is pseudocomplemented lower-bounded Lattice;
func x* -> Element of L means :def3:
it is_a_pseudocomplement_of x;
existence by def2,A1;
uniqueness by A1,Th2;
end;
theorem ThD:
for L being pseudocomplemented lower-bounded Lattice, x being Element of L
holds x* "/\" x = Bottom L
proof
let L be pseudocomplemented lower-bounded Lattice, x be Element of L;
x* is_a_pseudocomplement_of x by def3;
hence thesis;
end;
reserve L for lower-bounded pseudocomplemented Lattice;
:: LTRS: Lemma 42(a)
theorem Th5:
for a being Element of L holds a [= a**
proof
let a be Element of L;
a1: a** is_a_pseudocomplement_of a* by def3;
a* is_a_pseudocomplement_of a by def3;
hence thesis by a1;
end;
:: LTRS: Lemma 42(b)
theorem Th6:
for a, b being Element of L holds a [= b implies b* [= a*
proof
let a, b be Element of L;
assume a [= b; then
a1: a "/\" b = a by LATTICES:4;
a2: a* is_a_pseudocomplement_of a by def3;
a "/\" (b*) = (b* "/\" b) "/\" a by a1,LATTICES:def 7
.= Bottom L "/\" a by ThD .= Bottom L;
hence b* [= a* by a2;
end;
:: LTRS: Lemma 42(c)
theorem Th7:
for a being Element of L holds a* = a***
proof
let a be Element of L;
a1: a*** [= a* by Th6,Th5;
a* [= a*** by Th5;
hence thesis by a1,LATTICES:8;
end;
:: LTRS: Example 43(c)
theorem Th11:
for L being pseudocomplemented bounded Lattice holds (Bottom L)* = Top L
proof
let L be pseudocomplemented bounded Lattice;
Top L is_a_pseudocomplement_of Bottom L by LATTICES:19;
hence (Bottom L)* = Top L by def3;
end;
:: LTRS: Example 43(c)
theorem Th11a:
for L being pseudocomplemented bounded Lattice holds (Top L)* = Bottom L
proof
let L be pseudocomplemented bounded Lattice;
for x be Element of L st Top L "/\" x = Bottom L holds x [= Bottom L; then
Bottom L is_a_pseudocomplement_of Top L;
hence (Top L)* = Bottom L by def3;
end;
:: Pseudocomplements are precisely complements in complemented lattices.
:: A complement (if exists) is always a pseudocomplement.
:: LTRS: Example 43(a)
theorem ThE:
for L being Boolean Lattice, x being Element of L holds x` = x*
proof
let L be Boolean Lattice, x be Element of L;
a1: x* [= x`
proof
x* "/\" x = Bottom L by ThD;
hence thesis by LATTICES:25;
end;
x` [= x*
proof
a1: (x`) "/\" x = Bottom L by LATTICES:20;
x* is_a_pseudocomplement_of x by def3;
hence thesis by a1;
end;
hence thesis by a1,LATTICES:8;
end;
:: Connection between pseudocomplements and the functor from LATTICEA
theorem ThF:
for L being pseudocomplemented bounded Lattice, x, y being Element of L st
y is_a_pseudocomplement_of x holds y in PseudoComplements x
proof
let L be pseudocomplemented bounded Lattice, x, y be Element of L;
assume y is_a_pseudocomplement_of x; then
x "/\" y = Bottom L; then
y in { xx where xx is Element of L : x "/\" xx = Bottom L };
hence thesis by LATTICEA:def 8;
end;
theorem
for L being pseudocomplemented bounded Lattice, x being Element of L holds
x* in PseudoComplements x
proof
let L be pseudocomplemented bounded Lattice, a be Element of L;
a* is_a_pseudocomplement_of a by def3;
hence thesis by ThF;
end;
begin :: Skeleton of a Pseudocomplemented Lattice
definition let L be lower-bounded pseudocomplemented Lattice;
func Skeleton L -> Subset of L equals
the set of all a* where a is Element of L;
coherence
proof
{ a* where a is Element of L : not contradiction }
c= the carrier of L
proof
let x be object;
assume x in { a* where a is Element of L : not contradiction };
then consider a being Element of L such that
A1: x = a*;
thus thesis by A1;
end;
hence thesis;
end;
end;
theorem
for L being lower-bounded pseudocomplemented Lattice holds
Skeleton L = { a where a is Element of L : a** = a }
proof
let L be lower-bounded pseudocomplemented Lattice;
thus Skeleton L c= { a where a is Element of L : a** = a }
proof
let x be object;
assume x in Skeleton L; then
consider a being Element of L such that
a1: x = a*;
a*** = a* by Th7;
hence thesis by a1;
end;
let x be object;
assume x in { a where a is Element of L : a** = a };
then consider a being Element of L such that
a3: x = a & a** = a;
thus thesis by a3;
end;
:: LTRS: Lemma 44(a), p. 424
theorem Th13:
for L being lower-bounded pseudocomplemented Lattice, x being Element of L
holds x in Skeleton L iff x** = x
proof
let L be lower-bounded pseudocomplemented Lattice, x be Element of L;
hereby
assume x in Skeleton L; then
consider a being Element of L such that
a1: x = a*;
thus x** = x by a1,Th7;
end;
assume x** = x;
hence x in Skeleton L;
end;
registration let L be bounded pseudocomplemented Lattice;
cluster Skeleton L -> non empty;
coherence
proof
set x = the Element of L;
x* in Skeleton L;
hence thesis;
end;
end;
theorem
for L being pseudocomplemented distributive lower-bounded Lattice holds
for a, b being Element of L st a in Skeleton L & b in Skeleton L holds
a "/\" b in Skeleton L
proof
let L be pseudocomplemented distributive lower-bounded Lattice;
let a, b be Element of L;
assume a in Skeleton L & b in Skeleton L; then
A2: a = a** & b = b** by Th13;
a* [= (a "/\" b)* by Th6,LATTICES:6; then
B2: (a "/\" b)** [= a by A2,Th6;
b* [= (a "/\" b)* by Th6,LATTICES:6; then
(a "/\" b)** [= b** by Th6; then
B1: (a "/\" b)** [= a "/\" b by B2,FILTER_0:7,A2;
a "/\" b [= (a "/\" b)** by Th5; then
a "/\" b = (a "/\" b)** by B1,LATTICES:8;
hence thesis;
end;
begin :: Stone Identity
definition let L be non empty LattStr;
attr L is satisfying_Stone_identity means :SatStone:
for x being Element of L holds x* "\/" (x**) = Top L;
end;
theorem Th4:
for L being Boolean Lattice holds L is satisfying_Stone_identity
proof
let L be Boolean Lattice, x be Element of L;
x* "\/" ((x*)*) = Top L
proof
x* "\/" ((x*)*) = x` "\/" ((x*)*) by ThE .= x` "\/" ((x`)*) by ThE
.= x` "\/" ((x`)`) by ThE .= Top L by LATTICES:21;
hence thesis;
end;
hence thesis;
end;
registration
cluster Boolean -> satisfying_Stone_identity for Lattice;
coherence by Th4;
end;
registration
cluster satisfying_Stone_identity pseudocomplemented Boolean for Lattice;
existence
proof
set L = the Boolean Lattice;
take L;
thus thesis;
end;
end;
:: LTRS: Lemma 44(b), p. 424
theorem Th12:
for L being pseudocomplemented distributive bounded Lattice holds
L is satisfying_Stone_identity iff for a, b being Element of L holds
(a "/\" b)* = (a*) "\/" (b*)
proof
let L be pseudocomplemented distributive bounded Lattice;
hereby
assume
a11: L is satisfying_Stone_identity;
let a, b be Element of L;
set g = a "/\" b;
a0: g "/\" (a* "\/" (b*))
= ((a*) "/\" g) "\/" (g "/\" (b*)) by LATTICES:def 11
.= (((a*) "/\" a) "/\" b) "\/" (g "/\" (b*))
by LATTICES:def 7
.= (((a*) "/\" a) "/\" b) "\/" (a "/\" (b "/\" (b*)))
by LATTICES:def 7
.= ((Bottom L) "/\" b) "\/" (a "/\" (b "/\" (b*))) by ThD
.= ((Bottom L) "/\" b) "\/" (a "/\" Bottom L) by ThD .= Bottom L;
for x be Element of L st g "/\" x = Bottom L holds x [= a* "\/" (b*)
proof
let x be Element of L;
set z = b "/\" x;
set w = x "/\" (a**);
assume a1: g "/\" x = Bottom L;
a0: Bottom L [= b "/\" x "/\" (a**) by LATTICES:16;
a2: (b "/\" x) "/\" a = Bottom L by a1,LATTICES:def 7;
a* is_a_pseudocomplement_of a by def3; then
b "/\" x "/\" (a**) [= a* "/\" (a**) by LATTICES:9,a2;
then b "/\" x "/\" (a**) [= Bottom L by ThD; then
b "/\" x "/\" (a**) = Bottom L by LATTICES:8,a0; then
a4: x "/\" (a**) "/\" b = Bottom L by LATTICES:def 7;
b* is_a_pseudocomplement_of b by def3; then
a5: x "/\" (a**) [= b* by a4;
a6: x "/\" (a*) [= a* by LATTICES:6;
x = x "/\" Top L .= x "/\" ((a*) "\/" (a**)) by a11
.= (x "/\" (a*)) "\/" (x "/\" (a**)) by LATTICES:def 11;
hence x [= a* "\/" (b*) by a5,a6,FILTER_0:4;
end; then
(a*) "\/" (b*) is_a_pseudocomplement_of (a "/\" b) by a0;
hence (a "/\" b)* = (a*) "\/" (b*) by def3;
end;
assume
a1: for a, b being Element of L holds (a "/\" b)* = (a*) "\/" (b*);
let x be Element of L;
x "/\" (x*) = Bottom L by ThD; then
x* "\/" (x**) = (Bottom L)* by a1;
hence thesis by Th11;
end;
::$N Stone algebra
definition
let L be Lattice;
attr L is Stone means
L is pseudocomplemented distributive bounded satisfying_Stone_identity;
end;
registration
cluster Stone ->
pseudocomplemented distributive bounded satisfying_Stone_identity
for Lattice;
coherence;
cluster pseudocomplemented distributive bounded satisfying_Stone_identity
-> Stone for Lattice;
coherence;
end;
:: LTRS:
theorem
for L being pseudocomplemented distributive bounded Lattice holds
L is satisfying_Stone_identity iff for a, b being Element of L st
a in Skeleton L & b in Skeleton L holds a "\/" b in Skeleton L
proof
let L be pseudocomplemented distributive bounded Lattice;
hereby
assume
a0: L is satisfying_Stone_identity;
let a, b be Element of L;
assume
X0: a in Skeleton L & b in Skeleton L; then
consider x being Element of L such that
X1: a = x*;
consider y being Element of L such that
X2: b = y* by X0;
(x "/\" y)* in Skeleton L;
hence a "\/" b in Skeleton L by X1,X2,Th12,a0;
end;
assume
a1: for a, b being Element of L st a in Skeleton L & b in Skeleton L
holds a "\/" b in Skeleton L;
let y be Element of L;
y* in Skeleton L & y** in Skeleton L; then
sp: (y* "\/" (y**))** = y* "\/" (y**) by Th13,a1;
r1: y** [= (y* "/\" (y**))* by Th6,LATTICES:6;
y*** [= (y* "/\" (y**))* by LATTICES:6,Th6; then
y* [= (y* "/\" (y**))* by Th7; then
r: (y*) "\/" (y**) [= (y* "/\" (y**))* "\/" ((y* "/\" (y**))*)
by r1,FILTER_0:4;
s1: (y* "\/" (y**))* [= y** by Th6,LATTICES:5;
(y* "\/" (y**))* [= y*** by Th6,LATTICES:5; then
s2: (y* "\/" (y**))* [= y* by Th7;
(y* "\/" (y**))* "/\" ((y* "\/" (y**))*) [= y* "/\" (y**)
by s1,s2,FILTER_0:5; then
(y* "/\" (y**))* [= y* "\/" (y**) by sp,Th6;then
(y* "/\" (y**))* = y* "\/" (y**) by r,LATTICES:8;
then (Bottom L)* = y* "\/" (y**) by ThD;
hence thesis by Th11;
end;
reserve L for Stone Lattice;
:: LTRS: Proposition 45
theorem Lm1:
Top L in Skeleton L & Bottom L in Skeleton L
proof
(Top L)* in Skeleton L & (Bottom L)* in Skeleton L;
hence thesis by Th11a,Th11;
end;
definition let L be Stone Lattice, a be Element of L;
attr a is skeletal means
a in Skeleton L;
end;
registration
let L be Stone Lattice;
cluster Top L -> skeletal;
coherence by Lm1;
cluster Bottom L -> skeletal;
coherence by Lm1;
end;
:: LTRS: Proposition 45
registration let L be Stone Lattice;
cluster Skeleton L -> join-closed meet-closed;
coherence
proof
set S = Skeleton L;
tt: for p,q being Element of L st p in S & q in S holds p "\/" q in S
proof
let p,q be Element of L;
assume p in S & q in S; then
p** = p & q** = q by Th13; then
p "\/" q = (p* "/\" (q*))* by Th12;
hence thesis;
end;
for p,q being Element of L st p in S & q in S holds p "/\" q in S
proof
let p,q be Element of L;
assume
T1: p in S & q in S; then
T3: p = p** by Th13;
p * [= (p "/\" q)* by Th6,LATTICES:6; then
T2: (p "/\" q)** [= p** by Th6;
T4: q = q** by T1,Th13;
q* [= (p "/\" q)* by Th6,LATTICES:6; then
(p "/\" q)** [= q** by Th6; then
T5: (p "/\" q)** [= p "/\" q by T3,T4,T2,FILTER_0:7;
p "/\" q [= (p "/\" q)** by Th5; then
p "/\" q = (p "/\" q)** by T5,LATTICES:8;
hence thesis;
end;
hence thesis by tt,LATTICES:def 24,def 25;
end;
end;
definition let L be Stone Lattice;
redefine func Skeleton L -> ClosedSubset of L;
coherence;
end;
:: Skeleton equipped with the lattice structure
definition let L be Stone Lattice;
func SkelLatt L -> Sublattice of L equals
latt (L,Skeleton L);
coherence;
end;
registration let L be Stone Lattice;
cluster SkelLatt L -> distributive;
coherence;
end;
theorem
Bottom L = Bottom SkelLatt L & Top L = Top SkelLatt L
proof
Bottom L in Skeleton L & Top L in Skeleton L by Lm1;
hence thesis by ThB,ThC;
end;
LattIsComplemented:
latt (L,Skeleton L) is complemented
proof
set P = Skeleton L;
A0: Bottom L in P & Top L in P by Lm1;
set LL = latt (L,P);
for b being Element of LL
ex a being Element of LL st a is_a_complement_of b
proof
let b be Element of LL;
reconsider bb = b as Element of L by FILTER_2:68;
set aa = bb*;
bb* in Skeleton L; then
reconsider a = aa as Element of LL by FILTER_2:72;
b in the carrier of LL; then
z1: b in Skeleton L by FILTER_2:72;
aa "\/" bb = bb*** "\/" bb by z1,Th13 .= bb*** "\/" (bb**) by z1,Th13
.= Top L by SatStone; then
A1: a "\/" b = Top L by FILTER_2:73 .= Top LL by A0,ThC;
aa "/\" bb = Bottom L by ThD; then
a "/\" b = Bottom L by FILTER_2:73 .= Bottom LL by A0,ThB;
hence thesis by LATTICES:def 18,A1;
end;
hence thesis by LATTICES:def 19;
end;
:: LTRS: Proposition 45
registration let L be Stone Lattice;
cluster SkelLatt L -> Boolean;
coherence
proof
Top L in Skeleton L by Lm1; then
A0: latt (L, Skeleton L) is upper-bounded by ThC;
Bottom L in Skeleton L by Lm1; then
a2: latt (L, Skeleton L) is lower-bounded by ThB;
latt (L, Skeleton L) is complemented by LattIsComplemented;
hence thesis by a2,A0;
end;
end;
begin :: Dense Elements in Lattices
definition let L be lower-bounded Lattice;
func DenseElements L -> Subset of L equals
{ a where a is Element of L : a* = Bottom L };
coherence
proof
{ a where a is Element of L : a* = Bottom L } c= the carrier of L
proof
let x be object;
assume x in { a where a is Element of L : a* = Bottom L }; then
consider aa being Element of L such that
A1: aa = x & aa* = Bottom L;
thus thesis by A1;
end;
hence thesis;
end;
end;
theorem TopIsDense:
Top L in DenseElements L
proof
(Top L)* = Bottom L by Th11a;
hence thesis;
end;
registration let L be Stone Lattice;
cluster DenseElements L -> non empty;
coherence by TopIsDense;
end;
definition let L be Stone Lattice, a be Element of L;
attr a is dense means
a in DenseElements L;
end;
registration let L be Stone Lattice;
cluster Top L -> dense;
coherence by TopIsDense;
end;
theorem DenseIsBot:
for L being Stone Lattice, x being Element of L st x in DenseElements L holds
x* = Bottom L
proof
let L be Stone Lattice, x be Element of L;
assume x in DenseElements L; then
consider aa being Element of L such that
A1: x = aa & aa* = Bottom L;
thus thesis by A1;
end;
registration let L be Stone Lattice;
cluster DenseElements L -> join-closed meet-closed;
coherence
proof
A1: for x,y being Element of L st x in DenseElements L & y in DenseElements L
holds x "\/" y in DenseElements L
proof
let x,y be Element of L;
assume x in DenseElements L & y in DenseElements L; then
A2: x* = Bottom L & y* = Bottom L by DenseIsBot;
F1: (x "\/" y)* [= x* by Th6,LATTICES:5;
Bottom L [= (x "\/" y)* by LATTICES:16; then
(x "\/" y)* = Bottom L by LATTICES:8,A2,F1;
hence x "\/" y in DenseElements L;
end;
for x,y being Element of L st x in DenseElements L & y in DenseElements L
holds x "/\" y in DenseElements L
proof
let x,y be Element of L;
assume x in DenseElements L & y in DenseElements L; then
A2: x* = Bottom L & y* = Bottom L by DenseIsBot;
(x "/\" y)* = x* "\/" (y*) by Th12;
hence x "/\" y in DenseElements L by A2;
end;
hence thesis by A1,LATTICES:def 25,def 24;
end;
end;
definition let L be Stone Lattice;
redefine func DenseElements L -> ClosedSubset of L;
coherence;
end;
:: Set of dense elements equipped with the lattice structure
definition let L be Stone Lattice;
func DenseLatt L -> Sublattice of L equals
latt (L, DenseElements L);
coherence;
end;
registration let L be Stone Lattice;
cluster DenseLatt L -> distributive;
coherence;
end;
:: Meet-decomposition of elements of a lattice in terms
:: of skeletal and dense elements
theorem
for L being Stone Lattice, a being Element of L holds
ex b, c being Element of L st a = b "/\" c & b in Skeleton L &
c in DenseElements L
proof
let L be Stone Lattice, a be Element of L;
A1: a = (a** "/\" a) "\/" Bottom L by LATTICES:4,Th5
.= (a** "/\" a) "\/" (a** "/\" (a*)) by ThD
.= a** "/\" (a "\/" (a*)) by LATTICES:def 11;
take b = a**;
take c = a "\/" (a*);
G1: (a "\/" (a*))* [= a* by Th6,LATTICES:5;
(a "\/" (a*))* [= a** by Th6,LATTICES:5; then
(a "\/" (a*))* [= a* "/\" (a**) by FILTER_0:7,G1; then
(a "\/" (a*))* [= Bottom L by ThD; then
(a "\/" (a*))* = Bottom L by BOOLEALG:9;
hence thesis by A1;
end;
begin :: An Example: Lattice of Natural Divisors
theorem
for p being Prime holds NatDivisors p = {1, p}
proof
let p be Prime;
a1: NatDivisors (p|^1)={ p|^k where k is Element of NAT:k <= 1 } by NAT_5:19;
{ p|^k where k is Element of NAT : k <= 1 } = {1, p}
proof
thus { p|^k where k is Element of NAT : k <= 1 } c= {1, p}
proof
let x be object;
assume x in { p|^k where k is Element of NAT : k <= 1 }; then
consider kk being Element of NAT such that
A2: x = p |^ kk & kk <= 1;
kk = 0 or ... or kk = 1 by A2; then
x = 1 or x = p |^ 1 by NEWTON:4,A2;
hence thesis by TARSKI:def 2;
end;
let x be object;
assume x in {1, p}; then
x = 1 or x = p by TARSKI:def 2; then
x = p |^ 0 or x = p |^ 1 by NEWTON:4;
hence thesis;
end;
hence thesis by a1;
end;
theorem DivisorsSquare:
for p being Prime holds NatDivisors (p * p) = {1, p, p*p}
proof
let p be Prime;
a1: NatDivisors (p|^2)={p|^k where k is Element of NAT : k <= 2} by NAT_5:19;
{ p|^k where k is Element of NAT : k <= 2 } = {1, p, p * p}
proof
thus { p|^k where k is Element of NAT : k <= 2 } c= {1, p, p * p}
proof
let x be object;
assume x in { p|^k where k is Element of NAT : k <= 2 }; then
consider kk being Element of NAT such that
A2: x = p |^ kk & kk <= 2;
kk = 0 or ... or kk = 2 by A2; then
x = 1 or x = p or x = p * p by A2,NEWTON:81,4;
hence thesis by ENUMSET1:def 1;
end;
let x be object;
assume x in {1, p, p * p}; then
x = 1 or x = p or x = p * p by ENUMSET1:def 1; then
x = p |^ 0 or x = p |^ 1 or x = p |^ 2 by NEWTON:4,81;
hence thesis;
end;
hence thesis by a1,NEWTON:81;
end;
registration let n be non zero Nat;
cluster Divisors_Lattice n -> finite;
coherence
proof
NatDivisors n is finite;
hence thesis by MOEBIUS2:def 10;
end;
end;
registration
cluster complete for Boolean Lattice;
existence
proof
reconsider n = 1 as non zero Nat;
set L = Divisors_Lattice n;
L is Boolean by MOEBIUS1:22;
hence thesis;
end;
end;
registration let p be Prime;
cluster Divisors_Lattice (p) -> Boolean;
coherence
proof
p is square-free by MOEBIUS1:23;
hence thesis;
end;
end;
:: Hence it is pseudocomplemented, and Stone
registration let p be Prime;
cluster Divisors_Lattice (p * p) -> pseudocomplemented;
coherence
proof
set L = Divisors_Lattice (p * p);
the carrier of Divisors_Lattice (p * p) =
NatDivisors (p*p) by MOEBIUS2:def 10; then
A1: the carrier of L = {1, p, p * p} by DivisorsSquare;
for x being Element of L
ex y being Element of L st y is_a_pseudocomplement_of x
proof
let x be Element of L;
per cases by ENUMSET1:def 1,A1;
suppose
C1: x = 1;
reconsider pp = p * p as Element of L by A1,ENUMSET1:def 1;
x = Bottom L by C1,MOEBIUS2:64; then
B1: pp "/\" x = Bottom L;
for y being Element of L st x "/\" y = Bottom L holds y [= pp
proof
let y be Element of L;
assume x "/\" y = Bottom L;
pp = Top L by MOEBIUS2:64;
hence thesis by LATTICES:19;
end; then
pp is_a_pseudocomplement_of x by B1;
hence thesis;
end;
suppose
B0: x = p;
reconsider pp = Bottom L as Element of L;
for y being Element of L st x "/\" y = Bottom L holds y [= pp
proof
let y be Element of L;
w3: y = 1 or y = p or y = p * p by A1,ENUMSET1:def 1;
assume
W1: x "/\" y = Bottom L;
W2: y <> Top L
proof
assume y = Top L; then
p = 1 by B0,MOEBIUS2:64,W1;
hence thesis by INT_2:def 4;
end;
y <> p
proof
assume y = p; then
1 = p by W1,MOEBIUS2:64,B0;
hence thesis by INT_2:def 4;
end;
hence thesis by W2,w3,MOEBIUS2:64;
end; then
pp is_a_pseudocomplement_of x;
hence thesis;
end;
suppose
C1: x = p * p;
reconsider pp = 1 as Element of L by A1,ENUMSET1:def 1;
b1: pp = Bottom L by MOEBIUS2:64;
for y being Element of L st x "/\" y = Bottom L holds y [= pp
proof
let y be Element of L;
assume
S1: x "/\" y = Bottom L;
s3: y = 1 or y = p * p or y = p by A1,ENUMSET1:def 1;
R2: Top L = p * p by MOEBIUS2:64;
y <> p
proof
assume y = p; then
p = 1 by MOEBIUS2:64,S1,R2,C1;
hence thesis by INT_2:def 4;
end;
hence thesis by s3,C1,MOEBIUS2:64,S1;
end; then
pp is_a_pseudocomplement_of x by b1;
hence thesis;
end;
end;
hence thesis;
end;
end;
theorem PsCompl:
for L being Lattice, p being Prime, x being Element of L st
L = Divisors_Lattice (p * p) & x = p holds x* = Bottom L
proof
let L be Lattice, p be Prime, x be Element of L;
assume that
A1: L = Divisors_Lattice (p * p) and
B0: x = p;
reconsider pp = Bottom L as Element of L;
for y being Element of L st x "/\" y = Bottom L holds y [= pp
proof
let y be Element of L;
y in the carrier of L; then
y in NatDivisors (p*p) by A1,MOEBIUS2:def 10; then
y in {1, p, p*p} by DivisorsSquare; then
w3: y = 1 or y = p or y = p * p by ENUMSET1:def 1;
assume
W1: x "/\" y = Bottom L;
W2: y <> Top L
proof
assume y = Top L; then
x = 1 by MOEBIUS2:64,A1,W1;
hence thesis by INT_2:def 4,B0;
end;
y <> p
proof
assume y = p; then
1 = p by A1,MOEBIUS2:64,B0,W1;
hence thesis by INT_2:def 4;
end;
hence thesis by W2,w3,A1,MOEBIUS2:64;
end; then
pp is_a_pseudocomplement_of x by A1;
hence thesis by def3,A1;
end;
:: The example of a lattice which is Stone but not Boolean
registration let p be Prime;
cluster Divisors_Lattice (p * p) -> satisfying_Stone_identity;
coherence
proof
set L = Divisors_Lattice (p * p);
let x be Element of L;
x in the carrier of L; then
x in NatDivisors (p * p) by MOEBIUS2:def 10; then
x in {1, p, p * p} by DivisorsSquare; then
per cases by ENUMSET1:def 1;
suppose
x = 1; then
x = Bottom L by MOEBIUS2:64; then
x* = Top L by Th11;
hence thesis;
end;
suppose
x = p; then
x* = Bottom L by PsCompl;
hence thesis by Th11;
end;
suppose
x = p * p; then
x = Top L by MOEBIUS2:64; then
x* = Bottom L by Th11a;
hence thesis by Th11;
end;
end;
end;
registration let p be Prime;
cluster Divisors_Lattice (p * p) -> non Boolean Stone;
coherence
proof
p |^ 2 divides (p * p) by NEWTON:81; then
p * p is square-containing by MOEBIUS1:def 1;
hence thesis by MOEBIUS2:65;
end;
end;
registration
cluster Stone non Boolean for Lattice;
existence
proof
set p = the Prime;
take Divisors_Lattice (p * p);
thus thesis;
end;
end;
:: Every squarefree number generates Stone lattice as it is Boolean
begin :: Products of Pseudocomplemented Lattices
reserve L1, L2 for Lattice;
reserve p1, q1 for Element of L1;
reserve p2, q2 for Element of L2;
theorem ProductPsCompl:
L1 is 01_Lattice & L2 is 01_Lattice implies
(p1 is_a_pseudocomplement_of q1 & p2 is_a_pseudocomplement_of q2 iff
[p1,p2] is_a_pseudocomplement_of [q1,q2])
proof
assume
A0: L1 is 01_Lattice & L2 is 01_Lattice;
set L = [:L1,L2:];
thus p1 is_a_pseudocomplement_of q1 & p2 is_a_pseudocomplement_of q2
implies [p1,p2] is_a_pseudocomplement_of [q1,q2]
proof
assume
A1: p1 is_a_pseudocomplement_of q1 & p2 is_a_pseudocomplement_of q2;
set a = [p1,p2], b = [q1,q2];
a2: a "/\" b = [Bottom L1, Bottom L2] by A1,FILTER_1:35;
for x being Element of L st b "/\" x = Bottom L holds x [= a
proof
let x be Element of L;
assume
Z1: b "/\" x = Bottom L;
consider x1,x2 being object such that
Z2: x1 in the carrier of L1 & x2 in the carrier of L2 & x = [x1,x2]
by ZFMISC_1:def 2;
reconsider x1 as Element of L1 by Z2;
reconsider x2 as Element of L2 by Z2;
[q1,q2] "/\" [x1,x2] = [Bottom L1,Bottom L2] by FILTER_1:42,A0,Z2,Z1;
then
Z3: [q1 "/\" x1,q2 "/\" x2] = [Bottom L1,Bottom L2] by FILTER_1:35; then
T1: q1 "/\" x1 = Bottom L1 by XTUPLE_0:1;
T2: q2 "/\" x2 = Bottom L2 by Z3,XTUPLE_0:1;
Z4: x1 [= p1 by T1,A1;
x2 [= p2 by A1,T2;
hence thesis by Z2,FILTER_1:36,Z4;
end;
hence thesis by a2,FILTER_1:42,A0;
end;
assume
W0: [p1,p2] is_a_pseudocomplement_of [q1,q2]; then
[p1,p2] "/\" [q1,q2] = [Bottom L1,Bottom L2] by A0,FILTER_1:42; then
w1: [p1 "/\" q1, p2 "/\" q2] = [Bottom L1,Bottom L2] by FILTER_1:35; then
W1: p1 "/\" q1 = Bottom L1 & p2 "/\" q2 = Bottom L2 by XTUPLE_0:1;
w2: for xx1 being Element of L1 st q1 "/\" xx1 = Bottom L1 holds xx1 [= p1
proof
let xx1 be Element of L1;
assume q1 "/\" xx1 = Bottom L1; then
[q1 "/\" xx1, q2 "/\" p2] = Bottom [:L1, L2:]
by W1,FILTER_1:42,A0; then
[q1,q2] "/\" [xx1, p2] = Bottom [:L1, L2:] by FILTER_1:35; then
[xx1,p2] [= [p1,p2] by W0;
hence thesis by FILTER_1:36;
end;
for xx2 being Element of L2 st q2 "/\" xx2 = Bottom L2 holds xx2 [= p2
proof
let xx2 be Element of L2;
assume q2 "/\" xx2 = Bottom L2; then
[q1 "/\" p1, q2 "/\" xx2] = Bottom [:L1, L2:]
by W1,FILTER_1:42,A0; then
[q1,q2] "/\" [p1, xx2] = Bottom [:L1, L2:] by FILTER_1:35; then
[p1,xx2] [= [p1,p2] by W0;
hence thesis by FILTER_1:36;
end;
hence thesis by w2,w1,XTUPLE_0:1;
end;
theorem PsComplProduct:
L1 is 01_Lattice & L2 is 01_Lattice implies (L1 is pseudocomplemented &
L2 is pseudocomplemented iff [:L1,L2:] is pseudocomplemented)
proof
assume
A0: L1 is 01_Lattice & L2 is 01_Lattice;
thus L1 is pseudocomplemented & L2 is pseudocomplemented implies
[:L1,L2:] is pseudocomplemented
proof
assume
A1: L1 is pseudocomplemented & L2 is pseudocomplemented;
for x being Element of [:L1,L2:] holds ex y being Element of [:L1,L2:]
st y is_a_pseudocomplement_of x
proof
let x be Element of [:L1,L2:];
consider x1, x2 being object such that
B1: x1 in the carrier of L1 & x2 in the carrier of L2 & x = [x1,x2]
by ZFMISC_1:def 2;
reconsider x1 as Element of L1 by B1;
reconsider x2 as Element of L2 by B1;
consider xx1 being Element of L1 such that
A2: xx1 is_a_pseudocomplement_of x1 by A1;
consider xx2 being Element of L2 such that
A3: xx2 is_a_pseudocomplement_of x2 by A1;
take y = [xx1,xx2];
thus thesis by B1,A2,A3,ProductPsCompl,A0;
end;
hence thesis;
end;
assume
aa: [:L1,L2:] is pseudocomplemented;
for x being Element of L1 holds ex y being Element of L1 st
y is_a_pseudocomplement_of x
proof
let x be Element of L1;
set x2 = the Element of L2;
consider y being Element of [:L1,L2:] such that
B0: y is_a_pseudocomplement_of [x,x2] by aa;
consider y1, y2 being object such that
B1: y1 in the carrier of L1 & y2 in the carrier of L2 & y = [y1,y2]
by ZFMISC_1:def 2;
reconsider y1 as Element of L1 by B1;
reconsider y2 as Element of L2 by B1;
take y1;
thus thesis by B1,B0,ProductPsCompl,A0;
end;
hence L1 is pseudocomplemented;
for x being Element of L2 holds ex y being Element of L2 st
y is_a_pseudocomplement_of x
proof
let x be Element of L2;
set x2 = the Element of L1;
consider y being Element of [:L1,L2:] such that
B0: y is_a_pseudocomplement_of [x2,x] by aa;
consider y1, y2 being object such that
B1: y1 in the carrier of L1 & y2 in the carrier of L2 & y = [y1,y2]
by ZFMISC_1:def 2;
reconsider y1 as Element of L1 by B1;
reconsider y2 as Element of L2 by B1;
take y2;
thus thesis by B1,B0,ProductPsCompl,A0;
end;
hence thesis;
end;
registration let L1, L2 be pseudocomplemented 01_Lattice;
cluster [:L1,L2:] -> pseudocomplemented;
coherence by PsComplProduct;
end;
theorem ProductPCompl:
L1 is pseudocomplemented 01_Lattice & L2 is pseudocomplemented 01_Lattice
implies [p1,p2]* = [p1*,p2*]
proof
assume
A1: L1 is pseudocomplemented 01_Lattice &
L2 is pseudocomplemented 01_Lattice;
A3: p1* is_a_pseudocomplement_of p1 by def3,A1;
p2* is_a_pseudocomplement_of p2 by def3,A1; then
[p1*,p2*] is_a_pseudocomplement_of [p1,p2] by A3,ProductPsCompl,A1;
hence thesis by def3,A1;
end;
reserve L1, L2 for non empty Lattice;
theorem ProductIsSStone:
L1 is satisfying_Stone_identity pseudocomplemented 01_Lattice &
L2 is satisfying_Stone_identity pseudocomplemented 01_Lattice implies
[:L1,L2:] is satisfying_Stone_identity
proof
assume that
A1: L1 is satisfying_Stone_identity pseudocomplemented 01_Lattice and
A2: L2 is satisfying_Stone_identity pseudocomplemented 01_Lattice;
set L = [:L1,L2:];
for x being Element of L holds x* "\/" (x**) = Top L
proof
let x be Element of L;
consider x1,x2 being object such that
A3: x1 in the carrier of L1 & x2 in the carrier of L2 & x = [x1,x2]
by ZFMISC_1:def 2;
reconsider x1 as Element of L1 by A3;
reconsider x2 as Element of L2 by A3;
X1: [x1*,x2*] = x* by ProductPCompl,A3,A1,A2;
A4: x1* "\/" (x1**) = Top L1 by A1,SatStone;
[x1* "\/" (x1**), x2* "\/" (x2**)] = [Top L1, Top L2]
by A4,A2,SatStone
.= Top L by FILTER_1:43,A1,A2; then
Top L = [x1*,x2*] "\/" [x1**,x2**] by FILTER_1:35
.= x* "\/" (x**) by X1,ProductPCompl,A1,A2;
hence thesis;
end;
hence thesis;
end;
theorem
L1 is Stone & L2 is Stone implies [:L1,L2:] is Stone
by ProductIsSStone,FILTER_1:38;
registration let L1, L2 be Stone Lattice;
cluster [:L1,L2:] -> Stone;
coherence by ProductIsSStone,FILTER_1:38;
end;
begin :: Special Construction: B squared
reserve B for Boolean Lattice;
:: LTRS: Lemma 46 - definition
definition let B be Boolean Lattice;
func B squared -> Subset of [:B,B:] equals
{ [a,b] where a, b is Element of B : a [= b };
coherence
proof
{ [a,b] where a, b is Element of B : a [= b } c= the carrier of [:B,B:]
proof
let x be object;
assume x in { [a,b] where a, b is Element of B : a [= b }; then
consider a1, b1 being Element of B such that
A1: x = [a1,b1] & a1 [= b1;
thus thesis by A1;
end;
hence thesis;
end;
end;
registration let B be Boolean Lattice;
cluster B squared -> non empty;
coherence
proof
set x = Bottom B;
take [x,x];
thus thesis;
end;
end;
registration let B be Boolean Lattice;
cluster B squared -> join-closed meet-closed;
coherence
proof
set S = B squared;
AA: for p,q being Element of [:B,B:] st p in S & q in S holds p "\/" q in S
proof
let p,q be Element of [:B,B:];
assume
A0: p in S & q in S; then
consider a1, b1 being Element of B such that
A1: p = [a1,b1] & a1 [= b1;
consider a2, b2 being Element of B such that
A2: q = [a2,b2] & a2 [= b2 by A0;
A3: p "\/" q = [a1 "\/" a2, b1 "\/" b2] by FILTER_1:35,A1,A2;
a1 "\/" a2 [= b1 "\/" b2 by A1,A2,FILTER_0:4;
hence thesis by A3;
end;
for p,q being Element of [:B,B:] st p in S & q in S holds p "/\" q in S
proof
let p,q be Element of [:B,B:];
assume
A0: p in S & q in S; then
consider a1, b1 being Element of B such that
A1: p = [a1,b1] & a1 [= b1;
consider a2, b2 being Element of B such that
A2: q = [a2,b2] & a2 [= b2 by A0;
A3: p "/\" q = [a1 "/\" a2, b1 "/\" b2] by FILTER_1:35,A1,A2;
a1 "/\" a2 [= b1 "/\" b2 by A1,A2,FILTER_0:5;
hence thesis by A3;
end;
hence thesis by LATTICES:def 24,def 25,AA;
end;
end;
definition let B be Boolean Lattice;
redefine func B squared -> non empty ClosedSubset of [:B,B:];
coherence;
end;
definition let B be Boolean Lattice;
func B squared-latt -> Lattice equals
latt ([:B,B:], B squared);
coherence;
end;
theorem SquaredCarrier:
the carrier of (B squared-latt) = B squared
proof
set L = [:B,B:];
set P = B squared;
consider o1,o2 being BinOp of P such that
A1: o1 = (the L_join of L)||P & o2 = (the L_meet of L)||P &
latt ([:B,B:], P) = LattStr (#P, o1, o2#) by FILTER_2:def 14;
thus thesis by A1;
end;
theorem
[Bottom B, Bottom B] in the carrier of (B squared-latt)
proof
[Bottom B, Bottom B] in B squared;
hence thesis by SquaredCarrier;
end;
theorem
[Top B, Top B] in the carrier of (B squared-latt)
proof
[Top B, Top B] in B squared;
hence thesis by SquaredCarrier;
end;
registration let B be Boolean Lattice;
cluster B squared-latt -> lower-bounded;
coherence
proof
Bottom [:B,B:] = [Bottom B, Bottom B] by FILTER_1:42; then
Bottom [:B,B:] in B squared;
hence thesis by ThB;
end;
cluster B squared-latt -> upper-bounded;
coherence
proof
Top [:B,B:] = [Top B, Top B] by FILTER_1:43; then
Top [:B,B:] in B squared;
hence thesis by ThC;
end;
end;
theorem SquaredBottom:
Bottom (B squared-latt) = [Bottom B, Bottom B]
proof
[Bottom B, Bottom B] in B squared; then
Bottom [:B,B:] in B squared by FILTER_1:42; then
Bottom latt ([:B,B:], B squared) = Bottom [:B,B:] by ThB;
hence thesis by FILTER_1:42;
end;
theorem SquaredTop:
Top (B squared-latt) = [Top B, Top B]
proof
[Top B, Top B] in B squared; then
Top [:B,B:] in B squared by FILTER_1:43; then
Top latt ([:B,B:], B squared) = Top [:B,B:] by ThC;
hence thesis by FILTER_1:43;
end;
registration let B be Boolean Lattice;
cluster B squared-latt -> pseudocomplemented;
coherence
proof
set L = B squared-latt;
for x being Element of L holds ex y being Element of L st
y is_a_pseudocomplement_of x
proof
let x be Element of L;
x in the carrier of L; then
x in B squared by SquaredCarrier; then
consider x1,x2 being Element of B such that
A1: x = [x1,x2] & x1 [= x2;
I1: [x2`,x2`] in B squared;
reconsider y = [x2`,x2`] as Element of L by I1,SquaredCarrier;
take y;
Z1: x "/\" y = [x1,x2] "/\" [x2`,x2`] by A1,MSUALG_7:11
.= [x1 "/\" x2`, x2 "/\" x2`] by FILTER_1:35
.= [x1 "/\" x2`, Bottom B] by LATTICES:20;
x2` [= x1` by A1,LATTICES:26; then
x1 "/\" x2` [= x1 "/\" x1` by FILTER_0:5; then
x1 "/\" x2` [= Bottom B by LATTICES:20; then
tt: x1 "/\" x2` = Bottom B by BOOLEALG:9;
for w being Element of L st x "/\" w = Bottom L holds w [= y
proof
let w be Element of L;
assume
O1: x "/\" w = Bottom L;
w in the carrier of L; then
w in B squared by SquaredCarrier; then
consider w1,w2 being Element of B such that
Y1: w = [w1,w2] & w1 [= w2;
[x1,x2] "/\" [w1,w2] = Bottom L by O1,Y1,A1,MSUALG_7:11; then
[x1,x2] "/\" [w1,w2] = [Bottom B, Bottom B] by SquaredBottom; then
[x1 "/\" w1,x2 "/\" w2] = [Bottom B, Bottom B] by FILTER_1:35; then
x1 "/\" w1 = Bottom B & x2 "/\" w2 = Bottom B by XTUPLE_0:1; then
Y2: w2 [= x2` by LATTICES:25; then
w1 [= x2` by Y1,LATTICES:7; then
[w1,w2] "/\" [x2`,x2`] = [w1,w2] by LATTICES:4,Y2,FILTER_1:36; then
w "/\" y = w by Y1,MSUALG_7:11;
hence thesis by LATTICES:4;
end;
hence thesis by tt,Z1,SquaredBottom;
end;
hence thesis;
end;
end;
theorem PseudoInSquared:
for L being Lattice, x1,x2 being Element of B, x being Element of L st
L = B squared-latt & x = [x1,x2] holds x* = [x2`,x2`]
proof
let L be Lattice, x1,x2 be Element of B, x be Element of L;
assume
A0: L = B squared-latt & x = [x1,x2];
x in the carrier of L; then
x in B squared by SquaredCarrier,A0; then
consider xx1,xx2 being Element of B such that
W1: x = [xx1,xx2] & xx1 [= xx2;
aa: xx1 = x1 & xx2 = x2 by W1,A0,XTUPLE_0:1;
[x2`,x2`] in B squared; then
reconsider y = [x2`,x2`] as Element of L by A0,SquaredCarrier;
Z1: x "/\" y = [x1,x2] "/\" [x2`,x2`] by A0,MSUALG_7:11
.= [x1 "/\" x2`, x2 "/\" x2`] by FILTER_1:35
.= [x1 "/\" x2`, Bottom B] by LATTICES:20;
x2` [= x1` by aa,W1,LATTICES:26; then
x1 "/\" x2` [= x1 "/\" x1` by FILTER_0:5; then
x1 "/\" x2` [= Bottom B by LATTICES:20; then
tt: x1 "/\" x2` = Bottom B by BOOLEALG:9;
for w being Element of L st x "/\" w = Bottom L holds w [= y
proof
let w be Element of L;
assume
O1: x "/\" w = Bottom L;
w in the carrier of L; then
w in B squared by A0,SquaredCarrier; then
consider w1,w2 being Element of B such that
Y1: w = [w1,w2] & w1 [= w2;
[x1,x2] "/\" [w1,w2] = Bottom L by O1,Y1,A0,MSUALG_7:11; then
[x1,x2] "/\" [w1,w2] = [Bottom B, Bottom B]
by A0,SquaredBottom; then
[x1 "/\" w1,x2 "/\" w2] = [Bottom B, Bottom B]
by FILTER_1:35; then
x1 "/\" w1 = Bottom B & x2 "/\" w2 = Bottom B by XTUPLE_0:1; then
Y2: w2 [= x2` by LATTICES:25; then
w1 [= x2` by Y1,LATTICES:7; then
[w1,w2] "/\" [x2`,x2`] = [w1,w2] by LATTICES:4,Y2,FILTER_1:36; then
w "/\" y = w by Y1,MSUALG_7:11,A0;
hence thesis by LATTICES:4;
end; then
y is_a_pseudocomplement_of x by tt,Z1,A0,SquaredBottom;
hence thesis by def3,A0;
end;
registration let B be Boolean Lattice;
cluster B squared-latt -> satisfying_Stone_identity;
coherence
proof
set L = B squared-latt;
for x being Element of L holds x* "\/" (x**) = Top L
proof
let x be Element of L;
x in the carrier of L; then
x in B squared by SquaredCarrier; then
consider x1,x2 being Element of B such that
W1: x = [x1,x2] & x1 [= x2;
[x2`,x2`] in B squared; then
reconsider y = [x2`,x2`] as Element of L by SquaredCarrier;
W0: x* = [x2`,x2`] by W1,PseudoInSquared; then
x** = [x2``,x2``] by PseudoInSquared; then
x* "\/" (x**) = [x2`,x2`] "\/" [x2,x2] by W0,MSUALG_7:11
.= [x2` "\/" x2, x2` "\/" x2] by FILTER_1:35
.= [Top B, x2` "\/" x2] by LATTICES:21
.= [Top B, Top B] by LATTICES:21 .= Top L by SquaredTop;
hence thesis;
end;
hence thesis;
end;
end;
:: LTRS: Lemma 46, p. 425
registration let B be Boolean Lattice;
cluster B squared-latt -> Stone;
coherence;
end;
:: Skeleton and dense elements in the lattice B squared
theorem
Skeleton (B squared-latt) = the set of all [a,a] where a is Element of B
proof
set L = B squared-latt;
Skeleton (B squared-latt) = the set of all [a,a] where a is Element of B
proof
thus Skeleton (B squared-latt)
c= the set of all [a,a] where a is Element of B
proof
let x be object;
assume x in Skeleton L; then
consider a being Element of L such that
A1: x = a*;
a in the carrier of L; then
a in B squared by SquaredCarrier; then
consider a1,a2 being Element of B such that
A2: a = [a1,a2] & a1 [= a2;
a* = [a2`,a2`] by A2,PseudoInSquared;
hence thesis by A1;
end;
let x be object;
assume x in the set of all [a,a] where a is Element of B; then
consider a being Element of B such that
B1: x = [a,a];
set b = [a`,a`];
b in B squared; then
reconsider b as Element of L by FILTER_2:72;
reconsider a1 = b* as Element of L;
b* = [a``,a``] by PseudoInSquared .= [a,a];
hence thesis by B1;
end;
hence thesis;
end;
theorem
DenseElements (B squared-latt) = the set of all [a,Top B]
where a is Element of B
proof
set L = B squared-latt;
thus DenseElements L c= the set of all [a,Top B] where a is Element of B
proof
let x be object;
assume x in DenseElements L; then
consider a being Element of L such that
A1: x = a & a* = Bottom L;
x in the carrier of L by A1; then
x in B squared by SquaredCarrier; then
consider a1, a2 being Element of B such that
A2: x = [a1,a2] & a1 [= a2;
A3: a* = [Bottom B, Bottom B] by A1,SquaredBottom;
a* = [a2`,a2`] by A1,A2,PseudoInSquared; then
a2`` = (Bottom B)` by A3,XTUPLE_0:1; then
a2 = Top B by LATTICE4:30;
hence thesis by A2;
end;
let x be object;
assume x in the set of all [a,Top B] where a is Element of B; then
consider a being Element of B such that
A1: x = [a, Top B];
a [= Top B by LATTICES:19; then
x in B squared by A1; then
reconsider y = x as Element of L by SquaredCarrier;
y* = [(Top B)`,(Top B)`] by A1,PseudoInSquared; then
y* = [Bottom B, (Top B)`] by LATTICE4:29
.= [Bottom B, Bottom B] by LATTICE4:29
.= Bottom L by SquaredBottom;
hence thesis;
end;