Journal of Formalized Mathematics
Volume 3, 1991
University of Bialystok
Copyright (c) 1991 Association of Mizar Users

### Filters - Part II. Quotient Lattices Modulo Filters and Direct Product of Two Lattices

by
Grzegorz Bancerek

MML identifier: FILTER_1
[ Mizar article, MML identifier index ]

```environ

vocabulary LATTICES, FILTER_0, RELAT_1, EQREL_1, BINOP_1, BOOLE, FUNCT_1,
LATTICE2, ZF_LANG, SUBSET_1, FUNCT_4, WELLORD1, FUNCT_3, PARTFUN1,
FILTER_1;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2,
BINOP_1, EQREL_1, DOMAIN_1, WELLORD1, FUNCT_3, FUNCT_4, STRUCT_0,
LATTICES, FILTER_0, LATTICE2;
constructors BINOP_1, DOMAIN_1, WELLORD1, FUNCT_3, FUNCT_4, FILTER_0,
LATTICE2, MEMBERED, EQREL_1, XBOOLE_0;
clusters LATTICES, FILTER_0, LATTICE2, RELSET_1, STRUCT_0, SUBSET_1, MEMBERED,
ZFMISC_1, XBOOLE_0, PARTFUN1;
requirements SUBSET, BOOLE;

begin

reserve L,L1,L2 for Lattice, F1,F2 for Filter of L,
p,q,r,s for Element of L,
p1,q1,r1,s1 for Element of L1,
p2,q2,r2,s2 for Element of L2,
X,x,x1,x2,y,y1,y2 for set,
D,D1,D2 for non empty set, R for Relation,
R_D for Equivalence_Relation of D,
a,b,d for Element of D,
a1,b1,c1 for Element of D1,
a2,b2,c2 for Element of D2,
B for B_Lattice, F_B for Filter of B,
I for I_Lattice, F_I for Filter of I,
i,i1,i2,j,j1,j2,k for Element of I,
f1,g1 for BinOp of D1, f2,g2 for BinOp of D2;

theorem :: FILTER_1:1
F1 /\ F2 is Filter of L;

theorem :: FILTER_1:2
<.p.) = <.q.) implies p = q;

definition let L,F1,F2;
redefine func F1 /\ F2 -> Filter of L;
end;

definition let D,R;
mode UnOp of D,R -> UnOp of D means
:: FILTER_1:def 1
for x,y being Element of D st [x,y] in R
holds [it.x,it.y] in R;

mode BinOp of D,R -> BinOp of D means
:: FILTER_1:def 2
for x1,y1, x2,y2 being Element of D
st [x1,y1] in R & [x2,y2] in R holds
[it.(x1,x2),it.(y1,y2)] in R;
end;

reserve F,G for BinOp of D,R_D;

definition let D; let R be Equivalence_Relation of D;
mode UnOp of R is UnOp of D,R;
mode BinOp of R is BinOp of D,R;
cluster Class R -> non empty;
end;

definition let D; let R be Equivalence_Relation of D;
let d be Element of D;
redefine func Class(R,d) -> Element of Class R;
end;

definition let D; let R be Equivalence_Relation of D; let u be UnOp of D;
assume
u is UnOp of D,R;
func u /\/ R -> UnOp of Class R means
:: FILTER_1:def 3
for x,y st x in Class R & y in x holds it.x = Class(R,u.y);
end;

definition let D; let R be Equivalence_Relation of D; let b be BinOp of D;
assume
b is BinOp of D,R;
func b /\/ R -> BinOp of Class R means
:: FILTER_1:def 4
for x,y, x1,y1 st x in Class R & y in Class R & x1 in x & y1 in y holds
it.(x,y) = Class(R,b.(x1,y1));
end;

theorem :: FILTER_1:3
(F /\/ R_D).(Class(R_D,a), Class(R_D,b)) = Class(R_D, F.(a,b));

theorem :: FILTER_1:4
F is commutative implies F/\/R_D is commutative;

theorem :: FILTER_1:5
F is associative implies F/\/R_D is associative;

theorem :: FILTER_1:6
d is_a_left_unity_wrt F implies Class(R_D,d) is_a_left_unity_wrt F/\/R_D;

theorem :: FILTER_1:7
d is_a_right_unity_wrt F implies
Class(R_D,d) is_a_right_unity_wrt F/\/R_D;

theorem :: FILTER_1:8
d is_a_unity_wrt F implies Class(R_D,d) is_a_unity_wrt F/\/R_D;

theorem :: FILTER_1:9
F is_left_distributive_wrt G implies F/\/R_D is_left_distributive_wrt G/\/R_D;

theorem :: FILTER_1:10
F is_right_distributive_wrt G implies
F/\/R_D is_right_distributive_wrt G/\/R_D;

theorem :: FILTER_1:11
F is_distributive_wrt G implies F/\/R_D is_distributive_wrt G/\/R_D;

theorem :: FILTER_1:12
F absorbs G implies F/\/R_D absorbs G/\/R_D;

theorem :: FILTER_1:13
the L_join of I is BinOp of the carrier of I, equivalence_wrt F_I;

theorem :: FILTER_1:14
the L_meet of I is BinOp of the carrier of I, equivalence_wrt F_I;

definition let L be Lattice, F be Filter of L;
assume

L is I_Lattice;
func L /\/ F -> strict Lattice means
:: FILTER_1:def 5

for R being Equivalence_Relation of the carrier of L st
R = equivalence_wrt F holds
it = LattStr (#Class R, (the L_join of L)/\/R, (the L_meet of L)/\/R#);
end;

definition let L be Lattice, F be Filter of L,
a be Element of L;
assume

L is I_Lattice;
func a /\/ F -> Element of L /\/ F means
:: FILTER_1:def 6

for R being Equivalence_Relation of the carrier of L st
R = equivalence_wrt F holds it = Class(R, a);
end;

theorem :: FILTER_1:15
(i/\/F_I) "\/" (j/\/F_I) = (i"\/"j)/\/F_I &
(i/\/F_I) "/\" (j/\/F_I) = (i"/\"j)/\/F_I;

theorem :: FILTER_1:16
i/\/F_I [= j/\/F_I iff i => j in F_I;

theorem :: FILTER_1:17
(i"/\"j) => k = i => (j => k);

theorem :: FILTER_1:18
I is lower-bounded
implies I/\/F_I is 0_Lattice & Bottom (I/\/F_I) = (Bottom I)/\/F_I;

theorem :: FILTER_1:19
I/\/F_I is 1_Lattice & Top (I/\/F_I) = (Top I)/\/F_I;

theorem :: FILTER_1:20
I/\/F_I is implicative;

theorem :: FILTER_1:21
B/\/F_B is B_Lattice;

definition let D1,D2 be set;
let f1 be BinOp of D1; let f2 be BinOp of D2;
redefine func |:f1,f2:| -> BinOp of [:D1,D2:];
end;

theorem :: FILTER_1:22
|:f1,f2:|.([a1,a2],[b1,b2]) = [f1.(a1,b1),f2.(a2,b2)];

theorem :: FILTER_1:23
f1 is commutative & f2 is commutative iff |:f1,f2:| is commutative;

theorem :: FILTER_1:24
f1 is associative & f2 is associative iff |:f1,f2:| is associative;

theorem :: FILTER_1:25
a1 is_a_left_unity_wrt f1 & a2 is_a_left_unity_wrt f2 iff
[a1,a2] is_a_left_unity_wrt |:f1,f2:|;

theorem :: FILTER_1:26
a1 is_a_right_unity_wrt f1 & a2 is_a_right_unity_wrt f2 iff
[a1,a2] is_a_right_unity_wrt |:f1,f2:|;

theorem :: FILTER_1:27
a1 is_a_unity_wrt f1 & a2 is_a_unity_wrt f2 iff
[a1,a2] is_a_unity_wrt |:f1,f2:|;

theorem :: FILTER_1:28
f1 is_left_distributive_wrt g1 & f2 is_left_distributive_wrt g2 iff
|:f1,f2:| is_left_distributive_wrt |:g1,g2:|;

theorem :: FILTER_1:29
f1 is_right_distributive_wrt g1 & f2 is_right_distributive_wrt g2 iff
|:f1,f2:| is_right_distributive_wrt |:g1,g2:|;

theorem :: FILTER_1:30
f1 is_distributive_wrt g1 & f2 is_distributive_wrt g2 iff
|:f1,f2:| is_distributive_wrt |:g1,g2:|;

theorem :: FILTER_1:31
f1 absorbs g1 & f2 absorbs g2 iff |:f1,f2:| absorbs |:g1,g2:|;

definition let L1,L2 be non empty LattStr;
func [:L1,L2:] -> strict LattStr equals
:: FILTER_1:def 7

LattStr (#[:the carrier of L1, the carrier of L2:],
|:the L_join of L1, the L_join of L2:|,
|:the L_meet of L1, the L_meet of L2:|#);
end;

definition let L1,L2 be non empty LattStr;
cluster [:L1,L2:] -> non empty;
end;

definition let L be Lattice;
func LattRel L -> Relation equals
:: FILTER_1:def 8

{ [p,q] where p is Element of L,
q is Element of L: p [= q };
end;

theorem :: FILTER_1:32
[p,q] in LattRel L iff p [= q;

theorem :: FILTER_1:33
dom LattRel L = the carrier of L &
rng LattRel L = the carrier of L & field LattRel L = the carrier of L;

definition let L1,L2 be Lattice;
pred L1,L2 are_isomorphic means
:: FILTER_1:def 9
LattRel L1, LattRel L2 are_isomorphic;
reflexivity;
symmetry;
cluster [:L1,L2:] -> Lattice-like;
end;

theorem :: FILTER_1:34
for L1,L2,L3 being Lattice st L1,L2 are_isomorphic & L2,L3 are_isomorphic
holds L1,L3 are_isomorphic;

theorem :: FILTER_1:35
for L1,L2 being non empty LattStr st [:L1,L2:] is Lattice holds
L1 is Lattice & L2 is Lattice;

definition let L1,L2 be Lattice;
let a be Element of L1, b be Element of L2;
redefine func [a,b] -> Element of [:L1,L2:];
end;

theorem :: FILTER_1:36
[p1,p2] "\/" [q1,q2] = [p1"\/"q1,p2"\/"q2] &
[p1,p2] "/\" [q1,q2] = [p1"/\"q1,p2"/\"q2];

theorem :: FILTER_1:37
[p1,p2] [= [q1,q2] iff p1 [= q1 & p2 [= q2;

theorem :: FILTER_1:38
L1 is modular & L2 is modular iff [:L1,L2:] is modular;

theorem :: FILTER_1:39
L1 is D_Lattice & L2 is D_Lattice iff [:L1,L2:] is D_Lattice;

theorem :: FILTER_1:40
L1 is lower-bounded & L2 is lower-bounded
iff [:L1,L2:] is lower-bounded;

theorem :: FILTER_1:41
L1 is upper-bounded & L2 is upper-bounded
iff [:L1,L2:] is upper-bounded;

theorem :: FILTER_1:42
L1 is bounded & L2 is bounded
iff [:L1,L2:] is bounded;

theorem :: FILTER_1:43
L1 is 0_Lattice & L2 is 0_Lattice implies Bottom [:L1,L2:] = [Bottom L1,
Bottom L2];

theorem :: FILTER_1:44
L1 is 1_Lattice & L2 is 1_Lattice implies Top [:L1,L2:] = [Top L1,Top
L2];

theorem :: FILTER_1:45
L1 is 01_Lattice & L2 is 01_Lattice implies
(p1 is_a_complement_of q1 & p2 is_a_complement_of q2 iff
[p1,p2] is_a_complement_of [q1,q2]);

theorem :: FILTER_1:46
L1 is C_Lattice & L2 is C_Lattice iff [:L1,L2:] is C_Lattice;

theorem :: FILTER_1:47
L1 is B_Lattice & L2 is B_Lattice iff [:L1,L2:] is B_Lattice;

theorem :: FILTER_1:48
L1 is implicative & L2 is implicative
iff [:L1,L2:] is implicative;

theorem :: FILTER_1:49
[:L1,L2:].: = [:L1.:,L2.: :];

theorem :: FILTER_1:50
[:L1,L2:], [:L2,L1:] are_isomorphic;

reserve B for B_Lattice,
a,b,c,d for Element of B;

theorem :: FILTER_1:51
a <=> b = (a"/\"b)"\/"(a`"/\"b`);

theorem :: FILTER_1:52
(a => b)` = a "/\" b` & (a <=> b)` = (a "/\" b`) "\/" (a` "/\" b) &
(a <=> b)` = a <=> b` & (a <=> b)` = a` <=> b;

theorem :: FILTER_1:53
a <=> b = a <=> c implies b = c;

theorem :: FILTER_1:54
a <=> (a <=> b) = b;

theorem :: FILTER_1:55
(i"\/"j) => i = j => i & i => (i"/\"j) = i => j;

theorem :: FILTER_1:56
i => j [= i => (j"\/"k) & i => j [= (i"/\"k) => j &
i => j [= i => (k"\/"j) & i => j [= (k"/\"i) => j;

theorem :: FILTER_1:57
(i => k)"/\"(j => k) [= (i"\/"j) => k;

theorem :: FILTER_1:58
(i => j)"/\"(i => k) [= i => (j"/\"k);

theorem :: FILTER_1:59
i1 <=> i2 in F_I & j1 <=> j2 in F_I implies
(i1"\/"j1) <=> (i2"\/"j2) in F_I & (i1"/\"j1) <=> (i2"/\"j2) in F_I;

theorem :: FILTER_1:60
i in Class(equivalence_wrt F_I,k) & j in Class(equivalence_wrt F_I,k) implies
i"\/"j in Class(equivalence_wrt F_I,k) & i"/\"
j in Class(equivalence_wrt F_I,k);

theorem :: FILTER_1:61
c"\/"(c <=>d) in Class(equivalence_wrt <.d.),c) &
for b st b in Class(equivalence_wrt <.d.),c) holds b [= c"\/"(c <=>d);

theorem :: FILTER_1:62
B, [:B/\/<.a.),latt <.a.):] are_isomorphic;
```