Journal of Formalized Mathematics
Volume 1, 1989
University of Bialystok
Copyright (c) 1989 Association of Mizar Users

### Introduction to Lattice Theory

by
Stanislaw Zukowski

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

```environ

vocabulary BINOP_1, BOOLE, FINSUB_1, FUNCT_1, SUBSET_1, LATTICES;
notation XBOOLE_0, ZFMISC_1, SUBSET_1, STRUCT_0, BINOP_1, FINSUB_1;
constructors STRUCT_0, BINOP_1, FINSUB_1, XBOOLE_0;
clusters FINSUB_1, STRUCT_0, SUBSET_1, ZFMISC_1, XBOOLE_0;
requirements SUBSET, BOOLE;

begin

definition
struct(1-sorted) /\-SemiLattStr
(# carrier -> set, L_meet -> BinOp of the carrier #);
end;

definition
struct(1-sorted) \/-SemiLattStr
(# carrier -> set, L_join -> BinOp of the carrier #);
end;

definition
struct(/\-SemiLattStr,\/-SemiLattStr) LattStr
(# carrier -> set, L_join, L_meet -> BinOp of the carrier #);
end;

definition
cluster strict non empty \/-SemiLattStr;
cluster strict non empty /\-SemiLattStr;
cluster strict non empty LattStr;
end;

definition let G be non empty \/-SemiLattStr,
p, q be Element of G;
func p"\/"q -> Element of G equals
:: LATTICES:def 1
(the L_join of G).(p,q);
end;

definition let G be non empty /\-SemiLattStr,
p, q be Element of G;
func p"/\"q -> Element of G equals
:: LATTICES:def 2
(the L_meet of G).(p,q);
end;

definition let G be non empty \/-SemiLattStr,
p, q be Element of G;
pred p [= q means
:: LATTICES:def 3
p"\/"q = q;
end;

definition let IT be non empty \/-SemiLattStr;
attr IT is join-commutative means
:: LATTICES:def 4
for a,b being Element of IT holds a"\/"b = b"\/"a;
attr IT is join-associative means
:: LATTICES:def 5
for a,b,c being Element of IT holds a"\/"(b"\/"c) = (a"\/"b)
"\/"c;
end;

definition let IT be non empty /\-SemiLattStr;
attr IT is meet-commutative means
:: LATTICES:def 6
for a,b being Element of IT holds a"/\"b = b"/\"a;
attr IT is meet-associative means
:: LATTICES:def 7
for a,b,c being Element of IT holds a"/\"(b"/\"c) = (a"/\"b)
"/\"c;
end;

definition let IT be non empty LattStr;
attr IT is meet-absorbing means
:: LATTICES:def 8
for a,b being Element of IT holds (a"/\"b)"\/"b = b;
attr IT is join-absorbing means
:: LATTICES:def 9
for a,b being Element of IT holds a"/\"(a"\/"b)=a;
end;

definition let IT be non empty LattStr;
attr IT is Lattice-like means
:: LATTICES:def 10
IT is join-commutative join-associative meet-absorbing
meet-commutative meet-associative join-absorbing;
end;

definition
cluster Lattice-like ->
join-commutative join-associative meet-absorbing
meet-commutative meet-associative join-absorbing
(non empty LattStr);
cluster join-commutative join-associative meet-absorbing
meet-commutative meet-associative join-absorbing
-> Lattice-like (non empty LattStr);
end;

definition
cluster strict join-commutative join-associative (non empty \/-SemiLattStr);
cluster strict meet-commutative meet-associative (non empty /\-SemiLattStr);
cluster strict Lattice-like (non empty LattStr);
end;

definition
mode Lattice is Lattice-like (non empty LattStr);
end;

definition let L be join-commutative (non empty \/-SemiLattStr),
a, b be Element of L;
redefine func a"\/"b;
commutativity;
end;

definition let L be meet-commutative (non empty /\-SemiLattStr),
a, b be Element of L;
redefine func a"/\"b;
commutativity;
end;

definition let IT be non empty LattStr;
attr IT is distributive means
:: LATTICES:def 11
for a,b,c being Element of IT
holds a"/\"(b"\/"c) = (a"/\"b)"\/"(a"/\"c);
end;

definition let IT be non empty LattStr;
attr IT is modular means
:: LATTICES:def 12
for a,b,c being Element of IT st a [= c
holds a"\/"(b"/\"c) = (a"\/"b)"/\"c;
end;

definition let IT be non empty /\-SemiLattStr;
attr IT is lower-bounded means
:: LATTICES:def 13
ex c being Element of IT st
for a being Element of IT holds c"/\"a = c & a"/\"c = c;
end;

definition let IT be non empty \/-SemiLattStr;
attr IT is upper-bounded means
:: LATTICES:def 14
ex c being Element of IT st
for a being Element of IT holds c"\/"a = c & a"\/"c = c;
end;

definition
cluster strict distributive lower-bounded upper-bounded modular Lattice;
end;

definition
mode D_Lattice is distributive Lattice;
mode M_Lattice is modular Lattice;
mode 0_Lattice is lower-bounded Lattice;
mode 1_Lattice is upper-bounded Lattice;
end;

definition let IT be non empty LattStr;
attr IT is bounded means
:: LATTICES:def 15
IT is lower-bounded upper-bounded;
end;

definition
cluster lower-bounded upper-bounded -> bounded (non empty LattStr);
cluster bounded -> lower-bounded upper-bounded (non empty LattStr);
end;

definition
cluster bounded strict Lattice;
end;

definition
mode 01_Lattice is bounded Lattice;
end;

definition let L be non empty /\-SemiLattStr;
assume  L is lower-bounded;
func Bottom L -> Element of L means
:: LATTICES:def 16
for a being Element of L holds it "/\" a = it & a "/\"
it = it;
end;

definition let L be non empty \/-SemiLattStr;
assume  L is upper-bounded;
func Top L -> Element of L means
:: LATTICES:def 17
for a being Element of L holds it "\/" a = it & a "\/"
it = it;
end;

definition let L be non empty LattStr,
a, b be Element of L;
pred a is_a_complement_of b means
:: LATTICES:def 18
a"\/"b = Top L & b"\/"a = Top L & a"/\"b = Bottom L & b"/\"a = Bottom L;
end;

definition let IT be non empty LattStr;
attr IT is complemented means
:: LATTICES:def 19
for b being Element of IT
ex a being Element of IT st a is_a_complement_of b;
end;

definition
cluster bounded complemented strict Lattice;
end;

definition
mode C_Lattice is complemented 01_Lattice;
end;

definition let IT be non empty LattStr;
attr IT is Boolean means
:: LATTICES:def 20
IT is bounded complemented distributive;
end;

definition
cluster Boolean -> bounded complemented distributive (non empty LattStr);
cluster bounded complemented distributive -> Boolean (non empty LattStr);
end;

definition
cluster Boolean strict Lattice;
end;

definition
mode B_Lattice is Boolean Lattice;
end;

reserve L for meet-absorbing join-absorbing meet-commutative
(non empty LattStr);
reserve a for Element of L;

canceled 16;

theorem :: LATTICES:17
a"\/"a = a;

theorem :: LATTICES:18
a"/\"a = a;

reserve L for Lattice;
reserve a, b, c for Element of L;

theorem :: LATTICES:19
(for a,b,c holds a"/\"(b"\/"c) = (a"/\"b)"\/"(a"/\"c))
iff
(for a,b,c holds a"\/"(b"/\"c) = (a"\/"b)"/\"(a"\/"c));

canceled;

theorem :: LATTICES:21
for L being meet-absorbing join-absorbing (non empty LattStr),
a, b being Element of L holds
a [= b iff a"/\"b = a;

theorem :: LATTICES:22
for L being meet-absorbing join-absorbing join-associative meet-commutative
(non empty LattStr),
a, b being Element of L holds
a [= a"\/"b;

theorem :: LATTICES:23
for L being meet-absorbing meet-commutative (non empty LattStr),
a, b being Element of L holds
a"/\"b [= a;

definition
let L be meet-absorbing join-absorbing meet-commutative (non empty LattStr),
a, b be Element of L;
redefine pred a [= b;
reflexivity;
end;

canceled;

theorem :: LATTICES:25
for L being join-associative (non empty \/-SemiLattStr),
a, b, c being Element of L holds
a [= b & b [= c implies a [= c;

theorem :: LATTICES:26
for L being join-commutative (non empty \/-SemiLattStr),
a, b being Element of L holds
a [= b & b [= a implies a=b;

theorem :: LATTICES:27
for L being meet-absorbing join-absorbing meet-associative (non empty LattStr),
a, b, c being Element of L holds
a [= b implies a"/\"c [= b"/\"c;

canceled;

theorem :: LATTICES:29
for L being Lattice holds
(for a,b,c being Element of L holds
(a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a) = (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a))
implies L is distributive;

reserve L for D_Lattice;
reserve a, b, c for Element of L;

canceled;

theorem :: LATTICES:31
a"\/"(b"/\"c) = (a"\/"b)"/\"(a"\/"c);

theorem :: LATTICES:32
c"/\"a = c"/\"b & c"\/"a = c"\/"b implies a=b;

canceled;

theorem :: LATTICES:34
(a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a) = (a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a);

definition
cluster distributive -> modular Lattice;
end;

reserve L for 0_Lattice;
reserve a for Element of L;

canceled 4;

theorem :: LATTICES:39
Bottom L"\/"a = a;

theorem :: LATTICES:40
Bottom L"/\"a = Bottom L;

theorem :: LATTICES:41
Bottom L [= a;

reserve L for 1_Lattice;
reserve a for Element of L;

canceled;

theorem :: LATTICES:43
Top L"/\"a = a;

theorem :: LATTICES:44
Top L"\/"a = Top L;

theorem :: LATTICES:45
a [= Top L;

definition let L be non empty LattStr,
x be Element of L;
assume  L is complemented D_Lattice;
func x` -> Element of L means
:: LATTICES:def 21
it is_a_complement_of x;
end;

reserve L for B_Lattice;
reserve a, b for Element of L;

canceled;

theorem :: LATTICES:47
a`"/\"a = Bottom L;

theorem :: LATTICES:48
a`"\/"a = Top L;

theorem :: LATTICES:49
a`` = a;

theorem :: LATTICES:50
( a"/\"b )` = a`"\/" b`;

theorem :: LATTICES:51
( a"\/"b )` = a`"/\" b`;

theorem :: LATTICES:52
b"/\"a = Bottom L iff b [= a`;

theorem :: LATTICES:53
a [= b implies b` [= a`;
```