:: MSUALG_7 semantic presentation

theorem :: MSUALG_7:1

for x, X being set holds
( x in the carrier of (EqRelLatt X) iff x is Equivalence_Relation of X )

proof end;

theorem Th2: :: MSUALG_7:2

for I being non empty set
for M being ManySortedSet of I holds id M is Equivalence_Relation of M

proof end;

theorem Th3: :: MSUALG_7:3

for I being non empty set
for M being ManySortedSet of I holds [|M,M|] is Equivalence_Relation of M

proof end;

registration

let I be non empty set ;
let M be ManySortedSet of I;
cluster EqRelLatt M -> bounded ;
coherence

proof end;

end;

theorem :: MSUALG_7:4

for I being non empty set
for M being ManySortedSet of I holds Bottom (EqRelLatt M) = id M

proof end;

theorem :: MSUALG_7:5

for I being non empty set
for M being ManySortedSet of I holds Top (EqRelLatt M) = [|M,M|]

proof end;

theorem Th6: :: MSUALG_7:6

for I being non empty set
for M being ManySortedSet of I
for X being Subset of (EqRelLatt M) holds X is SubsetFamily of [|M,M|]

proof end;

theorem Th7: :: MSUALG_7:7

for I being non empty set
for M being ManySortedSet of I
for a, b being Element of (EqRelLatt M)
for A, B being Equivalence_Relation of M st a = A & b = B holds
( a [= b iff A c= B )

proof end;

theorem Th8: :: MSUALG_7:8

for I being non empty set
for M being ManySortedSet of I
for X being Subset of (EqRelLatt M)
for X1 being SubsetFamily of [|M,M|] st X1 = X holds
for a, b being Equivalence_Relation of M st a = meet |:X1:| & b in X holds
a c= b

proof end;

theorem Th9: :: MSUALG_7:9

for I being non empty set
for M being ManySortedSet of I
for X being Subset of (EqRelLatt M)
for X1 being SubsetFamily of [|M,M|] st X1 = X & not X is empty holds
meet |:X1:| is Equivalence_Relation of M

proof end;

definition

let L be non empty LattStr ;
redefine attr L is complete means :Def1: :: MSUALG_7:def 1
for X being Subset of L ex a being Element of L st
( X is_less_than a & ( for b being Element of L st X is_less_than b holds
a [= b ) );
compatibility

proof end;

end;

:: deftheorem Def1 defines complete MSUALG_7:def 1 :

theorem Th10: :: MSUALG_7:10

for I being non empty set
for M being ManySortedSet of I holds EqRelLatt M is complete

proof end;

registration

let I be non empty set ;
let M be ManySortedSet of I;
cluster EqRelLatt M -> bounded complete ;
coherence by Th10;

end;

theorem :: MSUALG_7:11

for I being non empty set
for M being ManySortedSet of I
for X being Subset of (EqRelLatt M)
for X1 being SubsetFamily of [|M,M|] st X1 = X & not X is empty holds
for a, b being Equivalence_Relation of M st a = meet |:X1:| & b = "/\" X,(EqRelLatt M) holds
a = b

proof end;

definition

let L be Lattice;
let IT be SubLattice of L;
attr IT is /\-inheriting means :Def2: :: MSUALG_7:def 2
for X being Subset of IT holds "/\" X,L in the carrier of IT;
attr IT is \/-inheriting means :Def3: :: MSUALG_7:def 3
for X being Subset of IT holds "\/" X,L in the carrier of IT;

end;

:: deftheorem Def2 defines /\-inheriting MSUALG_7:def 2 :

:: deftheorem Def3 defines \/-inheriting MSUALG_7:def 3 :

theorem Th12: :: MSUALG_7:12

for L being Lattice
for L' being SubLattice of L
for a, b being Element of L
for a', b' being Element of L' st a = a' & b = b' holds
( a "\/" b = a' "\/" b' & a "/\" b = a' "/\" b' )

proof end;

theorem Th13: :: MSUALG_7:13

for L being Lattice
for L' being SubLattice of L
for X being Subset of L'
for a being Element of L
for a' being Element of L' st a = a' holds
( a is_less_than X iff a' is_less_than X )

proof end;

theorem Th14: :: MSUALG_7:14

for L being Lattice
for L' being SubLattice of L
for X being Subset of L'
for a being Element of L
for a' being Element of L' st a = a' holds
( X is_less_than a iff X is_less_than a' )

proof end;

theorem Th15: :: MSUALG_7:15

for L being complete Lattice
for L' being SubLattice of L st L' is /\-inheriting holds
L' is complete

proof end;

theorem Th16: :: MSUALG_7:16

for L being complete Lattice
for L' being SubLattice of L st L' is \/-inheriting holds
L' is complete

proof end;

registration

let L be complete Lattice;
cluster complete SubLattice of L;
existence

proof end;

end;

registration

let L be complete Lattice;
cluster /\-inheriting -> complete SubLattice of L;
coherence by Th15;
cluster \/-inheriting -> complete SubLattice of L;
coherence by Th16;

end;

theorem Th17: :: MSUALG_7:17

for L being complete Lattice
for L' being SubLattice of L st L' is /\-inheriting holds
for A' being Subset of L' holds "/\" A',L = "/\" A',L'

proof end;

theorem Th18: :: MSUALG_7:18

for L being complete Lattice
for L' being SubLattice of L st L' is \/-inheriting holds
for A' being Subset of L' holds "\/" A',L = "\/" A',L'

proof end;

theorem :: MSUALG_7:19

for L being complete Lattice
for L' being SubLattice of L st L' is /\-inheriting holds
for A being Subset of L
for A' being Subset of L' st A = A' holds
"/\" A = "/\" A' by Th17;

theorem :: MSUALG_7:20

for L being complete Lattice
for L' being SubLattice of L st L' is \/-inheriting holds
for A being Subset of L
for A' being Subset of L' st A = A' holds
"\/" A = "\/" A' by Th18;

definition

let r1, r2 be Real;
assume A1: r1 <= r2 ;
func RealSubLatt r1,r2 -> strict Lattice means :Def4: :: MSUALG_7:def 4
( the carrier of it = [.r1,r2.] & the L_join of it = maxreal || [.r1,r2.] & the L_meet of it = minreal || [.r1,r2.] );
existence

proof end;

uniqueness ;

end;

:: deftheorem Def4 defines RealSubLatt MSUALG_7:def 4 :

theorem Th21: :: MSUALG_7:21

for r1, r2 being Real st r1 <= r2 holds
RealSubLatt r1,r2 is complete

proof end;

theorem Th22: :: MSUALG_7:22

ex L' being SubLattice of RealSubLatt 0,1 st
( L' is \/-inheriting & not L' is /\-inheriting )

proof end;

theorem :: MSUALG_7:23

ex L being complete Lattice ex L' being SubLattice of L st
( L' is \/-inheriting & not L' is /\-inheriting )

proof end;

theorem Th24: :: MSUALG_7:24

ex L' being SubLattice of RealSubLatt 0,1 st
( L' is /\-inheriting & not L' is \/-inheriting )

proof end;

theorem :: MSUALG_7:25

ex L being complete Lattice ex L' being SubLattice of L st
( L' is /\-inheriting & not L' is \/-inheriting )

proof end;