:: LATTICE7 semantic presentation
:: deftheorem Def1 defines c= LATTICE7:def 1 :
for b
1 being
1-sorted for b
2, b
3 being
Subset of b
1 holds
( b
2 c= b
3 iff for b
4 being
Element of b
1 holds
( b
4 in b
2 implies b
4 in b
3 ) );
:: deftheorem Def2 defines Chain LATTICE7:def 2 :
for b
1 being
LATTICEfor b
2, b
3 being
Element of b
1 holds
( b
2 <= b
3 implies for b
4 being non
empty Chain of b
1 holds
( b
4 is
Chain of b
2,b
3 iff ( b
2 in b
4 & b
3 in b
4 & ( for b
5 being
Element of b
1 holds
( b
5 in b
4 implies ( b
2 <= b
5 & b
5 <= b
3 ) ) ) ) ) );
theorem Th1: :: LATTICE7:1
:: deftheorem Def3 defines height LATTICE7:def 3 :
theorem Th2: :: LATTICE7:2
theorem Th3: :: LATTICE7:3
theorem Th4: :: LATTICE7:4
theorem Th5: :: LATTICE7:5
theorem Th6: :: LATTICE7:6
theorem Th7: :: LATTICE7:7
:: deftheorem Def4 defines <(1) LATTICE7:def 4 :
for b
1 being
LATTICEfor b
2, b
3 being
Element of b
1 holds
( b
2 <(1) b
3 iff ( b
2 < b
3 & ( for b
4 being
Element of b
1 holds
not ( b
2 < b
4 & b
4 < b
3 ) ) ) );
theorem Th8: :: LATTICE7:8
:: deftheorem Def5 defines max LATTICE7:def 5 :
theorem Th9: :: LATTICE7:9
:: deftheorem Def6 defines Join-IRR LATTICE7:def 6 :
theorem Th10: :: LATTICE7:10
theorem Th11: :: LATTICE7:11
Lemma15:
for b1 being finite distributive LATTICE
for b2 being Element of b1 holds
( ( for b3 being Element of b1 holds
( b3 < b2 implies sup ((downarrow b3) /\ (Join-IRR b1)) = b3 ) ) implies sup ((downarrow b2) /\ (Join-IRR b1)) = b2 )
theorem Th12: :: LATTICE7:12
:: deftheorem Def7 defines LOWER LATTICE7:def 7 :
theorem Th13: :: LATTICE7:13
theorem Th14: :: LATTICE7:14
:: deftheorem Def8 defines Ring_of_sets LATTICE7:def 8 :
Lemma20:
for b1, b2 being non empty RelStr
for b3 being Function of b1,b2 holds
( b3 is infs-preserving implies b3 is meet-preserving )
Lemma21:
for b1, b2 being non empty RelStr
for b3 being Function of b1,b2 holds
( b3 is sups-preserving implies b3 is join-preserving )
theorem Th15: :: LATTICE7:15
theorem Th16: :: LATTICE7:16