:: LATTICE6 semantic presentation
Lemma25:
for L being finite Lattice
for X being Subset of L holds
( X is empty or ex a being Element of (LattPOSet L) st
( a in X & ( for b being Element of (LattPOSet L) st b in X & b <> a holds
not b <= a ) ) )
Lemma72:
for L being finite Lattice
for X being Subset of L holds
( X is empty or ex a being Element of (LattPOSet L) st
( a in X & ( for b being Element of (LattPOSet L) st b in X & b <> a holds
not a <= b ) ) )
Lemma73:
for L being finite Lattice
for X being Subset of L holds
( X is empty or ex a being Element of (LattPOSet L) st
for b being Element of (LattPOSet L) st b in X holds
b <= a )
Lemma75:
for L being finite Lattice ex a being Element of (LattPOSet L) st
for b being Element of (LattPOSet L) holds b <= a
Lemma77:
for L being finite Lattice holds L is complete
:: deftheorem Def1 defines % LATTICE6:def 1 :
:: deftheorem Def2 defines % LATTICE6:def 2 :
Lemma93:
for L being finite Lattice holds (LattPOSet L) ~ is well_founded
:: deftheorem Def3 defines noetherian LATTICE6:def 3 :
:: deftheorem Def4 defines co-noetherian LATTICE6:def 4 :
theorem Th1: :: LATTICE6:1
Lemma99:
for L being finite Lattice holds
( L is noetherian & L is co-noetherian )
:: deftheorem Def5 defines is-upper-neighbour-of LATTICE6:def 5 :
theorem Th2: :: LATTICE6:2
theorem Th3: :: LATTICE6:3
theorem Th4: :: LATTICE6:4
theorem Th5: :: LATTICE6:5
theorem Th6: :: LATTICE6:6
theorem Th7: :: LATTICE6:7
theorem Th8: :: LATTICE6:8
:: deftheorem Def6 defines *' LATTICE6:def 6 :
:: deftheorem Def7 defines *' LATTICE6:def 7 :
:: deftheorem Def8 defines completely-meet-irreducible LATTICE6:def 8 :
:: deftheorem Def9 defines completely-join-irreducible LATTICE6:def 9 :
theorem Th9: :: LATTICE6:9
theorem Th10: :: LATTICE6:10
theorem Th11: :: LATTICE6:11
theorem Th12: :: LATTICE6:12
theorem Th13: :: LATTICE6:13
theorem Th14: :: LATTICE6:14
theorem Th15: :: LATTICE6:15
theorem Th16: :: LATTICE6:16
Lemma120:
for L being Lattice
for a, b being Element of L holds
( a "/\" b = (a % ) "/\" (b % ) & a "\/" b = (a % ) "\/" (b % ) )
theorem Th17: :: LATTICE6:17
theorem Th18: :: LATTICE6:18
theorem Th19: :: LATTICE6:19
theorem Th20: :: LATTICE6:20
:: deftheorem Def10 defines atomic LATTICE6:def 10 :
:: deftheorem Def11 defines co-atomic LATTICE6:def 11 :
theorem Th21: :: LATTICE6:21
theorem Th22: :: LATTICE6:22
:: deftheorem Def12 defines atomic LATTICE6:def 12 :
Lemma129:
ex L being Lattice st
( L is atomic & L is complete )
:: deftheorem Def13 defines supremum-dense LATTICE6:def 13 :
:: deftheorem Def14 defines infimum-dense LATTICE6:def 14 :
theorem Th23: :: LATTICE6:23
theorem Th24: :: LATTICE6:24
theorem Th25: :: LATTICE6:25
:: deftheorem Def15 defines MIRRS LATTICE6:def 15 :
:: deftheorem Def16 defines JIRRS LATTICE6:def 16 :
theorem Th26: :: LATTICE6:26
theorem Th27: :: LATTICE6:27
Lemma137:
for L being complete co-noetherian Lattice holds MIRRS L is infimum-dense
Lemma144:
for L being complete noetherian Lattice holds JIRRS L is supremum-dense