:: LATTICE7 semantic presentation
:: deftheorem Def1 defines c= LATTICE7:def 1 :
:: deftheorem Def2 defines Chain LATTICE7:def 2 :
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
L being
LATTICE for
x,
y being
Element of
L holds
(
x <(1) y iff (
x < y & ( for
z being
Element of
L holds
( not
x < z or not
z < y ) ) ) );
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
Lemma107:
for L being finite distributive LATTICE
for a being Element of L st ( for b being Element of L st b < a holds
sup ((downarrow b) /\ (Join-IRR L)) = b ) holds
sup ((downarrow a) /\ (Join-IRR L)) = a
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 :
Lemma129:
for L1, L2 being non empty RelStr
for f being Function of L1,L2 st f is infs-preserving holds
f is meet-preserving
Lemma132:
for L1, L2 being non empty RelStr
for f being Function of L1,L2 st f is sups-preserving holds
f is join-preserving
theorem Th15: :: LATTICE7:15
theorem Th16: :: LATTICE7:16