:: MSUALG_2 semantic presentation
theorem Th1: :: MSUALG_2:1
canceled;
theorem Th2: :: MSUALG_2:2
:: deftheorem Def1 MSUALG_2:def 1 :
canceled;
:: deftheorem Def2 defines with_const_op MSUALG_2:def 2 :
:: deftheorem Def3 defines all-with_const_op MSUALG_2:def 3 :
:: deftheorem Def4 defines Constants MSUALG_2:def 4 :
:: deftheorem Def5 defines Constants MSUALG_2:def 5 :
:: deftheorem Def6 defines is_closed_on MSUALG_2:def 6 :
:: deftheorem Def7 defines opers_closed MSUALG_2:def 7 :
theorem Th3: :: MSUALG_2:3
:: deftheorem Def8 defines /. MSUALG_2:def 8 :
:: deftheorem Def9 defines Opers MSUALG_2:def 9 :
theorem Th4: :: MSUALG_2:4
theorem Th5: :: MSUALG_2:5
:: deftheorem Def10 defines MSSubAlgebra MSUALG_2:def 10 :
Lemma14:
for b1 being non empty non void ManySortedSign
for b2 being MSAlgebra of b1 holds
MSAlgebra(# the Sorts of b2,the Charact of b2 #) is MSSubAlgebra of b2
theorem Th6: :: MSUALG_2:6
theorem Th7: :: MSUALG_2:7
theorem Th8: :: MSUALG_2:8
theorem Th9: :: MSUALG_2:9
theorem Th10: :: MSUALG_2:10
theorem Th11: :: MSUALG_2:11
theorem Th12: :: MSUALG_2:12
theorem Th13: :: MSUALG_2:13
:: deftheorem Def11 defines SubSort MSUALG_2:def 11 :
Lemma20:
for b1 being non empty non void ManySortedSign
for b2 being MSAlgebra of b1
for b3 being MSSubset of b2 holds the Sorts of b2 in SubSort b3
:: deftheorem Def12 defines SubSort MSUALG_2:def 12 :
:: deftheorem Def13 defines @ MSUALG_2:def 13 :
theorem Th14: :: MSUALG_2:14
theorem Th15: :: MSUALG_2:15
:: deftheorem Def14 defines SubSort MSUALG_2:def 14 :
:: deftheorem Def15 defines MSSubSort MSUALG_2:def 15 :
theorem Th16: :: MSUALG_2:16
theorem Th17: :: MSUALG_2:17
theorem Th18: :: MSUALG_2:18
theorem Th19: :: MSUALG_2:19
theorem Th20: :: MSUALG_2:20
theorem Th21: :: MSUALG_2:21
:: deftheorem Def16 defines | MSUALG_2:def 16 :
:: deftheorem Def17 defines /\ MSUALG_2:def 17 :
:: deftheorem Def18 defines GenMSAlg MSUALG_2:def 18 :
theorem Th22: :: MSUALG_2:22
theorem Th23: :: MSUALG_2:23
theorem Th24: :: MSUALG_2:24
:: deftheorem Def19 defines "\/" MSUALG_2:def 19 :
theorem Th25: :: MSUALG_2:25
theorem Th26: :: MSUALG_2:26
theorem Th27: :: MSUALG_2:27
theorem Th28: :: MSUALG_2:28
theorem Th29: :: MSUALG_2:29
:: deftheorem Def20 defines MSSub MSUALG_2:def 20 :
definition
let c
1 be non
empty non
void ManySortedSign ;
let c
2 be
non-empty MSAlgebra of c
1;
func MSAlg_join c
2 -> BinOp of
MSSub a
2 means :
Def21:
:: MSUALG_2:def 21
for b
1, b
2 being
Element of
MSSub a
2for b
3, b
4 being
strict MSSubAlgebra of a
2 holds
( b
1 = b
3 & b
2 = b
4 implies a
3 . b
1,b
2 = b
3 "\/" b
4 );
existence
ex b1 being BinOp of MSSub c2 st
for b2, b3 being Element of MSSub c2
for b4, b5 being strict MSSubAlgebra of c2 holds
( b2 = b4 & b3 = b5 implies b1 . b2,b3 = b4 "\/" b5 )
uniqueness
for b1, b2 being BinOp of MSSub c2 holds
( ( for b3, b4 being Element of MSSub c2
for b5, b6 being strict MSSubAlgebra of c2 holds
( b3 = b5 & b4 = b6 implies b1 . b3,b4 = b5 "\/" b6 ) ) & ( for b3, b4 being Element of MSSub c2
for b5, b6 being strict MSSubAlgebra of c2 holds
( b3 = b5 & b4 = b6 implies b2 . b3,b4 = b5 "\/" b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def21 defines MSAlg_join MSUALG_2:def 21 :
definition
let c
1 be non
empty non
void ManySortedSign ;
let c
2 be
non-empty MSAlgebra of c
1;
func MSAlg_meet c
2 -> BinOp of
MSSub a
2 means :
Def22:
:: MSUALG_2:def 22
for b
1, b
2 being
Element of
MSSub a
2for b
3, b
4 being
strict MSSubAlgebra of a
2 holds
( b
1 = b
3 & b
2 = b
4 implies a
3 . b
1,b
2 = b
3 /\ b
4 );
existence
ex b1 being BinOp of MSSub c2 st
for b2, b3 being Element of MSSub c2
for b4, b5 being strict MSSubAlgebra of c2 holds
( b2 = b4 & b3 = b5 implies b1 . b2,b3 = b4 /\ b5 )
uniqueness
for b1, b2 being BinOp of MSSub c2 holds
( ( for b3, b4 being Element of MSSub c2
for b5, b6 being strict MSSubAlgebra of c2 holds
( b3 = b5 & b4 = b6 implies b1 . b3,b4 = b5 /\ b6 ) ) & ( for b3, b4 being Element of MSSub c2
for b5, b6 being strict MSSubAlgebra of c2 holds
( b3 = b5 & b4 = b6 implies b2 . b3,b4 = b5 /\ b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def22 defines MSAlg_meet MSUALG_2:def 22 :
theorem Th30: :: MSUALG_2:30
theorem Th31: :: MSUALG_2:31
theorem Th32: :: MSUALG_2:32
theorem Th33: :: MSUALG_2:33
:: deftheorem Def23 defines MSSubAlLattice MSUALG_2:def 23 :
theorem Th34: :: MSUALG_2:34
theorem Th35: :: MSUALG_2:35
theorem Th36: :: MSUALG_2:36
theorem Th37: :: MSUALG_2:37
theorem Th38: :: MSUALG_2:38
theorem Th39: :: MSUALG_2:39
canceled;
theorem Th40: :: MSUALG_2:40
theorem Th41: :: MSUALG_2:41