:: MSUALG_5 semantic presentation
theorem Th1: :: MSUALG_5:1
:: deftheorem Def1 defines EqCl MSUALG_5:def 1 :
theorem Th2: :: MSUALG_5:2
theorem Th3: :: MSUALG_5:3
theorem Th4: :: MSUALG_5:4
definition
let c
1 be
set ;
func EqRelLatt c
1 -> strict Lattice means :: MSUALG_5:def 2
( the
carrier of a
2 = { b1 where B is Relation of a1,a1 : b1 is Equivalence_Relation of a1 } & ( for b
1, b
2 being
Equivalence_Relation of a
1 holds
( the
L_meet of a
2 . b
1,b
2 = b
1 /\ b
2 & the
L_join of a
2 . b
1,b
2 = b
1 "\/" b
2 ) ) );
existence
ex b1 being strict Lattice st
( the carrier of b1 = { b2 where B is Relation of c1,c1 : b2 is Equivalence_Relation of c1 } & ( for b2, b3 being Equivalence_Relation of c1 holds
( the L_meet of b1 . b2,b3 = b2 /\ b3 & the L_join of b1 . b2,b3 = b2 "\/" b3 ) ) )
uniqueness
for b1, b2 being strict Lattice holds
( the carrier of b1 = { b3 where B is Relation of c1,c1 : b3 is Equivalence_Relation of c1 } & ( for b3, b4 being Equivalence_Relation of c1 holds
( the L_meet of b1 . b3,b4 = b3 /\ b4 & the L_join of b1 . b3,b4 = b3 "\/" b4 ) ) & the carrier of b2 = { b3 where B is Relation of c1,c1 : b3 is Equivalence_Relation of c1 } & ( for b3, b4 being Equivalence_Relation of c1 holds
( the L_meet of b2 . b3,b4 = b3 /\ b4 & the L_join of b2 . b3,b4 = b3 "\/" b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines EqRelLatt MSUALG_5:def 2 :
:: deftheorem Def3 defines EqCl MSUALG_5:def 3 :
theorem Th5: :: MSUALG_5:5
:: deftheorem Def4 defines "\/" MSUALG_5:def 4 :
theorem Th6: :: MSUALG_5:6
theorem Th7: :: MSUALG_5:7
theorem Th8: :: MSUALG_5:8
theorem Th9: :: MSUALG_5:9
theorem Th10: :: MSUALG_5:10
theorem Th11: :: MSUALG_5:11
theorem Th12: :: MSUALG_5:12
theorem Th13: :: MSUALG_5:13
definition
let c
1 be non
empty set ;
let c
2 be
ManySortedSet of c
1;
func EqRelLatt c
2 -> strict Lattice means :
Def5:
:: MSUALG_5:def 5
( ( for b
1 being
set holds
( b
1 in the
carrier of a
3 iff b
1 is
Equivalence_Relation of a
2 ) ) & ( for b
1, b
2 being
Equivalence_Relation of a
2 holds
( the
L_meet of a
3 . b
1,b
2 = b
1 /\ b
2 & the
L_join of a
3 . b
1,b
2 = b
1 "\/" b
2 ) ) );
existence
ex b1 being strict Lattice st
( ( for b2 being set holds
( b2 in the carrier of b1 iff b2 is Equivalence_Relation of c2 ) ) & ( for b2, b3 being Equivalence_Relation of c2 holds
( the L_meet of b1 . b2,b3 = b2 /\ b3 & the L_join of b1 . b2,b3 = b2 "\/" b3 ) ) )
uniqueness
for b1, b2 being strict Lattice holds
( ( for b3 being set holds
( b3 in the carrier of b1 iff b3 is Equivalence_Relation of c2 ) ) & ( for b3, b4 being Equivalence_Relation of c2 holds
( the L_meet of b1 . b3,b4 = b3 /\ b4 & the L_join of b1 . b3,b4 = b3 "\/" b4 ) ) & ( for b3 being set holds
( b3 in the carrier of b2 iff b3 is Equivalence_Relation of c2 ) ) & ( for b3, b4 being Equivalence_Relation of c2 holds
( the L_meet of b2 . b3,b4 = b3 /\ b4 & the L_join of b2 . b3,b4 = b3 "\/" b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines EqRelLatt MSUALG_5:def 5 :
theorem Th14: :: MSUALG_5:14
theorem Th15: :: MSUALG_5:15
for b
1 being non
empty non
void ManySortedSign for b
2 being
non-empty MSAlgebra of b
1for b
3 being
OperSymbol of b
1for b
4, b
5 being
MSCongruence of b
2for b
6 being
MSEquivalence-like ManySortedRelation of b
2 holds
( b
6 = b
4 "\/" b
5 implies for b
7, b
8 being
set for b
9 being
Natfor b
10, b
11, b
12 being
FinSequence holds
(
len b
10 = b
9 &
len b
10 = len b
11 & ( for b
13 being
Nat holds
( b
13 in dom b
10 implies
[(b10 . b13),(b11 . b13)] in b
6 . ((the_arity_of b3) /. b13) ) ) &
[((Den b3,b2) . ((b10 ^ <*b7*>) ^ b12)),((Den b3,b2) . ((b11 ^ <*b7*>) ^ b12))] in b
6 . (the_result_sort_of b3) &
[b7,b8] in b
6 . ((the_arity_of b3) /. (b9 + 1)) implies for b
13 being
Element of
Args b
3,b
2 holds
( b
13 = (b10 ^ <*b7*>) ^ b
12 implies
[((Den b3,b2) . b13),((Den b3,b2) . ((b11 ^ <*b8*>) ^ b12))] in b
6 . (the_result_sort_of b3) ) ) )
theorem Th16: :: MSUALG_5:16
theorem Th17: :: MSUALG_5:17
theorem Th18: :: MSUALG_5:18
:: deftheorem Def6 defines CongrLatt MSUALG_5:def 6 :
theorem Th19: :: MSUALG_5:19
theorem Th20: :: MSUALG_5:20
theorem Th21: :: MSUALG_5:21
theorem Th22: :: MSUALG_5:22