:: ORDERS_3 semantic presentation
:: deftheorem Def1 defines discrete ORDERS_3:def 1 :
Lemma2:
for b1 being empty RelStr holds the InternalRel of b1 = {}
Lemma3:
for b1 being RelStr holds
( b1 is empty implies b1 is discrete )
:: deftheorem Def2 defines disconnected ORDERS_3:def 2 :
:: deftheorem Def3 defines disconnected ORDERS_3:def 3 :
theorem Th1: :: ORDERS_3:1
theorem Th2: :: ORDERS_3:2
theorem Th3: :: ORDERS_3:3
theorem Th4: :: ORDERS_3:4
theorem Th5: :: ORDERS_3:5
:: deftheorem Def4 defines POSet_set-like ORDERS_3:def 4 :
:: deftheorem Def5 defines monotone ORDERS_3:def 5 :
for b
1, b
2 being
RelStr for b
3 being
Function of b
1,b
2 holds
( b
3 is
monotone iff for b
4, b
5 being
Element of b
1 holds
( b
4 <= b
5 implies for b
6, b
7 being
Element of b
2 holds
( b
6 = b
3 . b
4 & b
7 = b
3 . b
5 implies b
6 <= b
7 ) ) );
Lemma10:
for b1, b2, b3 being non empty RelStr
for b4 being Function of b1,b2
for b5 being Function of b2,b3 holds
not ( b4 is monotone & b5 is monotone & ( for b6 being Function of b1,b3 holds
not ( b6 = b5 * b4 & b6 is monotone ) ) )
Lemma11:
for b1 being non empty RelStr holds id b1 is monotone
definition
let c
1, c
2 be
RelStr ;
func MonFuncs c
1,c
2 -> set means :
Def6:
:: ORDERS_3:def 6
for b
1 being
set holds
( b
1 in a
3 iff ex b
2 being
Function of a
1,a
2 st
( b
1 = b
2 & b
2 in Funcs the
carrier of a
1,the
carrier of a
2 & b
2 is
monotone ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ex b3 being Function of c1,c2 st
( b2 = b3 & b3 in Funcs the carrier of c1,the carrier of c2 & b3 is monotone ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ex b4 being Function of c1,c2 st
( b3 = b4 & b4 in Funcs the carrier of c1,the carrier of c2 & b4 is monotone ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4 being Function of c1,c2 st
( b3 = b4 & b4 in Funcs the carrier of c1,the carrier of c2 & b4 is monotone ) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines MonFuncs ORDERS_3:def 6 :
theorem Th6: :: ORDERS_3:6
theorem Th7: :: ORDERS_3:7
:: deftheorem Def7 defines Carr ORDERS_3:def 7 :
Lemma16:
for b1 being non empty POSet_set
for b2 being Element of b1 holds the carrier of b2 in Carr b1
by Def7;
theorem Th8: :: ORDERS_3:8
theorem Th9: :: ORDERS_3:9
theorem Th10: :: ORDERS_3:10
theorem Th11: :: ORDERS_3:11
definition
let c
1 be non
empty POSet_set;
func POSCat c
1 -> strict with_triple-like_morphisms Category means :: ORDERS_3:def 8
( the
Objects of a
2 = a
1 & ( for b
1, b
2 being
Element of a
1for b
3 being
Element of
Funcs (Carr a1) holds
( b
3 in MonFuncs b
1,b
2 implies
[[b1,b2],b3] is
Morphism of a
2 ) ) & ( for b
1 being
Morphism of a
2 holds
ex b
2, b
3 being
Element of a
1ex b
4 being
Element of
Funcs (Carr a1) st
( b
1 = [[b2,b3],b4] & b
4 in MonFuncs b
2,b
3 ) ) & ( for b
1, b
2 being
Morphism of a
2for b
3, b
4, b
5 being
Element of a
1for b
6, b
7 being
Element of
Funcs (Carr a1) holds
( b
1 = [[b3,b4],b6] & b
2 = [[b4,b5],b7] implies b
2 * b
1 = [[b3,b5],(b7 * b6)] ) ) );
existence
ex b1 being strict with_triple-like_morphisms Category st
( the Objects of b1 = c1 & ( for b2, b3 being Element of c1
for b4 being Element of Funcs (Carr c1) holds
( b4 in MonFuncs b2,b3 implies [[b2,b3],b4] is Morphism of b1 ) ) & ( for b2 being Morphism of b1 holds
ex b3, b4 being Element of c1ex b5 being Element of Funcs (Carr c1) st
( b2 = [[b3,b4],b5] & b5 in MonFuncs b3,b4 ) ) & ( for b2, b3 being Morphism of b1
for b4, b5, b6 being Element of c1
for b7, b8 being Element of Funcs (Carr c1) holds
( b2 = [[b4,b5],b7] & b3 = [[b5,b6],b8] implies b3 * b2 = [[b4,b6],(b8 * b7)] ) ) )
uniqueness
for b1, b2 being strict with_triple-like_morphisms Category holds
( the Objects of b1 = c1 & ( for b3, b4 being Element of c1
for b5 being Element of Funcs (Carr c1) holds
( b5 in MonFuncs b3,b4 implies [[b3,b4],b5] is Morphism of b1 ) ) & ( for b3 being Morphism of b1 holds
ex b4, b5 being Element of c1ex b6 being Element of Funcs (Carr c1) st
( b3 = [[b4,b5],b6] & b6 in MonFuncs b4,b5 ) ) & ( for b3, b4 being Morphism of b1
for b5, b6, b7 being Element of c1
for b8, b9 being Element of Funcs (Carr c1) holds
( b3 = [[b5,b6],b8] & b4 = [[b6,b7],b9] implies b4 * b3 = [[b5,b7],(b9 * b8)] ) ) & the Objects of b2 = c1 & ( for b3, b4 being Element of c1
for b5 being Element of Funcs (Carr c1) holds
( b5 in MonFuncs b3,b4 implies [[b3,b4],b5] is Morphism of b2 ) ) & ( for b3 being Morphism of b2 holds
ex b4, b5 being Element of c1ex b6 being Element of Funcs (Carr c1) st
( b3 = [[b4,b5],b6] & b6 in MonFuncs b4,b5 ) ) & ( for b3, b4 being Morphism of b2
for b5, b6, b7 being Element of c1
for b8, b9 being Element of Funcs (Carr c1) holds
( b3 = [[b5,b6],b8] & b4 = [[b6,b7],b9] implies b4 * b3 = [[b5,b7],(b9 * b8)] ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines POSCat ORDERS_3:def 8 :
for b
1 being non
empty POSet_setfor b
2 being
strict with_triple-like_morphisms Category holds
( b
2 = POSCat b
1 iff ( the
Objects of b
2 = b
1 & ( for b
3, b
4 being
Element of b
1for b
5 being
Element of
Funcs (Carr b1) holds
( b
5 in MonFuncs b
3,b
4 implies
[[b3,b4],b5] is
Morphism of b
2 ) ) & ( for b
3 being
Morphism of b
2 holds
ex b
4, b
5 being
Element of b
1ex b
6 being
Element of
Funcs (Carr b1) st
( b
3 = [[b4,b5],b6] & b
6 in MonFuncs b
4,b
5 ) ) & ( for b
3, b
4 being
Morphism of b
2for b
5, b
6, b
7 being
Element of b
1for b
8, b
9 being
Element of
Funcs (Carr b1) holds
( b
3 = [[b5,b6],b8] & b
4 = [[b6,b7],b9] implies b
4 * b
3 = [[b5,b7],(b9 * b8)] ) ) ) );
scheme :: ORDERS_3:sch 1
s1{ F
1()
-> non
empty set , F
2(
set ,
set )
-> functional set } :
ex b
1 being
strict AltCatStr st
( the
carrier of b
1 = F
1() & ( for b
2, b
3 being
Element of F
1() holds
( the
Arrows of b
1 . b
2,b
3 = F
2(b
2,b
3) & ( for b
4, b
5, b
6 being
Element of F
1() holds the
Comp of b
1 . b
4,b
5,b
6 = FuncComp F
2(b
4,b
5),F
2(b
5,b
6) ) ) ) )
provided
E19:
for b
1, b
2, b
3 being
Element of F
1()
for b
4, b
5 being
Function holds
( b
4 in F
2(b
1,b
2) & b
5 in F
2(b
2,b
3) implies b
5 * b
4 in F
2(b
1,b
3) )
scheme :: ORDERS_3:sch 2
s2{ F
1()
-> non
empty set , F
2(
set ,
set )
-> functional set } :
for b
1, b
2 being
strict AltCatStr holds
( the
carrier of b
1 = F
1() & ( for b
3, b
4 being
Element of F
1() holds
( the
Arrows of b
1 . b
3,b
4 = F
2(b
3,b
4) & ( for b
5, b
6, b
7 being
Element of F
1() holds the
Comp of b
1 . b
5,b
6,b
7 = FuncComp F
2(b
5,b
6),F
2(b
6,b
7) ) ) ) & the
carrier of b
2 = F
1() & ( for b
3, b
4 being
Element of F
1() holds
( the
Arrows of b
2 . b
3,b
4 = F
2(b
3,b
4) & ( for b
5, b
6, b
7 being
Element of F
1() holds the
Comp of b
2 . b
5,b
6,b
7 = FuncComp F
2(b
5,b
6),F
2(b
6,b
7) ) ) ) implies b
1 = b
2 )
definition
let c
1 be non
empty POSet_set;
func POSAltCat c
1 -> strict AltCatStr means :
Def9:
:: ORDERS_3:def 9
( the
carrier of a
2 = a
1 & ( for b
1, b
2 being
Element of a
1 holds
( the
Arrows of a
2 . b
1,b
2 = MonFuncs b
1,b
2 & ( for b
3, b
4, b
5 being
Element of a
1 holds the
Comp of a
2 . b
3,b
4,b
5 = FuncComp (MonFuncs b3,b4),
(MonFuncs b4,b5) ) ) ) );
existence
ex b1 being strict AltCatStr st
( the carrier of b1 = c1 & ( for b2, b3 being Element of c1 holds
( the Arrows of b1 . b2,b3 = MonFuncs b2,b3 & ( for b4, b5, b6 being Element of c1 holds the Comp of b1 . b4,b5,b6 = FuncComp (MonFuncs b4,b5),(MonFuncs b5,b6) ) ) ) )
uniqueness
for b1, b2 being strict AltCatStr holds
( the carrier of b1 = c1 & ( for b3, b4 being Element of c1 holds
( the Arrows of b1 . b3,b4 = MonFuncs b3,b4 & ( for b5, b6, b7 being Element of c1 holds the Comp of b1 . b5,b6,b7 = FuncComp (MonFuncs b5,b6),(MonFuncs b6,b7) ) ) ) & the carrier of b2 = c1 & ( for b3, b4 being Element of c1 holds
( the Arrows of b2 . b3,b4 = MonFuncs b3,b4 & ( for b5, b6, b7 being Element of c1 holds the Comp of b2 . b5,b6,b7 = FuncComp (MonFuncs b5,b6),(MonFuncs b6,b7) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines POSAltCat ORDERS_3:def 9 :
for b
1 being non
empty POSet_setfor b
2 being
strict AltCatStr holds
( b
2 = POSAltCat b
1 iff ( the
carrier of b
2 = b
1 & ( for b
3, b
4 being
Element of b
1 holds
( the
Arrows of b
2 . b
3,b
4 = MonFuncs b
3,b
4 & ( for b
5, b
6, b
7 being
Element of b
1 holds the
Comp of b
2 . b
5,b
6,b
7 = FuncComp (MonFuncs b5,b6),
(MonFuncs b6,b7) ) ) ) ) );
theorem Th12: :: ORDERS_3:12