ÿþ( 1 ) A . B i a By n i c k i - B i r u l a , A l g e b r a L i n i o w a z G e o m e t r i ,
P W N , W a r s z a w a 1 9 7 6 .
( 2 ) J . G a n c a r z e w i c z , A l g e b r a L i n i o w a z E l e m e n t a m i G e o m e t r i i ,
U J S k r y p t y U c z e l n i a n e , K r a k ó w 1 9 9 3 .
( 3 ) N . J e f i m o w , E . R o z e n d o r n , A l g e b r a L i n i o w a w r a z z G e o m e t r i
W i e l o w y m i a r o w ,
P W N , W a r s z a w a 1 9 7 4 .
( 4 ) T . L a w s o n , L i n e a r A l g e b r a ,
J o h n W i l e y & S o n s , I n c . 1 9 9 6 .
( 5 ) Z . O p i a l , A l g e b r a W y |s z a ,
P W N , W a r s z a w a 1 9 6 7 .
( 6 ) A . M o s t o w s k i , M . S t a r k , A l g e b r a L i n i o w a ,
P W N , W a r s z a w a 1 9 6 9 .
1
R O Z D Z I A A 1
W s t p
1 . 1 . R e l a c j e
R e l a c j o k r e s l o n n a z b i o r a c h A i B n a z y w a m y d o w o l n y p o d z b i ó r
R i l o c z y n u k a r t e z j a Ds k i e g o A × B .
R ‚" A × B
a R b Ð!Ò! ( a , b ) " R .
R e l a c j d w u a r g u m e n t o w R o k r e [l o n w z b i o r z e A ( R ‚" A 2 ) n a z y w a m y
( 1 ) z w r o t n j e |e l i "a " A
a R a
( 2 ) s y m e t r y c z n j e |e l i "a , b " A
a R b = Ò! b R a
( 3 ) p r z e c h o d n i j e |e l i "a , b , c " A
a R b , b R c = Ò! a R c
R e l a c j R ‚" A 2 n a z y w a m y r e l a c j r ó w n o w a |n o [c i , j e |e l i j e s t
" z w r o t n a ,
" s y m e t r y c z n a ,
" p r z e c h o d n i a .
3
R ‚" A 2 , a " A - d o w o l n y e l e m e n t .
K l a s r ó w n o w a |n o [c i e l e m e n t u a w z g l d e m r e l a c j i R n a z y w a m y
z b i ó r w s z y s t k i c h e l e m e n t ó w z b i o r u A , k t ó r e s p e Bn i a j z a r e l a c j R .
[ a ] R = { b " A : a R b }
= { b " A : ( a , b ) " R }
Z b i o r e m i l o r a z o w y m l u b i l o r a z e m z b i o r u A p r z e z r e l a c j R
n a z y w a m y z b i ó r w s z y s t k i c h k l a s r ó w n o w a |n o [c i w z g l d e m r e l a c j i R .
A / R = { [ a ] R : a " A }
[ a ] R ‚" A i [ a ] R = "
P r z y k Ba d
A - z b i ó r p u n k t ó w n a m a p i e ,
"x , y " A x R y Ô! x i y m a j t s a m w y s k o [ n a d p o z i o m e m m o r z a
k l a s y r ó w n o w a |n o [c i - p o z i o m i c e ,
Z b i ó r i l o r a z o w y - z b i ó r p o z i o m i c
T W I E R D Z E N I E 1 . 1 . 1 . R ‚" A 2 - r e l a c j a r ó w n o w a |n o [c i
"a , b " A
[ a ] R = [ b ] R Ô! a R b
[ a ] R )" [ b ] R = " Ô! [ a ] R = [ b ] R
B - d o w o l n y , n i e p u s t y z b i ó r ,
{ B i } i "I - r o d z i n a n i e p u s t y c h p o d z b i o r ó w B
"i " I B i ‚" B , B i = "
B = B i
i "I
"i , j " I i = j Ò! B i )" B j = "
{ B i } i "I - p o d z i a B z b i o r u B
W N I O S E K 1 . 1 . 1 .
( 1 ) K l a s y r ó w n o w a |n o [c i t w o r z p o d z i a B z b i o r u A .
( 2 ) K a |d y p o d z i a B { A i } i "I z b i o r u A w y z n a c z a r e l a c j r ó w n o w a |n o -
[c i
a R b Ô! "i " I : a , b " A i
R e l a c j a p r z y s t a w a n i a m o d u l o m
Z - z b i ó r l i c z b c a Bk o w i t y c h ,
m "N , <"m ‚"Z 2 ,
N = { 1 , 2 , . . . }
"a , b "Z
a <"m b Ô! m | a - b Ô!
a - b
"p " Z : a - b = p m Ô! " Z
m
<"m - r e l a c j a r ó w n o w a |n o [c i
[ 0 ] = { . . . - 2 m , - m , 0 , m , 2 m . . . }
[ 1 ] = { . . . - m + 1 , 1 , m + 1 , 2 m + 1 . . . }
. . . . . . . . . . . . . . .
[ m - 1 ] = { . . . - m - 1 , - 1 , m - 1 , . . . }
Z / <" = { [ 0 ] , [ 1 ] , . . . , [ m - 1 ] }
m
Z / <" o z n Z m o z n { 0 , 1 , . . . , m - 1 }
= =
m
1 . 2 . O d w z o r o w a n i a
R e l a c j R o k r e [l o n n a z b i o r a c h A i B n a z y w a m y o d w z o r o w a n i e m ,
j e |e l i
"a " A "b " B : a R b
"a " A , "b , c " B a R b , a R c Ò! b = c
"a " A "! b " B : a R b
B
B
B
A
A
A
B
A
B
A
B
A
B
B
A
A
f : A - ’! B
f
A B
" A - d z i e d z i n a
" B - p r z e c i w d z i e d z i n a
" b = f ( a ) ( a f b ) - w a r t o [, o b r a z e l e m e n t u a p o p r z e z o d -
w z o r o w a n i e f
o b r a z z b i o r u A
f ( A ) = { b " B : "a " A , f ( a ) = b } ‚" B
" o b r a z p o d z b i o r u A ‚" A
f ( A ) = { b " B : "a " A , f ( a ) = b } ‚" B
" p r z e c i w o b r a z p o d z b i o r u B ‚" B
f - 1 ( B ) = { a " A : f ( a ) " B } ‚" A
" z a c i e [n i e n i e n i e m o d w z o r o w a n i a f : A - ’! B d o p o d z b i o r u
A ‚" A n a z y w a m y o d w z o r o w a n i e g : A - ’! B t a k i e , |e
"a " A
g ( a ) = f ( a )
o z n
g = f | A
O d w z o r o w a n i e f : A - ’! B n a z y w a m y :
( 1 ) i n i e k c j ( o d w z o r o w a n i e m r ó |n o w a r t o [c i o w y m ) , j e |e l i
"a , a " A a = a Ò! f ( a ) = f ( a )
"a , a " A f ( a ) = f ( a ) Ò! a = a
( 2 ) s u r i e k c j ( o d w z o r o w a n i e m n a c a By z b i ó r B ) , j e |e l i
"b " B "a " A : f ( a ) = b
f ( A ) = B
( 3 ) b i j e k c j ( o d w z o r o w a n i e m w z a j e m n i e j e d n o z n a c z n y m ) , j e |e l i
j e s t i n i e k c j i s u r i e k c j .
O d w z o r o w a n i e m o d w r o t n y m d o o d w z o r o w a n i a b i j e k t y w n e g o
f : A - ’! B n a z y w a m y o d g : B - ’! A t a k i e , |e "b " B
g ( b ) = a Ô! f ( a ) = b
o z n
g = f - 1 .
O d w z o r o w a n i e o d w r o t n e j e s t r ó w n i e | b i j e k c j .
f
A B
g
e A : A - ’! A - i d e n t y c z n o [
"a " A
e A ( a ) = a .
e A
A A
h : A - ’! B , g : B - ’! C
O d w z o r o w a n i e
f : A - ’! C
t a k i e , |e "a " A
o z n
f ( a ) = g ( h ( a ) ) = ( g æ% h ) ( a )
n a z y w a m y z Bo |e n i e m ( i l o c z y n e m ) o d w z o r o w a D h i g .
f
A C
h g
B
f æ% e A = f e B æ% f = f
f f
A B A B
e A f f e B
A B
" S k Ba d a n i e o d w z o r o w a D n i e j e s t p r z e m i e n n e .
" S k Ba d a n i e o d w z o r o w a D j e s t Bc z n e .
h : A - ’! B , g : B - ’! C ,
f : C - ’! D
f æ% ( g æ% h ) = ( f æ% g ) æ% h
"a " A
[ f æ% ( g æ% h ) ] ( a ) = f ( ( g æ% h ) ( a ) ) =
= f ( ( g ( h ( a ) ) ) = [ ( f æ% g ) æ% h ] ( a )
f g h
A D
h f g
B
g h f
g
C
" D l a d o w o l n e g o o d w z . b i j e k t y w n e g o
f : A - ’! B
e A e B
A A B B
f f - 1 f - 1 f
B A
f - 1 æ% f = e A f æ% f - 1 = e B
T W I E R D Z E N I E 1 . 2 . 1 . O d w z o r o w a n i e
f : A - ’! B
j e s t b i j e k c j w t w g d y i s t n i e j e o d w z o r o w a n i e
g : B - ’! A
t a k i e , |e
g æ% f = e A , f æ% g = e B .
O d w z o r o w a n i e g s p e Bn i a j c e t e n w a r u n e k j e s t j e d n o z n a c z n i e w y z n a c z o -
n e w z o r e m g = f - 1 .
T W I E R D Z E N I E 1 . 2 . 2 . Z Bo |e n i e o d w z o r o w a D b i j e k t y w n y c h j e s t
b i j e k c j i z a c h o d z i z w i z e k
( g h ) - 1 = h - 1 g - 1 .
T W I E R D Z E N I E 1 . 2 . 3 . N i e c h f : A - ’! B b d z i e b i j e k c j . W ó w -
c z a s
( f - 1 ) - 1 = f .
T W I E R D Z E N I E 1 . 2 . 4 . ( o f a k t o r y z a c j i )
D o w o l n e o d w z o r o w a n i e f : A - ’! B m o |n a p r z e d s t a w i w p o s t a c i
z Bo |e n i a s u r i e k c j i i i n i e k c j i .
( f ) ‚" A 2 - r e l a c j a , t a k a |e :
"a , b " A
a ( f ) b Ô! f ( a ) = f ( b )
( f ) - r e l a c j a r ó w n o w a |n o [c i
f
A B
k f "
A / ( f )
f = f " æ% k
"a " A k ( a ) = [ a ] ( f ) ,
"[ a ] ( f ) " A / ( f ) f "( [ a ] ( f ) ) = f ( a ) .
f
k f "
W N I O S E K 1 . 2 . 1 . J e [l i d o d a t k o w o z a Bo |y m y , |e f j e s t s u r i e k c j ,
t o w ó w c z a s f " j e s t b i j e k c j .
f
k f "
P r z y k Ba d
f
R
k f "
A / ( f )
1 . 3 . D z i a Ba n i a
D z i a Ba n i e m w e w n t r z n y m w z b i o r z e A n a z y w a m y d o w o l n e o d -
w z o r o w a n i e
h : A × A - ’! A .
( 1 ) p r z e m i e n n o [ "a , b " A
h ( a , b ) = h ( b , a )
( 2 ) Bc z n o [ "a , b , c " A
h ( h ( a , b ) , c ) = h ( a , h ( b , c ) )
( 3 ) e " A - e l e m e n t n e u t r a l n y , j e [l i "a " A
h ( a , e ) = h ( e , a ) = a
( 4 ) j e [l i i s t n i e j e e l e m e n t n e u t r a l n y e , t o e l e m e n t e m o d w r o t n y m
d o d o w o l n e g o e l e m e n t u a " A , n a z y w a m y e l e m e n t b " A
s p e Bn i a j c y w a r u n e k :
h ( a , b ) = h ( b , a ) = e
( 5 ) h , g : A × A - ’! A - d w a d z i a Ba n i a - r o z d z i e l n o [ d z i a Ba n i a
h w z g l d e m g o z n a c z a
"a , b , c "A
h ( g ( a , b ) , c ) = g ( h ( a , c ) , h ( b , c ) ) .
D z i a Ba n i e m z e w n t r z n y m w z b i o r z e A n a z y w a m y d o w o l n e o d -
w z o r o w a n i e
g : F × A - ’! A .
F - z b i ó r o p e r a t o r ó w .
R ‚" A 2 - r e l a c j a r ó w n o w a |n o [c i ,
h : A × A - ’! A - d z i a Ba n i e w e w n t r z n e ,
g : F × A - ’! A - d z i a Ba n i e z e w n t r z n e .
R e l a c j R n a z y w a m y :
( 1 ) z g o d n z d z i a Ba n i e m w e w n t r z n y m h ( k o n g r u e n c j ) ,
j e [l i
"a , b , a , b " A
a R a
b R b Ò! h ( a , b ) R h ( a , b )
( 2 ) z g o d n z d z i a Ba n i e m z e w n t r z n y m g ( k o n g r u e n c j ) , j e [l i
"a , b " A , "x " F
a R b Ò! g ( x , a ) R g ( x , b )
o z n o z n
h ( a , b ) = a b , g ( x , a ) = x a
a R a
a R b Ò! x a R x b
b R b Ò! a b R a b
o z n
h ( a , b ) = a + b
a R a
b R b Ò! ( a + b ) R ( a + b )
R - r e l a c j a r ó w n o w a |n o [c i z g o d n a z d z i a Ba n i e m
" w e w n t r z n y m h ,
" z e w n t r z n y m g .
W z b i o r z e i l o r a z o w y m A / R o k r e [l a m y d z i a Ba n i e i n d u k o w a n e
" w e w n t r z n e h ",
" z e w n t r z n e g ".
"a , b " A , "x " F
h " : A / R × A / R - ’! A / R
g " : F × A / R - ’! A / R
h "( [ a ] , [ b ] ) = [ h ( a , b ) ]
g "( x , [ a ] ) = [ g ( x , a ) ]
[ a ] [ b ] = [ a b ] , x [ a ] = [ x a ] ,
[ a ] + [ b ] = [ a + b ]
f : A - ’! B , g : C - ’! D ,
( f × g ) : A × C - ’! B × D ,
( f × g ) ( a , c ) = ( f ( a ) , g ( c ) )
h
A × A A
k × k k
A / R × A / R
h " A / R
h " æ% ( k × k ) = k æ% h
g
F × A A
e F × k k
F × A / R
g " A / R
g " æ% ( e F × k ) = k æ% g
D z i a Ba n i a i n d u k o w a n e m a j t e s a m e w Ba s n o [c i c o d z i a Ba n i a w y j [c i o -
w e .
R e l a c j a <"m †" Z 2 j e s t z g o d n a z d o d a w a n i e m i m n o |e n i e m w Z .
W z b i o r z e Z m m a m y d w a d z i a Ba n i a i n d u k o w a n e d o d a w a n i e i m n o -
|e n i e m o d m .
1 . 4 . S t r u k t u r y A l g e b r a i c z n e
S t r u k t u r a l g e b r a i c z n o k r e [l o n n a z b i o r z e A n a z y w a m y s y s -
t e m
( A , F 1 , . . . , F m , h 1 , . . . , h n , g 1 , . . . , g m )
A = " - z b i ó r ,
h i : A × A - ’! A
i = 1 , . . . , n - d z i a Ba n i a w e w n t r z n e ,
g j : F j × A - ’! A
j = 1 , . . . , m - d z i a Ba n i a z e w n t r z n e .
( 1 ) P ó Bg r u p a - ( G , · )
· - Bc z n e
( 2 ) G r u p a - ( G , · )
( G , · ) - p ó Bg r u p a
i s t n i e j e e l e m e n t n e u t r a l n y ,
k a |d y e l e m e n t p o s i a d a o d w r o t n y .
· - p r z e m i e n n e , t o g r u p n a z y w a m y a b e l o w ( p r z e m i e n n )
( 3 ) P i e r [c i e D - ( P , + , · )
( P , + ) - g r u p a a b e l o w a ,
( P , · ) - p ó Bg r u p a
p r a w a r o z d z i e l n o [c i : "a , b , c "P
a ( b + c ) = a b + a c
( a + b ) c = a c + b c
( 4 ) C i a Bo - ( F , + , · )
( F , + ) - g r u p a a b e l o w a ,
( F \ { 0 } , · ) - g r u p a a b e l o w a
p r a w o r o z d z i e l n o [c i : "a , b , c "F
a ( b + c ) = a b + a c
( 5 ) P r z e s t r z e D w e k t o r o w a - ( V , F , + , · )
( V , + ) - g r u p a a b e l o w a ,
( F , + , · , 0 , 1 ) - c i a Bo ,
"a , b " F , "X , Y " V
a ( X + Y ) = a X + a Y
( a + b ) X = a X + b X
( a b ) X = a ( b X )
1 X = X
( 6 ) A l g e b r a - ( A , F , + , æ%, · )
( A , + , æ%) - p i e r [c i e D,
( A , F , + , · ) - p r z e s t r z e D w e k t o r o w a ,
"a " F , "X , Y " A
X æ% ( a Y ) = ( a X ) æ% Y = a ( X æ% Y )
( A , F j , h i , g j ) - s t r u k t u r a a l g e b r a i c z n a ,
i = 1 , . . . , n , j = 1 , . . . , m
S y s t e m ( A , F j , h i , g j ) n a z y w a m y p o d s t r u k t u r , j e [l i
( 1 ) " = A ‚" A
( 2 ) h i , g j s d z i a Ba n i a m i w A , t z n
h i ( A × A ) ‚" A
g j ( F j × A ) ‚" A
( 3 ) d z i a Ba n i a p o s i a d a j t e s a m e w Ba s n o [c i , ( s t r u k t u r a j e s t t e g o s a -
m e g o t y p u ) .
R e l a c j r ó w n o w a |n o [c i R ‚" A 2 n a z y w a m y z g o d n z e s t r u k t u r
( k o n g r u e n c j ) , j e [l i R j e s t z g o d n a z e w s z y s t k i m i d z i a Ba n i a m i w e -
w n t r z n y m i i z e w n t r z n y m i .
R - k o n g r u e n c j a w s t r u k t u r z e ( A , F j , h i , g j ) . S t r u k t u r a l g e b r a i c z n
"
( A / R , F j , h ", g j ) n a z y w a m y s t r u k t u r i l o r a z o w .
i
1 . 5 . H o m o m o r f i z m y
( A , F j , h i , g j ) , ( A , F j , h , g j )
i
s t r u k t u r y o n d z i a Ba n i a c h w e w n t r z n y c h , m d z i a Ba n i a c h z e w n t r z -
n y c h i t y c h s a m y c h z b i o r a c h o p e r a t o r ó w .
O d w z o r o w a n i e
f : A - ’! A
n a z y w a m y h o m o m o r f i z m e m s t r u k t u r y A w s t r u k t u r A , j e [l i
"i = 1 , . . . , n , "j = 1 , . . . , m
"a , b " A , "x " F j
h ( f ( a ) , f ( b ) ) = f ( h i ( a , b ) )
i
g j ( x , f ( a ) ) = f ( g j ( x , a ) )
h i
A × A A
f × f f
A × A A
h
i
h æ% ( f × f ) = f æ% h i
i
g j
F j × A A
e F × f f
j
F j × A A
g j
g j æ% ( e F × f ) = f æ% g j
j
H o m o m o r f i z m f : A - ’! A n a z y w a m y :
" e p i m o r f i z m e m - j e [l i j e s t s u r j e k c j ,
" m o n o m o r f i z m e m - j e [l i j e s t i n j e k c j
" i z o m o r f i z m e m - j e [l i j e s t b i j e k c j ,
" e n d o m o r f i z m e m - j e [l i j e s t h o m o m o r f i z m e m w s i e b i e ,
" a u t o m o r f i z m e m - j e [l i j e s t i z o m o r f i z m e m w s i e b i e
T W I E R D Z E N I E 1 . 5 . 1 . Z Bo |e n i e h o m o m o r f i z m ó w j e s t h o m o m o r -
f i z m e m .
W N I O S E K 1 . 5 . 1 . Z Bo |e n i e i z o m o r f i z m ó w j e s t i z o m o r f i z m e m .
T W I E R D Z E N I E 1 . 5 . 2 . O d w z o r o w a n i e o d w r o t n e d o i z o m o r f i z m u
j e s t i z o m o r f i z m e m .
S t r u k t u r a a l g e b r a i c z n a ( A , F j , h i , g j ) j e s t i z o m o r f i c z n a z e s t r u k -
t u r ( A , F j , h , g j ) , j e [l i i s t n i e j e i z o m o r f i c z n e o d w z o r o w a n i e j e d n e j s t r u k -
i
t u r y w d r u g .
( A , F j , h i , g j ) <" ( A , F j , h , g j )
i
U W A G A 1 . 5 . 1 . I z o m o r f i z m s t r u k t u r j e s t r e l a c j r ó w n o w a |n o [c i .
T W I E R D Z E N I E 1 . 5 . 3 . J e [l i
f : A - ’! A
j e s t h o m o r f i z m e m s t r u k t u r y A w A , t o r e l a c j a
( f ) ‚" A 2
- j e s t k o n g r u e n c j w s t r u k t u r z e ( A , F j , h i , g j ) .
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W N I O S E K 1 . 5 . 2 . ( A / ( f ) , F j , h ", g j ) j e s t s t r u k t u r i l o r a z o w .
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T W I E R D Z E N I E 1 . 5 . 4 . P o d s t a w o w e t w i e r d z e n i e o h o m o -
m o r f i zm i e .
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w A , t o
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( A , F j , h , g j ) <" ( A / ( f ) , F j , h ", g j ) .
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T e o r i a m n o g o [c i A l g e b r a
z b i ó r s t r u k t u r a
p o d z b i ó r p o d s t r u k t u r a
o d w z o r o w a n i e h o m o m o r f i z m
s u r i e k c j a e p i m o r f i z m
i n i e k c j a m o n o m o r f i z m
b i j e k c j a i z o m o r f i z m
r e l a c j a r ó w n o w a |n o [c i k o n g r u e n c j a
z b i ó r i l o r a z o w y s t r u k t u r a i l o r a z o w a
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