Лекции по упорядоченным множествам и универсальной алгебре. Гуров С.И. - 102 стр.

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x + x = (x + x)
2
= x
2
+ x
2
+ x
2
+ x
2
= x + x + x + x ⇒ x + x = 0
x = −x
x + y = (x + y)
2
= x
2
+ xy + yx + y
2
= x + xy + yx + y ⇒ xy + yx = 0
xy = xy + (yx + yx) = (xy + yx) + yx = 0 + yx = yx .
¤
B = h B, t, u,
0
, o, ι i x, y ∈ B
x + y = (x u y
0
) t (x
0
u y) , x · y = x u y .
B
∗
= h B, +, ·, o, ι i
·
+ x + y = xy
0
t x
0
y
x
2
= x
(x + y) + z = (xy
0
t x
0
y)z
0
t (xy
0
t x
0
y)
0
z =
= xy
0
z
0
t x
0
yz
0
t [(xy
0
)
0
(x
0
y)
0
]z =
= xy
0
z
0
t x
0
yz
0
t (x
0
t y)(x t y
0
)z+ =
= xy
0
z
0
t x
0
yz
0
t x
0
y
0
z t xyz ,
x + (y + z) = x(yz
0
t y
0
z)
0
t x
0
(y
0
z t yz
0
) =
= x(y t z
0
)(y
0
z) t x
0
y
0
z t x
0
yz
0
=
= x(yz t y
0
z
0
) t x
0
y
0
z t x
0
yz
0
=
= xyz t xy
0
z
0
t x
0
y
0
z t x
0
yz
0
=
= (x + y) + z ,
x + o = xo
0
t x
0
o = xι = x ,
x + x = xx
0
t x
0
x = o ,
B +
(x + y)z = (xy
0
t x
0
y)z = xy
0
z t x
0
yz .
xz + yz = xz(yz)
0
t (xz)
0
(yz) =
= xz(y
0
t z
0
) t (x
0
t z
0
)yz = xy
0
z + x
0
yz
· + ¤
h P(A), ⊕, ∩, ∅, A i
R = h R, +, ·, 0, 1 i
x, y ∈ R
x t y = x + y + x · y, x u y = x · y , x
0
= x + 1 .
R
∗
= h R, t, u,
0
, 0, 1 i
102                                                  Ãëàâà 5. Áóëåâû àëãåáðû (ïðîäîëæåíèå)


Äîêàçàòåëüñòâî. Äîêàæåì ñíà÷àëà âòîðîå óòâåðæäåíèå:

          x + x = (x + x)2 = x2 + x2 + x2 + x2 = x + x + x + x ⇒ x + x = 0
(ò.å. x = −x). Îòñþäà

        x + y = (x + y)2 = x2 + xy + yx + y 2 = x + xy + yx + y ⇒ xy + yx = 0
è äàëåå ïîëó÷àåì

                 xy = xy + (yx + yx) = (xy + yx) + yx = 0 + yx = yx .
                                                                                         ¤


Òåîðåìà 5.4. Ïóñòü B = h B, t, u, 0 , o, ι i  áóëåâà àëãåáðà. Äëÿ ëþáûõ x, y ∈ B
ïîëîæèì
                        x + y = (x u y 0 ) t (x 0 u y) ,   x·y = xuy.
Òîãäà ÀÑ B∗ = h B, +, ·, o, ι i åñòü áóëåâî êîëüöî ñ åäèíèöåé.
Äîêàçàòåëüñòâî. Ñèìâîë · óìíîæåíèÿ áóäåì, êàê îáû÷íî, îïóñêàòü è ñ÷èòàòü åãî ïðèî-
ðèòåò âûøå ïðèîðèòåòà ñëîæåíèÿ +. Òîãäà x + y = xy 0 t x 0 y .
   Ðàâåíñòâî x2 = x, àññîöèàòèâíîñòü óìíîæåíèÿ, íàëè÷èå åäèíèöû è êîììóòàòèâíîñòü
îáåèõ îïåðàöèé î÷åâèäíû. Äàëåå, ïðèíèìàÿ âî âíèìàíèå çàêîíû áóëåâîé àëãåáðû, ïîëó-
÷èì

                   (x + y) + z = (xy 0 t x 0 y)z 0 t (xy 0 t x 0 y) 0 z =
                               = xy 0 z 0 t x 0 yz 0 t [(xy 0 ) 0 (x 0 y) 0 ]z =
                               = xy 0 z 0 t x 0 yz 0 t (x 0 t y)(x t y 0 )z+ =
                               = xy 0 z 0 t x 0 yz 0 t x 0 y 0 z t xyz ,
                   x + (y + z) = x(yz 0 t y 0 z) 0 t x 0 (y 0 z t yz 0 ) =
                               = x(y t z 0 )(y 0 z) t x 0 y 0 z t x 0 yz 0 =
                               = x(yz t y 0 z 0 ) t x 0 y 0 z t x 0 yz 0 =
                               = xyz t xy 0 z 0 t x 0 y 0 z t x 0 yz 0 =
                               = (x + y) + z ,
                         x + o = xo 0 t x 0 o = xι = x ,
                         x + x = xx 0 t x 0 x = o ,
ò.å. B îêàçûâàåòñÿ àáåëåâîé ãðóïïîé ïî ñëîæåíèþ +. Íàêîíåö,

                   (x + y)z = (xy 0 t x 0 y)z = xy 0 z t x 0 yz .
                    xz + yz = xz(yz) 0 t (xz) 0 (yz) =
                            = xz(y 0 t z 0 ) t (x 0 t z 0 )yz = xy 0 z + x 0 yz
è äèñòðèáóòèâíûé çàêîí · îòíîñèòåëüíî + äîêàçàí.                                         ¤

   Îñíîâíûì ïðèìåðîì áóëåâà êîëüöà è ÿâëÿåòñÿ êàê ðàç êîëüöî h P(A), ⊕, ∩, ∅, A i,
ïîëó÷àåìîå óêàçàííûì ñïîñîáîì èç òîòàëüíîé àëãåáðû ìíîæåñòâ.
Òåîðåìà 5.5. Ïóñòü R = h R, +, ·, 0, 1 i  áóëåâî êîëüöî ñ åäèíèöåé. Äëÿ ëþáûõ
x, y ∈ R ïîëîæèì
                   x t y = x + y + x · y,      xuy = x·y,          x0 = x + 1.
Òîãäà ÀÑ R∗ = h R, t, u, 0 , 0, 1 i  áóëåâà àëãåáðà.

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