Лекции по упорядоченным множествам и универсальной алгебре. Гуров С.И. - 135 стр.

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МГУ | Москва

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σ = h +, ·, −, 0, 1 i τ = h 2, 2, 1, 0, 0 i
Σ : (1
+
) − (4
+
), (5), (6), (1), (2) ,
(3
0
) ∀x 6= 0 ∃x
−1
(x
−1
x = xx
−1
= 1) .
σ = h +, ·, −, 0 i τ = h 2, 2, 1, 0 i
Σ : (1
+
) − (4
+
), (5), (6), (1), (2), (3
0
), (4) .
σ = h 6 i τ = h 2 i
Σ : (R) x 6 x ,
(AS) (x 6 y) N (y 6 x) ⇒ x = y ,
(T ) (x 6 y) N (y 6 z) ⇒ x 6 z ,
σ = h +, · i τ = h 2, 2 i
Σ : (1), (4), (1
+
), (4
+
), (9) ,
(9
+
) x + x = x ,
(10) x(x + y) = x , (10
+
) x + yz = x .
σ = h +, · i τ = h 2, 2 i
Σ : (1), (4), (1
+
), (4
+
), (9), (9
+
), (10), (10
+
) ,
(14) x(x + y + z) = xy + xz .
σ = h +, · i τ = h 2, 2 i
Σ : (1), (4), (1
+
), (4
+
), (9), (9
+
), (10), (10
+
) ,
(15) x(y + z) = xy + xz .
σ = h +, ·,
0
, 0, 1 i τ = h 2, 2, 1, 0, 0 i
Σ : (1), (4), (1
+
), (4
+
), (9), (9
+
),
(10), (10
+
), (5) ,
(11) x0 = 0 ; (11
+
) x + 1 = 1 ,
(12) xx
0
= 0 ; (12
+
) x + x
0
= 1 ,
(13) x
00
= x .
σ = h +, ·,
0
, 6, 0, 1 i τ = h 2, 2, 1, 2, 0, 0 i
Σ : (1), (4), (1
+
), (4
+
), (9), (9
+
),
(10), (10
+
), (5), (11), (11
+
), (12), (12
+
), (13),
(R), (AS), (T ) .
Ïðèëîæåíèå                                                                                     135


Òåëà.   σ = h +, ·, −, 0, 1 i ; τ = h 2, 2, 1, 0, 0 i ;

                    Σ:      (1+ ) − (4+ ), (5), (6), (1), (2) ,
                            (30 ) ∀x 6= 0 ∃x−1 (x−1 x = xx−1 = 1) .

Ïîëÿ.   σ = h +, ·, −, 0 i ; τ = h 2, 2, 1, 0 i ;

                    Σ:     (1+ ) − (4+ ), (5), (6), (1), (2), (30 ), (4) .

×.ó. ìíîæåñòâà.          σ = h6i; τ = h2i;

                    Σ:        (R) x 6 x ,
                              (AS) (x 6 y) N (y 6 x) ⇒ x = y ,
                              (T ) (x 6 y) N (y 6 z) ⇒ x 6 z ,

Ðåø¼òêè.     σ = h +, · i ; τ = h 2, 2 i ;

             Σ:     (1), (4), (1+ ), (4+ ), (9) ,
                    (9+ ) x + x = x ,
                    (10) x(x + y) = x ,                       (10+ ) x + yz = x .

Ìîäóëÿðíûå ðåø¼òêè.               σ = h +, · i ; τ = h 2, 2 i ;

                  Σ:      (1), (4), (1+ ), (4+ ), (9), (9+ ), (10), (10+ ) ,
                          (14) x(x + y + z) = xy + xz .

Äèñòðèáóòèâíûå ðåø¼òêè.                 σ = h +, · i ; τ = h 2, 2 i ;

                  Σ:      (1), (4), (1+ ), (4+ ), (9), (9+ ), (10), (10+ ) ,
                          (15) x(y + z) = xy + xz .

Áóëåâû àëãåáðû.           σ = h +, ·, 0 , 0, 1 i ; τ = h 2, 2, 1, 0, 0 i ;

               Σ:      (1), (4), (1+ ), (4+ ), (9), (9+ ),
                       (10), (10+ ), (5) ,
                       (11) x0 = 0 ;                                      (11+ ) x + 1 = 1 ,
                       (12) xx0 = 0 ;                                    (12+ ) x + x0 = 1 ,
                       (13) x00 = x .

Áóëåâû ñòðóêòóðû.             σ = h +, ·, 0 , 6, 0, 1 i ; τ = h 2, 2, 1, 2, 0, 0 i ;

                         Σ:     (1), (4), (1+ ), (4+ ), (9), (9+ ),
                                (10), (10+ ), (5), (11), (11+ ), (12), (12+ ), (13),
                                (R), (AS), (T ) .

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