一类完备格的直积分解与Fuzzy格

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_ 1 2r = @ x ∈ L, c(x) =↓ x ∩ CP (L), F (L) = {c(x)|x ∈ L}, F (L) U CP (L) E 2 - q 6 9 % G ϕ : L → F (L) Q 1 ϕ(x) = c(x), ∀x ∈ L, ϕ U` < 2 k W % G ? 6 q6 x (CP (L), F (L)) L = @ x ∈ CP (L), ( {x} = c(x). bt 1.8 I L U _ 1 K 2 r Q B ` ; h { E (1) = @ x ∈ CP (L), ' 6 m = 0 Q m x( 29 z [3] 9 I-1.1); (2)L Ur> Heyting * Y (z [3] 9 0–2.6), q6 x (CP (L), F (L)) 2 & T i J B < U r y (1) I A w (CP (L), F (L)) 2 i J B B = CP (L) \ A, I A B U r p = sup B, ? B ⊂ c(p). ; ( A ) c(p) ⊂ B. TPE C ' 6 x ∈ c(p) Q x ∈ B, x ∈ A. ' h { (1) ' 6 m = 0 Q m x. ' L U _ 1 K 2 ' 6 q ∈ CP (L) Q q ≤ m, %? q p = sup B. ' [3] I-1.5, ' 6 B 2 ( Q r {b1 , · · · , bn } Q q ≤ ∨n i=1 bi . 5' q U ¯=A: _ 1 ' 6 i (1 ≤ i ≤ n) Q q ≤ bi ∈ B, t q ∈ c(bi ). + q ∈ c(x) = {x} ⊂ A c(bi ) = {bi } U i 2 ` bi ∈ c(bi ) ⊂ A, >. A ∩ B = ∅ $ > ' $ A ) B = c(p), $ B U (CP (L), F (L)) L 2 r (2) A, B, p, x 5uc2 z (1) 2 A ) a C = @ b ∈ B ( x ∧ b = 0, ' L wr> Heyting * Y x = x ∧ p = x ∧ (∨B ) = ∨b∈B (x ∧ b) = 0, > U / 2 $' 6 b ∈ B Q x ∧ b = 0. ', L U _ 1 K 2 ' 6 q ∈ CP (L) Q q ≤ x ∧ b, %? q ∈ c(x), q ∈ c(b). ', A U i J B ` b ∈ c(b) ⊂ A, >. A ∩ B = ∅ $ > ' $ A ) B = c( p ) : B U r ' 9 1.7, 1.8 . p! 1.5 1" bt 1.9 6 9 1.8 2 h { ; L ' 6 v 4 J : L 2 @ Tv 4 J < U 5 |'
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z~ 1.2 I L U (X ( 1 1 .X 1 0 2J 3 Q L U 4 2- : - L L ' 6 I 1 a, b Q a ∨ b = 1 : a ∧ b = 0. y U ?2 A J = @ x ∈ X, u ↓ x = {y ∈ L|y ≤ x}. C ' L1 =↓ a, L2 =↓ b. 9 % G 6 a, b ∈ L, a = 0, b = 0 Q a ∨ b = 1 : a ∧ b = 0, f : L → L1 × L2 Q f (x) = (a ∧ x, b ∧ x), ∀x ∈ L. A A f UQkW $ L U 4 2 6 ; 2 ; J T q6 xu w (X, F ), 8 L F U X 2 r >g (- q6). ', F = @ ~\ : (X ( 1 $ (F, ⊂) Ur Q (F, ⊂)(yu w F ) w X 2 rQ z~ 1.3 I (X, F ) wq6 x (X, F ) U i x- : - (F, ⊂) U v 4 Q y ' i x9 . +f 1.2 F 1 bt 1.4 I (X, F ) wq6 x {Xi }i∈I w X 2 i J B 2 >g Fi w Q x Xi E 2 - q6 (i ∈ I ). C F ( v 4 J {Lj }j ∈J , (1) ' 6 \G g : I → J Q Fi . Lg(i) Q k W (i ∈ I ). (2) & T Xi 6 X L U 2 y (1) I ϕ : j∈J Lj → F Uk W % G 9 ek ∈ j∈J Lj # V Pk (ek ) = 1k : j = k N Pj (ek ) = 0j , 8 L 1k . 0k J S Lk 2 X ( 1.X 1 Pk : j ∈J Lj → Lk Un $ = @ j ∈ J, Yj = ϕ(ej ), Ej = {A ∈ F |A ⊂ Yj }, A " ( Y j , Ej ) U (X, F ) 2 Q x : Ej . Lj U Q k W2 w Lj U v 4 2 ' +f 1.2 (Yj , Ej ) U i x +', j = k N ϕ(ej ) ∩ ϕ(ek ) = ϕ(ej ∧ ek ) = ∅, ` = @ j ∈ J, ϕ(ej ) ∩ ϕ(∨k=j ek ) = ∪k=j (ϕ(ej ) ∩ ϕ(ek )) = ∅, ϕ(ej ) ∪ ϕ(∨k=j ek ) = ϕ(∨j ∈J ej ) = X. ' , ϕ(∨k=j ek ) U X 2 r $ Yj = ϕ(ej ) U X 2 r > A ) {Yj }j∈J U (X, F ) 2 ` ( i J B ', q6 x2 i J B J 2 v ' 6 \G g : I → J Q Xi = Yg(i) , Fi = Eg(i) . $ Fi . Lg(i) k W (i ∈ I ). $ & T Xi 6 X L (2) ' (1) 2 A ) D = @ j ∈ J, Yj = ϕ(ej ) U X 2 r U 2 F 2@ Tv 4 J < U ; x 1.5 I (X, F ) wq6 x C F ( v 4 J 5w2 B PY x R 2 ( Y> v 1.6 I & TQ (J J r Q) <' 6 v 4 J g W R 2 Q x u w Q. I Q E 2 - q6w F, (Q, F ) 2 & T i J B < U +8 r {r}, r ∈ Q. , {r} 6 (Q, F ) L U r $ ' 9 1.4 F ' 6 v 4 J bt 1.7 I (X, F ) wq6 x B ` X 2 & T i J B < U r F ( v 4 J y I {Xi }i∈I w X 2 i J B 2 >g Fi = {A ∩ Xi |A ∈ F }, (Xi , Fi ) U (X, F ) 2 i Q x 9 % G f : F → i∈I Fi # V = @ A ∈ F, f (A) = {A ∩ Xi }i∈I . ' {Xi }i∈I U X 2v + 2 Q r W : ~ D {Xi }i∈I U X 2 D r W $= @ {Ai }i∈I ∈ i∈I Fi , A = ∪i∈I Ai U X 2 r ([2] 9 1.1.11) ` f U# 2+ G: f −1 U 2 $ f U Q k W ' (Xi , Fi ) U i 2 D G ?fU Fi U v 4 2 ` {Fi }i∈I U F 2v 4 J I L w Q L L 2 1 p w _ 1 (m - _ 1 [3] ) B ` p ≤ x ∨ y U /p " p ≤ x m p ≤ y. B ` L L & T 1U Sw L L _12 L U _1K 2 y [4] A ) Cr Q L U _ 1 K 2 L . - T q6 x2 rQ k W w 9
a 36 pa 1 | 1993 { 1
ACTA MATHEMATICA SINICA
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Z
Vol.36, No. 1 Jan., 1993
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f_ kdn Fuzzy f_h *1 Z%E ( NME 3 N 221009) T`A (^ LH l 3 ^N 510050)
]rc4,\ 1. eXF93 <U$p\(7 o@>i QjM YH Heyting P m0k q/$5 GR Jf='0kq/$5N !2 #W*> [1] rc4, 2. hYH$D& 0kq/$5 Fuzzy & d')n": m - LS? Fuzzy & q/$5 o@> Jd Fuzzy & m0kq/$5 maV) fg^N d jm^ <U$p QjM YH Heyting P Fuzzy & q/$ 5 w e s \g` leo r> J 3 Q (J Q Q) 2 F oJ 9 ; y [1] i a9 OQ Q2GA| L |' | q6 x2 i J B. Q2 F oJ E x2 ~ %? d! V]F2 r Q2v 4F oJ 2' 6 . v W k ;n 9 b 1.1 (1) I L U TQ {Lx }x∈X U W Q C L ∼ = x∈X Lx ( ∼ = S Q k W), {Lx }x∈X U L 2 T F oJ y J (2) I {Lx }x∈X U L 2 TJ C {Lx }x∈X U L 2 T 4 CJ ' 6 x0 ∈ X, - x ∈ X : x = x0 N ( Lx = {0}, (3) C L I( 4 CJ L Uv42 M L U 42 (4) I {Lx }x∈X U L 2 TJ C = @ x ∈ X ( Lx = {0}, : Lx U v 4 2 {Lx }x∈X U L 2v 4 (5) I {Lx }x∈X . {Ly }y∈Y U L 2 TJ C ' 6 \G g : X → Y Q = @ J ∼ x ∈ X, Lx = Lg(x) , {Lx }x∈X . {Ly }y∈Y U L 25wJ ; T TPU) 2 (1) C {Lx}x∈X U L 2J : L U J 3 Q;r Q; Q;r> J 3 Q55 = & x ∈ X, Lx ( " 2 W (2) L U 4 2- : -' 6J F b 1 2Q L1 . L2 Q L ∼ = L1 × L2 .