❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ❚❍➚ ❍➬◆● ❚❍➁▼ ❱➋ ✣■➋❯ ❑■➏◆ (1 − C2) ❈Õ❆ ▼➷✣❯◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ✣❸■ ❙➮ ❱⑨ ▲Þ ❚❍❯❨➌❚ ❙➮ ▼➣ sè✿ ✽✹ ✻✵ ữớ ữợ P ò ề ◆❣❤➺ ❆♥ ✕ ✷✵✶✽ ▼ư❝ ❧ư❝ ❈⑩❈ ❑Þ ❍■➏❯ ❉Ị◆● ❚❘❖◆● ▲❯❾◆ ❱❿◆ ✸ ▲❮■ ◆➶■ ✣❺❯ ✹ ❈❍×❒◆● ■ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ✼ ✶ ▼æ✤✉♥ ❝♦♥ ❝èt ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷ ❈→❝ ✤✐➲✉ ❦✐➺♥ (Ci ) ❝õ❛ ♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ❈❍×❒◆● ■■ ✣■➋❯ ❑■➏◆ (1 − C2) ❈Õ❆ ▼➷✣❯◆ ✶✼ ❑➌❚ ▲❯❾◆ ✷✽ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✷✾ ✷ ❈⑩❈ ❑Þ ❍■➏❯ ❉Ị◆● ❚❘❖◆● ▲❯❾◆ ❱❿◆ N ❚➟♣ ❤đ♣ ❝→❝ sè tü ♥❤✐➯♥✳ N∗ ❚➟♣ ❤ñ♣ ❝→❝ sè tü ♥❤✐➯♥ ❦❤→❝ ✵✳ Z∗ ❚➟♣ ❤ñ♣ ❝→❝ sè ♥❣✉②➯♥✳ Q ❚➟♣ ❤ñ♣ ❝→❝ sè ❤ú✉ t✛✳ A≤M ❆ ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♠æ✤✉♥ ▼✳ A ≤e M ❆ ❧➔ ♠æ✤✉♥ ❝♦♥ ❝èt ②➳✉ ❝õ❛ ♠æ✤✉♥ ▼✳ A≤ ❆ ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ▼✳ M A⊆M ❆ ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❝õ❛ t➟♣ ▼✳ ❚ê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ♠æ✤✉♥✳ N M ▼ỉ✤✉♥ ◆ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ▼✳ u − dim(M ) ❈❤✐➲✉ ✤➲✉ ❝õ❛ ♠æ ✤✉♥ ▼✳ ❚r❛♥❣ ✸ ▲❮■ ◆➶■ ự ỵ tt ổ ✤÷đ❝ ♣❤→t tr✐➸♥ ♠↕♥❤ ♠➩ ✈➔ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ q trồ tr ự ỵ tt ✈➔♥❤ ❘ ❧➔ ♠ët ♠æ✤✉♥ tr➯♥ ❝❤➼♥❤ ♥â ✭♠æ✤✉♥ RR ✮ ♥➯♥ ♥❣÷í✐ t❛ ❝â t❤➸ ①→❝ ✤à♥❤ ❝➜✉ tró❝✱ t➼♥❤ ❝❤➜t✱ ✤➦❝ tr÷♥❣ ✈➔♥❤ ❘ q✉❛ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♠ët ❧ỵ♣ ♥➔♦ ✤â ❝→❝ ♠ỉ✤✉♥ tr➯♥ ✈➔♥❤ ❘✳ ❑❤✐ ♥❣❤✐➯♥ ❝ù✉ ♠ỉ✤✉♥ ♥❣÷í✐ t❛ ❝â ✤÷❛ r❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ (C1 ), (C2 ) ✈➔ (C3 ) ❝õ❛ ♠æ✤✉♥✳ ợ ổ tọ (C1 ) ữủ ❧➔ ❈❙✲♠æ✤✉♥ ✭❤❛② ❊①t❡♥❞✐♥❣ ♠♦❞✉❧❡✮ ✈➔ ❝â ♥❤✐➲✉ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❧ỵ♣ ♠ỉ✤✉♥ ♥➔②✳ ◆➠♠ ✶✾✾✹ ◆✳ ❱✳ ❉✉♥❣✱ ❉✳❱✳ ❍✉②♥❤✱ ❋✳ ❙♠✐t❤✱ ❘✳ ❲✐s❤❜❛✉❡r ✤➣ ✈✐➳t ❝✉è♥ s→❝❤ ✧ ❊①t❡♥❞✐♥❣ ♠♦❞✉❧❡✧ ♥❤➡♠ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ♠ỉ✤✉♥ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ (C1 ) ✭①❡♠ ❬✹❪✮✳ ▼ët ♠ỉ✤✉♥ ▼ ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (1 − C2 ) ♥➳✉ ✈ỵ✐ ♠å✐ ♠ỉ✤✉♥ ❝♦♥ ✤➲✉ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ▼ t❤➻ ♥â ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ▼ ✳ ❘ã r➔♥❣ ✤✐➲✉ ❦✐➺♥ (C2 ) s✉② r❛ ✤✐➲✉ ❦✐➺♥ (1 − C2 )✳ ✣➣ ❝â ♠ët sè t→❝ ❣✐↔ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ❧ỵ♣ ♠ỉ✤✉♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (C2 ) ✈➔ (1 − C2 )t✉② ♥❤✐➯♥ ❝→❝ ❦➳t q✉↔ ✤↕t ✤÷đ❝ ❝❤÷❛ ♥❤✐➲✉✳ ◆➠♠ ✷✵✶✼ ❝â ❝ỉ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❧ỵ♣ ♠ỉ✤✉♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (1 − C2 ) ❝õ❛ ▲➯ ❱➠♥ ❆♥✱ ◆❣✉②➵♥ ❚❤à ❍↔✐ ❆♥❤✱ ◆❣ỉ ❙ÿ ❚ị♥❣ ✭①❡♠ ❬✷❪✮✳ ❉ü❛ ✈➔♦ t➔✐ ❧✐➺✉ ❝❤➼♥❤ ❧➔ ❬✷❪✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉✱ t➻♠ ❤✐➸✉ ✈➔ t÷í♥❣ ❚r❛♥❣ ✹ ♠✐♥❤ ♠ët sè ❦➳t q✉↔ ❝õ❛ ❧ỵ♣ ♠ỉ✤✉♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (1 − C2 ) ✈➔ ❝→❝ ♠æ✤✉♥ ❧✐➯♥ q✉❛♥✳ ❱➻ ✈➟② ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ❧➔✿ ❦✐➺♥ (1 − C2) ❝õ❛ ♠æ✤✉♥ ❱➲ ✤✐➲✉ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t t trú ỗ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝ì sð ✶✳✶ ▼ỉ✤✉♥ ❝♦♥ ❝èt ②➳✉ ✶✳✷ ❈→❝ ✤✐➲✉ ❦✐➺♥ (Ci ) ❝õ❛ ♠æ✤✉♥ ❈❤÷ì♥❣ ✷✿ ✣✐➲✉ ❦✐➺♥ (1 − C2) ❝õ❛ ♠ỉ✤✉♥ ▲✉➟♥ ữủ tỹ t trữớ ữợ sỹ ữợ P ổ ũ ♥➔② t→❝ ❣✐↔ ①✐♥ ✤÷đ❝ tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s t ữợ trỹ t ✈✐➯♥✱ ❞➻✉ ❞➢t t➟♥ t➻♥❤✱ ❝❤➾ ❜↔♦ ♥❣❤✐➯♠ tó❝ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✱ t→❝ ❣✐↔ ❝ơ♥❣ ♥❤➟♥ ✤÷đ❝ sü ❣✐ó♣ ✤ï t➟♥ t➻♥❤ ❝õ❛ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ tr♦♥❣ tê ✣↕✐ sè tr÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❈ơ♥❣ tr♦♥❣ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❝↔♠ ì♥ sü ❣✐ó♣ ✤ï ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ❦❤♦❛ ❙❛✉ ✣↕✐ ❤å❝ trữớ ợ õ số ỵ tt số ỗ sỹ ú ù ✤ë♥❣ ✈✐➯♥✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ♥➔②✳ ❈✉è✐ ❝ò♥❣✱ ❞♦ ❦❤↔ ♥➠♥❣ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❚r❛♥❣ ✺ s❛✐ sât✱ t→❝ ❣✐↔ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü õ ỵ qỵ t ổ ũ tt ❝→❝ ❜↕♥✳ ◆❣❤➺ ❆♥✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❚r❛♥❣ ✻ ❈❍×❒◆● ■ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ❚r♦♥❣ s✉èt ❧✉➟♥ ✈➠♥ ♥➔② ♥➳✉ ❦❤æ♥❣ t❤➜② ❜ê s✉♥❣ ❣➻ t❤➯♠ t❤➻ ❝❤ó♥❣ tỉ✐ ❧✉ỉ♥ ❣✐↔ t❤✐➳t r➡♥❣ ❘ ❧➔ ✈➔♥❤ t ủ õ ỡ ỵ ổ ổ t ỵ ❝→❝ t➼♥❤ ❝❤➜t ❝❤ó♥❣ tỉ✐ ❞ü❛ ✈➔♦ ▲❡ ❱❛♥ ❆♥✱ ◆❣✉②❡♥ ❍❛✐ ❆♥❤✱ ◆❣♦ ❙② ❚✉♥❣ ❬✷❪❀ ❉✉♥❣ ◆✳❱✱ ❍✉②♥❤ ❉✳❱✱ ❙♠✐t❤ P✳❋ ❛♥❞ ❲✐s❜❛✇❡r ❘ ❬✹❪✱ ❋✳❑❛s❝❤ ❬✺❪✳ ✶ ▼æ✤✉♥ ❝♦♥ ❝èt ②➳✉ ✶✳✶ ✣à♥❤ ♥❣❤➽❛ ❈❤♦ ✈➔♥❤ ❘ ✈➔ ❘✲♠ỉ✤✉♥ ▼✳ ▼ỉ✤✉♥ ❝♦♥ ❆ ❝õ❛ ▼ ✤÷đ❝ ❣å✐ ❧➔ ♠ỉ✤✉♥ ❝♦♥ ❝èt ②➳✉ ❝õ❛ ▼ ♥➳✉ ✈ỵ✐ ♠å✐ ♠æ✤✉♥ ❝♦♥ B = 0, B ≤ M t❤➻ A B = ỵ A e M ✶✳✷ ❍➺ q✉↔ A ≤e M ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐ ♠ỉ✤✉♥ ❝♦♥ ❇ ❝õ❛ ▼ ♠➔ A∩B = t❤➻ B = 0✳ ✶✳✸ ❱➼ ❞ư ✶✮ ❱ỵ✐ ổ t ý tM e ữợ ≤e M ⇔ M = 0✳ ✷✮ ❈❤♦ Z ❧➔ ✈➔♥❤ ❝→❝ sè ♥❣✉②➯♥ ✱ ①➨t Z ✲♠æ✤✉♥ Z ✭♠æ✤✉♥ Z tr➯♥ ❝❤➼♥❤ ❚r❛♥❣ ✼ ♥â✮✳ ❑❤✐ ✤â ♠å✐ ♠æ✤✉♥ ❦❤→❝ ✵ ❝õ❛ Z ❧➔ ♠æ✤✉♥ ❝♦♥ ❝èt ②➳✉ tr Z ự rữợ t t t r ♠æ✤✉♥ ❝♦♥ A = ❝õ❛ Z ✤➲✉ ❝â ❞↕♥❣ A = k.Z(k ∈ N∗ )✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❆ ❝èt ②➳✉ tr♦♥❣ Z✳ ▲➜② ❜➜t ❦ý = B ⊆ Z ❚❛ ❝â B = n.Z, n ∈ N∗ ✳ ❑❤✐ ✤â kn ∈ kZ ∩ nZ ✈➔ kn = ❉♦ ✤â A ∩ B = ✸✮ ❳➨t Z ✲♠æ✤✉♥ Q ✭tù❝ ♥❤â♠ ❝ë♥❣ ❝→❝ sè ❤ú✉ t✛ Q✮✳ ❑❤✐ ✤â ♠å✐ ♠æ✤✉♥ ❝♦♥ ❦❤→❝ ✵ ❝õ❛ Q ❧➔ ❝èt ②➳✉ tr♦♥❣ Q✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② ❳ ❧➔ ♠æ✤✉♥ ❝♦♥ ❦❤→❝ ✵ ❜➜t ❦ý ❝õ❛ Q✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ X ≤e Q ▲➜② = B ổ Q õ tỗ t = a/b ∈ A tr♦♥❣ ✤â a, b ∈ Z∗ = p/q ∈ B tr♦♥❣ ✤â p, q ∈ Z∗ ✳ ❚❛ ❝â ap = pb.a/b ∈ A; ap = aq.p/q ∈ B; ap = 0✳ ❈❤ù♥❣ tä A ∩ B = 0✳ ✶✳✹ ✣à♥❤ ♥❣❤➽❛ ▼ỉ✤✉♥ ❯ ✤÷đ❝ ❣å✐ ❧➔ ♠æ✤✉♥ ✤➲✉ ♥➳✉ ♠å✐ ♠æ✤✉♥ ❝♦♥ ❦❤→❝ ❦❤æ♥❣ ❝õ❛ ❯ ✤➲✉ ❝èt ②➳✉ tr♦♥❣ ❯✳ ❚r❛♥❣ ✽ ✶✳✺ ❍➺ q✉↔ ❯ ❧➔ ♠æ✤✉♥ ✤➲✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ❤❛✐ ♠ỉ✤✉♥ ❝♦♥ ❦❤→❝ ❦❤ỉ♥❣ ❜➜t ❦ý ✤➲✉ ❝â ❣✐❛♦ ❦❤→❝ ❦❤ỉ♥❣✳ ✶✳✻ ❱➼ ❞ư Z ✈➔ Q ❧➔ ♠æ✤✉♥ ✤➲✉ ✭①➨t tr➯♥ ✈➔♥❤ Z✮✳ ✶✳✼ ✣à♥❤ ♥❣❤➽❛ ❈❤♦ ❆ ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ▼✱ t❛ ♥â✐ ❆ ❧➔ ♠æ✤✉♥ ❝♦♥ ✤â♥❣ tr♦♥❣ ▼ ♥➳✉ ❆ ❦❤æ♥❣ ❝â ♠ð rë♥❣ ❝èt ②➳✉ t❤ü❝ sü tr♦♥❣ ▼✳ ❚ù❝ ❧➔ A ≤e K ⊆ M ⇒ A = K ✳ ✶✳✽ ❱➼ ❞ö ❆ ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ▼ t õ tr ú ỵ A M tỗ t ổ ❝õ❛ ▼ ♠➔ A ∩ B = ✈➔ A + B = M ✳ ❑❤✐ ✤â t❛ ✈✐➳t A B = M✳ ❈❤ù♥❣ ♠✐♥❤ ✈➼ ❞ö ✶✳✶✳✽✳ ❈❤♦ ❑ ❧➔ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ ▼ ♠➔ A ≤e K ✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❆❂❑✳ ❉♦ A ≤ M tỗ t ổ s ❝❤♦ M = A B(∗)✳ ❱➻ A ≤ K ♥➯♥ ❞ị♥❣ ❧✉➟t ♠♦❞✉❧❡✳ ●✐❛♦ ❤❛✐ ✈➳ ❝õ❛ ✭✯✮ ✈ỵ✐ ❑✱ t❛ ❝â ✿ K=A (B ∩ K)✳ ❚r❛♥❣ ✾ ◆➳✉ tỗ t = x B K t ①❘ ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ B ∩ K ✳ ❙✉② r❛ = xR ≤ K ✳ ❉♦ A ≤e K ♥➯♥ xR ∩ A = 0✳ ✣✐➲✉ ✤â ❝❤ù♥❣ tọ tỗ t = r R s = xr ∈ A ♠➔ = xr ∈ xR ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ❇✳ ❉♦ ✤â = xr A B ổ ỵ ❝❤ù♥❣ tä ♠å✐ x ∈ B ∩ K t❤➻ x = 0✱ ❤❛② ❆❂❑✳ ◆❤➢❝ ❧↕✐ ❧✉➟t ♠♦❞✉❧❡✿ ❈❤♦ ▼✱❆✱❇✱❳ ❝ị♥❣ ❧➔ ♠ët ❝➜✉ tró❝ ✤↕✐ sè ♥➔♦ ✤â ✭♥❤â♠✱ ✈➔♥❤✱ tr÷í♥❣✱ ♠ỉ✤✉♥✱✳✳✮ ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ❧✉➟t ▼♦❞✉❧❡ ♥➳✉ M = A + B ✈➔ X ≤ M, A ≤ X t❤➻ X = A + (X ∩ B)✳ ❚ù❝ ❧➔ X = M ∩ X = (A + B) ∩ X = A + (X ∩ B) ✶✳✶✵ ❍➺ q✉↔ ❈➜✉ tró❝ ♠ỉ✤✉♥ t❤ä❛ ♠➣♥ ❧✉➟t ▼♦❞✉❧❡✳ ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ❝♦♥ ❝èt ②➳✉ ✳ ✶✳✶✶ ▼➺♥❤ ✤➲ ❈❤♦ ♠æ✤✉♥ ▼✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ ①↔② r❛✿ ✭✐✮ ❈❤♦ ❆ ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ▼✳ ❑❤✐ ✤â t❤➻ A ≤e M ⇔ ✈ỵ✐ ♠å✐ 0=x∈M xR ∩ A = 0✳ ✭✐✐✮ ❈❤♦ ❝❤➾ ❦❤✐ A≤B ≤C A ≤e B ✭✐✐✐✮ ❈❤♦ ✈➔ ❧➔ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ▼✳ ❑❤✐ ✤â B ≤e C ✳ Ai ≤e Mi ∀i = (1, n) ✱ ❦❤✐ ✤â ❚r❛♥❣ ✶✵ A ≤e C ❦❤✐ ✈➔ ✷ ❈→❝ ✤✐➲✉ ❦✐➺♥ (C ) ❝õ❛ ♠æ✤✉♥ i ✷✳✶ ✣à♥❤ ♥❣❤➽❛ ❈❤♦ ♠æ✤✉♥ ▼ tr➯♥ ✈➔♥❤ ❘✳ ❚❛ ①➨t ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤è✐ ✈ỵ✐ ♠ỉ✤✉♥ ▼✿ (C1 )✿ ❱ỵ✐ ♠å✐ ♠ỉ✤✉♥ ❝♦♥ ❆ ❝õ❛ ▼✱ tỗ t B (C2 ) ổ ❝♦♥ ❝õ❛ ▼✱ A ≤ (C3 )✿ ❈❤♦ A, B ≤ M, A ∩ B = t❤➻ A M, A B≤ M ✤➸ A ≤e B ✳ B t❤➻ B ≤ M✳ (1 − C1 )✿ ▼å✐ ♠æ✤✉♥ ❝♦♥ tỗ t B (1 C2 )✿ ❈❤♦ ❆✱❇ ❧➔ ♠æ✤✉♥ ❝♦♥ ✤➲✉ ❝õ❛ ▼✱ A ≤ B≤ M ✷✳✷ ❚➼♥❤ ❝❤➜t ❳➨t Z✲♠æ✤✉♥ Z✳ ❚❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✿ ❑❤➥♥❣ ✤à♥❤ ✶✳ Z ❝â ✤✐➲✉ ❦✐➺♥ (C1 )✳ ❑❤➥♥❣ ✤à♥❤ ✷✳ Z ❦❤æ♥❣ ❝â ✤✐➲✉ ❦✐➺♥ (C2 )✳ ❑❤➥♥❣ ✤à♥❤ ✸✳ Z ❝â ✤✐➲✉ ❦✐➺♥ (C3 )✳ ❑❤➥♥❣ ✤à♥❤ ✹✳ Z ❝â ✤✐➲✉ ❦✐➺♥ (1 − C1 )✳ ✷✳✸ ❈❤ù♥❣ ♠✐♥❤ ❑❤➥♥❣ ✤à♥❤ ✶✳ Z ❝â ✤✐➲✉ ❦✐➺♥ (C1 )✳ ❈❤ù♥❣ ♠✐♥❤✿ ❱ỵ✐ ♠å✐ ♠ỉ✤✉♥ ❝♦♥ ❆ ❝õ❛ Z t❤➻✿ ✰✮ ◆➳✉ A = ⇒ A ≤e ≤ Z ✰✮ ◆➳✉ A = ⇒ A = kZ ≤e Z ≤ Z, k = ❚r❛♥❣ ✶✺ M M ✤➸ U ≤e B ✳ M, A B t❤➻ ❑❤➥♥❣ ✤à♥❤ ✷✳ ❩ ❦❤æ♥❣ ❝â ✤✐➲✉ ❦✐➺♥ (C2 )✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ t❛ ❝â nZ Z, ∀n ∈ N ∗ ♠➔ Z ≤ Z ♥❤÷♥❣ nZ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ Z ❑❤➥♥❣ ✤à♥❤ ✸✳ Z ❝â ✤✐➲✉ ❦✐➺♥ (C3 )✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ Z ❝❤➾ ❝â ✵ ✈➔ Z ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ✱ ♠➔ Z ∩ = ✈➔ Z 0≤ Z ❑❤➥♥❣ ✤à♥❤ ✹✳ Z ❝â ✤✐➲✉ ❦✐➺♥ (1 − C1 )✳ ❈❤ù♥❣ ♠✐♥❤✳ ❍✐➸♥ ♥❤✐➯♥ ✈➻ (C1 ) ⇒ (1 − C1 ) ✷✳✹ ✣à♥❤ ♥❣❤➽❛ ❛✳ ▼æ✤✉♥ ▼ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (C1 ) ❣å✐ ❧➔ ❈❙✲♠æ✤✉♥ ✭❤❛② ❊①t❡♥❞✐♥❣ ♠♦❞✉❧❡✮✳ ❜✳ ▼æ✤✉♥ ▼ ❝â ✤✐➲✉ ❦✐➺♥ (C1 ) ✈➔ (C2 ) ✤÷đ❝ ❣å✐ ❧➔ ♠ỉ✤✉♥ ❧✐➯♥ tư❝✳ ❝✳ ▼ỉ✤✉♥ ▼ ❝â ✤✐➲✉ ❦✐➺♥ (C1 ) ✈➔ (C3 ) ✤÷đ❝ ❣å✐ ❧➔ ♠ỉ✤✉♥ tü❛ ❧✐➯♥ tư❝✳ ✷✳✺ ❱➼ ❞ư • Z✲♠ỉ✤✉♥ Z ổ tỹ tử ữ ổ tử ã Z✲♠ỉ✤✉♥ Q ❧➔ ❧✐➯♥ tư❝ ✈➔ tü❛ ❧✐➯♥ tư❝✳ ❚r❛♥❣ ✶✻ ❈❍×❒◆● ■■ ✣■➋❯ ❑■➏◆ (1 − C2) ❈Õ❆ ▼➷✣❯◆ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ❝❤õ ②➳✉ tr➻♥❤ ❜➔② ❝❤✐ tt ởt số t q rữợ t ú t❛ ①❡♠ ①➨t ✤✐➲✉ ❦✐➺♥ s❛✉ ❝❤♦ ♠ët ♠æ✤✉♥ ▼✳ • (1 − C2 ) ✿ ▼é✐ ♠æ✤✉♥ ❝♦♥ ✤➲✉ ❝õ❛ ▼✱ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♠ët ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ▼ t❤➻ ❜↔♥ t❤➙♥ ♥â ❝ô♥❣ ❧➔ ♠ët ❤↕♥❣ tỷ trỹ t ã ởt ổ ữủ ✤à♥❤ ♥❣❤➽❛ ❧➔ ♠ët ♠æ✤✉♥ (1 − C2 ) ♥➳✉ ♥â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (1 − C2 )✳ ◆➳✉ ▼ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (1 − C1 ) ✈➔ (1 − C2 ) t❤➻ ▼ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠ỉ✤✉♥ ✶✲❧✐➯♥ tư❝ ♠↕♥❤✳ ▼ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠ỉ✤✉♥ ✶✲❧✐➯♥ tö❝ ♠↕♥❤ ♥➳✉ ♥â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (C1 ) ✈➔ (1 − C2 )✳ ▼ët ✈➔♥❤ ❘ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✈➔♥❤ ♣❤↔✐ ✭ tr→✐ ✮ ✶✲❧✐➯♥ tö❝ ♥➳✉ RR ✭ t÷ì♥❣ ù♥❣ R R✮ ❧➔ ♠ët ♠ỉ✤✉♥ ✶✲❧✐➯♥ tư❝✳ ❈❤ó♥❣ t❛ ❝â ❝→❝ ♣❤➨♣ ❦➨♦ t❤❡♦ s❛✉✿ ▲✐➯♥ tö❝ ⇒ ✶✲❧✐➯♥ tö❝ ♠↕♥❤ ⇒ ✶✲❧✐➯♥ tö❝✱ ✈➔ (C2 ) (1 C2 ) ữ ỵ ❚❤❡♦ ❬✹✱ ❍➺ q✉↔ ✼✳✽❪✱ ✈ỵ✐ ▼ ❧➔ ♠ët ❘✲♠ỉ✤✉♥ ♣❤↔✐ ✈ỵ✐ ❝❤✐➲✉ ✤➲✉ ❤ú✉ ❤↕♥✱ ▼ ❧➔ ♠ët ♠ỉ✤✉♥ (1 − C1 ) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ▼ ❧➔ ❈❙✳ ❱➻ t❤➳ ▼ ❝â ❝❤✐➲✉ ✤➲✉ ❤ú✉ ❤↕♥ ❦❤✐ ➜② ▼ ❧➔ ♠ët ♠ỉ✤✉♥ ✶✲❧✐➯♥ tư❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ▼ ❧➔ ♠ët ♠ỉ✤✉♥ ✶✲❧✐➯♥ tư❝ ♠↕♥❤✳ ◆â✐ ❝❤✉♥❣✱ ♥➳✉ ▼ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (1 − C2 )✱ ▼ ❝â t❤➸ ❦❤æ♥❣ t❤ä❛ ✤✐➲✉ ❦✐➺♥ (C2 )✳ ❚❤❡♦ ❝→❝ ✤à♥❤ ♥❣❤➽❛ (1 − C2 ) ♠æ✤✉♥✱ ♠æ✤✉♥ ✶✲❧✐➯♥ tư❝ ✈➔ ♠ỉ✤✉♥ ✶✲❧✐➯♥ tư❝ ♠↕♥❤✱ ❝❤ó♥❣ t❛ ❝â✿ ❚r❛♥❣ ✶✼ ❇ê ✤➲ ✷✳✷ ❱ỵ✐ ▼ ❧➔ ♠ët ❘✲♠ỉ✤✉♥ ♣❤↔✐ ✈➔ ◆ ❧➔ ♠ët ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ▼✳ ◆➳✉ ▼ ❧➔ ♠ët (1 − C2 ) (1 − C2 ) ♠ỉ✤✉♥ ✭ ✶✲❧✐➯♥ tư❝✱ ✶✲❧✐➯♥ tư❝ ♠↕♥❤✮ t❤➻ ◆ ❝ơ♥❣ ❧➔ ♠ỉ✤✉♥ ✭ t÷ì♥❣ ù♥❣ ✶✲❧✐➯♥ tư❝✱ ✶✲❧✐➯♥ tử ỵ ợ U= n i=1 Ui tr♦♥❣ ✤â ♠é✐ Ui ❧➔ ♠ët ♠æ✤✉♥ ✤➲✉✱ ❦❤✐ ✤â ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭✐✮ ❯ ❧➔ ♠ët ♠æ✤✉♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✐✐✮ ❯ ❧➔ ♠ët ♠æ✤✉♥ (1 − C2) (C2)❀ ✈➔ ❯ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (C3)✳ ❈❤ù♥❣ ♠✐♥❤✿ (i) ⇒ (ii)✳ ✭❍✐➸♥ ♥❤✐➯♥✮ (ii) ⇒ (i) : ❈❤ó♥❣ t❛ ❜✐➵✉ ❞✐➵♥ ❯ ❧➔ ♠ët ♠æ✤✉♥ (C2)✱ tù❝ ❧➔✱ ❝❤♦ ❤❛✐ ♠æ✤✉♥ ❝♦♥ ❳✱❨ ❝õ❛ ❯✱ ✈ỵ✐ X ∼ = Y ✈➔ ❨ ❧➔ ♠ët ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ❯✱ ❳ ❝ô♥❣ ❧➔ ♠ët tờ trỹ t ữ ỵ ởt ổ õ tỗ t ởt t ❝♦♥ ❋ ❝õ❛ 1, , n s❛♦ ❝❤♦ Y ❝♦♥ ❝èt ②➳✉ ❝õ❛ ❯✳ ◆❤÷♥❣ Y ( i∈F ( i∈F Ui ) ❧➔ ♠ët ♠æ✤✉♥ Ui ) ❧➔ ❝→❝ tê♥❣ trü❝ t✐➳♣ ❝ò❛ ❯ ✈➔ ❯ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (C3) ✱ ❝❤ó♥❣ t❛ s✉② r❛ Y ( i∈F Ui ) = U ✳ ◆➳✉ F = 1, , n t❤➻ X = Y = ♥❤÷ ♠♦♥❣ ✤đ✐✳ ◆➳✉ F = 1, , n ✈➔ t➟♣ = 1, , n\F ✱ ❦❤✐ ✤â U = Y ❚r❛♥❣ ✶✽ ( i∈F Ui ) = ( i∈F Ui ) ( i∈F Ui )✳ ◆❤÷ ✈➟② X ∼ = U/( =Y ∼ i∈F Ui ) ∼ = ●✐↔ sû J = 1, , k ✈ỵ✐ ≤ k ≤ n ✱ tù❝ ❧➔✱ Z = U1 i∈J Ui = Z Uk ✳ ❱ỵ✐ ϕ : Z → X ✈➔ t➟♣ Xi = ϕ(Ui ) ❦❤✐ ✤â Xi ∼ = Ui ❝❤♦ ❜➜t ❦ý i = 1, , k ✳ ❈❤ó♥❣ t❛ t❤ø❛ ♥❤➟♥ X = ϕ(Z) = ϕ(U1 X1 Uk ) = ϕ(U1 ) ϕ(Uk ) = Xk ✳ ❱➻ Xi ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ✤➲✉ ❝õ❛ ❯✱ Xi ∼ = Ui ✈ỵ✐ Ui ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯ ✈➔ ❯ ❧➔ ♠ët ♠æ✤✉♥ (1 − C2)✱ Xi ❝ô♥❣ ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯ ❝❤♦ ❜➜t ❦ý i = 1, , k ✳ ❚✉② ♥❤✐➯♥ ❯ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (C3)✱ X = X1 Xk ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯✳ ❉♦ ✤â ởt ổ (C2)( (i)) ỵ U= n i=1 Ui tr♦♥❣ ✤â ♠é✐ Ui ❧➔ ♠ët ♠æ✤✉♥ ✤➲✉ ✱ ❦❤✐ ✤â ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭✐✮ ❯ ❧➔ ♠ët ♠ỉ✤✉♥ ❧✐➯♥ tư❝❀ ✭✐✐✮ ❯ ❧➔ ♠ët ♠ỉ✤✉♥ ❧✲❧✐➯♥ tư❝✳ ❈❤ù♥❣ ♠✐♥❤✿ (i) ⇒ (ii) ✿ ❤✐➸♥ ♥❤✐➯♥✳ (ii) ⇒ (i)✿ ❈❤ó♥❣ t❛ ❜✐➸✉ ❞✐➵♥ S = End(Ui ) ❧➔ ♠ët ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ❝❤♦ ❜➜t ❦ý = 1, , n✳ ✣➛✉ t✐➯♥✱ ❝❤ó♥❣ t❛ ❣✐↔ sû Ui ❦❤æ♥❣ ♥➡♠ tr♦♥❣ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Ui ✳ ❚❛ ❝â f : Ui → Ui ❧➔ ♠ët ✤ì♥ →♥❤ ✈ỵ✐ f (Ui ) ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Ui ✳ ❚➟♣ f (Ui ) = V ✱ ✈➔ V = 0✱ ♠æ✤✉♥ ❝♦♥ t❤➟t ❚r❛♥❣ ✶✾ sü ❝õ❛ Ui ✈➔ V ∼ = Ui ✳ ❚❤❡♦ ❣✐↔ t❤✐➳t ✱ Ui ❧➔ ♠ët ♠æ✤✉♥ (1 − C2 ) ❞♦ ✤â ❱ ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ Ui ✱ ❝ư t❤➸ ❤ì♥✱ Ui ❦❤æ♥❣ ♣❤↔✐ ♠æ✤✉♥ ✤➲✉✱ ✭♠➙✉ t❤✉➝♥✮✳ ❉♦ ✤â Ui ❦❤æ♥❣ ♥➡♠ tr♦♥❣ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Ui ❱ỵ✐ g ∈ S ✈➔ ❣✐↔ sû ❣ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ ❚ø ✤â s✉② r❛ ✶✲❣ ởt ữ ỵ ổ ♠ët ✤ì♥ →♥❤✳ ❉♦ ✤â ✈ỵ✐ Keg(g) ❧➔ ♠ët ♠ỉ✤✉♥ ❝♦♥ ❦❤→❝ ✵✱ ♥â ❝➛♥ ð tr♦♥❣ ♠æ✤✉♥ ✤➲✉ Ui ✳ ❈❤ó♥❣ t❛ ❧✉ỉ♥ ❝â Keg(g) ∩ Keg(1 − g) = 0✱ t❤❡♦ s❛✉ ✤â Ker(1 − g) = ✱ ❝ư t❤➸✱ ✶✲❣ ❧➔ ♠ët ✤ì♥ →♥❤✳ ❚✉② ♥❤✐➯♥ Ui ❦❤æ♥❣ ♥➡♠ tr♦♥❣ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Ui ✱ ✶✲❣ ♣❤↔✐✱ ✈➔ ❞♦ ✤â ✶✲❣ ❧➔ ♠ët ữ ủ ợ Uij = Ui Uj ✈ỵ✐ i, j ∈ 1, , n ✈➔ i = j ✳ ❈❤ó♥❣ t❛ ❝❤♦ t❤➜② Uij t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❈✸✮ ✱ ❝ö t❤➸✱ ❝❤♦ ❤❛✐ tê♥❣ trü❝ t✐➳♣ S1 , S2 ❝õ❛ Uij ✈ỵ✐ S1 ∩ S2 = 0✱ S1 S2 ❝ô♥❣ ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ Uij ữ ỵ u dim(Uij ) = 2✱ ♥❤ú♥❣ tr÷í♥❣ ❤đ♣ s❛✉ ❧➔ t❤÷í♥❣✿ ✶✮ ▼ët tr♦♥❣ ♥❤ú♥❣ Si ❝â ❝❤✐➲✉ ✤➲✉✱ ❦➳t q✉↔ ❧➔ ♥❤ú♥❣ Si ❝á♥ ❧↕✐ ❧➔ ✵✱ ❤♦➦❝ ✷✮ ▼ët tr♦♥❣ ❝→❝ Si ❧➔ ✵ ❉♦ ✤â ❝❤ó♥❣ t❛ ①❡♠ ①➨t tr÷í♥❣ ❤đ♣ ❝↔ ❤❛✐ S1 , S2 ❧➔ ✤➲✉✳ ❈❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ Ui ❦❤æ♥❣ ♥➡♠ tr♦♥❣ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Uj ✳ ❱ỵ✐ h : Ui → Uj ❧➔ ♠ët ✤ì♥ →♥❤ ✈ỵ✐ h(Ui ) ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Uj ✳ ❚➟♣ h : Uxi → Uj ❧➔ ♠ët ✤ì♥ →♥❤ ✈ỵ✐ h(Ui ) ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t ❚r❛♥❣ ✷✵ sü ❝õ❛ Uj ✳ ❚➟♣ h(Uj ) = L, L = 0✱ ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Uj ✈➔ L ∼ = Ui ✳ ❚❤❡♦ ❣✐↔ t❤✐➳t ✱ ❯ ❧➔ ♠ët ♠æ✤✉♥ (1 − C2 ) ✈➔ Uij ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯✱ Uij ❝ơ♥❣ ❧➔ ♠ỉ✤✉♥ (1 − C2 )✳ ữ ỵ r ởt ổ ừUij ✈➔ L ∼ = Ui ✈ỵ✐ Ui ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ Uij ✱ ▲ ❝ô♥❣ ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ Uij ✳ ❚➟♣ Uij = L ❝❤ó♥❣ t❛ ❝â Ui = L L ✳ ❉♦ ✤â t❤❡♦ t➼♥❤ ❝❤➜t ♠ỉ✤✉♥ L ✈ỵ✐ L = Uj ∩ L ữ ỵ r L ụ ởt ổ t❤➟t sü ❝õ❛ Uj ✈➔ L = ✱ ❞♦ ✤â Uj ❦❤æ♥❣ ❧➔ ♠æ✤✉♥ ✤➲✉✱ ✭♠ët sü ♠➙✉ t❤✉➝♥✮✳ ❉♦ ✤â Ui ❦❤æ♥❣ ♥➡♠ tr♦♥❣ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Uj ✳ ❚÷ì♥❣ tü Uj ❦❤ỉ♥❣ ♥➡♠ tr♦♥❣ ởt ổ tt sỹ Uj ữ ỵ r➡♥❣ Ui ✭✈➔ Uj ✮ ❦❤æ♥❣ ♥➡♠ tr♦♥❣ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Ui ✭❚÷ì♥❣ tü ❝❤♦ Uj ✮ ữ ỵ End(Ui ) End(Uj ) ỳ ✤à❛ ♣❤÷ì♥❣✱ t❤❡♦ ❜ê ✤➲ ❝õ❛ ❆③✉♠❛②❛ ✭❬✶✱✶✷✳✻✱✶✷✳✼❪✮✱ ❝❤ó♥❣ t❛ ❝â Uij = S2 S2 K = S2 K = S2 Ui ❤♦➦❝ Uj ✳ ❉♦ ✐ ✈➔ ❥ ❝â t❤➸ t❤❛② t❤➳ ❧➝♥ ♥❤❛✉✳ ❈❤ó♥❣ t❛ ❝❤➾ ❝➛♥ ①❡♠ ①➨t ✶ tr♦♥❣ ✷ ❦❤↔ ♥➠♥❣✳ ❈❤ó♥❣ t❛ ❤➣② ①❡♠ ①➨t tr÷í♥❣ ❤đ♣ Uij = S2 S1 K = S2 H = S1 ◆➳✉ Uij = S1 S1 S2 = S1 Ui = Ui Ui ❤♦➦❝ S1 H = S1 Uj ✳ ❚✐➳♣ ✤â S2 ∼ = Uj ✳ ❱✐➳t Uij = H = S1 Uj Ui ✱ t❤❡♦ t➼♥❤ ❝❤➜t ♠ỉ✤✉♥ ❝❤ó♥❣ t❛ ❝â W tr♦♥❣ ✤â W = (S1 S2 ) ∩ Ui ✳ ❚ø ✤➙② ❝❤ó♥❣ t❛ ❝â W ∼ = Uj ✳ ❉♦ = S2 ✱ ✤✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔ Ui ❝❤ù❛ ♠ët ❜↔♥ s❛♦ ❝õ❛ S2 ∼ Ui ❦❤æ♥❣ ♥➡♠ tr♦♥❣ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Ui ✱ ❝❤ó♥❣ t❛ ♣❤↔✐ ❝â W = Ui ✈➔ ❞♦ ✤â S1 S2 = Ui Uj = Uij ✳ ❚r❛♥❣ ✷✶ ◆➳✉ Uij = S1 S1 S2 = S1 H = S1 Uj ✱ t❤❡♦ t➼♥❤ ❝❤➜t ♠ỉ✤✉♥ ❝❤ó♥❣ t❛ ❝â W tr♦♥❣ ✤â W = (S1 S2 ) ∩ Uj ✳ ❚ø ✤➙② ❝❤ó♥❣ t❛ ❝â W ∼ = Uj ✳ ❉♦ = S2 ✱ ✤✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ Uj ❝❤ù❛ ♠ët ❜↔♥ s❛♦ ❝õ❛ S2 ∼ Uj ❦❤æ♥❣ ♥➡♠ tr♦♥❣ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Uj ✱ ❝❤ó♥❣ t❛ ♣❤↔✐ ❝â W = Uj ✈➔ ❞♦ ✤â S1 S2 = Uij ✳ ữ t Uij tọ ữ ỵ r Uij ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯ ✈➔ ❯ ởt ổ õ tữợ ợ ❤↕♥ ✈➔ ❯ ❧➔ ♠ët ✭✶✲❈✶ ♠æ✤✉♥ ✮ ♥➯♥ ❯ ❧➔ ♠ët ❈❙ ♠ỉ✤✉♥ ✮✱ Ui j ❝ơ♥❣ ❧➔ ♠ët ❈❙✲♠æ✤✉♥ ✱ ✈➔ ❞♦ ✤â Uij ❧➔ ♠ët ♠æ✤✉♥ tü❛ ❧✐➯♥ tư❝ ✈ỵ✐ ❜➜t ❦ý i, j ∈ 1, , n, i = j ✳ ❇➙② ❣✐í✱ t❤❡♦ ❬✻✱ ❤➺ q✉↔ ✶✶❪✱ ❯ ❧➔ ♠ët ♠ët ♠ỉ✤✉♥ tü❛ ❧✐➯♥ tư❝✳ ỵ ởt ổ tử ự ữủ q ợ U = n i=1 Ui tr♦♥❣ ✤â ♠é✐ Ui ❧➔ ♠ët ♠ỉ✤✉♥ ✤➲✉✱ t❤➻ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭✐✮ ❯ ❧➔ ♠ët ✭✐✐✮ ❯ ❧➔ ♠ët ✲♠ỉ✤✉♥ tü❛ ♥ë✐ ①↕❀ ✲✤➳♠ ✤÷đ❝ (1 − C1 )✲♠ỉ✤✉♥✳ ❈❤ù♥❣ ♠✐♥❤✿ (i) ⇒ (ii)✿ ❤✐➸♥ ♥❤✐➯♥✳ (ii) ⇒ (i) : ỵ ởt ổ tử ❚❤❡♦ ❬✼✱ ♠➺♥❤ ✤➲ ✷✳✺❪✱ ❯ ❧➔ ♠ët ✲♠æ✤✉♥ tü❛ ♥ë✐ ①↕ ✭✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ✭✐✮✮✳ ▼ët ❘✲♠ỉ✤✉♥ ♣❤↔✐ ữủ ố ợ t ý ♠æ✤✉♥ ❝♦♥ ❆✱❇✱❈ ❝õ❛ ▼ t❤➻ A ∩ (B + C) = A ∩ B + A ∩ C ✳ ❈❤ó♥❣ t❛ ♥â✐ r➡♥❣ ▼ ❧➔ ❚r❛♥❣ ✷✷ ♠ët ❯❈✲♠ỉ✤✉♥ ♥➳✉ ♠é✐ ♠æ✤✉♥ ❝♦♥ ❝õ❛ ♥â ❝â ♠ët ❜❛♦ ✤â♥❣ t tr ỵ ợ U = n i=1 Ui tr♦♥❣ ✤â ♠é✐ Ui ❧➔ ♠ët ♠æ✤✉♥ ✤➲✉✳ ●✐↔ sû ❯ ❧➔ ♠ët ♠æ✤✉♥ ♣❤➙♥ ♣❤è✐ ✱ t❤➻ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭✐✮ ❯ ❧➔ ♠ët ♠æ✤✉♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✐✐✮ ❯ ❧➔ ♠ët (1 − C2) (C2 )❀ ♠æ✤✉♥✳ ❈❤ù♥❣ ♠✐♥❤✿ (i) ⇒ (ii) ✿ ❤✐➸♥ ♥❤✐➯♥✳ (ii) ⇒ (i)✿ ❚÷ì♥❣ tü ự ỵ Ui ổ tr ♠ët ♠ỉ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ Uj ✈ỵ✐ ❜➜t ❦ý i, j ∈ 1, , n ✈➔ S = End(Ui ) ❧➔ ♠ët ♠ỉ✤✉♥ ✤➲✉ ✈ỵ✐ ❜➜t ❦ý i ∈ 1, , n✳ ❈❤ó♥❣ t❛ ✤➛✉ t✐➯♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ S1 ✈➔ S2 ❧➔ ♥❤ú♥❣ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯ ✈ỵ✐ u−dim(S1 ) = 1, u−dim(S2 ) = n−1 ✈➔ S1 ∩ S2 = ✱ t❤➻ S1 U = S2 ( K = S2 n i=2 Ui ) ❱✐➳t U = S1 S2 = U ✳ ❚❤❡♦ ❜ê ✤➲ ❆③✉♠❛②❛✱ ❝❤ó♥❣ t❛ ❝â Ui ✳ ●✐↔ sû i = ✱ t❤❡♦ ✤â U = S2 U1 = U1 ✳ H = S1 ( i∈I Ui ) ✈ỵ✐ ■ ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ 1, , n ✈➔ card(I) = n − 1✳ ❈â ♥❤ú♥❣ tr÷í♥❣ ❤đ♣ s❛✉✿ ❚r÷í♥❣ ❤đ♣ ✶✿ ◆➳✉ ∈/ I, U = S1 (U2 Un ) = U1 (U2 t❤➻ ♥â ✤➳♥ tø S1 ∼ = U1 ✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ♠æ✤✉♥✱ ❝❤ó♥❣ t❛ ❝â S1 S2 V tr♦♥❣ ✤â V = (S1 S2 = S2 ) ∩ U1 ✳ ❚ø ✤➙② ❝❤ó♥❣ t❛ ❝â V ∼ = S1 ✱ ✤✐➲✉ ❚r❛♥❣ ✷✸ Un ) ♥➔② ❝â ♥❣❤➽❛ U1 ❝❤ù❛ ♠ët ❜↔♥ s❛♦ ❝õ❛ S1 ∼ = U1 ✳ ❉♦ U1 ❦❤æ♥❣ ♥➡♠ tr♦♥❣ ♠ët ♠æ✤✉♥ ❝♦♥ t❤➟t sü ❝õ❛ U1 ✱ ❝❤ó♥❣ t❛ ♣❤↔✐ ❝â V = U1 ✱ ✈➔ ❞♦ ✤â S1 S2 = S2 U1 = U ✳ ❚r÷í♥❣ ủ I s tỗ t k = ✱ s❛♦ ❝❤♦ k = 1, , n\I, U = S1 ( i∈I Ui ) = Uk ♠æ✤✉♥ t❛ ❝â S1 ( S2 = S2 i∈I Ui )✳ ❚❤❡♦ ✤â S1 ∼ = Uk ✳ ❚❤❡♦ t➼♥❤ ❝❤➜t V tr♦♥❣ ✤â V = (S1 S2 ) ∩ U1 ✳ ❚ø ✤➙② t❛ ❝â V ∼ = S1 ✱ ✤✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ U1 ❝❤ù❛ ♠ët ❜↔♥ s❛♦ ❝õ❛ S1 ∼ = Uk ✳ ❉♦ Uk ❦❤æ♥❣ ♥➡♠ tr♦♥❣ ♠ët ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ U1 ✱ ❝❤ó♥❣ t❛ ♣❤↔✐ ❝â V = U1 ✈➔ ❞♦ ✤â S1 S2 = U ✱ ♥❤÷ ♠♦♥❣ ♠✉è♥ ❝❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ t t ữợ ự tọ ❦✐➺♥ ✭❈✸✮✱ ❝ö t❤➸✱ ❝❤♦ ❤❛✐ tê♥❣ trü❝ t✐➳♣ ❝õ❛ X1 , X2 ❝õ❛ ❯ ✈ỵ✐ X1 ∩X2 = 0, X1 X2 ❝ô♥❣ ❧➔ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯ ✳ ❚❤❡♦ ❜ê ✤➲ ❆③✉♠❛②❛✱ ❝❤ó♥❣ t❛ ❝â U = X1 K = X1 ( F = 1, , n\J ✮ ✈➔ U = X2 i∈J Ui ) = ( L = X2 i∈F ( Ui ) j∈D ( Uj ) = ( i∈J Ui ) ✭tr♦♥❣ ✤â j∈E Uj ) ✭tr♦♥❣ ✤â E = 1, , n\D✮✳ ❈❤ó♥❣ t❛ t❤ø❛ ♥❤➟♥ X1 ∼ = X2 ∼ = j∈E Uj ✳ ●✐↔ sû E = 1, , t ✈➔ ❝❤♦ ϕ : t j=1 Uj i∈F ( j∈D Ui ✈➔ → X2 ❧➔ ♠ët ✤➥♥❣ ❝➜✉ ✈➔ t➟♣ Yj = ϕ(Uj )✱ ❝❤ó♥❣ t❛ ❝â Yj ∼ = Uj ✈➔ X2 = t j=1 Yj ✳ ❚❤❡♦ ❣✐↔ t❤✐➳t X2 ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯✱ ♥❤÷ t❤➳ Yj ❝ơ♥❣ ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯ ❝❤♦ ❜➜t ❦ý j ∈ 1, , t✳ ❈❤ó♥❣ t❛ ❝❤♦ t❤➜② X1 X = X1 (Y1 Yt ) ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯✳ ❈❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ X1 Y1 ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯✳ ❚❤❡♦ ❜ê ✤➲ ❆③✉♠❛②❛✱ ❝❤ó♥❣ t❛ ❝â U = Y1 ❚r❛♥❣ ✷✹ W = Y1 ( p∈P Up )✱ ✈ỵ✐ P ❧➔ Uj ) ♠ët t➟♣ ❝♦♥ ❝õ❛ 1, , n✱ ♥❤÷ t❤➳ card(P ) = n − ✈➔ α = 1, , n\P ữ ỵ r card(P J) ≥ card(J) − = m✳ ●✐↔ sû r➡♥❣ 1, , m ⊆ (P ∩ J)✱ ❝ö t❤➸✱ U = (X1 Z = X1 (U1 (U1 Uβ = Z Uβ ✈ỵ✐ β = J\1, , m ✈➔ Um )✳ ❉♦ ❯ ❧➔ ♠ët ♠ỉ✤✉♥ ♣❤➙♥ ♣❤è✐ ✱ ❝❤ó♥❣ t❛ ❝â Z ∩ Y1 = (X1 Um )) (U1 Um )) ∩ Y1 = (X1 ∩ Y1 ) ((U1 Um ) ∩ Y1 ) = 0✳ ữ ỵ r Z, Y1 tờ trỹ t ❝õ❛ ❯ ✈ỵ✐ u − dim(Z) = n − ✈➔ u − dim(Y1 ) = 1, U = Z (X1 Y1 ) (U1 Y1 = (X1 Um )✳ ❉♦ ✤â X1 ❯✳ ❚❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ ✱ t❛ ❝â X1 (X1 Y1 Yt−1 ) (U1 Um )) Y1 = Y1 ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ X = X1 (Y1 Yt ) = Yt ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯✳ ◆❤÷ t❤➳ ❯ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❈✸✮ ❈✉è✐ ❝ị♥❣ ✱ ❝❤ó♥❣ t❛ ❝❤♦ t❤➜② ❯ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❈✷✮ ✳ ❚❤❡♦ ❣✐↔ t❤✐➳t ✭✐✐✮ ✈➔ ❯ t❤ä❛ ♠➣♥ ✭❈✸✮✱ ❞➝♥ ✤➳♥ ❯ ❧➔ ♠ët (1 − C2) ♠♦❞✉❧❡ ✭①❡♠ ✤à♥❤ ỵ ự ữủ ỵ ❱ỵ✐ U1 , , Un J(Ui ) ❧➔ ♥❤ú♥❣ ♠ỉ✤✉♥ ❝♦♥ ✤➲✉✱ tr♦♥❣ ✤â ✈ỵ✐ ❜➜t ❦ý i, j = 1, , n✳ ◆➳✉ U = Ui ❦❤æ♥❣ ♥➡♠ tr♦♥❣ n i=1 Ui ❧➔ ♠ët ✏❯❈ ♠æ✤✉♥ ✈➔ ❝â t➼♥❤ ❝❤➜t ♣❤➙♥ ♣❤è✐✑ t❤➻ ♥â ❧➔ ♠ët ♠ỉ✤✉♥ ❧✐➯♥ tư❝✳ ❈❤ù♥❣ ♠✐♥❤✿ ❈❤ó♥❣ t❛ ✤➛✉ t✐➯♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ❯ ❧➔ ♠ët ❈❙✲♠æ✤✉♥✳ ❈❤♦ ❆ ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ✤â♥❣ ✤➲✉ ❝õ❛ ❯✳ ✣➦t Xi = A ∩ Ui ∀i ∈ 1, , n✳ ●✐↔ sû r➡♥❣ Xi = 0∀i ∈ 1, , n✳ ❚❤❡♦ ❣✐↔ t❤✐➳t✱ ❯ ❧➔ ♠ët ♠ỉ✤✉♥ ❝â t➼♥❤ ❚r❛♥❣ ✷✺ ❝❤➜t ♣❤➙♥ ♣❤è✐ ✱ ❝❤ó♥❣ t❛ ❝â = A ∩ (U1 Un ) = X1 Xn = ✱ ✭♠➙✉ t❤✉➝♥✮✳ ❉♦ ✤â tỗ t ởt Xt = tự A Ut = 0✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ❆ ✈➔ Ut ❧➔ ♠æ✤✉♥ ❝♦♥ ✤â♥❣ ✤➲✉ ❝õ❛ ❯✱ t❤➻ Xt ❧➔ ♠ët ♠ỉ✤✉♥ ❝♦♥ ❝èt ②➳✉ ❝õ❛ ❆ ✈➔ Xt ❝ơ♥❣ ❧➔ ♠ët ♠ỉ✤✉♥ ❝♦♥ ❝èt ②➳✉ ❝õ❛ Ut ✳ ◆❤÷ t❤➳ ❆ ✈➔ Ut ❧➔ ✤â♥❣ ❝õ❛ Xt tr➯♥ ❯✱ ❯ ❧➔ ♠ët ❯❈✲♠ỉ✤✉♥ ❝❤ó♥❣ t❛ ❝â A = Ut ✳ ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ❆ ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯✱ ❤❛② ❯ ❧➔ ♠ët ✭✶✲❈✶✮ ♠æ✤✉♥ ✳ õ tữợ ợ ởt ❈❙✲♠ỉ✤✉♥ ✭①❡♠ ❬✹✱ ❤➺ q✉↔ ✼✳✽❪✮ ✭✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ✤✐➲✉ ✤➛✉ t✐➯♥✮✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ S = End(Ul ) ❧➔ ♠ët ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ❝❤♦ ❜➜t ❦ý ∈ 1, , n ✳ ❱ỵ✐ f ∈ S ✈➔ ❣✐↔ sû r➡♥❣ ❢ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♠ët ✤➥♥❣ ❝➜✉ ✳ ❙✉② r❛ ✶✲❢ ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ ●✐↔ sû ❢ ❧➔ ♠ët ✤ì♥ →♥❤ ✳ ◆❤÷ t❤➳ ❢ ❦❤æ♥❣ t♦➔♥ →♥❤ ✈➔ f : Ul → J(Ul ) ❧➔ ♠ët ♣❤➨♣ ♥❤ó♥❣✱ ✭♠➙✉ t❤✉➝♥ ✮✳ ◆❤÷ t❤➳ ❢ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♠ët ✤ì♥ →♥❤✳ ◆❤÷ t❤➳ ❑❡r✭❢✮ ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❦❤→❝ ✵✱ ♥â ❧➔ ❝➛♥ t❤✐➳t tr♦♥❣ ♠ỉ✤✉♥ ❝♦♥ ✤➲✉ Ut ✳ ◆❤÷ t❤➳✱ ❝❤ó♥❣ t❛ ❧✉ỉ♥ ❝â Ker(f ) ∩ Ker(1 − f ) = 0✱ t❤❡♦ ✤â Ker(1 − f ) = ✱ ❤❛② ✶✲❢ ❧➔ ♠ët ✤ì♥ →♥❤✳ ◆❤÷♥❣ ❞♦ Ul ❦❤ỉ♥❣ ♥➡♠ tr♦♥❣ J(Ul )✱ ✶✲❢ ♣❤↔✐ ð tr➯♥ ✈➔ ♥❤÷ t❤➳ ✶✲❢ ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ ❙✉② r❛ ❙ ❧➔ ♠ët ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣✳ ❇➙② ❣✐í ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ❯ ❧➔ ♠ët ✭✶✲❈✷✮ ♠æ✤✉♥ ✳ ❈❤♦ ❤❛✐ ♠æ✤✉♥ ❝♦♥ ✤➲✉ ❱✱❲ ❝õ❛ ❯✱ ✈ỵ✐ V ∼ = W ✈➔ ❲ ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯✱ ❱ ❝ô♥❣ ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯✳ ❚❤❡♦ ❜ê ✤➲ ❝õ❛ ❆③✉♠❛②❛✱ ❝❤ó♥❣ t❛ ❝â U = W W =W ( j∈J Uj ) = Uk ❚r❛♥❣ ✷✻ ( j∈J Uj ) tr♦♥❣ ✤â ❏ ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ 1, , n ✈ỵ✐ card(J) = n − ✈➔ k = 1, , n\J ✳ ❉♦ ✤â V ∼ = Uk ✳ ❱ỵ✐ V ∗ ❧➔ ♠ët sü ✤â♥❣ ❝õ❛ ❱ tr♦♥❣ ❯✳ ❉♦ ❯ ❧➔ ♠ët = W ∼ ❈❙✲♠æ✤✉♥ ✱ ♥➯♥ V ∗ ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ữỡ tỹ tỗ t s 1, , n s❛♦ ❝❤♦ V ∗ = Us ✱ ♥❣❤➽❛ ❧➔ Us ❝❤ù❛ ♠ët ❜↔♥ s❛♦ ❝õ❛ W ∼ = Uk ✳ ◆➳✉ ❱ ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ r✐➯♥❣ ❝õ❛ Us ✱ t❤➻ Uk ♥➡♠ tr♦♥❣ J(Us ) ✱ ✭♠➙✉ t❤✉➝♥✮✳ ❈❤ó♥❣ t❛ ♣❤↔✐ ❝â V = Us ✳ ❱➔ ❞♦ ✤â ❱ ❧➔ ♠ët tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❯✳ ◆➯♥ ❯ ❧➔ ♠ët ✭✶✲❈✷✮ ♠æ✤✉♥ ✱ s✉② r❛ ❯ ❧➔ ♠ët ♠ỉ✤✉♥ ❧✲❧✐➯♥ tư❝ ✭❞♦ ❯ ❧➔ ♠ët ❈❙✲♠ỉ✤✉♥ ố ũ t ỵ t ♠ët ♠ỉ✤✉♥ ❧✐➯♥ tư❝✳ ❇ê ✤➲ ✷✳✽ ❱ỵ✐ U1 , Un ỳ ổ ỗ J(Ui )i, j = 1, , n✳ ◆➳✉ U = Ui ❦❤æ♥❣ ♥➡♠ tr♦♥❣ n i=1 Ui ❧➔ ♠ët ❯❈✲ ♠æ✤✉♥ ❝â t➼♥❤ ❝❤➜t ♣❤➙♥ ♣❤è✐ t❤➻ ♥❤ú♥❣ ✤✐➲✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭✐✮ ❯ ❧➔ ♠ët ✕♠ỉ✤✉♥ tü❛ ♥ë✐ ①↕❀ ✭✐✐✮ ❯ ❧➔ ♠ët −(1 − C1 )✲♠ỉ✤✉♥ ✤➳♠ ✤÷đ❝✳ ❈❤ù♥❣ ♠✐♥❤✿ (i) ⇒ (ii)✿ ❤✐➸♥ ♥❤✐➯♥✳ (ii) ⇒ (i)✿ t❤❡♦ ỵ ởt ổ tử ❚❤❡♦ ❬✼✳♠➺♥❤ ✤➲ ✷✳✺❪ ❯ ❧➔ ♠ët − ♠æ✤✉♥ tü❛ ♥ë✐ ①↕ ✭❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ✭✐✮✮✳ ❚r❛♥❣ ✷✼ ❑➌❚ ▲❯❾◆ ❉ü❛ ✈➔♦ ❝→❝ t➔✐ ❧✐➺✉ ❝❤➼♥❤ ❧➔ ❬✷❪✱ ❬✹❪✱ ❧✉➟♥ ✈➠♥ ✤➣ ✤➲ ❝➟♣ ✈➔ ❣✐↔✐ q✉②➳t ❞÷đ❝ ❝→❝ ✈➜♥ ✤➲ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ♠æ✤✉♥ ❝♦♥ ❝èt ②➳✉✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ (Ci ) ❝õ❛ ♠æ✤✉♥ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ✷✳ ❚r➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥✱ ❝→❝ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ♠æ✤✉♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (C2 ) ✈➔ (1 − C2 ) ♠æ ✤✉♥✳ ✸✳ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ✤➦❝ tr÷♥❣ ❝õ❛ ♠ỉ ✤✉♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (C2 ) q ổ (1C2 ) ỵ ỵ ởt số t t ổ tử ổ tử ỵ ỵ r t [1] ◆❣ỉ ❙ÿ ❚ị♥❣✱ ▲➯ ❱➠♥ ❆♥✱ ◆❣✉②➵♥ ❚❤à ❍↔✐ ❆♥❤ ✭✷✵✶✵✮✱ ❚ê♥❣ trü❝ t✐➳♣ ❝→❝ ♠æ✤✉♥ ✤➲✉ ✈➔ ♠æ✤✉♥ ❧✐➯♥ tư❝✱ ❚↕♣ ❝❤➼ ❦❤♦❛ ❤å❝ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✱ ❚➟♣ ✸✾ sè ✷❆✳ ❚✐➳♥❣ ❆♥❤✿ [2] ▲❡ ❱❛♥ ❆♥✱ ◆❣✉②❡♥ ❍❛✐ ❆♥❤✱ ◆❣♦ ❙② ❚✉♥❣ ✭✷✵✶✼✮✱ ❖♥ t❤❡ (1 − C2 ) ❝♦♥❞✐t✐♦♥✱ ▼❛t❤✳❏✳ ♦❢ ❖❦❛②❛♠❛ ❯♥✐✳✺✾ ✭✷✵✶✼✮✱ ✶✹✶✲✶✹✼✳ [3] ❋✳❲✳ ❆♥❞❡rs♦♥ ❛♥❞ ❑✳❘✳❋✉❧❧❡r ✭✶✾✼✹✮✱ ❘✐♥❣ ❛♥❞ ❈❛t❡❣♦r✐❡s ♦❢ ▼♦❞✲ ✉❧❡s✱ ❙♣r✐♥❣❡r ✕ ❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦ ✕ ❍❡✐❞❡❧❜❡r❣ ✕ ❇❡r❧✐♥✳ [4] ❉✉♥❣ ◆✳❱✱ ❍✉②♥❤ ❉✳❱✱ ❙♠✐t❤ P✳❋ ❛♥❞ ❲✐s❜❛✇❡r ❘ ✭✶✾✾✹✮✱ ❊①t❡♥❞✲ ✐♥❣ ▼♦❞✉❧❡s✱ P✐❝♠❛♥ ✕ ▲♦♥❞♦♥✳ [5] ❋✳❑❛s❝❤ ✭✶✾✽✷✮✱ ▼♦❞✉❧❡s ❛♥❞ ❘✐♥❣s✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ▲♦♥❞♦♥ ✕ ◆❡✇②♦r❦✳ [6] ❙✳❍✳▼♦❤❛♠❡❞ ❛♥❞ ❇✳❏✳▼✉❧❧❡r ✭✶✾✾✵✮✱ ❈♦♥t✐♥✉♦✉s ❛♥❞ ❉✐s❝r❡t❡ ▼♦❞✲ ✉❧❡s✱ ▲♦♥❞♦♥ ▼❛t❤✳❙♦❝✳▲❡❝t✉r❡ ◆♦t❡ ❙❡r✳❱♦❧✳✶✹✼✱❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❚r❛♥❣ ✷✾