rtt sr r - - PowerPoint PPT Presentation

• ❆r✐t❤♠❡t✐❝ ♦❢

s❡♠✐♥♦r♠❛❧ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s ❛♥❞ ❞♦♠❛✐♥s

❆. ●❡r♦❧❞✐♥❣❡r ∗ ❛♥❞ ❋✳ ❑❛✐♥r❛t❤ ❛♥❞ ❆✳ ❘❡✐♥❤❛rt

■♥t❡r♥❛t✐♦♥❛❧ ▼❡❡t✐♥❣ ♦♥ ◆✉♠❡r✐❝❛❧ ❙❡♠✐❣r♦✉♣s

❈♦rt♦♥❛✱ ❙❡♣t❡♠❜❡r ✷✵✶✹

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❖✉t❧✐♥❡

✭❯♥✐♦♥s ♦❢✮ ❙❡ts ♦❢ ▲❡♥❣t❤s ❑r✉❧❧ ❛♥❞ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s ❆r✐t❤♠❡t✐❝✿ Pr❡❝✐s❡ ✈❡rs✉s ◗✉❛❧✐t❛t✐✈❡ ❘❡s✉❧ts ❙❡♠✐♥♦r♠❛❧ ▼♦♥♦✐❞s ❛♥❞ ❉♦♠❛✐♥s ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s✿ ❚r❛♥s❢❡r ❍♦♠♦♠♦r♣❤✐s♠s

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❙❡ts ♦❢ ❧❡♥❣t❤s ✐♥ ♠♦♥♦✐❞s

▲❡t ❍ ❜❡ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ✇r✐tt❡♥✱ ❝♦♠♠✉t❛t✐✈❡✱ ❝❛♥❝❡❧❧❛t✐✈❡ s❡♠✐❣r♦✉♣✱ ❛♥❞ ❧❡t ❛ ∈ ❍ ❜❡ ❛ ♥♦♥✲✉♥✐t✳

• ■❢ ❛ = ✉✶ · . . . · ✉❦

✇❤❡r❡ ✉✶, . . . , ✉❦ ❛r❡ ✐rr❡❞✉❝✐❜❧❡s ✭❛t♦♠s✮✱ t❤❡♥ ❦ ✐s ❝❛❧❧❡❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❢❛❝t♦r✐③❛t✐♦♥✳

• ▲❍(❛) = {❦ | ❛ ❤❛s ❛ ❢❛❝t♦r✐③❛t✐♦♥ ♦❢ ❧❡♥❣t❤ ❦} ⊂ N

✐s t❤❡ s❡t ♦❢ ❧❡♥❣t❤s ♦❢ ❛✳

• ■❢ ▲(❛) = {❦✶, ❦✷, ❦✸, . . .} ✇✐t❤ ❦✶ < ❦✷ < ❦✸ < . . .✱ t❤❡♥

∆

• ▲(❛)
• = {❦✷ − ❦✶, ❦✸ − ❦✷, . . .}

✐s t❤❡ s❡t ♦❢ ❞✐st❛♥❝❡s ♦❢ ▲(❛)✳

• ■❢ |▲(❛)| ≥ ✷✱ t❤❡♥ |▲(❛♠)| > ♠ ❢♦r ❡❛❝❤ ♠ ∈ N✳

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❙❡ts ♦❢ ❞✐st❛♥❝❡s ❛♥❞ ✉♥✐♦♥s ♦❢ s❡ts ♦❢ ❧❡♥❣t❤s

❲❡ ❝❛❧❧ ∆(❍) =

• ❛∈❍

∆

• ▲(❛)
• ⊂ N

t❤❡ s❡t ♦❢ ❞✐st❛♥❝❡s ♦❢ ❍✳ ❋♦r ❦ ∈ N✱ ✇❡ ❝❛❧❧ U❦(❍) =

• ❦∈▲(❛)

▲(❛) = {ℓ ∈ N | t❤❡r❡ ✐s ❛♥ ❡q✉❛t✐♦♥ ✉✶ · . . . · ✉❦ = ✈✶ · . . . · ✈ℓ} t❤❡ ✉♥✐♦♥ ♦❢ s❡ts ♦❢ ❧❡♥❣t❤s ❝♦♥t❛✐♥✐♥❣ ❦✳ ❆♥ ❛t♦♠✐❝ ♠♦♥♦✐❞ ❍ ✐s ❝❛❧❧❡❞ ❤❛❧❢✲❢❛❝t♦r✐❛❧ ✐❢ ♦♥❡ ❢♦❧❧✳ ❡q✉✐✈✳ ❤♦❧❞s✿ ✭❛✮ |▲(❛)| = ✶ ❢♦r ❡❛❝❤ ❛ ∈ ❍✳ ✭❜✮ ∆(❍) = ∅✳ ✭❝✮ U❦(❍) = {❦} ❢♦r ❡❛❝❤ ❦ ∈ N✳

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❖✉t❧✐♥❡

✭❯♥✐♦♥s ♦❢✮ ❙❡ts ♦❢ ▲❡♥❣t❤s ❑r✉❧❧ ❛♥❞ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s ❆r✐t❤♠❡t✐❝✿ Pr❡❝✐s❡ ✈❡rs✉s ◗✉❛❧✐t❛t✐✈❡ ❘❡s✉❧ts ❙❡♠✐♥♦r♠❛❧ ▼♦♥♦✐❞s ❛♥❞ ❉♦♠❛✐♥s ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s✿ ❚r❛♥s❢❡r ❍♦♠♦♠♦r♣❤✐s♠s

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❉❡✜♥✐t✐♦♥ ♦❢ ❑r✉❧❧ ♠♦♥♦✐❞s

❍ ✐s ❝❛❧❧❡❞ ❛ ❑r✉❧❧ ♠♦♥♦✐❞ ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧✳ ❡q✉✐✈✳ ❤♦❧❞s ✿ ✭❛✮ ❍ ✐s ✈✲♥♦❡t❤❡r✐❛♥ ❛♥❞ ❝♦♠♣❧❡t❡❧② ✐♥t❡❣r❛❧❧② ❝❧♦s❡❞✳ ✭❜✮ ❍ ❤❛s ❛ ❞✐✈✐s♦r t❤❡♦r② ϕ: ❍ → F(P) = ❋✿

• ϕ ✐s ❛ ❞✐✈✐s♦r ❤♦♠♦♠♦r♣❤✐s♠✿

❋♦r ❛❧❧ ❛, ❜ ∈ ❍ ✇❡ ❤❛✈❡ ❛ | ❜ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ϕ(❛) | ϕ(❜) .

• ❋♦r ❛❧❧ ♣ ∈ P t❤❡r❡ ✐s ❛ s❡t ❳ ⊂ ❍ s✉❝❤ t❤❛t ♣ = ❣❝❞(ϕ(❳))✮✳

✭❝✮ ❚❤❡r❡ ✐s ❛ ❞✐✈✐s♦r ❤♦♠♦♠♦r♣❤✐s♠ ✐♥t♦ ❛♥② ❢r❡❡ ❛❜❡❧✐❛♥ ♠♦♥♦✐❞✳ ❚❤❡ ❞✐✈✐s♦r ❝❧❛ss ❣r♦✉♣ ● ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ✈✲❝❧❛ss ❣r♦✉♣✿

• = q(❋)/q
• ϕ(❍)
• = {❛q
• ϕ(❍)
• = [❛] | ❛ ∈ ❋} ∼

= C✈(❍) . ▲❡t ❘ ❜❡ ❛ ❞♦♠❛✐♥✳

• ❘ ✐s ❛ ❑r✉❧❧ ❞♦♠❛✐♥ ✐✛ • ✐s ❛ ❑r✉❧❧ ♠♦♥♦✐❞✳
• ■♥t❡❣r❛❧❧② ❝❧♦s❡❞ ♥♦❡t❤❡r✐❛♥ ❞♦♠❛✐♥s ❛r❡ ❑r✉❧❧ ❜② Pr♦♣❡rt② ✭❛✮✳

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

Pr✐♠❛r② ♠♦♥♦✐❞s ❛♥❞ ❞♦♠❛✐♥s

✶✳ ❆♥ ❡❧❡♠❡♥t q ∈ ❍ ✐s ❝❛❧❧❡❞ ♣r✐♠❛r② ✐❢ q / ∈ ❍× ❛♥❞✱ ❢♦r ❛❧❧ ❛, ❜ ∈ ❍✱ ✐❢ q | ❛❜ ❛♥❞ q ∤ ❛✱ t❤❡♥ q | ❜♥ ❢♦r s♦♠❡ ♥ ∈ N✳ ✷✳ ❍ ✐s ❝❛❧❧❡❞ ♣r✐♠❛r② ✐❢ m = ❍ \ ❍× = ∅ ❛♥❞ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉✐✈❛❧❡♥t st❛t❡♠❡♥ts ❛r❡ s❛t✐s✜❡❞ ✿

✭❛✮ s✲s♣❡❝(❍) = {∅, ❍ \ ❍×}✳ ✭❜✮ ❊✈❡r② q ∈ m ✐s ♣r✐♠❛r②✳ ✭❝✮ ❋♦r ❛❧❧ ❛, ❜ ∈ m t❤❡r❡ ❡①✐sts s♦♠❡ ♥ ∈ N s✉❝❤ t❤❛t ❛ | ❜♥✳

✸✳ ▲❡t ❘ ❜❡ ❛ ❞♦♠❛✐♥✳ ❚❤❡♥ ❘• ✐s ♣r✐♠❛r② ✐✛ ❘ ✐s ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❧♦❝❛❧✳

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❋✐♥✐t❡❧② ♣r✐♠❛r② ♠♦♥♦✐❞s ❛♥❞ ❞♦♠❛✐♥s

❆ ♠♦♥♦✐❞ ❍ ✐s ❝❛❧❧❡❞ ✜♥✐t❡❧② ♣r✐♠❛r② ✭♦❢ r❛♥❦ s ❛♥❞ ❡①♣♦♥❡♥t α✮ ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉✐✈❛❧❡♥t ❝♦♥❞✐t✐♦♥s ❤♦❧❞s✿ ✭❛✮ ❚❤❡r❡ ❡①✐st s, α ∈ N ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ✿

❍ ✐s ❛ s✉❜♠♦♥♦✐❞ ♦❢ ❛ ❢❛❝t♦r✐❛❧ ♠♦♥♦✐❞ ❋ = ❋ ××[♣✶, . . . , ♣s] ✇✐t❤ s ♣❛✐r✇✐s❡ ♥♦♥✲❛ss♦❝✐❛t❡❞ ♣r✐♠❡ ❡❧❡♠❡♥ts ♣✶, . . . , ♣s s✳t✳ ❍ \ ❍× ⊂ ♣✶ · . . . · ♣s❋ ❛♥❞ (♣✶ · . . . · ♣s)α❋ ⊂ ❍ .

✭❜✮ ❍ ✐s ♣r✐♠❛r②✱ (❍ : ❍) = ∅ ❛♥❞ ❍r❡❞ ∼ = (Ns

✵, +)✳

❈❧❡❛r❧②✱ ♥✉♠❡r✐❝❛❧ ♠♦♥♦✐❞s ❛r❡ ✜♥✐t❡❧② ♣r✐♠❛r② ♦❢ r❛♥❦ ✶✳ ▲❡t ❘ ❜❡ ❛ ❞♦♠❛✐♥✳

• ■❢ ❘ ✐s ❛ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❧♦❝❛❧ ▼♦r✐ ❞♦♠❛✐♥ s✉❝❤ t❤❛t

(❘ : ❘) = {✵}✱ t❤❡♥ ❘• ✐s ✜♥✐t❡❧② ♣r✐♠❛r②✳

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❲❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s ❛♥❞ ❞♦♠❛✐♥s

❆ ♠♦♥♦✐❞ ❍ ✐s ✇❡❛❦❧② ❑r✉❧❧ ✐❢ ❍ =

• p∈X(❍)

❍p ❛♥❞ {p ∈ X(❍) | ❛ ∈ p} ✐s ✜♥✐t❡ ❢♦r ❛❧❧ ❛ ∈ ❍ , ❲❡❛❦❧② ❑r✉❧❧ ❞♦♠❛✐♥s✿ ❆♥❞❡rs♦♥✱ ▼♦tt✱ ❛♥❞ ❩❛❢r✉❧❧❛❤✱ ✶✾✾✷ ❲❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s✿ ❍❛❧t❡r✲❑♦❝❤✱ ❇♦❧❧✳ ❯▼■ ✶✾✾✺

• ❆ ❞♦♠❛✐♥ ❘ ✐s ✇❡❛❦❧② ❑r✉❧❧ ✐✛ ❘• ✐s ❛ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞✳
• ❍ ✈✲♥♦❡t❤❡r✐❛♥✿ ❍ ✇❡❛❦❧② ❑r✉❧❧ ⇐

⇒ ✈✲♠❛①(❍) = X(❍)✳

• ❍ ❑r✉❧❧ ⇒ ❍ s❡♠✐♥♦r♠❛❧ ✈✲♥♦❡t❤❡r✐❛♥ ✇❡❛❦❧② ❑r✉❧❧ ❛✳ ❍ =

❍✳

• ❲❡ s✉♣♣♦s❡ t❤❛t ❛❧❧ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s ❛r❡
• ✈✲♥♦❡t❤❡r✐❛♥
• ❍p ❛r❡ ✜♥✐t❡❧② ♣r✐♠❛r② ❢♦r ❡❛❝❤ p ∈ X(❍)✳
• (❍ :

❍) = f = ∅✳

• ❊①❛♠♣❧❡✿ ✶✲❞✐♠✳ ♥♦❡t❤✳ ❞♦♠❛✐♥s ❘ s✳t✳ ❘ ✐s ❛ ❢✳❣✳ ❘✲♠♦❞✉❧❡

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❖✉t❧✐♥❡

✭❯♥✐♦♥s ♦❢✮ ❙❡ts ♦❢ ▲❡♥❣t❤s ❑r✉❧❧ ❛♥❞ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s ❆r✐t❤♠❡t✐❝✿ Pr❡❝✐s❡ ✈❡rs✉s ◗✉❛❧✐t❛t✐✈❡ ❘❡s✉❧ts ❙❡♠✐♥♦r♠❛❧ ▼♦♥♦✐❞s ❛♥❞ ❉♦♠❛✐♥s ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s✿ ❚r❛♥s❢❡r ❍♦♠♦♠♦r♣❤✐s♠s

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❆r✐t❤♠❡t✐❝ ♦❢ ❑r✉❧❧ ♠♦♥♦✐❞s✿ Pr❡❝✐s❡ ❘❡s✉❧ts

▲❡t ❍ ❜❡ ❛ ❑r✉❧❧ ♠♦♥♦✐❞ ✇✐t❤ ❝❧❛ss ❣r♦✉♣ ● s✉❝❤ t❤❛t ❡❛❝❤ ❝❧❛ss ❝♦♥t❛✐♥s ❛ ♣r✐♠❡ ❞✐✈✐s♦r✳ ✶✳ ✭❈❛r❧✐t③ ✶✾✻✵✮ ❍ ✐s ❤❛❧❢✲❢❛❝t♦r✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ |●| ≤ ✷✳ ✷✳ ▲❡t ✷ < |●| < ∞✳ ❚❤❡♥

• ∆(❍) ✐s ❛ ✜♥✐t❡ ✐♥t❡r✈❛❧ ✇✐t❤ ♠✐♥ ∆(❍) = ✶✳
• ❆❧❧ U❦(❍) ❛r❡ ✜♥✐t❡ ✐♥t❡r✈❛❧s✳
• ✳✳✳✳ ❛♥❞ ♠✉❝❤ ♠♦r❡ ✳✳✳✳ ❢♦r ❡①❛♠♣❧❡ ✳✳✳✳
• ■❢ ● ✐s ❝②❝❧✐❝ ♦❢ ♦r❞❡r ♥✱ t❤❡♥ ∆(❍) = [✶, ♥ − ✷]✱

♠❛① U✷❦(❍) = ❦♥✱ ❛♥❞ ♠❛① U✷❦+✶(❍) = ❦♥ + ✶✳

✸✳ ■❢ ● ✐s ✐♥✜♥✐t❡✱ t❤❡♥ ∆(❍) = U❦(❍) = N≥✷ ❢♦r ❛❧❧ ❦ ∈ N✳

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❆r✐t❤♠❡t✐❝ ♦❢ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s✿ ◗✉❛❧✐t❛t✐✈❡ ❘❡s✉❧ts

▲❡t ❘ ❜❡ ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ♦r❞❡r ✐♥ ❛♥ ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡r ✜❡❧❞s ✇✐t❤ P✐❝❛r❞ ❣r♦✉♣ ●✳

• ❆♣❛rt ❢r♦♠ q✉❛❞r❛t✐❝ ♥✉♠❜❡r ✜❡❧❞s ✭❍❛❧t❡r✲❑♦❝❤ ✶✾✽✸✮✱

t❤❡r❡ ✐s ♥♦ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❤❛❧❢✲❢❛❝t♦r✐❛❧✐t②✳

• ∆(❘) ✐s ✜♥✐t❡✳ ■❢ |●| ≤ ✷✱ t❤❡♥ ✐t ✐s ♦♣❡♥ ✇❤❡t❤❡r ✶ ∈ ∆(❘)✳
• ❋♦r ❡❛❝❤ ❦ ∈ N≥✷ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿
• U❦(❘) ✐s ✜♥✐t❡✳
• ❚❤❡ ♥❛t✉r❛❧ ♠❛♣ X(

❘) → X(❘) ✐s ❜✐❥❡❝t✐✈❡✳

• ❚❤❡r❡ ✐s ♥♦ ✐♥❢♦r♠❛t✐♦♥
• ♦♥ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ s❡t ♦❢ ❞✐st❛♥❝❡s ∆(❘)
• ♥♦r ♦♥ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ✉♥✐♦♥s U❦(❘)✳

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❖✉t❧✐♥❡

✭❯♥✐♦♥s ♦❢✮ ❙❡ts ♦❢ ▲❡♥❣t❤s ❑r✉❧❧ ❛♥❞ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s ❆r✐t❤♠❡t✐❝✿ Pr❡❝✐s❡ ✈❡rs✉s ◗✉❛❧✐t❛t✐✈❡ ❘❡s✉❧ts ❙❡♠✐♥♦r♠❛❧ ▼♦♥♦✐❞s ❛♥❞ ❉♦♠❛✐♥s ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s✿ ❚r❛♥s❢❡r ❍♦♠♦♠♦r♣❤✐s♠s

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❙❡♠✐♥♦r♠❛❧✐t②✿ ❉❡✜♥✐t✐♦♥s ❛♥❞ ❘❡♠❛r❦s

❚❤❡ s❡♠✐♥♦r♠❛❧✐③❛t✐♦♥ ❍′ ♦❢ ❍ ✐s ❞❡✜♥❡❞ ❜② ❍′ = {① ∈ q(❍) | t❤❡r❡ ✐s s♦♠❡ ◆ ∈ N s✉❝❤ t❤❛t ①♥ ∈ ❍ ❢♦r ❛❧❧ ♥ ≥ ◆} ❚❤❡♥

• ❍ ⊂ ❍′ ⊂

❍ ⊂ q(❍)✳

• ❍ ✐s s❡♠✐♥♦r♠❛❧ ✐❢ ❍ = ❍′✳ ❊q✉✐✈❛❧❡♥t❧②✱

✐❢ ① ∈ q(❍) ❛♥❞ ①✷, ①✸ ∈ ❍✱ t❤❡♥ ① ∈ ❍✳ ❆ ❞♦♠❛✐♥ ❘ ✐s s❡♠✐♥♦r♠❛❧ ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧✳ ❡q✉✐✈✳ ❤♦❧❞s✿ ✭❛✮ ❘• ✐s s❡♠✐♥♦r♠❛❧✳ ✭❜✮ P✐❝(❘) → P✐❝

• ❘[❳]
• ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳

❚r❛✈❡rs♦ ✭✶✾✼✵✮✱ ❙✇❛♥ ✭✶✾✽✵✮❀ ❙✉r✈❡② ❜② ❱✐t✉❧❧✐ ✭✷✵✶✵✮

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❙❡♠✐♥♦r♠❛❧ ✜♥✐t❡❧② ♣r✐♠❛r② ♠♦♥♦✐❞s

▲❡t ❍ ⊂ ❍ = ❋ = ❋ ××[♣✶, . . . , ♣s] ❜❡ ✜♥✐t❡❧② ♣r✐♠❛r②✳

• ❍′ = ♣✶ · . . . · ♣s❋ ∪ ❍′×✳
• ■❢ ❋ × = {✶}✱ t❤❡♥ ❍′ ∼

= (Ns ∪ {✵}, +) ⊂ (Ns

✵, +)✳

• A(❍′) =
• ε♣❦✶

✶ · . . . · ♣❦s s | ε ∈ ❋ ×, ♠✐♥{❦✶, . . . , ❦s} = ✶

• ✳
• ❍′ ✐s s❡♠✐♥♦r♠❛❧✱ ✈✲♥♦❡t❤❡r✐❛♥✱ ❛♥❞

✜♥✐t❡❧② ♣r✐♠❛r② ♦❢ r❛♥❦ s ❛♥❞ ❡①♣♦♥❡♥t ✶✳ ❋♦r ❛ ❞♦♠❛✐♥ ❘ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t ✿ ✭❛✮ ❘ ✐s ❛ s❡♠✐♥♦r♠❛❧ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❧♦❝❛❧ ▼♦r✐ ❞♦♠❛✐♥✳ ✭❜✮ ❘• ✐s s❡♠✐♥♦r♠❛❧ ✜♥✐t❡❧② ♣r✐♠❛r②✳

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❖✉t❧✐♥❡

✭❯♥✐♦♥s ♦❢✮ ❙❡ts ♦❢ ▲❡♥❣t❤s ❑r✉❧❧ ❛♥❞ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s ❆r✐t❤♠❡t✐❝✿ Pr❡❝✐s❡ ✈❡rs✉s ◗✉❛❧✐t❛t✐✈❡ ❘❡s✉❧ts ❙❡♠✐♥♦r♠❛❧ ▼♦♥♦✐❞s ❛♥❞ ❉♦♠❛✐♥s ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s✿ ❚r❛♥s❢❡r ❍♦♠♦♠♦r♣❤✐s♠s

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❆❧❣❡❜r❛✐❝ ❙tr✉❝t✉r❡ ♦❢ s❡♠✐♥♦r♠❛❧ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s

▲❡t ❍ ❜❡ ❛ s❡♠✐♥♦r♠❛❧ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞ ✇✐t❤ ♥♦♥tr✐✈✐❛❧ ❝♦♥❞✉❝t♦r f = (❍ : ❍) ❍✱ ❛♥❞ ❧❡t P∗ = {p ∈ X(❍) | p ⊃ f}✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ✶✳ ❍ ✐s ❑r✉❧❧ ❛♥❞ P∗ ✐s ✜♥✐t❡✳ ✷✳ ❚❤❡ ♠♦♥♦✐❞ I∗

✈ (❍) ♦❢ ✈✲✐♥✈❡rt✐❜❧❡ ✈✲✐❞❡❛❧s s❛t✐s✜❡s

I∗

✈(❍) ∼

= F(P) ×

• p∈P∗

(❍p)r❡❞ , ❛♥❞ ✐t ✐s s❡♠✐♥♦r♠❛❧✱ ✈✲♥♦❡t❤❡r✐❛♥✱ ❛♥❞ ✇❡❛❦❧② ❢❛❝t♦r✐❛❧✱ ✸✳ ❚❤❡r❡ ✐s ❛♥ ❡①❛❝t s❡q✉❡♥❝❡ ✶ → ❍×/❍× →

• p∈X(❍)
• ❍×

p /❍× p ε

→ C✈(❍) ϑ → C✈( ❍) → ✵ .

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❆r✐t❤♠❡t✐❝ ❙tr✉❝t✉r❡

❙✉♣♣♦s❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t ● = C✈(❍) ✐s ✜♥✐t❡✱ ❛♥❞ t❤❛t ❡✈❡r② ❝❧❛ss ❝♦♥t❛✐♥s ❛ p ∈ X(❍) ✇✐t❤ p ⊃ f✳ ✶✳ ❙✉♣♣♦s❡ t❤❡ ♥❛t✉r❛❧ ♠❛♣ X( ❍) → X(❍) ✐s ❜✐❥❡❝t✐✈❡✳

✶✳✶ U❦(❍) ✐s ❛ ✜♥✐t❡ ✐♥t❡r✈❛❧ ❢♦r ❛❧❧ ❦ ≥ ✷✳ ✶✳✷ ❙✉♣♣♦s❡ t❤❛t ϑ: C✈(❍) → C✈( ❍) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ tr❛♥s❢❡r ❤♦♠♦♠♦r♣❤✐s♠ θ: ❍ → B(●)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✭✉♥✐♦♥s ♦❢✮ s❡ts ♦❢ ❧❡♥❣t❤s ❛♥❞ ✭♠♦♥♦t♦♥❡✮ ❝❛t❡♥❛r② ❞❡❣r❡❡s ♦❢ ❍ ❛♥❞ B(●) ❝♦✐♥❝✐❞❡✳

✷✳ ❙✉♣♣♦s❡ t❤❡ ♥❛t✉r❛❧ ♠❛♣ X( ❍) → X(❍) ✐s ♥♦t ❜✐❥❡❝t✐✈❡✳ ❚❤❡♥ ❢♦r ❛❧❧ ❦ ≥ ✸✱ ✇❡ ❤❛✈❡ N≥✸ ⊂ U❦(❍) ⊂ N≥✷ .

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❍❛❧❢✲❋❛❝t♦r✐❛❧✐t②

❙✉♣♣♦s❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t t❤❡ ❝❧❛ss ❣r♦✉♣ ● = C(❍) ✐s ✜♥✐t❡✱ ❛♥❞ t❤❛t ❡✈❡r② ❝❧❛ss ❝♦♥t❛✐♥s ❛ p ∈ X(❍) ✇✐t❤ p ⊃ f✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t ✿ ✭❛✮ ❝(❍) ≤ ✷✳ ✭❜✮ ❍ ✐s ❤❛❧❢✲❢❛❝t♦r✐❛❧✳ ✭❝✮ |●| ≤ ✷✱ t❤❡ ♥❛t✉r❛❧ ♠❛♣ X( ❍) → X(❍) ✐s ❜✐❥❡❝t✐✈❡✱ ❛♥❞ t❤❡ ❤♦♠♦♠♦r♣❤✐s♠ ϑ: C✈(❍) → C✈( ❍) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳ ✇❤❡r❡ π: X( ❍) → X(❍)✱ ✐s ❞❡✜♥❡❞ ❜② π(P) = P ∩ ❍ ❢♦r ❛❧❧ P ∈ X( ❍)

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❖✉t❧✐♥❡

✭❯♥✐♦♥s ♦❢✮ ❙❡ts ♦❢ ▲❡♥❣t❤s ❑r✉❧❧ ❛♥❞ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s ❆r✐t❤♠❡t✐❝✿ Pr❡❝✐s❡ ✈❡rs✉s ◗✉❛❧✐t❛t✐✈❡ ❘❡s✉❧ts ❙❡♠✐♥♦r♠❛❧ ▼♦♥♦✐❞s ❛♥❞ ❉♦♠❛✐♥s ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s✿ ❚r❛♥s❢❡r ❍♦♠♦♠♦r♣❤✐s♠s

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❚r❛♥s❢❡r ❍♦♠♦♠♦r♣❤✐s♠s

❈♦♥s✐❞❡r ❍ − − − − → ❉ = F(P)×❚ ∼ = I∗

✈(❍) β

 

• β

 

• ❇ = B(●, ❚, ι) −

− − − → ❋ = F(●)×❚ ✇❤❡r❡

• ❍ ֒

→ ❉ ✐s s❛t✉r❛t❡❞✱ ❛♥❞ t❤❡ ❝❧❛ss ❣r♦✉♣ ● = C(❍, ❉) s❛t✐s✜❡s ● = {[♣] | ♣ ∈ P} ⊂ ●✳

• ι: ❚ → ● ✐s ❞❡✜♥❡❞ ❜② ι(t) = [t]✳

β: ❉ → ❋ ❜❡ t❤❡ ✉♥✐q✉❡ ❤♦♠♦♠♦r♣❤✐s♠ s❛t✐s❢②✐♥❣ β(♣) = [♣] ❢♦r ❛❧❧ ♣ ∈ P ❛♥❞ β | ❚ = ✐❞❚✳ ✶✳ ❚❤❡ r❡str✐❝t✐♦♥ β = β | ❍ : ❍ → ❇ ✐s ❛ tr❛♥s❢❡r ❤♦♠✳ ✷✳ ❚r❛♥s❢❡r ❤♦♠♦♠♦r♣❤✐s♠s ♣r❡s❡r✈❡ s❡ts ♦❢ ❧❡♥❣t❤s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✉♥✐♦♥s ♦❢ s❡ts ♦❢ ❧❡♥❣t❤s ❛♥❞ ❤❛❧❢✲❢❛❝t♦r✐❛❧✐t②✳

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❈♦♠❜✐♥❛t♦r✐❛❧ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞s✿ B(●, ❚, ι)

▲❡t ● ❜❡ ❛ ✜♥✐t❡ ❛❜❡❧✐❛♥ ❣r♦✉♣ ❛♥❞ ❚ = ❉✶ × . . . × ❉♥ ❛ ♠♦♥♦✐❞✳ ▲❡t

• ι: ❚ → ● ❛ ❤♦♠♦♠♦r♣❤✐s♠✱ ❛♥❞
• σ: F(●) → ● s❛t✐s❢②✐♥❣ σ(❣) = ❣✳

❚❤❡♥ B(●, ❚, ι) = {❙ t ∈ F(●)×❚ | σ(❙) + ι(t) = ✵ } ⊂ F(●)×❚ t❤❡ ❚✲❜❧♦❝❦ ♠♦♥♦✐❞ ♦✈❡r ● ❞❡✜♥❡❞ ❜② ι✳ ❙♣❡❝✐❛❧ ❈❛s❡s✿

• ■❢ ● = {✵}✱ t❤❡♥

B(●, ❚, ι) = ❚ = ❉✶ × . . . × ❉♥ ✐s ❛ ✜♥✐t❡ ♣r♦❞✉❝t ♦❢ ✜♥✐t❡❧② ♣r✐♠❛r② ♠♦♥♦✐❞s✳

• ■❢ ❚ = {✶}✱ t❤❡♥

B(●, ❚, ι) = B(●) = {❙ ∈ F(●) | σ(❙) = ✵} ⊂ F(●) ✐s t❤❡ ♠♦♥♦✐❞ ♦❢ ③❡r♦✲s✉♠ s❡q✉❡♥❝❡s ♦✈❡r ●✳

❙❡ts ♦❢ ▲❡♥❣t❤s ❲❡❛❦❧② ❑r✉❧❧ ❆r✐t❤♠❡t✐❝ ❙❡♠✐♥♦r♠❛❧✐t② ▼❛✐♥ ❘❡s✉❧ts ▼❡t❤♦❞s

❙❛t✉r❛t❡❞ s✉❜♠♦♥♦✐❞s ✐♥❤❡r✐t t❤❡ ♣r♦♣❡rt✐❡s ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥

❈♦♥s✐❞❡r ❛ s❛t✉r❛t❡❞ s✉❜♠♦♥♦✐❞ ❍ ⊂ ❉ = F(P)×

♥

• ✐=✶

❉✐ , ✇❤❡r❡ P ⊂ ❉ ✐s ❛ s❡t ♦❢ ♣r✐♠❡s✱ ♥ ∈ N✵✱ ❛♥❞ ❉✶, . . . , ❉♥ ❛r❡ ♣r✐♠❛r② ♠♦♥♦✐❞s✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ✳ ✶✳ ■❢ C(❍, ❉) ✐s ❛ t♦rs✐♦♥ ❣r♦✉♣✱ t❤❡♥ ❍ ✐s ❛ ✇❡❛❦❧② ❑r✉❧❧ ♠♦♥♦✐❞✳ ✷✳ ■❢ ❉✶, ..., ❉♥ ❛r❡ s❡♠✐♥♦r♠❛❧ ✜♥✐t❡❧② ♣r✐♠❛r②✱ t❤❡♥ ❍ ✐s s❡♠✐♥♦r♠❛❧ ❛♥❞ ✈✲♥♦❡t❤❡r✐❛♥ ✇✐t❤ (❍ : ❍) = ∅✳