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❖♥ ▼♦❞❛❧ ❘❡✜♥❡♠❡♥t ❛♥❞ ❈♦♥s✐st❡♥❝② ❑✐♠ ●✳ ▲❛rs❡♥✱ ❯❧r✐❦ ◆②♠❛♥ ❛♥❞ ❆♥❞r③❡❥ ❲→s♦✇s❦✐ ❆❛❧❜♦r❣ ❯♥✐✈❡rs✐t②

▼♦❞❛❧ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s t❡❛ ❝♦✐♥ ❝♦✛❡❡

▼♦❞❛❧ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s t❡❛ ❝♦✐♥ ❝♦✛❡❡

▼♦❞❛❧ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s ❝♦✐♥ ❝♦✛❡❡ ❆♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥✳

❖✉t❧✐♥❡ ❼ ▼♦❞❛❧ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s ❼ P❛rt ■✿ ❘❡✜♥❡♠❡♥t ✈s ■♠♣❧❡♠❡♥t❛t✐♦♥s ❼ P❛rt ■■✿ ❈♦♥s✐st❡♥❝② ❼ ❈♦♥❥❡❝t✉r❡s ✫ ❙✉♠♠❛r②

P❛rt ■ ❘❡✜♥❡♠❡♥t ✈s ■♠♣❧❡♠❡♥t❛t✐♦♥ ■♥❝❧✉s✐♦♥

❉❡❢✳ ▼♦❞❛❧ tr❛♥s✐t✐♦♥ s②st❡♠ S = ( st❛t❡s S , Σ, − → S , ��� S ) ❼ Σ ✿ ❛♥ ❛❧♣❤❛❜❡t ♦❢ ❛❝t✐♦♥s ❼ st❛t❡s S ✿ ❛ ✜♥✐t❡ s❡t ♦❢ st❛t❡s ❼ − → S ⊆ st❛t❡s S × Σ × st❛t❡s S ✭♠✉st✮ ❼ ��� S ⊆ st❛t❡s S × Σ × st❛t❡s S ✭♠❛②✮ ❚r❛♥s✐t✐♦♥ r❡❧❛t✐♦♥s ❛r❡ ✜♥✐t❡✳

❉❡❢✳ ▼♦❞❛❧ tr❛♥s✐t✐♦♥ s②st❡♠ S = ( st❛t❡s S , Σ, − → S , ��� S ) ❼ Σ ✿ ❛♥ ❛❧♣❤❛❜❡t ♦❢ ❛❝t✐♦♥s ❼ st❛t❡s S ✿ ❛ ✜♥✐t❡ s❡t ♦❢ st❛t❡s ❼ − → S ⊆ st❛t❡s S × Σ × st❛t❡s S ✭♠✉st✮ ❼ ��� S ⊆ st❛t❡s S × Σ × st❛t❡s S ✭♠❛②✮ ❚r❛♥s✐t✐♦♥ r❡❧❛t✐♦♥s ❛r❡ ✜♥✐t❡✳

❉❡❢✳ ▼♦❞❛❧ ❘❡✜♥❡♠❡♥t S ≤ ♠ T ✐✛ ❢♦r ❛♥② a ∈ Σ ✿ ��� S ′ ❢♦r s♦♠❡ S ′ t❤❡♥ ✇❤❡♥❡✈❡r S a ��� T ′ ❛♥❞ S ′ ≤ ♠ T ′ ❢♦r s♦♠❡ T ′ ✿ T a → T ′ ❢♦r s♦♠❡ T ′ t❤❡♥ ✇❤❡♥❡✈❡r T a − → S ′ ❛♥❞ S ′ ≤ ♠ T ′ ❢♦r s♦♠❡ S ′ ✿ S a − ●❡♥❡r❛❧✐③❡s s✐♠✉❧❛t✐♦♥✴❜✐s✐♠✉❧❛t✐♦♥

❉❡❢✳ ▼♦❞❛❧ ❘❡✜♥❡♠❡♥t S ≤ ♠ T ✐✛ ❢♦r ❛♥② a ∈ Σ ✿ ��� S ′ ❢♦r s♦♠❡ S ′ t❤❡♥ ✇❤❡♥❡✈❡r S a ��� T ′ ❛♥❞ S ′ ≤ ♠ T ′ ❢♦r s♦♠❡ T ′ ✿ T a → T ′ ❢♦r s♦♠❡ T ′ t❤❡♥ ✇❤❡♥❡✈❡r T a − → S ′ ❛♥❞ S ′ ≤ ♠ T ′ ❢♦r s♦♠❡ S ′ ✿ S a − ●❡♥❡r❛❧✐③❡s s✐♠✉❧❛t✐♦♥✴❜✐s✐♠✉❧❛t✐♦♥

■♠♣❧❡♠❡♥t❛t✐♦♥s ❉❡❢✳ ❆ ♠♦❞❛❧ tr❛♥s✐t✐♦♥ s②st❡♠ I ✐s ❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐✛ − → I = ��� I ✳ ◆♦t❡✿ r❡✜♥❡♠❡♥ts ♦❢ I ❛r❡ ❜✐s✐♠✐❧❛r✳ ❉❡❢✳ ■♠♣❧❡♠❡♥t❛t✐♦♥ ■♥❝❧✉s✐♦♥ S ⊆ m T ✐✛ ∀ ✐♠♣❧❡♠❡♥t❛t✐♦♥s I. I ≤ ♠ S ✐♠♣❧✐❡s I ≤ ♠ T ✳

■♠♣❧❡♠❡♥t❛t✐♦♥s ❉❡❢✳ ❆ ♠♦❞❛❧ tr❛♥s✐t✐♦♥ s②st❡♠ I ✐s ❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐✛ − → I = ��� I ✳ ◆♦t❡✿ r❡✜♥❡♠❡♥ts ♦❢ I ❛r❡ ❜✐s✐♠✐❧❛r✳ ❉❡❢✳ ■♠♣❧❡♠❡♥t❛t✐♦♥ ■♥❝❧✉s✐♦♥ S ⊆ m T ✐✛ ∀ ✐♠♣❧❡♠❡♥t❛t✐♦♥s I. I ≤ ♠ S ✐♠♣❧✐❡s I ≤ ♠ T ✳

❉❡❢✳ ❆ r❡✜♥❡♠❡♥t R ✐s s♦✉♥❞ ❛♥❞ ❝♦♠♣❧❡t❡ ✇rt ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥❝❧✉s✐♦♥ ✐❢ S R T ✐✛ S ⊆ m T . ❚❤♠✳ ▼♦❞❛❧ r❡✜♥❡♠❡♥t ✐s s♦✉♥❞✿ S ≤ ♠ T ✐♠♣❧✐❡s S ⊆ m T . Pr♦♦❢✳ ❙✐♠♣❧❡✳

❉❡❢✳ ❆ r❡✜♥❡♠❡♥t R ✐s s♦✉♥❞ ❛♥❞ ❝♦♠♣❧❡t❡ ✇rt ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥❝❧✉s✐♦♥ ✐❢ S R T ✐✛ S ⊆ m T . ❚❤♠✳ ▼♦❞❛❧ r❡✜♥❡♠❡♥t ✐s s♦✉♥❞✿ S ≤ ♠ T ✐♠♣❧✐❡s S ⊆ m T . Pr♦♦❢✳ ❙✐♠♣❧❡✳

❚❤♠✳ ▼♦❞❛❧ r❡✜♥❡♠❡♥t ✐s ✐♥❝♦♠♣❧❡t❡ Pr♦♦❢✳ s a b a b t a b s �≤ ♠ t, ✇❤✐❧❡ ∀ i.i ≤ ♠ s ✐✛ i ≤ ♠ t

❚❤❡♦r❡♠✳ ❼ ❊st❛❜❧✐s❤✐♥❣ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥❝❧✉s✐♦♥ ✐s ❝♦✲◆P ❤❛r❞ ❼ ❡✈❡♥ ❢♦r s②♥t❛❝t✐❝❛❧❧② ❝♦♥s✐st❡♥t s②st❡♠s ✭ ��� S = − → S ✮✳ ❙✐❞❡ ♥♦t❡✳ ▼♦❞❛❧ r❡✜♥❡♠❡♥t ✐s ✐♥ P✳ Pr♦♦❢✳ ❜② r❡❞✉❝t✐♦♥ ❢r♦♠ ✈❛❧✐❞✐t② ❝❤❡❝❦✐♥❣ ✭ ✸✲❉◆❋✲❚❆❯❚❖▲❖●❨ ✮✳

❘❡♣r❡s❡♥t✐♥❣ x i ❘❡♣r❡s❡♥t✐♥❣ x i x n x n ✳ ✳ ✳ x i + 1 ✳ ✳ x 2 ✳ x i − 1 x 1 ✳ ✳ ✳ x i x 1 x i ¬ x i 0 0 ❈♦♠❜✐♥❡ t♦ r❡♣r❡s❡♥t ❛♥② s❛t✐s✜❛❜❧❡ t❡r♠✳

❆ ❉◆❋ ❢♦r♠✉❧❛✿ c 1 ∨ c 2 ∨ . . . ∨ c m − 1 ∨ c m ✳ c m a c m − 1 a ϕ ✳ a ✳ ✳ c 2 a c 1

❆ tr✉❡ ❢♦r♠✉❧❛ ♦✈❡r t❤❡ s❛♠❡ ✈❛r✐❛❜❧❡s✳ x n x n − 1 a a ✳ 0 ✳ t ✳ tr✉❡ x 2 x 1

■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥❝❧✉s✐♦♥ → φ ✐s ✈❛❧✐❞✳ c m x n a c m − 1 x n − 1 a a ϕ a ✳ a ✳ ✳ 0 ✳ ✳ t ✳ tr✉❡ c 2 x 2 a c 1 x 1

P❛rt ■■ ❈♦♥s✐st❡♥❝②

( ∗ ) ❙②♥t❛❝t✐❝ ❝♦♥s✐st❡♥❝②✿ − → ⊆ ��� ❼ ◆♦ s✉♣♣♦rt ❢♦r ❝♦♥tr❛❞✐❝t✐♦♥s✳ ❼ ▲♦❣✐❝✿ ❝♦♥s✐st❡♥❝② ≡ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ✉♥❞❡r ❛ s❛t✐s❢❛❝t✐♦♥ r❡❧❛t✐♦♥✳ ❍❡r❡✿ ◮ r❡✜♥❡♠❡♥t ✐s s❛t✐s❢❛❝t✐♦♥ ◮ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❛r❡ s♦❧✉t✐♦♥s✳ ◮ ❝♦♥s✐st❡♥❝②✿ ❡①✐st❡♥❝❡ ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❼ ❈❤❛r❛❝t❡r✐③❡ ❝♦♥s✐st❡♥❝② ✉s✐♥❣ ❛ ❝♦♠♣✉t❛❜❧❡ ❝r✐t❡r✐♦♥✱ ❧✐❦❡ ( ∗ )

( ∗ ) ❙②♥t❛❝t✐❝ ❝♦♥s✐st❡♥❝②✿ − → ⊆ ��� ❼ ◆♦ s✉♣♣♦rt ❢♦r ❝♦♥tr❛❞✐❝t✐♦♥s✳ ❼ ▲♦❣✐❝✿ ❝♦♥s✐st❡♥❝② ≡ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ✉♥❞❡r ❛ s❛t✐s❢❛❝t✐♦♥ r❡❧❛t✐♦♥✳ ❍❡r❡✿ ◮ r❡✜♥❡♠❡♥t ✐s s❛t✐s❢❛❝t✐♦♥ ◮ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❛r❡ s♦❧✉t✐♦♥s✳ ◮ ❝♦♥s✐st❡♥❝②✿ ❡①✐st❡♥❝❡ ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❼ ❈❤❛r❛❝t❡r✐③❡ ❝♦♥s✐st❡♥❝② ✉s✐♥❣ ❛ ❝♦♠♣✉t❛❜❧❡ ❝r✐t❡r✐♦♥✱ ❧✐❦❡ ( ∗ )

❉❡❢✳ ❙tr♦♥❣ ❈♦♥s✐st❡♥❝② ❆ st❛t❡ S ✐s str♦♥❣❧② ❝♦♥s✐st❡♥t ✐✛ t❤❡r❡ ❡①✐sts ❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ I s✉❝❤ t❤❛t I ≤ ♠ S .

❈♦♠♣✉t✐♥❣ ❈♦♥s✐st❡♥❝② ❋♦r σ, σ ′ ⊆ st❛t❡s S ✇❡ ✇r✐t❡✿ ⌊ S ⌋ σ ′ ∃ s ∈ σ. ∃ s ′ ∈ σ ′ . s a → s ′ σ a ✐✛ − − → ⌊ S ⌋ σ ′ ∀ s ∈ σ. ∃ s ′ ∈ σ ′ . s a σ a ��� s ′ ✐✛ ��� ✭st❛t❡ s❡ts ❛r❡ ❝♦♥❥✉♥❝t✐♦♥s ♦❢ ❝♦♥str❛✐♥ts✮

❈♦♠♣✉t✐♥❣ ❈♦♥s✐st❡♥❝② ❉❡❢✳ B ⊆ P ( st❛t❡s S ) ✐s ❛ str♦♥❣ ❝♦♥s✐st❡♥❝② r❡❧❛t✐♦♥ ✐✛ ❢♦r ❛❧❧ a ∈ ❛❝t ❛♥❞ σ ∈B ✿ → s ′ ∃ σ ′ ∈B . ∀ s ∈ σ ✳ s a − ⌊ S ⌋ σ ′ ❛♥❞ σ a ⌊ S ⌋ σ ′ ❛♥❞ s ′ ∈ σ ′ ✳ σ a − ��� →

❚❤♠✳ ❆ st❛t❡ S ✐s ✭str♦♥❣❧②✮ ❝♦♥s✐st❡♥t ✐✛ t❤❡r❡ ❡①✐sts ❛ ❝♦♥s✐st❡♥❝② r❡❧❛t✐♦♥ ✇✐t❤ ❛ ❝❧❛ss σ s s✉❝❤ t❤❛t S ∈ σ s ✳ ❚❤♠✳ ❊st❛❜❧✐s❤✐♥❣ str♦♥❣ ❝♦♥s✐st❡♥❝② ✐s ◆P✲❤❛r❞✳ Pr♦♦❢✳ ❘❡❞✉❝t✐♦♥ ❢r♦♠ ✸✲❈♥❢✲❙❛t ✳

❚❤♠✳ ❆ st❛t❡ S ✐s ✭str♦♥❣❧②✮ ❝♦♥s✐st❡♥t ✐✛ t❤❡r❡ ❡①✐sts ❛ ❝♦♥s✐st❡♥❝② r❡❧❛t✐♦♥ ✇✐t❤ ❛ ❝❧❛ss σ s s✉❝❤ t❤❛t S ∈ σ s ✳ ❚❤♠✳ ❊st❛❜❧✐s❤✐♥❣ str♦♥❣ ❝♦♥s✐st❡♥❝② ✐s ◆P✲❤❛r❞✳ Pr♦♦❢✳ ❘❡❞✉❝t✐♦♥ ❢r♦♠ ✸✲❈♥❢✲❙❛t ✳