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❆ ❙❛❤❧q✈✐st t❤❡♦r❡♠ ❢♦r s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s

▲❛✉r❡♥t ❉❡ ❘✉❞❞❡r ❛♥❞ ●❡♦r❣❡s ❍❛♥s♦✉❧ P❤❉s ✐♥ ▲♦❣✐❝ ❳■ ✲ ❆♣r✐❧ ✷✵✶✾

❙❛❤❧q✈✐st t❤❡♦r❡♠

❊①❛♠♣❧❡

❋♦r ❛ ❑r✐♣❦❡ ❢r❛♠❡ (X, R) ✇❡ ❤❛✈❡ (X, R) | = p → p ✐✛ (X, R) | = x R y ∧ y R z → x R z.

❚❤❡♦r❡♠

■❢ ✐s ❛ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✜rst ♦r❞❡r ❢♦r♠✉❧❛ ✭✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥✮ s✉❝❤ t❤❛t✱ ❢♦r ❛ ❑r✐♣❦❡ ❢r❛♠❡ ✱ ✐✛

❊①❛♠♣❧❡

✭r❡✢❡①✐✈✐t②✮✱ ✭s②♠♠❡tr②✮✱ ✭r✐❣❤t ✉♥❜♦✉♥❞♥❡ss✮✱ ✳✳✳

❙❛❤❧q✈✐st t❤❡♦r❡♠

❊①❛♠♣❧❡

❋♦r ❛ ❑r✐♣❦❡ ❢r❛♠❡ (X, R) ✇❡ ❤❛✈❡ (X, R) | = p → p ✐✛ (X, R) | = x R y ∧ y R z → x R z.

❚❤❡♦r❡♠

■❢ ϕ ✐s ❛ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛ ϕ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✜rst ♦r❞❡r ❢♦r♠✉❧❛ α(ϕ) ✭✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥✮ s✉❝❤ t❤❛t✱ ❢♦r ❛ ❑r✐♣❦❡ ❢r❛♠❡ (X, R)✱ (X, R) | = ϕ ✐✛ (X, R) | = α(ϕ).

❊①❛♠♣❧❡

✭r❡✢❡①✐✈✐t②✮✱ ✭s②♠♠❡tr②✮✱ ✭r✐❣❤t ✉♥❜♦✉♥❞♥❡ss✮✱ ✳✳✳

❙❛❤❧q✈✐st t❤❡♦r❡♠

❊①❛♠♣❧❡

❋♦r ❛ ❑r✐♣❦❡ ❢r❛♠❡ (X, R) ✇❡ ❤❛✈❡ (X, R) | = p → p ✐✛ (X, R) | = x R y ∧ y R z → x R z.

❚❤❡♦r❡♠

■❢ ϕ ✐s ❛ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛ ϕ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✜rst ♦r❞❡r ❢♦r♠✉❧❛ α(ϕ) ✭✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥✮ s✉❝❤ t❤❛t✱ ❢♦r ❛ ❑r✐♣❦❡ ❢r❛♠❡ (X, R)✱ (X, R) | = ϕ ✐✛ (X, R) | = α(ϕ).

❊①❛♠♣❧❡

p → ♦p ✭r❡✢❡①✐✈✐t②✮✱ p → ♦p ✭s②♠♠❡tr②✮✱ p → ♦p ✭r✐❣❤t ✉♥❜♦✉♥❞♥❡ss✮✱ ✳✳✳

❙✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥

❆ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ ✐s ❛ ♣❛✐r (B, ≺) ✇❤❡r❡ B ✐s ❛ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ❛♥❞ ≺ ❛ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥ B s✉❝❤ t❤❛t ✿ ◮ ✵ ≺ ✵ ❛♥❞ ✶ ≺ ✶✱ ◮ a ≺ b, c ✐♠♣❧✐❡s a ≺ b ∧ c✱ ◮ a, b ≺ c ✐♠♣❧✐❡s a ∨ b ≺ c✱ ◮ a ≤ b ≺ c ≤ d ✐♠♣❧✐❡s a ≺ d✳

❙✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s ❛s ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ ♠♦❞❛❧ ❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥ ✭❖♣t✐♦♥ ✶✮

▲❡t (B, ♦) ❜❡ ❛ ♠♦❞❛❧ ❛❧❣❡❜r❛✳ ❉❡✜♥❡ ♦♥ B t❤❡ r❡❧❛t✐♦♥ a ≺♦ b ✐✛ ♦a ≤ b. ❚❤❡♥✱ (B, ≺♦) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳

❉❡✜♥✐t✐♦♥ ✭❖♣t✐♦♥ ✷✮

▲❡t ❜❡ ❛ ♠♦❞❛❧ ❛❧❣❡❜r❛✳ ❉❡✜♥❡ ♦♥ t❤❡ r❡❧❛t✐♦♥ ✐✛ ❚❤❡♥ ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳

❙✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s ❛s ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ ♠♦❞❛❧ ❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥ ✭❖♣t✐♦♥ ✶✮

▲❡t (B, ♦) ❜❡ ❛ ♠♦❞❛❧ ❛❧❣❡❜r❛✳ ❉❡✜♥❡ ♦♥ B t❤❡ r❡❧❛t✐♦♥ a ≺♦ b ✐✛ ♦a ≤ b. ❚❤❡♥✱ (B, ≺♦) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳

❉❡✜♥✐t✐♦♥ ✭❖♣t✐♦♥ ✷✮

▲❡t (B, ) ❜❡ ❛ ♠♦❞❛❧ ❛❧❣❡❜r❛✳ ❉❡✜♥❡ ♦♥ B t❤❡ r❡❧❛t✐♦♥ a ≺ b ✐✛ a ≤ b. ❚❤❡♥ (B, ≤) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳

❙✉❜♦r❞✐♥❛t✐♦♥ ♠♦r♣❤✐s♠s

❉❡✜♥✐t✐♦♥

▲❡t B, C ❜❡ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s ❛♥❞ h : B − → C ❛ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛①✐♦♠s ✿ ✭✇✮ a ≺ b ✐♠♣❧✐❡s h(a) ≺ h(b)✱ ✭♦✮ h(a) ≺ c ✐♠♣❧✐❡s a ≺ b ❛♥❞ h(b) ≤ c ❢♦r s♦♠❡ b✱ ✭✮ a ≺ h(c) ✐♠♣❧✐❡s b ≺ c ❛♥❞ a ≤ h(b) ❢♦r s♦♠❡ b✳

❙✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s ❛s ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ ♠♦❞❛❧ ❛❧❣❡❜r❛s

Pr♦♣♦s✐t✐♦♥

■❢ h : (B, ♦) − → (C, ♦) ✐s ❛ ♠♦❞❛❧ ♠♦r♣❤✐s♠✱ t❤❡♥ h : (B, ≺♦) − → (C, ≺♦) ✐s ❛ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ ✈❡r✐❢②✐♥❣ ✭✇✮ ❛♥❞ (♦)✳

Pr♦♣♦s✐t✐♦♥

■❢ ✐s ❛ ♠♦❞❛❧ ♠♦r♣❤✐s♠✱ t❤❡♥ ✐s ❛ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ ✈❡r✐❢②✐♥❣ ✭✇✮ ❛♥❞

❙✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s ❛s ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ ♠♦❞❛❧ ❛❧❣❡❜r❛s

Pr♦♣♦s✐t✐♦♥

■❢ h : (B, ♦) − → (C, ♦) ✐s ❛ ♠♦❞❛❧ ♠♦r♣❤✐s♠✱ t❤❡♥ h : (B, ≺♦) − → (C, ≺♦) ✐s ❛ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ ✈❡r✐❢②✐♥❣ ✭✇✮ ❛♥❞ (♦)✳

Pr♦♣♦s✐t✐♦♥

■❢ h : (B, ) − → (C, ) ✐s ❛ ♠♦❞❛❧ ♠♦r♣❤✐s♠✱ t❤❡♥ h : (B, ≺) − → (C, ≺) ✐s ❛ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ ✈❡r✐❢②✐♥❣ ✭✇✮ ❛♥❞ ()

❙✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡s

❉❡✜♥✐t✐♦♥

❆ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡ ✐s ❛ ♣❛✐r (X, R) ✇❤❡r❡ X ❛ ❙t♦♥❡ s♣❛❝❡ ❛♥❞ R ❛ ❝❧♦s❡❞ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥ X✳

❉❡✜♥✐t✐♦♥

▲❡t ❜❡ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡s ❛♥❞ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛①✐♦♠s ✿ ✭✇✮ ✐♠♣❧✐❡s ✱ ✭ ✮ ✐♠♣❧✐❡s ❛♥❞ ❢♦r s♦♠❡ ✱ ✭ ✮ ✐♠♣❧✐❡s ❛♥❞ ❢♦r s♦♠❡ ✳

❙✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡s

❉❡✜♥✐t✐♦♥

❆ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡ ✐s ❛ ♣❛✐r (X, R) ✇❤❡r❡ X ❛ ❙t♦♥❡ s♣❛❝❡ ❛♥❞ R ❛ ❝❧♦s❡❞ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥ X✳

❉❡✜♥✐t✐♦♥

▲❡t X, Y ❜❡ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡s ❛♥❞ f : X − → C ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛①✐♦♠s ✿ ✭✇✮ x R y ✐♠♣❧✐❡s f (x) R f (y)✱ ✭♦✮ f (x) R y ✐♠♣❧✐❡s x R z ❛♥❞ f (z) = y ❢♦r s♦♠❡ z✱ ✭✮ x R f (y) ✐♠♣❧✐❡s z R y ❛♥❞ f (z) = x ❢♦r s♦♠❡ z✳

❉✉❛❧ ♦❢ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛

▲❡t (B, ≺) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳ ❲❡ ❞❡♥♦t❡ ✶✳ XB = Ult(B) t❤❡ ❙t♦♥❡ ❞✉❛❧ ♦❢ B✱ t❤❛t ✐s t❤❡ s❡t ♦❢ ✉❧tr❛✜❧t❡rs ♦❢ B ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ❣❡♥❡r❛t❡❞ ❜② t❤❡ s❡t η(a) = {x ∈ Ult(B) | x ∋ a}, ✷✳ t❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥ ❞❡✜♥❡❞ ❜②

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♣❛✐r ❢♦r♠s ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳

❉✉❛❧ ♦❢ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛

▲❡t (B, ≺) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳ ❲❡ ❞❡♥♦t❡ ✶✳ XB = Ult(B) t❤❡ ❙t♦♥❡ ❞✉❛❧ ♦❢ B✱ t❤❛t ✐s t❤❡ s❡t ♦❢ ✉❧tr❛✜❧t❡rs ♦❢ B ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ❣❡♥❡r❛t❡❞ ❜② t❤❡ s❡t η(a) = {x ∈ Ult(B) | x ∋ a}, ✷✳ R≺ t❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥ XB ❞❡✜♥❡❞ ❜② x R≺ y ⇔ ≺ (y, −) := {a | ∃b ∈ y : b ≺ a} ⊆ x.

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♣❛✐r ❢♦r♠s ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳

❉✉❛❧ ♦❢ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛

▲❡t (B, ≺) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳ ❲❡ ❞❡♥♦t❡ ✶✳ XB = Ult(B) t❤❡ ❙t♦♥❡ ❞✉❛❧ ♦❢ B✱ t❤❛t ✐s t❤❡ s❡t ♦❢ ✉❧tr❛✜❧t❡rs ♦❢ B ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ❣❡♥❡r❛t❡❞ ❜② t❤❡ s❡t η(a) = {x ∈ Ult(B) | x ∋ a}, ✷✳ R≺ t❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥ XB ❞❡✜♥❡❞ ❜② x R≺ y ⇔ ≺ (y, −) := {a | ∃b ∈ y : b ≺ a} ⊆ x.

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♣❛✐r (XB, R≺) ❢♦r♠s ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳

❉✉❛❧ ♦❢ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡

▲❡t (X, R) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❲❡ ❞❡♥♦t❡ ✶✳ BX = Clop(X) t❤❡ ❙t♦♥❡ ❞✉❛❧ ♦❢ X✱ t❤❛t ✐s t❤❡ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ♦❢ ❝❧♦♣❡♥ s❡ts ♦❢ X✱ ✷✳ t❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥ ❞❡✜♥❡❞ ❜②

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♣❛✐r ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳

❉✉❛❧ ♦❢ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡

▲❡t (X, R) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❲❡ ❞❡♥♦t❡ ✶✳ BX = Clop(X) t❤❡ ❙t♦♥❡ ❞✉❛❧ ♦❢ X✱ t❤❛t ✐s t❤❡ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ♦❢ ❝❧♦♣❡♥ s❡ts ♦❢ X✱ ✷✳ ≺R t❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥ BX ❞❡✜♥❡❞ ❜② O ≺R U ⇔ R(−, O) ⊆ U.

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♣❛✐r ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳

❉✉❛❧ ♦❢ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡

▲❡t (X, R) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❲❡ ❞❡♥♦t❡ ✶✳ BX = Clop(X) t❤❡ ❙t♦♥❡ ❞✉❛❧ ♦❢ X✱ t❤❛t ✐s t❤❡ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ♦❢ ❝❧♦♣❡♥ s❡ts ♦❢ X✱ ✷✳ ≺R t❤❡ ❜✐♥❛r② r❡❧❛t✐♦♥ ♦♥ BX ❞❡✜♥❡❞ ❜② O ≺R U ⇔ R(−, O) ⊆ U.

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♣❛✐r (BX, ≺R) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳

❉✉❛❧s ♦❢ ♠♦r♣❤✐s♠s

Pr♦♣♦s✐t✐♦♥

✶✳ ■❢ h : B − → C ✐s ❛ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ ✈❡r✐❢②✐♥❣ ✭✇✮ ✭r❡s♣✳ ✭♦✮ ❛♥❞ ✭✮✮ t❤❡♥ Ult(h) : Ult(C) − → Ult(B) : x − → h−✶(x) ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ t❤❛t ✈❡r✐✜❡s ✭✇✮ ✭r❡s♣✳ ✭♦✮ ❛♥❞ ✭✮ ✷✳ ■❢ ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✈❡r✐❢②✐♥❣ ✭✇✮ ✭r❡s♣✳ ✭ ✮ ❛♥❞ ✭ ✮✮ t❤❡♥

✶

✐s ❛ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ ✈❡r✐❢②✐♥❣ ✭✇✮ ✭r❡s♣✳ ✭ ✮ ❛♥❞ ✭ ✮✮✳

❉✉❛❧s ♦❢ ♠♦r♣❤✐s♠s

Pr♦♣♦s✐t✐♦♥

✶✳ ■❢ h : B − → C ✐s ❛ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ ✈❡r✐❢②✐♥❣ ✭✇✮ ✭r❡s♣✳ ✭♦✮ ❛♥❞ ✭✮✮ t❤❡♥ Ult(h) : Ult(C) − → Ult(B) : x − → h−✶(x) ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ t❤❛t ✈❡r✐✜❡s ✭✇✮ ✭r❡s♣✳ ✭♦✮ ❛♥❞ ✭✮ ✷✳ ■❢ f : X − → Y ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✈❡r✐❢②✐♥❣ ✭✇✮ ✭r❡s♣✳ ✭♦✮ ❛♥❞ ✭✮✮ t❤❡♥ Clop(f ) : Clop(Y ) − → Clop(Y ) : O − → f −✶(O) ✐s ❛ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ ✈❡r✐❢②✐♥❣ ✭✇✮ ✭r❡s♣✳ ✭♦✮ ❛♥❞ ✭✮✮✳

❉✉❛❧✐t②

❚❤❡♦r❡♠

✶✳ ■❢ (B, ≺) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ t❤❡♥ η : (B, ≺) − → (Clop(Ult(B)), ≺R≺) : a − → {x ∈ Ult(B) | x ∋ a} ✐s ❛ ❜✐❥❡❝t✐✈❡ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ t❤❛t ✈❡r✐✜❡s ✭✇✮✱ ✭♦✮ ❛♥❞ ✭✮ ❛♥❞ s✉❝❤ t❤❛t η(a) ≺ η(b) ⇒ a ≺ b. ✷✳ ■❢ ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡ t❤❡♥ ✐s ❛ ❜✐❥❡❝t✐✈❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ t❤❛t ✈❡r✐✜❡s ✭✇✮✱ ✭ ✮ ❛♥❞ ✭ ✮ ❛♥❞ s✉❝❤ t❤❛t

❉✉❛❧✐t②

❚❤❡♦r❡♠

✶✳ ■❢ (B, ≺) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ t❤❡♥ η : (B, ≺) − → (Clop(Ult(B)), ≺R≺) : a − → {x ∈ Ult(B) | x ∋ a} ✐s ❛ ❜✐❥❡❝t✐✈❡ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ t❤❛t ✈❡r✐✜❡s ✭✇✮✱ ✭♦✮ ❛♥❞ ✭✮ ❛♥❞ s✉❝❤ t❤❛t η(a) ≺ η(b) ⇒ a ≺ b. ✷✳ ■❢ (X, R) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡ t❤❡♥ ε : (X, R) − → (Ult(Clop(X)), R≺R) : x − → {O ∈ Clop(X) | O ∋ x} ✐s ❛ ❜✐❥❡❝t✐✈❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ t❤❛t ✈❡r✐✜❡s ✭✇✮✱ ✭♦✮ ❛♥❞ ✭✮ ❛♥❞ s✉❝❤ t❤❛t ε(x) R ε(y) ⇒ x R y.

❱❛❧✐❞✐t② ❢♦r s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡s

❉❡✜♥✐t✐♦♥

▲❡t (X, R) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❆ ✈❛❧✉❛t✐♦♥ ♦♥ X ✐s ❛ ♠❛♣ v : ❱❛r − → Clop(X)✳ ❚❤❡ ✈❛❧✉❛t✐♦♥ ✐s t❤❡♥ ❡①t❡♥❞ t♦ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s ✐♥ t❤❡ ✉s✉❛❧ ✇❛② ❛♥❞ ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ϕ ✐s ✈❛❧✐❞ ✐♥ X ❢♦r t❤❡ ✈❛❧✉❛t✐♦♥ v✱ ❞❡♥♦t❡❞ ❜② X | =v ϕ✱ ✐❢ v(ϕ) = X✳

❘❡♠❛r❦

❙✐♥❝❡ t❤❡ ❛❝❝❡ss r❡❧❛t✐♦♥ ♦❢ ✐s s♦❧❡❧② ❝❧♦s❡❞✱ t❤❡ ✈❛❧✉❛t✐♦♥ ♦❢ ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ♠❛② ❢❛✐❧ t♦ ❜❡ ❝❧♦♣❡♥✳ ❋♦r ✐♥st❛♥❝❡✱ ✐s ♥♦t ❣✉❛r❛♥t❡❡❞ t♦ ❜❡ ❝❧♦♣❡♥✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤✐s ♠❡❛♥s t❤❛t ✇❡ ❝❛♥♥♦t ❡①t❡♥❞ ❛ ✈❛❧✉❛t✐♦♥ ♦♥ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ ❱❛r t♦ ❛❧❧ t❤❡ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s✳

❱❛❧✐❞✐t② ❢♦r s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡s

❉❡✜♥✐t✐♦♥

▲❡t (X, R) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❆ ✈❛❧✉❛t✐♦♥ ♦♥ X ✐s ❛ ♠❛♣ v : ❱❛r − → Clop(X)✳ ❚❤❡ ✈❛❧✉❛t✐♦♥ ✐s t❤❡♥ ❡①t❡♥❞ t♦ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s ✐♥ t❤❡ ✉s✉❛❧ ✇❛② ❛♥❞ ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ϕ ✐s ✈❛❧✐❞ ✐♥ X ❢♦r t❤❡ ✈❛❧✉❛t✐♦♥ v✱ ❞❡♥♦t❡❞ ❜② X | =v ϕ✱ ✐❢ v(ϕ) = X✳

❘❡♠❛r❦

❙✐♥❝❡ t❤❡ ❛❝❝❡ss r❡❧❛t✐♦♥ ♦❢ X ✐s s♦❧❡❧② ❝❧♦s❡❞✱ t❤❡ ✈❛❧✉❛t✐♦♥ ♦❢ ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ♠❛② ❢❛✐❧ t♦ ❜❡ ❝❧♦♣❡♥✳ ❋♦r ✐♥st❛♥❝❡✱ v(♦p) = R(−, v(p)) = {x | ∃y ∈ v(p) : x R y} ✐s ♥♦t ❣✉❛r❛♥t❡❡❞ t♦ ❜❡ ❝❧♦♣❡♥✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤✐s ♠❡❛♥s t❤❛t ✇❡ ❝❛♥♥♦t ❡①t❡♥❞ ❛ ✈❛❧✉❛t✐♦♥ ♦♥ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ ❱❛r t♦ ❛❧❧ t❤❡ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s✳

❱❛❧✐❞✐t② ❢♦r s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡s

❉❡✜♥✐t✐♦♥

▲❡t (X, R) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❆ ✈❛❧✉❛t✐♦♥ ♦♥ X ✐s ❛ ♠❛♣ v : ❱❛r − → Clop(X)✳ ❚❤❡ ✈❛❧✉❛t✐♦♥ ✐s t❤❡♥ ❡①t❡♥❞ t♦ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s ✐♥ t❤❡ ✉s✉❛❧ ✇❛② ❛♥❞ ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ϕ ✐s ✈❛❧✐❞ ✐♥ X ❢♦r t❤❡ ✈❛❧✉❛t✐♦♥ v✱ ❞❡♥♦t❡❞ ❜② X | =v ϕ✱ ✐❢ v(ϕ) = X✳

❘❡♠❛r❦

❙✐♥❝❡ t❤❡ ❛❝❝❡ss r❡❧❛t✐♦♥ ♦❢ X ✐s s♦❧❡❧② ❝❧♦s❡❞✱ t❤❡ ✈❛❧✉❛t✐♦♥ ♦❢ ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ♠❛② ❢❛✐❧ t♦ ❜❡ ❝❧♦♣❡♥✳ ❋♦r ✐♥st❛♥❝❡✱ v(♦p) = R(−, v(p)) = {x | ∃y ∈ v(p) : x R y} ✐s ♥♦t ❣✉❛r❛♥t❡❡❞ t♦ ❜❡ ❝❧♦♣❡♥✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤✐s ♠❡❛♥s t❤❛t ✇❡ ❝❛♥♥♦t ❡①t❡♥❞ ❛ ✈❛❧✉❛t✐♦♥ ♦♥ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ v : ❱❛r − → B t♦ ❛❧❧ t❤❡ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s✳

❈❛♥♦♥✐❝❛❧ ❡①t❡♥s✐♦♥ ♦❢ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛

▲❡t B ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳ ❚❤❡♥ Ult(B) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✱ ❛♥❞ s♦✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❛ ❑r✐♣❦❡ ❢r❛♠❡✳ ■t ❢♦❧❧♦✇s t❤❛t Bδ = P(Ult(B)) ✐s ❛ ❝♦♠♣❧❡t❡ t❡♥s❡ ❜✐♠♦❞❛❧ ❛❧❣❡❜r❛ ✇✐t❤ ❢♦r ❡✈❡r② E ∈ P(Ult(B)) ♦(E) = R(−, E) ❛♥❞ (E) = R(E, −).

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♠❛♣ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ s✉❝❤ t❤❛t

❈❛♥♦♥✐❝❛❧ ❡①t❡♥s✐♦♥ ♦❢ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛

▲❡t B ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳ ❚❤❡♥ Ult(B) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✱ ❛♥❞ s♦✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❛ ❑r✐♣❦❡ ❢r❛♠❡✳ ■t ❢♦❧❧♦✇s t❤❛t Bδ = P(Ult(B)) ✐s ❛ ❝♦♠♣❧❡t❡ t❡♥s❡ ❜✐♠♦❞❛❧ ❛❧❣❡❜r❛ ✇✐t❤ ❢♦r ❡✈❡r② E ∈ P(Ult(B)) ♦(E) = R(−, E) ❛♥❞ (E) = R(E, −).

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♠❛♣ η : B − → Bδ : a − → {x ∈ Ult(B) | x ∋ a} ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇♦♦❧❡❛♥ ♠♦r♣❤✐s♠ s✉❝❤ t❤❛t a ≺ b ⇔ ♦η(a) ≤ η(b) ⇔ η(a) ≤ η(b).

❱❛❧✐❞✐t② ❢♦r s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛

❉❡✜♥✐t✐♦♥

▲❡t (B, ≺) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳ ❆ ✈❛❧✉❛t✐♦♥ ♦♥ B ✐s ❛ ♠❛♣ v : ❱❛r − → B✳ ❆ ✈❛❧✉❛t✐♦♥ ♦♥ B ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ✈❛❧✉❛t✐♦♥ η ◦ v : ❱❛r − → Bδ ♦♥ Bδ✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ✈❛❧✐❞ ✐♥ ✉♥❞❡r t❤❡ ✈❛❧✉❛t✐♦♥ ✐❢ ✶✶✱ t❤❛t ✐s ✐s ✈❛❧✐❞ ✐♥ ✉♥❞❡r t❤❡ ✈❛❧✉❛t✐♦♥ ✳

Pr♦♣♦s✐t✐♦♥

▲❡t ❜❡ ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛✳ ❲❡ ❤❛✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✳

✶❘❡♠❡♠❜❡r t❤❛t B ❛♥❞ Bδ s❤❛r❡ t❤❡ s❛♠❡ t♦♣ ❡❧❡♠❡♥t✳

❱❛❧✐❞✐t② ❢♦r s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛

❉❡✜♥✐t✐♦♥

▲❡t (B, ≺) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳ ❆ ✈❛❧✉❛t✐♦♥ ♦♥ B ✐s ❛ ♠❛♣ v : ❱❛r − → B✳ ❆ ✈❛❧✉❛t✐♦♥ ♦♥ B ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ✈❛❧✉❛t✐♦♥ η ◦ v : ❱❛r − → Bδ ♦♥ Bδ✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ϕ ✐s ✈❛❧✐❞ ✐♥ B ✉♥❞❡r t❤❡ ✈❛❧✉❛t✐♦♥ v ✐❢ η(v(ϕ)) = ✶✶✱ t❤❛t ✐s ϕ ✐s ✈❛❧✐❞ ✐♥ Bδ ✉♥❞❡r t❤❡ ✈❛❧✉❛t✐♦♥ η ◦ v✳

Pr♦♣♦s✐t✐♦♥

▲❡t ❜❡ ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛✳ ❲❡ ❤❛✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✳

✶❘❡♠❡♠❜❡r t❤❛t B ❛♥❞ Bδ s❤❛r❡ t❤❡ s❛♠❡ t♦♣ ❡❧❡♠❡♥t✳

❱❛❧✐❞✐t② ❢♦r s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛

❉❡✜♥✐t✐♦♥

▲❡t (B, ≺) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳ ❆ ✈❛❧✉❛t✐♦♥ ♦♥ B ✐s ❛ ♠❛♣ v : ❱❛r − → B✳ ❆ ✈❛❧✉❛t✐♦♥ ♦♥ B ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ✈❛❧✉❛t✐♦♥ η ◦ v : ❱❛r − → Bδ ♦♥ Bδ✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ϕ ✐s ✈❛❧✐❞ ✐♥ B ✉♥❞❡r t❤❡ ✈❛❧✉❛t✐♦♥ v ✐❢ η(v(ϕ)) = ✶✶✱ t❤❛t ✐s ϕ ✐s ✈❛❧✐❞ ✐♥ Bδ ✉♥❞❡r t❤❡ ✈❛❧✉❛t✐♦♥ η ◦ v✳

Pr♦♣♦s✐t✐♦♥

▲❡t ϕ ❜❡ ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛✳ ❲❡ ❤❛✈❡ B | = ϕ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Ult(B) | = ϕ✳

✶❘❡♠❡♠❜❡r t❤❛t B ❛♥❞ Bδ s❤❛r❡ t❤❡ s❛♠❡ t♦♣ ❡❧❡♠❡♥t✳

❚r❛♥s❧❛t✐♦♥s

❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♣r♦❜❧❡♠s t❤❛t ❝♦✉❧❞ ❜❡ st✉❞✐❡❞✳ ✶✳ ❚r❛♥s❧❛t✐♦♥ ♦❢ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s ♦♥ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ ✐♥t♦ ✜rst ♦r❞❡r ❢♦r♠✉❧❛s ♦❢ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥ ♦❢ t❤❡ ❞✉❛❧✳ (B, ≺) | = p → p ✐✛ (X, R) | = (∀x)(∃y)(y R x) ✷✳ ❚r❛♥s❧❛t✐♦♥ ♦❢ ✜rst ♦r❞❡r ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s ✐♥t♦ ✜rst ♦r❞❡r ❢♦r♠✉❧❛s ♦❢ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥ ♦❢ t❤❡ ❞✉❛❧✳ ✵ ✵ ✐✛ ✸✳ ❚r❛♥s❧❛t✐♦♥ ♦❢ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s ♦♥ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ ✐♥t♦ ✜rst ♦r❞❡r ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s✳ ✐✛ ✵ ✵

❚r❛♥s❧❛t✐♦♥s

❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♣r♦❜❧❡♠s t❤❛t ❝♦✉❧❞ ❜❡ st✉❞✐❡❞✳ ✶✳ ❚r❛♥s❧❛t✐♦♥ ♦❢ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s ♦♥ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ ✐♥t♦ ✜rst ♦r❞❡r ❢♦r♠✉❧❛s ♦❢ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥ ♦❢ t❤❡ ❞✉❛❧✳ (B, ≺) | = p → p ✐✛ (X, R) | = (∀x)(∃y)(y R x) ✷✳ ❚r❛♥s❧❛t✐♦♥ ♦❢ ✜rst ♦r❞❡r ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s ✐♥t♦ ✜rst ♦r❞❡r ❢♦r♠✉❧❛s ♦❢ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥ ♦❢ t❤❡ ❞✉❛❧✳ (B, ≺) | = p ≺ ✵ → p = ✵ ✐✛ (X, R) | = (∀x)(∃y)(y R x). ✸✳ ❚r❛♥s❧❛t✐♦♥ ♦❢ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s ♦♥ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ ✐♥t♦ ✜rst ♦r❞❡r ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s✳ ✐✛ ✵ ✵

❚r❛♥s❧❛t✐♦♥s

❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❝♦rr❡s♣♦♥❞❡♥❝❡ ♣r♦❜❧❡♠s t❤❛t ❝♦✉❧❞ ❜❡ st✉❞✐❡❞✳ ✶✳ ❚r❛♥s❧❛t✐♦♥ ♦❢ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s ♦♥ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ ✐♥t♦ ✜rst ♦r❞❡r ❢♦r♠✉❧❛s ♦❢ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥ ♦❢ t❤❡ ❞✉❛❧✳ (B, ≺) | = p → p ✐✛ (X, R) | = (∀x)(∃y)(y R x) ✷✳ ❚r❛♥s❧❛t✐♦♥ ♦❢ ✜rst ♦r❞❡r ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s ✐♥t♦ ✜rst ♦r❞❡r ❢♦r♠✉❧❛s ♦❢ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥ ♦❢ t❤❡ ❞✉❛❧✳ (B, ≺) | = p ≺ ✵ → p = ✵ ✐✛ (X, R) | = (∀x)(∃y)(y R x). ✸✳ ❚r❛♥s❧❛t✐♦♥ ♦❢ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛s ♦♥ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ ✐♥t♦ ✜rst ♦r❞❡r ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛s✳ (B, ≺) | = p → p ✐✛ (B, ≺) | = p ≺ ✵ → p = ✵.

s✲♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s

❉❡✜♥✐t✐♦♥

✶✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ❝❧♦s❡❞ ✭r❡s♣✳ ♦♣❡♥✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝♦♥st❛♥ts ⊤✱ ⊥✱ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r ♥❡❣❛t✐♦♥s ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ∨✱ ♦ ❛♥❞ ✭r❡s♣✳ ∧✱ ∨✱ ❛♥❞ ✮✳ ✷✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ♣♦s✐t✐✈❡ ✭r❡s♣✳ ♥❡❣❛t✐✈❡✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝♦♥st❛♥ts ✱ ❛♥❞ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ✭r❡s♣✳ ❛♥❞ ♥❡❣❛t✐♦♥ ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s✮ ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ✱ ✱ ✱ ✱ ❛♥❞ ✳ ✸✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s s✲♣♦s✐t✐✈❡ ✭r❡s♣✳ s✲♥❡❣❛t✐✈❡✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝❧♦s❡❞ ♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s ✭r❡s♣✳ ♦♣❡♥ ♥❡❣❛t✐✈❡ ❢♦r♠✉❧❛s✮ ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ✱ ✱ ❛♥❞ ✳

▲❡♠♠❛ ✭■♥t❡rs❡❝t✐♦♥ ❧❡♠♠❛✮

▲❡t

✶

❜❡ ❛ s✲♣♦s✐t✐✈❡ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ❛♥❞ ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② ❛♥❞ ❢♦r ❡✈❡r②

✶ ✶

❝❧♦s❡❞ s❡ts ♦❢ ✇❡ ❤❛✈❡

✶ ✶ ✶ ✶

s✲♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s

❉❡✜♥✐t✐♦♥

✶✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ❝❧♦s❡❞ ✭r❡s♣✳ ♦♣❡♥✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝♦♥st❛♥ts ⊤✱ ⊥✱ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r ♥❡❣❛t✐♦♥s ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ∨✱ ♦ ❛♥❞ ✭r❡s♣✳ ∧✱ ∨✱ ❛♥❞ ✮✳ ✷✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ♣♦s✐t✐✈❡ ✭r❡s♣✳ ♥❡❣❛t✐✈❡✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝♦♥st❛♥ts ⊤✱ ⊥ ❛♥❞ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ✭r❡s♣✳ ❛♥❞ ♥❡❣❛t✐♦♥ ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s✮ ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ∨✱ ♦✱ ✱ ❛♥❞ ✳ ✸✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s s✲♣♦s✐t✐✈❡ ✭r❡s♣✳ s✲♥❡❣❛t✐✈❡✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝❧♦s❡❞ ♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s ✭r❡s♣✳ ♦♣❡♥ ♥❡❣❛t✐✈❡ ❢♦r♠✉❧❛s✮ ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ✱ ✱ ❛♥❞ ✳

▲❡♠♠❛ ✭■♥t❡rs❡❝t✐♦♥ ❧❡♠♠❛✮

▲❡t

✶

❜❡ ❛ s✲♣♦s✐t✐✈❡ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ❛♥❞ ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② ❛♥❞ ❢♦r ❡✈❡r②

✶ ✶

❝❧♦s❡❞ s❡ts ♦❢ ✇❡ ❤❛✈❡

✶ ✶ ✶ ✶

s✲♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s

❉❡✜♥✐t✐♦♥

✶✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ❝❧♦s❡❞ ✭r❡s♣✳ ♦♣❡♥✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝♦♥st❛♥ts ⊤✱ ⊥✱ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r ♥❡❣❛t✐♦♥s ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ∨✱ ♦ ❛♥❞ ✭r❡s♣✳ ∧✱ ∨✱ ❛♥❞ ✮✳ ✷✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ♣♦s✐t✐✈❡ ✭r❡s♣✳ ♥❡❣❛t✐✈❡✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝♦♥st❛♥ts ⊤✱ ⊥ ❛♥❞ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ✭r❡s♣✳ ❛♥❞ ♥❡❣❛t✐♦♥ ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s✮ ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ∨✱ ♦✱ ✱ ❛♥❞ ✳ ✸✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s s✲♣♦s✐t✐✈❡ ✭r❡s♣✳ s✲♥❡❣❛t✐✈❡✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝❧♦s❡❞ ♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s ✭r❡s♣✳ ♦♣❡♥ ♥❡❣❛t✐✈❡ ❢♦r♠✉❧❛s✮ ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ∨✱ ❛♥❞ ✳

▲❡♠♠❛ ✭■♥t❡rs❡❝t✐♦♥ ❧❡♠♠❛✮

▲❡t

✶

❜❡ ❛ s✲♣♦s✐t✐✈❡ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ❛♥❞ ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② ❛♥❞ ❢♦r ❡✈❡r②

✶ ✶

❝❧♦s❡❞ s❡ts ♦❢ ✇❡ ❤❛✈❡

✶ ✶ ✶ ✶

s✲♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s

❉❡✜♥✐t✐♦♥

✶✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ❝❧♦s❡❞ ✭r❡s♣✳ ♦♣❡♥✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝♦♥st❛♥ts ⊤✱ ⊥✱ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r ♥❡❣❛t✐♦♥s ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ∨✱ ♦ ❛♥❞ ✭r❡s♣✳ ∧✱ ∨✱ ❛♥❞ ✮✳ ✷✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ♣♦s✐t✐✈❡ ✭r❡s♣✳ ♥❡❣❛t✐✈❡✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝♦♥st❛♥ts ⊤✱ ⊥ ❛♥❞ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ✭r❡s♣✳ ❛♥❞ ♥❡❣❛t✐♦♥ ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s✮ ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ∨✱ ♦✱ ✱ ❛♥❞ ✳ ✸✳ ❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s s✲♣♦s✐t✐✈❡ ✭r❡s♣✳ s✲♥❡❣❛t✐✈❡✮ ✐❢ ✐t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❝❧♦s❡❞ ♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s ✭r❡s♣✳ ♦♣❡♥ ♥❡❣❛t✐✈❡ ❢♦r♠✉❧❛s✮ ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ∨✱ ❛♥❞ ✳

▲❡♠♠❛ ✭■♥t❡rs❡❝t✐♦♥ ❧❡♠♠❛✮

▲❡t ϕ(p✶, ..., pk) ❜❡ ❛ s✲♣♦s✐t✐✈❡ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ❛♥❞ X ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② A ⊆ X ❛♥❞ ❢♦r ❡✈❡r② C✶, ..., Ck−✶ ❝❧♦s❡❞ s❡ts ♦❢ X ✇❡ ❤❛✈❡ ϕ(C✶, ..., A, ..., Ck−✶) = ∩{ϕ(C✶, ..., O, ..., Ck−✶) | A ⊆ O ∈ Clop(X)}.

❙❛❤❧q✈✐st ❢♦r♠✉❧❛s

❉❡✜♥✐t✐♦♥

✶✳ ❆ str♦♥❣❧② ♣♦s✐t✐✈❡ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ❛ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❢♦r♠✉❧❛s ♦❢ t❤❡ ❢♦r♠ µp = µ✶µ✷...µnp ✇❤❡r❡ p ∈ ❱❛r✱ n ∈ N ❛♥❞ µ ∈ Nn✳ ✷✳ ❆ s✲✉♥t✐❡❞ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ♦❜t❛✐♥❡❞ ❢r♦♠ s✲♥❡❣❛t✐✈❡ ❛♥❞ str♦♥❣❧② ♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ✱ ❛♥❞ ✳ ✸✳ ❆ s✲❙❛❤❧q✈✐st ❢♦r♠✉❧❛ ✐s ❛ ❢♦r♠✉❧❛ ♦❢ t❤❡ ❢♦r♠

✶ ✷

✇❤❡r❡

✶ ✐s s✲✉♥t✐❡❞ ❛♥❞ ✷ ✐s s✲♣♦s✐t✐✈❡✳

❙❛❤❧q✈✐st ❢♦r♠✉❧❛s

❉❡✜♥✐t✐♦♥

✶✳ ❆ str♦♥❣❧② ♣♦s✐t✐✈❡ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ❛ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❢♦r♠✉❧❛s ♦❢ t❤❡ ❢♦r♠ µp = µ✶µ✷...µnp ✇❤❡r❡ p ∈ ❱❛r✱ n ∈ N ❛♥❞ µ ∈ Nn✳ ✷✳ ❆ s✲✉♥t✐❡❞ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ♦❜t❛✐♥❡❞ ❢r♦♠ s✲♥❡❣❛t✐✈❡ ❛♥❞ str♦♥❣❧② ♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ♦ ❛♥❞ ✳ ✸✳ ❆ s✲❙❛❤❧q✈✐st ❢♦r♠✉❧❛ ✐s ❛ ❢♦r♠✉❧❛ ♦❢ t❤❡ ❢♦r♠

✶ ✷

✇❤❡r❡

✶ ✐s s✲✉♥t✐❡❞ ❛♥❞ ✷ ✐s s✲♣♦s✐t✐✈❡✳

❙❛❤❧q✈✐st ❢♦r♠✉❧❛s

❉❡✜♥✐t✐♦♥

✶✳ ❆ str♦♥❣❧② ♣♦s✐t✐✈❡ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ❛ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❢♦r♠✉❧❛s ♦❢ t❤❡ ❢♦r♠ µp = µ✶µ✷...µnp ✇❤❡r❡ p ∈ ❱❛r✱ n ∈ N ❛♥❞ µ ∈ Nn✳ ✷✳ ❆ s✲✉♥t✐❡❞ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s ❛ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ♦❜t❛✐♥❡❞ ❢r♦♠ s✲♥❡❣❛t✐✈❡ ❛♥❞ str♦♥❣❧② ♣♦s✐t✐✈❡ ❢♦r♠✉❧❛s ❜② ❛♣♣❧②✐♥❣ ♦♥❧② ∧✱ ♦ ❛♥❞ ✳ ✸✳ ❆ s✲❙❛❤❧q✈✐st ❢♦r♠✉❧❛ ✐s ❛ ❢♦r♠✉❧❛ ♦❢ t❤❡ ❢♦r♠ µ(ϕ✶ → ϕ✷) ✇❤❡r❡ ϕ✶ ✐s s✲✉♥t✐❡❞ ❛♥❞ ϕ✷ ✐s s✲♣♦s✐t✐✈❡✳

❙❛❤❧q✈✐st t❤❡♦r❡♠

❚❤❡♦r❡♠

▲❡t ϕ ❜❡ ❛ s✲❙❛❤❧q✈✐st ❢♦r♠✉❧❛✳ ❚❤❡r❡ ❡①✐sts ❛ ✜rst ♦r❞❡r ❢♦r♠✉❧❛ f (ϕ) ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ❛ ❜✐♥❛r② r❡❧❛t✐♦♥ s✉❝❤ t❤❛t ❢♦r ❛♥② s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ (B, ≺) ✇✐t❤ ❞✉❛❧ (X, R) ✇❡ ❤❛✈❡ (B, ≺) | = ϕ ✐✛ (X, R) | = f (ϕ).

❊①✐st❡♥❝❡ ♦❢ ❛ ❝♦✉♥t❡r❡①❛♠♣❧❡ ❄

❆ r❡♠❛r❦ ♦♥ s✉❜st✐t✉t✐♦♥

▲❡t X ❜❡ ❛ ❙t♦♥❡ s♣❛❝❡ ✇✐t❤ ❛♥ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥t x✵ ❛♥❞ ❞❡✜♥❡ R ⊆ X × X ❜② x R y ✐✛ x = x✵ ♦r x = y. ❚❤❡♥ ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡ s✉❝❤ t❤❛t ❇✉t ❢♦r ✱ ✇❡ ❤❛✈❡ t❤❛t

❆ r❡♠❛r❦ ♦♥ s✉❜st✐t✉t✐♦♥

▲❡t X ❜❡ ❛ ❙t♦♥❡ s♣❛❝❡ ✇✐t❤ ❛♥ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥t x✵ ❛♥❞ ❞❡✜♥❡ R ⊆ X × X ❜② x R y ✐✛ x = x✵ ♦r x = y. ❚❤❡♥ (X, R) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡ s✉❝❤ t❤❛t (X, R) | = p → ♦p. ❇✉t ❢♦r ✱ ✇❡ ❤❛✈❡ t❤❛t

❆ r❡♠❛r❦ ♦♥ s✉❜st✐t✉t✐♦♥

▲❡t X ❜❡ ❛ ❙t♦♥❡ s♣❛❝❡ ✇✐t❤ ❛♥ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥t x✵ ❛♥❞ ❞❡✜♥❡ R ⊆ X × X ❜② x R y ✐✛ x = x✵ ♦r x = y. ❚❤❡♥ (X, R) ✐s ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡ s✉❝❤ t❤❛t (X, R) | = p → ♦p. ❇✉t ❢♦r ϕ = p ∧ ¬p✱ ✇❡ ❤❛✈❡ t❤❛t (X, R) | = ϕ → ♦ϕ.

❙❝❤❡♠❡✲❡①t❡♥s✐❜❧❡ ❢♦r♠✉❧❛s

❉❡✜♥✐t✐♦♥

❆ ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ϕ ✐s s❛✐❞ t♦ ❜❡ s❝❤❡♠❡✲❡①t❡♥s✐❜❧❡ ✐❢ B | = ϕ(p) ✭✇❡ ✇r✐t❡ ϕ(p) t♦ ✐♥❞✐❝❛t❡ t❤❛t t❤❡ ✈❛r✐❛❜❧❡s ♦❢ ϕ ❛r❡ ❛♠♦♥❣ t❤❡ t✉♣❧❡ p✮ ✐♠♣❧✐❡s B | = ϕ(ψ) ❢♦r ❛❧❧ ψ✳

❚❤❡♦r❡♠

❆♥② s✲❙❛❤❧q✈✐st ❜✐♠♦❞❛❧ ❢♦r♠✉❧❛ ✐s s❝❤❡♠❡✲❡①t❡♥s✐❜❧❡✳

❈♦✉♥t❡r❡①❛♠♣❧❡

❚❤❡ ❢♦r♠✉❧❛ ϕ ≡ p − → ♦p ✐s ❛ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ (∀x)(∃y)(x R y ❛♥❞ R(y, −) ⊆ {x}, ❜✉t ϕ ✐s ♥♦t s❝❤❡♠❡✲❡①t❡♥s✐❜❧❡ ❛♥❞ ❤❡♥❝❡ ϕ ✐s ♥♦t ❛ s✲❙❛❤❧q✈✐st ❢♦r♠✉❧❛✳

• ♦r✐❧❧❛