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

❚❤❡♦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 .

❊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 ) .

❙❛❤❧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 ✳

❉❡✜♥✐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❛✳

❙✉❜♦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 ✳

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✐♥❣ ✭✇✮ ❛♥❞ ( ♦ ) ✳

❙✉❜♦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✐♥❣ ✭✇✮ ❛♥❞ ( � )

❉❡✜♥✐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 ✳

❙✉❜♦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 ✳

✷✳ 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❡ ✶✳ X B = 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 } ,

Pr♦♣♦s✐t✐♦♥ ❚❤❡ ♣❛✐r ❢♦r♠s ❛ s✉❜♦r❞✐♥❛t✐♦♥ s♣❛❝❡✳ ❉✉❛❧ ♦❢ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛ ▲❡t ( B , ≺ ) ❜❡ ❛ s✉❜♦r❞✐♥❛t✐♦♥ ❛❧❣❡❜r❛✳ ❲❡ ❞❡♥♦t❡ ✶✳ X B = 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✐♦♥ ♦♥ X B ❞❡✜♥❡❞ ❜② x R ≺ y ⇔ ≺ ( y , − ) := { a | ∃ b ∈ y : b ≺ a } ⊆ x .