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❆ ♣r♦❣r❡ss r❡♣♦rt✿ ❉❛❱✐③ ❛♥❞ ❉❛Pr♦♦❢

❍❛♥s✲❉✐❡t❡r ❆✳ ❍✐❡♣

▼❛st❡r ❙t✉❞❡♥t ✭❱r✐❥❡ ❬❯♥✐✈❡rs✐t❡✐t✮ ✈❛♥ ❆♠st❡r❞❛♠❪

❏✉❧② ✶✶✱ ✷✵✶✼

❖✉t❧✐♥❡ ♦❢ t❤✐s t❛❧❦

✶ P❛rt ✶✿ ❉❛❱✐③

▼♦t✐✈❛t✐♦♥ ❉❡♠♦♥str❛t✐♦♥

✷ P❛rt ✷✿ ❉❛Pr♦♦❢

▼♦♥♦✐❞s ❚r❛❝❡s ❈♦rr❡s♣♦♥❞✐♥❣ ❡✈❡♥ts ❖♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ❖✉t❧♦♦❦

• ♦❛❧s ♦❢ t❤✐s t❛❧❦

✶ ❉❡♠♦♥str❛t❡ ❉❛❱✐③ ✷ ❊①♣❧❛✐♥ ❢r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s ✸ ❙❤♦✇ ❉❛Pr♦♦❢✬s ♦♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s

❉❛❱✐③

❆ ♣r♦❥❡❝t t♦ ❞❡s✐❣♥ ❛♥❞ ❞❡✈❡❧♦♣ ❛ s②st❡♠ t❤❛t s✐♠✉❧❛t❡s ❞✐str✐❜✉t❡❞ ❛❧❣♦r✐t❤♠s ❛♥❞ ✈✐s✉❛❧✐③❡s t❤❡♠ ✐♥ ❛ str❛✐❣❤t✲❢♦r✇❛r❞ ❛♥❞ ✐♥t❡r❛❝t✐✈❡ ♠❛♥♥❡r✱ t♦ ♣r♦♠♦t❡ ❡①♣❡r✐♠❡♥t❛t✐♦♥✳

Pr❡❞❡❝❡ss♦r✿ ❇✐s✐♠✉❧❛t✐♦♥ ●❛♠❡s

❇✐s✐♠✉❧❛t✐♦♥ ●❛♠❡s ✭s✉♣❡r✈✐s❡❞ ❜② ❏❡r♦❡♥ ❑❡✐r❡♥✱ ❆♣r✐❧ ✷✵✶✻✮ ▼♦tt♦✿ ✏♣❧❛②✐♥❣ ❣❛♠❡s ⇒ ✐♥t✉✐t✐✈❡ ✉♥❞❡rst❛♥❞✐♥❣✑ ❘❡✉s❡ ❛♥ ✐♠♣❧❡♠❡♥t✐♦♥ ♦❢ ✭❜r❛♥❝❤✐♥❣✮ ❜✐s✐♠✉❧❛t✐♦♥ ✐♥ ❍❛s❦❡❧❧ P♦rt❡❞ ❍❛s❦❡❧❧ ❝♦❞❡ t♦ ❋r❡❣❡✱ ❛ ❍❛s❦❡❧❧✲❢♦r✲t❤❡✲❏❱▼ ❱✐s✉❛❧✐③❛t✐♦♥ ❛♥❞ ✐♥t❡r❛❝t✐♦♥ ✐♠♣❧❡♠❡♥t❡❞ ❛s ❛ ●❯■

❯s✐♥❣ ❆❲❚✴❙✇✐♥❣ ❢r❛♠❡✇♦r❦

❉❛❱✐③✿ ❛ r❡♣❧✐❝❛t❡❞ ❞❡s✐❣♥

❉❛❱✐③ ✭s✉♣❡r✈✐s❡❞ ❜② ❲❛♥ ❋♦❦❦✐♥❦✮ ▼♦tt♦✿ ✏❡①♣❧♦r❛t✐♦♥ ❜② ✉s❡r ⇒ ✐♥t✉✐t✐✈❡ ✉♥❞❡rst❛♥❞✐♥❣✑ ▲♦♥❣✲t❡r♠ ❣♦❛❧✿ ❛ t♦♦❧ ❢♦r t❡❛❝❤✐♥❣ ▲❡ss t❡❞✐♦✉s t❤❛♥ t♦ ✇♦r❦ ♦✉t ❡①❛♠♣❧❡s ❜② ❤❛♥❞ ❘❡♣❧✐❝❛t❡❞ ❞❡s✐❣♥✿ ❏❛✈❛ ❢♦r ●❯■ ❛♥❞ ❍❛s❦❡❧❧ ❢♦r s✐♠✉❧❛t✐♦♥ ■♠♣❧❡♠❡♥ts ∼✺ ❞✐str✐❜✉t❡❞ ❛❧❣♦r✐t❤♠s

❉❛❱✐③✿ ❞❡♠♦♥str❛t✐♦♥

❉ ❊ ▼ ❖

❉❛Pr♦♦❢

❉❛Pr♦♦❢ ✭s✉♣❡r✈✐s❡❞ ❜② ❆❧❜❛♥ P♦♥s❡✮ ❆ st✉❞② ♦❢ t❤❡ s❡♠❛♥t✐❝s ♦❢ ❉❛❱✐③ ▼♦t✐✈❛t❡❞ ❜② t✇♦ ❝♦♥❝❡r♥s✿

✶ ❯s❡❢✉❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞✐str✐❜✉t❡❞ ❛❧❣♦r✐t❤♠s ✷ ❉✐s❝✉ss✐♦♥ r❡❣❛r❞✐♥❣ ❈✐❞♦♥✬s ❞✐str✐❜✉t❡❞ ❞❡♣t❤✲✜rst s❡❛r❝❤

❍❡r❡ ❣✐✈❡♥ ❢r♦♠ t❤❡ ❜♦tt♦♠ ✉♣ ❋♦r♠✉❧❛t✐♦♥s ✐♥ ❈♦q ♣r♦♦❢ ❛ss✐st❛♥t

❖✉t❧✐♥❡

▼♦tt♦✿ ❛❜str❛❝t ❝♦♥❝r❡t❡ ❡q✉❛t✐♦♥❛❧ s♣❡❝✐✜❝❛t✐♦♥ r❡♣r❡s❡♥t❛t✐♦♥ ❛❧❣❡❜r❛ r❡❛❧✐③❛t✐♦♥ ♦♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s s✐♠✉❧❛t✐♦♥ ❉❛Pr♦♦❢ ✐s ❛❜str❛❝t✳ ❉❛❱✐③ ✐s ❝♦♥❝r❡t❡✳

▼♦♥♦✐❞s

❉❡✜♥✐t✐♦♥ ❆ ♠♦♥♦✐❞ M(S,·,✶) ✐s✿ s❡t S ❜✐♥❛r② ♦♣❡r❛t✐♦♥ · : S ×S → S ✉♥✐t ✶ : S   s✐❣♥❛t✉r❡ s✉❝❤ t❤❛t✿ ❛ss♦❝✐❛t✐✈✐t② ∀s,t,u ∈ S. (s ·t)·u = s ·(t ·u) ✐❞❡♥t✐t② ∀s ∈ S. ✶·s = s = s ·✶

• ❡q✉❛t✐♦♥❛❧ s♣❡❝✳

❆♥② str✉❝t✉r❡ ✇❤✐❝❤ ❛❞♠✐ts t❤❡s❡ t✇♦ ❛①✐♦♠s✱ ❛❧s♦ ❛❞♠✐ts ❛❧❧ ♠♦♥♦✐❞ t❤❡♦r❡♠s✳

❋r❡❡ ♠♦♥♦✐❞s

❲❡ ♥♦✇ ❞❡✜♥❡ ♠♦♥♦✐❞s ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② ❛ ❝❛rr✐❡r s❡t C✳ ❉❡✜♥✐t✐♦♥ ▲❡t T (C) ❜❡ t❤❡ s♠❛❧❧❡st s❡t✿ ❣❡♥❡r❛t❡❞ ❜② C✱ ✐✳❡✳ C ⊆ T (C) ❝❧♦s❡❞ ✉♥❞❡r ❜✐♥❛r② ♦♣❡r❛t✐♦♥ · : T (C)×T (C) → T (C) ❝❧♦s❡❞ ✉♥❞❡r ✉♥✐t ✶ : T (C) ◗✉❡st✐♦♥✿ ✐s t❤✐s ❛ ♠♦♥♦✐❞❄ ◆♦✱ ❛①✐♦♠s ❞♦ ♥♦t ❤♦❧❞✳ ❊✳❣✳ x ·✶ = x ♦r (x ·y)·z = x ·(y ·z) ❙✐♥❝❡ t❡r♠s ❛r❡ ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐❣♥❛t✉r❡

❋r❡❡ ♠♦♥♦✐❞s

❉❡✜♥✐t✐♦♥ ▲❡t ∼ = ❜❡ t❤❡ s♠❛❧❧❡st ❝♦♥❣r✉❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ T (C)✿ t❤❛t ✐s ❛ss♦❝✐❛t✐✈❡ (x ·y)·z ∼ = x ·(y ·z) t❤❛t r❡s♣❡❝ts ✐❞❡♥t✐t② ✶·x ∼ = x ∼ = x ·✶ ✭❘❡❝❛❧❧✿ ❝♦♥❣✉r❡♥❝❡ ✐s ❡q✉✐✈❛❧❡♥❝❡ ❛♥❞ r❡s♣❡❝ts ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡✮ ❲❡ ♥♦✇ ✐❞❡♥t✐❢② ❛❧❧ ❡❧❡♠❡♥ts r❡❧❛t❡❞ ❜② t❤❡ ❡q✉✐✈❛❧❡♥❝❡✱ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉♦t✐❡♥t✿ ❉❡✜♥✐t✐♦♥ ▲❡t C ∗ ❜❡ t❤❡ ❢r❡❡ ♠♦♥♦✐❞ t❤❛t ✐s T (C)\ ∼ =✳ ❲❡ ❝❛❧❧ C ❛ ❣❡♥❡r❛t♦r ♦❢ C ∗✳ ❲❤❛t ❛r❡ t❤❡ ❡❧❡♠❡♥ts ♦❢ C ∗❄ Pr❡❝✐s❡❧② ❛❧❧ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s [x] = {y ∈ T (C) | y ∼ = x}

❋r❡❡ ♠♦♥♦✐❞s

❲❤② ❢r❡❡ ♠♦♥♦✐❞❄ ■♥❢♦r♠❛❧❧②✱ ✐t ✐s t❤❡ s✐♠♣❧❡st str✉❝t✉r❡ t❤❛t ❝❛♣t✉r❡s t❤❡ ❛❧❣❡❜r❛✐❝ ❞❡✜♥✐t✐♦♥ ♦❢ ♠♦♥♦✐❞ ❋♦r ❡✈❡r② ♠♦♥♦✐❞ M(S,·,✶) t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ S∗ t♦ M(S,·,✶) ❈❛♣t✉r❡s ❛♣♣❧✐❝❛t✐♦♥ ♦♥ ❛r❜✐tr❛r② ♠♦♥♦✐❞ ❊q✉✐✈❛❧❡♥❝❡ ♦❢ t✇♦ ❡❧❡♠❡♥ts ✐♥ ❛ ❢r❡❡ ♠♦♥♦✐❞ ✐♠♣❧✐❡s ❡q✉✐✈❛❧❡♥❝❡ ✐♥ ❡✈❡r② ♠♦♥♦✐❞ ❍♦✇ t♦ ❞❡❝✐❞❡ ❡q✉✐✈❛❧❡♥❝❡❄ ❋✐♥❞ ❛ ✉♥✐q✉❡ r❡♣r❡s❡♥t❛t✐✈❡ ˆ x ✐♥ [x] ❢♦r ❡❛❝❤ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss ❋✐♥❞ ❛ ♥♦r♠❛❧✐③❛t✐♦♥ t❤❛t ♠❛♣s ❡✈❡r② ❡❧❡♠❡♥t ✐♥ [x] t♦ ˆ x

❋r❡❡ ♠♦♥♦✐❞s

❈❛♥ ✇❡ ❞♦ ❜❡tt❡r t❤❛♥ q✉♦t✐❡♥ts❄ ❆ ❝♦♥❝r❡t❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❢r❡❡ ♠♦♥♦✐❞s ❚❤❛t ❝♦♥s✐sts ♦♥❧② ♦❢ t❤❡ r❡♣r❡s❡♥t❛t✐✈❡s ❉❡✜♥✐t✐♦♥ ▲❡t S∗ ❜❡ t❤❡ s♠❛❧❧❡st s❡t s✉❝❤ t❤❛t✿ t❤❡ ❡♠♣t② ❧✐st ε ∈ S∗ ❣✐✈❡♥ ❧✐st Γ ∈ S∗ ❛♥❞ x ∈ S✱ t❤❡ ♣r❡✜① xΓ ∈ S∗ ◗✉❡st✐♦♥s✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✉♥✐t❄ ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❡❧❡♠❡♥ts ♦❢ ❣❡♥❡r❛t♦r❄ ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❜✐♥❛r② ♦♣❡r❛t♦r❄

▼♦♥♦✐❞ ♦✈❡r✈✐❡✇

❛❜str❛❝t M CM ICM ... ↓ ↓ ↓ ❣❡♥❡r❛t♦r C → C ∗ NC P(C)

• s❡q✉❡♥❝❡

♠✉❧t✐s❡t s❡t ↓ ↓ ↓ ❝♦♥❝r❡t❡ ❧✐st +s♦rt❡❞ +✉♥✐q✉❡

▼♦♥♦✐❞ ♦✈❡r✈✐❡✇

❘❡❝❛♣✿

✶ ●✐✈❡♥ ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡✜♥✐t✐♦♥ ✭s✐❣♥❛t✉r❡✱ ❡q✉❛t✐♦♥s✮ ✷ ❋✐① ❡❧❡♠❡♥ts ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ✸ ❚❡r♠ ❧❛♥❣✉❛❣❡ ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② s✐❣♥❛t✉r❡ ✹ ❈♦♥❣r✉❡♥❝❡ ❣❡♥❡r❛t❡❞ ❜② s♣❡❝✐✜❝❛t✐♦♥ ✺ ◗✉♦t✐❡♥t t❡r♠ ❧❛♥❣✉❛❣❡ ❜② ❝♦♥❣r✉❡♥❝❡ ✻ ❊❧❡❝t r❡♣r❡s❡♥t❛t✐✈❡s ♦❢ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ✼ ❉❡t❡r♠✐♥❡ t❡r♠ ❧❛♥❣✉❛❣❡ ❝♦♥s✐st✐♥❣ ♦♥❧② ♦❢ r❡♣r❡s❡♥t❛t✐✈❡s

▼♦♥♦✐❞ ♦✈❡r✈✐❡✇

❛❜str❛❝t M PCM CM ICM ... ↓ ↓ ↓ ↓ ❣❡♥❡r❛t♦r C → C ∗ C I∗ NC P(C)

• s❡q✉❡♥❝❡

tr❛❝❡ ♠✉❧t✐s❡t s❡t ↓ ↓ ↓ ↓ ❝♦♥❝r❡t❡ ❧✐st +t♦♣♦s♦rt +s♦rt❡❞ +✉♥✐q✉❡

P❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s

❉❡✜♥✐t✐♦♥ ❆♥ ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❧❛t✐♦♥ I ⊆ S ×S ✐s ❛♥ ✐rr❡✢❡①✐✈❡✱ s②♠♠❡tr✐❝ r❡❧❛t✐♦♥ ♦✈❡r S✳ ❚✇♦ ❡✈❡♥ts t❤❛t ❤❛♣♣❡♥ ❝♦♥❝✉rr❡♥t❧② ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳ ❲❡ ✇✐❧❧ ♥♦✇ ❧♦♦❦ ❛t ❝♦♠♣✉t❛t✐♦♥s✱ ✐✳❡✳ ♣❡r♠✉t❛t✐♦♥s ♦❢ ❝♦♥❝✉rr❡♥t ❡✈❡♥ts✳

P❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s

✶✳ ●✐✈❡ ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡✜♥✐t✐♦♥ ✭s✐❣♥❛t✉r❡✱ ❡q✉❛t✐♦♥s✮ ❉❡✜♥✐t✐♦♥ ❆ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ PCM(S,I,·,✶) ✐s✿ ♠♦♥♦✐❞ M(S,·,✶) ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❧❛t✐♦♥ I : P(S ×S) s✉❝❤ t❤❛t✿ ♣❛rt✐❛❧ ❝♦♠♠✉t❛t✐♦♥ ∀(s,t) ∈ I. s ·t = t ·s

❋r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s

❲❡ ♥♦✇ ❞❡✜♥❡ P❈▼ ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② ❛ ❝❛rr✐❡r s❡t C✳ ✷✳ ❋✐① ❡❧❡♠❡♥ts ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ❯♥❧✐❦❡ ❜❡❢♦r❡✱ ✇❡ ❛❧s♦ ✜① I ⊂ C ×C✳ ✸✳ ❚❡r♠ ❧❛♥❣✉❛❣❡ ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② s✐❣♥❛t✉r❡ ❚❤❡ s❡t ♦❢ t❡r♠s T (C) r❡♠❛✐♥s t❤❡ s❛♠❡✳ ✹✳ ❈♦♥❣r✉❡♥❝❡ ❣❡♥❡r❛t❡❞ ❜② s♣❡❝✐✜❝❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ▲❡t x ∼ =I y ❜❡ t❤❡ s♠❛❧❧❡st ❝♦♥❣r✉❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ T (C)✿ t❤❛t ✐s ❛ss♦❝✐❛t✐✈❡ (x ·y)·z ∼ =I x ·(y ·z) t❤❛t r❡s♣❡❝ts ✐❞❡♥t✐t② ✶·x ∼ =I x ∼ =I x ·✶ t❤❛t ❝♦♠♠✉t❡s x ·y ∼ =I y ·x ✐❢ (x,y) ∈ I

❋r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s

✺✳ ◗✉♦t✐❡♥t t❡r♠ ❧❛♥❣✉❛❣❡ ❜② ❝♦♥❣r✉❡♥❝❡ ❉❡✜♥✐t✐♦♥ ▲❡t C I∗ ❜❡ t❤❡ ❢r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ t❤❛t ✐s T (C)\ ∼ =I✳ ◗✉❡st✐♦♥✿ ❲❤❛t ❛r❡ t❤❡ ❡❧❡♠❡♥ts ♦❢ C I∗❄ ❊q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ✭❝❛❧❧❡❞ tr❛❝❡s✮✳ ✻✳ ❊❧❡❝t r❡♣r❡s❡♥t❛t✐✈❡s ♦❢ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ❲❡ ❤❛✈❡ t✇♦ ❝❛♥♦♥✐❝❛❧ ❝❤♦✐❝❡s✿ ❧❡①✐❝♦❣r❛♣❤✐❝ ♥♦r♠❛❧ ❢♦r♠ ❋♦❛t❛ ♥♦r♠❛❧ ❢♦r♠

❋r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s

▲❡t ♥♦r♠❛❧✐③❛t✐♦♥ ❛ss✉♠❡ ❛ t♦t❛❧ ♦r❞❡r < ❜❡t✇❡❡♥ ❡❧❡♠❡♥ts ✐♥ C✳ ▲❡t ≪ ❜❡ ❛ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r ✐♥❞✉❝❡❞ ❜② <✳ ❉❡✜♥✐t✐♦♥ ▲❡t ϕ : C ∗ → C I∗ ❜❡ t❤❡ ❝❛♥♦♥✐❝❛❧ ✐♥❥❡❝t✐♦♥✱ ϕ : ˆ x → [x]✳ ❲❡ ❝❛❧❧ ˆ x ❛ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ [x]✳ ❉❡✜♥✐t✐♦♥ ❆ ❧❡①✐❝♦❣r❛♣❤✐❝ ♥♦r♠❛❧ ❢♦r♠ ♦❢ t ∈ C I∗ ✐s t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝❛❧❧② s♠❛❧❧❡st ❡❧❡♠❡♥t x ∈ C ∗ s✉❝❤ t❤❛t ϕ(x) = t✳ ❆❧❣♦r✐t❤♠✿ s✇❛♣ ❛❞❥❛❝❡♥t ❡❧❡♠❡♥ts b ·a t♦ a·b ✐❢ a < b ❛♥❞ (a,b) ∈ I✱ ✉♥t✐❧ ♥♦ ❧♦♥❣❡r ♣♦ss✐❜❧❡✳

❋r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s

❊①❛♠♣❧❡ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡ tr❛❝❡s✿ [b a d c b a] ❛♥❞ [a b d a c b] ▲❡t (a,b) ∈ I ❛♥❞ (c,a) ∈ I✳ ❚❤❡s❡ ❛r❡ tr❛❝❡ ❡q✉✐✈❛❧❡♥t✳ ▲❡t a < b < c < d✱ ❡t ❝❡t❡r❛✳ ❚❤✐s tr❛❝❡ ✐s ✐♥ ❧❡①✐❝♦❣r❛♣❤✐❝ ♥♦r♠❛❧ ❢♦r♠✳

❋r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s

❊①❛♠♣❧❡ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡ tr❛❝❡s✿ [b a d c b a] ❛♥❞ [a b d a c b] ▲❡t (a,b) ∈ I ❛♥❞ (c,a) ∈ I✳ ❚❤❡s❡ ❛r❡ tr❛❝❡ ❡q✉✐✈❛❧❡♥t✳ ▲❡t a < b < c < d✱ ❡t ❝❡t❡r❛✳ ❚❤✐s tr❛❝❡ ✐s ✐♥ ❧❡①✐❝♦❣r❛♣❤✐❝ ♥♦r♠❛❧ ❢♦r♠✳

❋r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s

❊①❛♠♣❧❡ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡ tr❛❝❡s✿ [b a d c b a] ❛♥❞ [a b d a c b] ▲❡t (a,b) ∈ I ❛♥❞ (c,a) ∈ I✳ ❚❤❡s❡ ❛r❡ tr❛❝❡ ❡q✉✐✈❛❧❡♥t✳ ▲❡t a < b < c < d✱ ❡t ❝❡t❡r❛✳ ❚❤✐s tr❛❝❡ ✐s ✐♥ ❧❡①✐❝♦❣r❛♣❤✐❝ ♥♦r♠❛❧ ❢♦r♠✳

❋r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s

❊①❛♠♣❧❡ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡ tr❛❝❡s✿ [b a d c b a] ❛♥❞ [a b d a c b] ▲❡t (a,b) ∈ I ❛♥❞ (c,a) ∈ I✳ ❚❤❡s❡ ❛r❡ tr❛❝❡ ❡q✉✐✈❛❧❡♥t✳ ▲❡t a < b < c < d✱ ❡t ❝❡t❡r❛✳ ❚❤✐s tr❛❝❡ ✐s ✐♥ ❧❡①✐❝♦❣r❛♣❤✐❝ ♥♦r♠❛❧ ❢♦r♠✳

❋r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s

✼✳ ❉❡t❡r♠✐♥❡ t❡r♠ ❧❛♥❣✉❛❣❡ ❝♦♥s✐st✐♥❣ ♦♥❧② ♦❢ r❡♣r❡s❡♥t❛t✐✈❡s ❈♦♥❝r❡t❡❧②✱ ❋P❈▼ ❝♦rr❡s♣♦♥❞ t♦ ❧✐sts t❤❛t ❛r❡ t♦♣♦❧♦❣✐❝❛❧❧② s♦rt❡❞✳ ❋r❡❡ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞s ❛r❡ ❝❛❧❧❡❞ tr❛❝❡ ♠♦♥♦✐❞s✳ ■❢ I ♠❛①✐♠❛❧ s✉❜s❡t ♦❢ C ×C✱ t❤❡♥ P❈▼ ✐s ❛ ❈▼✳ ❘❡❝❛♣✿ ❋P❈▼ ✐s ✉s❡❞ t♦ ♠♦❞❡❧ ❝♦♥❝✉rr❡♥❝② ❋P❈▼ ❝❛♣t✉r❡s ❛❧❧ ♣♦ss✐❜❧❡ ♦♣❡r❛t✐♦♥s ♦♥ ❛ P❈▼✱ ✐✳❡✳ ❛ ✉♥✐q✉❡ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ SI∗ t♦ ❡✈❡r② PCM(S,I,·,✶)

P❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞

❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞✳ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣✉t❛t✐♦♥✿ P : a d x w Q : b c y z ❊①❛♠♣❧❡ ❧✐♥❡❛r✐③❛t✐♦♥s✿ [a b d x c w y z] [a b d x w c y z]

▲❛♠♣♦rt✬s ❧♦❣✐❝❛❧ ❝❧♦❝❦

❊①❛♠♣❧❡ P : a d x w Q : b c y z ❈♦♠♣✉t✐♥❣ ❧♦❣✐❝❛❧ ❝❧♦❝❦ ✈❛❧✉❡s ✭❝❢✳ ✈❡❝t♦r ❝❧♦❝❦s✮✿ [ ✶

✵

• ✵

✶

• ✷

✶

• ✸

✶

• ✸

✹

• ✹

✶

• ✸

✺

• ✸

✻

• ]

a b d x c w y z ❊✈❡r② ❈▼ ✐s P❈▼✱ s♦ ✏♠❡ss❛❣❡ ❝♦✉♥t✑ (N,+,✵) ✐s P❈▼✳ ❆❧s♦ t❤❡ ✏✉♥r❡❝❡✐✈❡❞ ♠❡ss❛❣❡s✑ ♠♦♥♦✐❞ ✐s P❈▼✳

❈♦rr❡s♣♦♥❞✐♥❣ ♠❡ss❛❣❡s

❆ ♣r♦❝❡ss p s❡♥❞s ❛ ♠❡ss❛❣❡ m t♦ ❛♥♦t❤❡r ♣r♦❝❡ss q✳ ❉❡✜♥✐t✐♦♥ ❚❤❡ s❡t ♦❢ ❡✈❡♥ts E ✐s✿ ❛ s❡♥❞ ❡✈❡♥t (p m → q) ∈ E✱ ❛ r❡❝❡✐✈❡ ❡✈❡♥t (q m ← p) ∈ E✳ ❲❡ ✐♥❢♦r♠❛❧❧② ✏❦♥♦✇✑ t❤❡ ♥♦t✐♦♥ ♦❢ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✈❡♥ts✳ ❆ss✉♠♣t✐♦♥s✿ ▲♦ss❧❡ss ❝❤❛♥♥❡❧s ❊✈❡r② s❡♥❞ ❝♦rr❡s♣♦♥❞s t♦ ♣r❡❝✐s❡❧② ♦♥❡ r❡❝❡✐✈❡✳ ❯♥❜♦✉♥❞❡❞ ❜✉✛❡rs ❚❤❡ ♥♦✳ ♦❢ ♠❡ss❛❣❡s ✐♥ tr❛♥s✐t ✐s ✉♥❜♦✉♥❞❡❞✳ ◗✉❡✉✐♥❣ ❝❤❛♥♥❡❧s ❚❤❡ ❝❤❛♥♥❡❧ ❛❝ts ❛s ❛ ❋■❋❖ q✉❡✉❡✳

❖♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s✿ ♦✈❡r✈✐❡✇

❖✈❡r❛❧❧ str❛t❡❣②✿

✶ ❆❜str❛❝t ♦✈❡r ♥❡t✇♦r❦✿

♥♦❞❡s✱ ♣r♦❝❡ss ✐❞❡♥t✐✜❡rs V s②♠♠❡tr✐❝ ❡❞❣❡ r❡❧❛t✐♦♥ E ⊆ V ×V ❝♦♥♥❡❝t❡❞

✷ ❆❜str❛❝t ♦✈❡r ❛❧❣♦r✐t❤♠✿

♠❡ss❛❣❡ s♣❛❝❡ µ✱ st❛t❡ s♣❛❝❡ σ

✸ ❘❡❛❧✐③❡ ❛❧❣♦r✐t❤♠ ❜② ❣✐✈✐♥❣ ♣r♦❝❡ss ❞❡s❝r✐♣t✐♦♥

▼❛♣ ❢r♦♠ st❛t❡ s♣❛❝❡ t♦ ❧♦❝❛❧ ❡✛❡❝t

✹ ▼♦❞❡❧ ❡①❡❝✉t✐♦♥s ❜② ✉♥r❛✈❡❧✐♥❣ ♣r♦❝❡ss ❞❡s❝r✐♣t✐♦♥s

❖♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s✿ ❜✐❣ ♣✐❝t✉r❡

... γ✶ γ✷ γn ... δ✶ δ✷ δn ❞✐r❡❝t❡❞ ❛❝②❝❧✐❝ ❣r❛♣❤ ♣♦t❡♥t✐❛❧❧② ✐♥✜♥✐t❡ r♦♦t❡❞ ✐♥ ✐♥✐t✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ❡❛❝❤ tr❛♥s✐t✐♦♥ ❧❛❜❡❧❡❞ ❜② ❡✈❡♥t ♣❛t❤ ❢r♦♠ r♦♦t t♦ ♥♦❞❡ ✐s ❛ ♣❛rt✐❛❧❧② ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞

Pr♦❝❡ss ❞❡s❝r✐♣t✐♦♥s✿ ❡✛❡❝ts

Pr♦❝❡ss ❞❡s❝r✐♣t✐♦♥s ❛r❡ ❧♦❝❛❧✳ ❆♣♣❧✐❡❞ ✉♥✐❢♦r♠❧② t♦ ❛❧❧ ♥♦❞❡s✳ ❉❡✜♥✐t✐♦♥ ❚❤❡ s❡t ♦❢ ❧♦❝❛❧ ❡✛❡❝ts L ✐s ❣✐✈❡♥ ❜②✿

✶ ❛ s❡♥❞ ❡✛❡❝t S : µ ×V ×σ → L ✷ ❛ r❡❝❡✐✈❡ ❡✛❡❝t R : (µ ×V → σ) → L ✸ ❛♥ ✐♥t❡r♥❛❧ ❡✛❡❝t I : P(σ) → L✳

■♥st❡❛❞ ♦❢ I(/ 0) ✇❡ ✇r✐t❡ ✱ ❛ t❡r♠✐♥❛t✐♦♥ ❡✛❡❝t✳ ❉❡✜♥✐t✐♦♥ ❇② P : σ → L ✇❡ ❞❡♥♦t❡ ❛ ♣r♦❝❡ss ❞❡s❝r✐♣t✐♦♥✳

Pr♦❝❡ss ❞❡s❝r✐♣t✐♦♥s✿ ❝♦♥✜❣✉r❛t✐♦♥s

❈♦♥✜❣✉r❛t✐♦♥s ❛r❡ ❣❧♦❜❛❧✳ ❆ ❝♦♥✜❣✉r❛t✐♦♥ C = (κ,λ) ✇❤❡r❡ ❈❤❛♥♥❡❧ ❧❛❜❡❧✐♥❣ κ : E → µ∗ ♠❛♣s ❝❤❛♥♥❡❧ t♦ tr❛✈❡❧✐♥❣ ♠❡ss❛❣❡s✳ Pr♦❝❡ss ❧❛❜❡❧✐♥❣ λ : V → σ ♠❛♣s ♥♦❞❡ t♦ ✐ts ❝✉rr❡♥t ❧♦❝❛❧ st❛t❡✳ ✉♣❞❛t❡ ❧♦❝❛❧ st❛t❡ λ[p := s](x)

def

=

• s

x = p λ(x) ♦t❤❡r✇✐s❡ ❛♣♣❡♥❞ ♠❡ss❛❣❡ κ[p m → q](x)

def

=

• κ(x)@(m#ε)

(p,q) = x κ(x) ♦t❤❡r✇✐s❡ t❡st ♠❡ss❛❣❡ κ[q m ← p]? ⇔ ∃xs!.κ(p,q) = m#xs r❡❝❡✐✈❡ ♠❡ss❛❣❡ κ[q ← p](x)

def

=

• xs

(p,q) = x κ(x) ♦t❤❡r✇✐s❡

Pr♦❝❡ss ❞❡s❝r✐♣t✐♦♥s✿ ❝♦♥✜❣✉r❛t✐♦♥s

❈♦♥✜❣✉r❛t✐♦♥s ❛r❡ ❣❧♦❜❛❧✳ ❆ ❝♦♥✜❣✉r❛t✐♦♥ C = (κ,λ) ✇❤❡r❡ ❈❤❛♥♥❡❧ ❧❛❜❡❧✐♥❣ κ : E → µ∗ ♠❛♣s ❝❤❛♥♥❡❧ t♦ tr❛✈❡❧✐♥❣ ♠❡ss❛❣❡s✳ Pr♦❝❡ss ❧❛❜❡❧✐♥❣ λ : V → σ ♠❛♣s ♥♦❞❡ t♦ ✐ts ❝✉rr❡♥t ❧♦❝❛❧ st❛t❡✳ ✉♣❞❛t❡ ❧♦❝❛❧ st❛t❡ λ[p := s](x)

def

=

• s

x = p λ(x) ♦t❤❡r✇✐s❡ ❛♣♣❡♥❞ ♠❡ss❛❣❡ κ[p m → q](x)

def

=

• κ(x)@(m#ε)

(p,q) = x κ(x) ♦t❤❡r✇✐s❡ t❡st ♠❡ss❛❣❡ κ[q m ← p]? ⇔ ∃xs!.κ(p,q) = m#xs r❡❝❡✐✈❡ ♠❡ss❛❣❡ κ[q ← p](x)

def

=

• xs

(p,q) = x κ(x) ♦t❤❡r✇✐s❡

Pr♦❝❡ss ❞❡s❝r✐♣t✐♦♥s✿ r✉❧❡s

∆ ⊢ (κ,λ) P(λ(p)) = I(X) s ∈ X ∆ ⊢ (κ,λ[p := s]) I ∆ ⊢ (κ,λ) P(λ(p)) = S(m,q,s) ∆,(p m → q) ⊢ (κ[p m → q],λ[p := s]) S ∆ ⊢ (κ,λ) P(λ(p)) = R(f ) κ[p m ← q]? f (m,q) = s ∆,(p m ← q) ⊢ (κ[p ← q],λ[p := s]) R

Pr♦❝❡ss ❞❡s❝r✐♣t✐♦♥s✿ r✉❧❡s

∆ ⊢ (κ,λ) P(λ(p)) = I(X) s ∈ X ∆ ⊢ (κ,λ[p := s]) I ∆ ✐s ❛ tr❛❝❡ ❝♦♥s✐st✐♥❣ ♦❢ ❡✈❡♥ts E ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡ ♦❢ ♥❡①t st❛t❡ s ∈ X ❞♦❡s ♥♦t ❛♣♣❧② ✐❢ ♥♦❞❡ t❡r♠✐♥❛t❡s✱ P(λ(p)) = = I(/ 0) ✉♣❞❛t❡s ❧♦❝❛❧ st❛t❡ λ[p := s]

Pr♦❝❡ss ❞❡s❝r✐♣t✐♦♥s✿ r✉❧❡s

∆ ⊢ (κ,λ) P(λ(p)) = S(m,q,s) ∆,(p m → q) ⊢ (κ[p m → q],λ[p := s]) S r❡❝♦r❞ s❡♥❞ ❡✈❡♥t p m → q ✉♣❞❛t❡s ❝❤❛♥♥❡❧ st❛t❡ κ[p m → q]✱ ❧♦❝❛❧ st❛t❡ λ[p := s]

Pr♦❝❡ss ❞❡s❝r✐♣t✐♦♥s✿ r✉❧❡s

∆ ⊢ (κ,λ) P(λ(p)) = R(f ) κ[p m ← q]? f (m,q) = s ∆,(p m ← q) ⊢ (κ[p ← q],λ[p := s]) R ❝♦♥❞✐t✐♦♥✿ κ[q m ← p]? ✐♠♣❧✐❡s ❝❤❛♥♥❡❧ st❛t❡ m#xs st❛t❡ f (m,q) = s ❞❡♣❡♥❞s ♦♥ r❡❝❡✐✈❡❞ ♠❡ss❛❣❡ ❛♥❞ s❡♥❞❡r r❡❝♦r❞ r❡❝❡✐✈❡ ❡✈❡♥t p m ← q ✉♣❞❛t❡s ❝❤❛♥♥❡❧ st❛t❡ κ[p ← q]✱ ❧♦❝❛❧ st❛t❡ λ[p := s]

❊①❛♠♣❧❡✿ ❈✐❞♦♥✬s ❛❧❣♦r✐t❤♠

❈✐❞♦♥✬s ❞✐str✐❜✉t❡❞ ❞❡♣t❤✲✜rst s❡❛r❝❤✳ µ

def

= {Token,Info} σel

def

= {Received(p),Replied(p),Undefined,Initiator | p ∈ V } σ

def

= {✵,✶}×σel ×(V ⊎{u})×✷V ×✷V Pr♦❝❡ss ❞❡s❝r✐♣t✐♦♥ ✐s t♦♦ ❧❛r❣❡ t♦ s❤♦✇ ❤❡r❡ ❆❧❣♦r✐t❤♠ ❛❞❞s ❛①✐♦♠✿ ✐♥✐t✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ❯s❡ t❤❡ t♦♦❧ t♦ ❡①♣❧♦r❡ ❡①❡❝✉t✐♦♥s

❖✉t❧♦♦❦✿ ♣r♦✈✐♥❣ ♣r♦♣❡rt✐❡s

❖❜s❡r✈❡ t❤❛t ✐♥ t❤❡ ❥✉❞❣❡♠❡♥t ∆ ⊢ (κ,λ)✱ κ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ∆✱ ✈✐③✳ ✏✉♥r❡❝❡✐✈❡❞ ♠❡ss❛❣❡s✑ P❈▼✳ ❲✐t❤ s❧✐❣❤t ❛❞❛♣t✐♦♥ ♦❢ ❞❡✜♥✐t✐♦♥s✱ ∆ ❛❧s♦ ❞❡t❡r♠✐♥❡s λ✳ ▲❡t ∆ ⊢ φ ❞❡♥♦t❡ t❤❡ ❥✉❞❣❡♠❡♥t t❤❛t φ ❤♦❧❞s ❢♦r (κ,λ)✳

✶ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ s②st❡♠ ⇒ t❡r♠✐♥❛t✐♦♥ ⇒ ✐♥❞✉❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ✷ ❙❤♦✇ ♣r♦♣❡rt✐❡s ∆ ⊢ φ ❜② ✐♥❞✉❝t✐♦♥

❖✉t❧♦♦❦✿ ❞❡❝♦♠♣♦s✐♥❣ ♣r♦❝❡ss❡s

Pr❡♠✐s❡✿ ❛❧❣♦r✐t❤♠s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ à✲❧❛ ♣r♦❝❡ss ❛❧❣❡❜r❛ Pr❡♠✐s❡✿ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛❧❣♦r✐t❤♠s ♣r❡s❡r✈❡ ❝♦rr❡❝t♥❡ss ♣r♦♣❡rt✐❡s ❊①❛♠♣❧❡

• ✐✈❡♥ t✇♦ ♣r♦❝❡ss ❞❡s❝r✐♣t✐♦♥s✿ µ✶,σ✶,P✶ ❛♥❞ µ✷,σ✷,P✷✳

s❡q✉❡♥t✐❛❧ ❝♦♠♣♦s✐t✐♦♥ P✶;P✷ ❜② µ = µ✶ ⊎ µ✷ ❛♥❞ σ = σ✶ ∪σ✷

❞♦❡s P✶ ❤❛✈❡ ❛♥② ❞❛♥❣❧✐♥❣ ♠❡ss❛❣❡s❄

♣❛r❛❧❧❡❧ ❝♦♠♣♦s✐t✐♦♥ P✶||P✷ ❜② µ = µ✶ ⊎ µ✷ ❛♥❞ σ = σ✶ ×σ✷

t✇♦ ♠♦❞❡s ♦❢ t❡r♠✐♥❛t✐♦♥✿ ❜♦t❤ ♦r ❥✉st ♦♥❡❄

❖✉t❧♦♦❦✿ ❞❡❝♦♠♣♦s✐♥❣ ♣r♦❝❡ss❡s

❊①❛♠♣❧❡

• ✐✈❡♥ ❛ ♣r♦❝❡ss ❞❡s❝r✐♣t✐♦♥✿ µ✶,σ✶,P✶

❧❡t P✷ ❞❡♣❡♥❞ ♦♥ ❛♥② ♦t❤❡r ♣r♦❝❡ss✱ t❤❡♥ ❡♠❜❡❞❞✐♥❣ P✶ ✐♥t♦ P✷

❢♦r ❡①❛♠♣❧❡✱ ❜❛s✐❝ ❛❧❣♦r✐t❤♠s ✫ ❝♦♥tr♦❧ ❛❧❣♦r✐t❤♠s

❧❡t P✷ ❞❡♣❡♥❞ ♦♥ t❡r♠✐♥❛❧ st❛t❡ σX✱ ❢✉♥❝t✐♦♥❛❧ ❝♦♠♣♦s✐t✐♦♥ P✶ ◦P✷

✇❤✐❝❤ st❛t❡s ❛r❡ r❡❧❡✈❛♥t❄ ❢❛♥✲✐♥ ♦r ❢❛♥✲♦✉t❄

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳ ◗✉❡st✐♦♥s❄

❈✐❞♦♥✬s ❉❉❋❙

P(⊤,s,⊥,F,I) = I({⊤,s,⌈x⌉,F\x,I\x} | x ∈ F) ✇❤❡r❡ s ∈ {Initiator,Received(p)} ❛♥❞ F = / P(⊤,s,⌈x⌉,F,I) = S(Info,i,(⊤,s,⌈x⌉,F,I\i)) ✇❤❡r❡ s ∈ {Initiator,Received(p)} ❛♥❞ i ∈ I ❛♥❞ I = / P(⊤,s,⌈x⌉,F, / 0) = S(Token,x,(⊥,s,⌈x⌉,F, / 0) ✇❤❡r❡ s ∈ {Initiator,Received(p)} P(⊤,s,⊥, / 0,I) = S(Info,i,(⊤,s,⊥, / 0,I\i)) ✇❤❡r❡ s = Received(p) ❛♥❞ i ∈ I ❛♥❞ I = / P(⊤,s,⊥, / 0, / 0) = S(Token,p,(⊥,s′,⊥, / 0, / 0)) ✇❤❡r❡ s = Received(p) ❛♥❞ s′ = Replied(p) P(⊤,Initiator,⊥, / 0, / 0) = P(⊥,Replied(p),⊥, / 0, / 0) =

❈✐❞♦♥✬s ❉❉❋❙

P(⊥,Undefined,⊥,F,I) = R(f ) ✇❤❡r❡ f (Token,p) = (⊤,Received(p),⊥,F\p,I\p) f (Info,p) = (⊥,Undefined,⊥,F\p,I) P(⊥,s,⌈x⌉,F,I) = R(f ) ✇❤❡r❡ s ∈ {Received(p),Initiator} ❛♥❞ f (m,x) = (⊤,s,⊥,F,I) f (m,q) = (⊥,s,⊥,F\q,I) P(⊥,s,⊥,F,I) = R(f ) ✇❤❡r❡ s ∈ {Received(p),Initiator} ❛♥❞ f (Token,q) = (⊤,s,⊥,F,I) f (Info,q) = (⊥,s,⊥,F\q,I)