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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  s✐❣♥❛t✉r❡ ✉♥✐t ✶ : S s✉❝❤ t❤❛t✿ ❛ss♦❝✐❛t✐✈✐t② ∀ s , t , u ∈ S . ( s · t ) · u = s · ( t · u ) � ❡q✉❛t✐♦♥❛❧ s♣❡❝✳ ✐❞❡♥t✐t② ∀ s ∈ S . ✶ · s = s = 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 x Γ ∈ 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 ... ↓ ↓ ↓ C ∗ N C ❣❡♥❡r❛t♦r C → 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