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t rt t sts t tr qt r

  1. ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ■❧❛r✐❛ ❈♦❧❛③③♦ ✐❧❛r✐❛✳❝♦❧❛③③♦❅✉♥✐s❛❧❡♥t♦✳✐t ❯♥✐✈❡rs✐tà ❞❡❧ ❙❛❧❡♥t♦ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❛♥❞ ♥♦♥✲❛ss♦❝✐❛t✐✈❡ str✉❝t✉r❡s✱ ❜r❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ▼❛r❝❤ ✶✹✱ ✷✵✶✽

  2. ❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ❚❤❡ ♠❛✐♥ r❡s✉❧ts ♦❢ t❤✐s t❛❧❦ ❛r❡ ❝♦♥t❡✐♥❡❞ ✐♥ ❋✳ ❈❛t✐♥♦✱ ■✳❈✳✱ P✳ ❙t❡❢❛♥❡❧❧✐✱ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ s❡t✲t❤❡♦r❡t✐❝❛❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳ ■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✶ ✴ ✶✷

  3. ❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ❚❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ■❢ X ✐s ❛ ♥♦♥✲❡♠♣t② s❡t✱ ❛ ✭s❡t✲t❤❡♦r❡t✐❝❛❧✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ r : X × X → X × X ✐s ❛ ♠❛♣ s✉❝❤ t❤❛t t❤❡ ✇❡❧❧✲❦♥♦✇♥ ❜r❛✐❞ ❡q✉❛t✐♦♥ r ✶ r ✷ r ✶ = r ✷ r ✶ r ✷ ✐s s❛t✐s✜❡❞✱ ✇❤❡r❡ r ✶ = r × id X ❛♥❞ r ✷ = id X × r ✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ♦❜t❛✐♥ ❛♥❞ ❝♦♥str✉❝t ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥❄ ❉❡t❡r♠✐♥✐♥❣ ❛❧❧ s❡t✲t❤❡♦r❡t✐❝ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✐s ❛ ✈❡r② ❞✐✣❝✉❧t t❛s❦✳ ❊✈❡♥ ✐❢ ✇❡ ❝❛♥ ✜♥❞ s❡✈❡r❛❧ ✇♦r❦s ❛❜♦✉t t❤✐s t♦♣✐❝✱ ✐t ✐s st✐❧❧ ❛♥ ♦♣❡♥ ♣r♦❜❧❡♠✳ ■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✷ ✴ ✶✷

  4. ❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ❚❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ■❢ X ✐s ❛ ♥♦♥✲❡♠♣t② s❡t✱ ❛ ✭s❡t✲t❤❡♦r❡t✐❝❛❧✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ r : X × X → X × X ✐s ❛ ♠❛♣ s✉❝❤ t❤❛t t❤❡ ✇❡❧❧✲❦♥♦✇♥ ❜r❛✐❞ ❡q✉❛t✐♦♥ r ✶ r ✷ r ✶ = r ✷ r ✶ r ✷ ✐s s❛t✐s✜❡❞✱ ✇❤❡r❡ r ✶ = r × id X ❛♥❞ r ✷ = id X × r ✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ♦❜t❛✐♥ ❛♥❞ ❝♦♥str✉❝t ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥❄ ❉❡t❡r♠✐♥✐♥❣ ❛❧❧ s❡t✲t❤❡♦r❡t✐❝ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✐s ❛ ✈❡r② ❞✐✣❝✉❧t t❛s❦✳ ❊✈❡♥ ✐❢ ✇❡ ❝❛♥ ✜♥❞ s❡✈❡r❛❧ ✇♦r❦s ❛❜♦✉t t❤✐s t♦♣✐❝✱ ✐t ✐s st✐❧❧ ❛♥ ♦♣❡♥ ♣r♦❜❧❡♠✳ ■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✷ ✴ ✶✷

  5. ❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ❚❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ■❢ X ✐s ❛ ♥♦♥✲❡♠♣t② s❡t✱ ❛ ✭s❡t✲t❤❡♦r❡t✐❝❛❧✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ r : X × X → X × X ✐s ❛ ♠❛♣ s✉❝❤ t❤❛t t❤❡ ✇❡❧❧✲❦♥♦✇♥ ❜r❛✐❞ ❡q✉❛t✐♦♥ r ✶ r ✷ r ✶ = r ✷ r ✶ r ✷ ✐s s❛t✐s✜❡❞✱ ✇❤❡r❡ r ✶ = r × id X ❛♥❞ r ✷ = id X × r ✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ♦❜t❛✐♥ ❛♥❞ ❝♦♥str✉❝t ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥❄ ❉❡t❡r♠✐♥✐♥❣ ❛❧❧ s❡t✲t❤❡♦r❡t✐❝ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✐s ❛ ✈❡r② ❞✐✣❝✉❧t t❛s❦✳ ❊✈❡♥ ✐❢ ✇❡ ❝❛♥ ✜♥❞ s❡✈❡r❛❧ ✇♦r❦s ❛❜♦✉t t❤✐s t♦♣✐❝✱ ✐t ✐s st✐❧❧ ❛♥ ♦♣❡♥ ♣r♦❜❧❡♠✳ ■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✷ ✴ ✶✷

  6. ❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ❚❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ■❢ X ✐s ❛ ♥♦♥✲❡♠♣t② s❡t✱ ❛ ✭s❡t✲t❤❡♦r❡t✐❝❛❧✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ r : X × X → X × X ✐s ❛ ♠❛♣ s✉❝❤ t❤❛t t❤❡ ✇❡❧❧✲❦♥♦✇♥ ❜r❛✐❞ ❡q✉❛t✐♦♥ r ✶ r ✷ r ✶ = r ✷ r ✶ r ✷ ✐s s❛t✐s✜❡❞✱ ✇❤❡r❡ r ✶ = r × id X ❛♥❞ r ✷ = id X × r ✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ♦❜t❛✐♥ ❛♥❞ ❝♦♥str✉❝t ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥❄ ❉❡t❡r♠✐♥✐♥❣ ❛❧❧ s❡t✲t❤❡♦r❡t✐❝ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✐s ❛ ✈❡r② ❞✐✣❝✉❧t t❛s❦✳ ❊✈❡♥ ✐❢ ✇❡ ❝❛♥ ✜♥❞ s❡✈❡r❛❧ ✇♦r❦s ❛❜♦✉t t❤✐s t♦♣✐❝✱ ✐t ✐s st✐❧❧ ❛♥ ♦♣❡♥ ♣r♦❜❧❡♠✳ ■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✷ ✴ ✶✷

  7. ❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ❚❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ■❢ X ✐s ❛ ♥♦♥✲❡♠♣t② s❡t✱ ❛ ✭s❡t✲t❤❡♦r❡t✐❝❛❧✮ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ r : X × X → X × X ✐s ❛ ♠❛♣ s✉❝❤ t❤❛t t❤❡ ✇❡❧❧✲❦♥♦✇♥ ❜r❛✐❞ ❡q✉❛t✐♦♥ r ✶ r ✷ r ✶ = r ✷ r ✶ r ✷ ✐s s❛t✐s✜❡❞✱ ✇❤❡r❡ r ✶ = r × id X ❛♥❞ r ✷ = id X × r ✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ♦❜t❛✐♥ ❛♥❞ ❝♦♥str✉❝t ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥❄ ❉❡t❡r♠✐♥✐♥❣ ❛❧❧ s❡t✲t❤❡♦r❡t✐❝ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✐s ❛ ✈❡r② ❞✐✣❝✉❧t t❛s❦✳ ❊✈❡♥ ✐❢ ✇❡ ❝❛♥ ✜♥❞ s❡✈❡r❛❧ ✇♦r❦s ❛❜♦✉t t❤✐s t♦♣✐❝✱ ✐t ✐s st✐❧❧ ❛♥ ♦♣❡♥ ♣r♦❜❧❡♠✳ ■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✷ ✴ ✶✷

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