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❚❤❡ ♠❛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❝❤ ✶✹✱ ✷✵✶✽

❙♦❧✉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✐♦♥ ✶ ✴ ✶✷

❙♦❧✉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 × idX ❛♥❞ r✷ = idX ×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✐♦♥ ✷ ✴ ✶✷

❙♦❧✉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 × idX ❛♥❞ r✷ = idX ×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✐♦♥ ✷ ✴ ✶✷

❙♦❧✉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 × idX ❛♥❞ r✷ = idX ×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✐♦♥ ✷ ✴ ✶✷

❙♦❧✉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 × idX ❛♥❞ r✷ = idX ×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✐♦♥ ✷ ✴ ✶✷

❙♦❧✉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 × idX ❛♥❞ r✷ = idX ×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✐♦♥ ✷ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥

■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ X ✐s ❛ s❡t✱ r : X × X → X × X ✐s ❛ s♦❧✉t✐♦♥ ❛♥❞ a, b ∈ X✱ t❤❡♥ ✇❡ ❞❡♥♦t❡ r (a, b) = (λa (b), ρb (a)) , ✇❤❡r❡ λa, ρb ❛r❡ ♠❛♣s ❢r♦♠ X ✐♥t♦ ✐ts❡❧❢✳ ❲❡ s❛② t❤❛t r ✐s

◮ ❧❡❢t ✭r❡s♣✳ r✐❣❤t✮ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ λa ✭r❡s♣✳ ρa✮ ✐s ❜✐❥❡❝t✐✈❡✱ ❢♦r ❡✈❡r②

a ∈ X❀

◮ ✐❞❡♠♣♦t❡♥t r ✷ (a, b) = r (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X ◮ ✐♥✈♦❧✉t✐✈❡ ✐❢ r ✷ (a, b) = (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✸ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥

■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ X ✐s ❛ s❡t✱ r : X × X → X × X ✐s ❛ s♦❧✉t✐♦♥ ❛♥❞ a, b ∈ X✱ t❤❡♥ ✇❡ ❞❡♥♦t❡ r (a, b) = (λa (b), ρb (a)) , ✇❤❡r❡ λa, ρb ❛r❡ ♠❛♣s ❢r♦♠ X ✐♥t♦ ✐ts❡❧❢✳ ❲❡ s❛② t❤❛t r ✐s

◮ ❧❡❢t ✭r❡s♣✳ r✐❣❤t✮ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ λa ✭r❡s♣✳ ρa✮ ✐s ❜✐❥❡❝t✐✈❡✱ ❢♦r ❡✈❡r②

a ∈ X❀

◮ ✐❞❡♠♣♦t❡♥t r ✷ (a, b) = r (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X ◮ ✐♥✈♦❧✉t✐✈❡ ✐❢ r ✷ (a, b) = (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✸ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥

■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ X ✐s ❛ s❡t✱ r : X × X → X × X ✐s ❛ s♦❧✉t✐♦♥ ❛♥❞ a, b ∈ X✱ t❤❡♥ ✇❡ ❞❡♥♦t❡ r (a, b) = (λa (b), ρb (a)) , ✇❤❡r❡ λa, ρb ❛r❡ ♠❛♣s ❢r♦♠ X ✐♥t♦ ✐ts❡❧❢✳ ❲❡ s❛② t❤❛t r ✐s

◮ ❧❡❢t ✭r❡s♣✳ r✐❣❤t✮ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ λa ✭r❡s♣✳ ρa✮ ✐s ❜✐❥❡❝t✐✈❡✱ ❢♦r ❡✈❡r②

a ∈ X❀

◮ ✐❞❡♠♣♦t❡♥t r ✷ (a, b) = r (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X ◮ ✐♥✈♦❧✉t✐✈❡ ✐❢ r ✷ (a, b) = (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✸ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥

■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ X ✐s ❛ s❡t✱ r : X × X → X × X ✐s ❛ s♦❧✉t✐♦♥ ❛♥❞ a, b ∈ X✱ t❤❡♥ ✇❡ ❞❡♥♦t❡ r (a, b) = (λa (b), ρb (a)) , ✇❤❡r❡ λa, ρb ❛r❡ ♠❛♣s ❢r♦♠ X ✐♥t♦ ✐ts❡❧❢✳ ❲❡ s❛② t❤❛t r ✐s

◮ ❧❡❢t ✭r❡s♣✳ r✐❣❤t✮ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ λa ✭r❡s♣✳ ρa✮ ✐s ❜✐❥❡❝t✐✈❡✱ ❢♦r ❡✈❡r②

a ∈ X❀

◮ ✐❞❡♠♣♦t❡♥t r ✷ (a, b) = r (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X ◮ ✐♥✈♦❧✉t✐✈❡ ✐❢ r ✷ (a, b) = (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✸ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥

■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ X ✐s ❛ s❡t✱ r : X × X → X × X ✐s ❛ s♦❧✉t✐♦♥ ❛♥❞ a, b ∈ X✱ t❤❡♥ ✇❡ ❞❡♥♦t❡ r (a, b) = (λa (b), ρb (a)) , ✇❤❡r❡ λa, ρb ❛r❡ ♠❛♣s ❢r♦♠ X ✐♥t♦ ✐ts❡❧❢✳ ❲❡ s❛② t❤❛t r ✐s

◮ ❧❡❢t ✭r❡s♣✳ r✐❣❤t✮ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ λa ✭r❡s♣✳ ρa✮ ✐s ❜✐❥❡❝t✐✈❡✱ ❢♦r ❡✈❡r②

a ∈ X❀

◮ ✐❞❡♠♣♦t❡♥t r ✷ (a, b) = r (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X ◮ ✐♥✈♦❧✉t✐✈❡ ✐❢ r ✷ (a, b) = (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✸ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥

■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ X ✐s ❛ s❡t✱ r : X × X → X × X ✐s ❛ s♦❧✉t✐♦♥ ❛♥❞ a, b ∈ X✱ t❤❡♥ ✇❡ ❞❡♥♦t❡ r (a, b) = (λa (b), ρb (a)) , ✇❤❡r❡ λa, ρb ❛r❡ ♠❛♣s ❢r♦♠ X ✐♥t♦ ✐ts❡❧❢✳ ❲❡ s❛② t❤❛t r ✐s

◮ ❧❡❢t ✭r❡s♣✳ r✐❣❤t✮ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐❢ λa ✭r❡s♣✳ ρa✮ ✐s ❜✐❥❡❝t✐✈❡✱ ❢♦r ❡✈❡r②

a ∈ X❀

◮ ✐❞❡♠♣♦t❡♥t r ✷ (a, b) = r (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X ◮ ✐♥✈♦❧✉t✐✈❡ ✐❢ r ✷ (a, b) = (a, b)✱ ❢♦r ❛❧❧ a, b ∈ X✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✸ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r ❛♥❞ ❙♦❧♦✈✐❡✈✱ ●❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ❱❛♥ ❞❡♥ ❇❡r❣❤ ❧❛✐❞ t❤❡ ❣r♦✉♥❞✇♦r❦ ❢♦r st✉❞②✐♥❣ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s✱ ♠❛✐♥❧② ✐♥ ❣r♦✉♣ t❤❡♦r② t❡r♠s✳ ▼❛♥② r❡s✉❧ts ❛r❡ ♦❜t❛✐♥❡❞ ❢♦r t❤✐s ❝❧❛ss ❜② s❡✈❡r❛❧ ❛✉t❤♦rs✳ ■♥ ✷✵✵✵✱ ▲✉✱ ❨❛♥ ❛♥❞ ❩❤✉ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❙♦❧♦✈✐❡✈ st❛rt❡❞ t♦ st✉❞② ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s ♥♦t ♥❡❝❡ss❛r✐❧② ✐♥✈♦❧✉t✐✈❡✳ ■♥ ✷✵✶✼✱ ●✉❛r♥✐❡r✐ ❛♥❞ ❱❡♥❞r❛♠✐♥ ♦❜t❛✐♥❡❞ ♥❡✇ r❡s✉❧ts ✐♥ t❤✐s ❝♦♥t❡①t✳ ❋✐♥❞✐♥❣ ❛♥❞ st✉❞②✐♥❣ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡s str✐❝t❧② ❧✐♥❦❡❞ ✇✐t❤ s♦❧✉t✐♦♥s ✐s ❛ ✇✐❞❡❧② ✉s❡❞ str❛t❡❣② t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ♦❢ ♦❜t❛✐♥✐♥❣ ♥❡✇ s♦❧✉t✐♦♥s✳ ❆❧t❤♦✉❣❤ ✐♥t❡r❡st✐♥❣ ❛♥❞ r❡♠❛r❦❛❜❧❡ r❡s✉❧ts ♦♥ ❝❧❛ss✐❢②✐♥❣ s♦❧✉t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♣r❡s❡♥t❡❞✱ t❤❡r❡ ❛r❡ st✐❧❧ ♠❛♥② ♦♣❡♥ r❡❧❛t❡❞ ♣r♦❜❧❡♠s✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ❝♦♥str✉❝t ♥❡✇ ❢❛♠✐❧✐❡s ♦❢ s♦❧✉t✐♦♥s st❛rt✐♥❣ ❢r♦♠ ♦t❤❡rs❄

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✹ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r ❛♥❞ ❙♦❧♦✈✐❡✈✱ ●❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ❱❛♥ ❞❡♥ ❇❡r❣❤ ❧❛✐❞ t❤❡ ❣r♦✉♥❞✇♦r❦ ❢♦r st✉❞②✐♥❣ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s✱ ♠❛✐♥❧② ✐♥ ❣r♦✉♣ t❤❡♦r② t❡r♠s✳ ▼❛♥② r❡s✉❧ts ❛r❡ ♦❜t❛✐♥❡❞ ❢♦r t❤✐s ❝❧❛ss ❜② s❡✈❡r❛❧ ❛✉t❤♦rs✳ ■♥ ✷✵✵✵✱ ▲✉✱ ❨❛♥ ❛♥❞ ❩❤✉ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❙♦❧♦✈✐❡✈ st❛rt❡❞ t♦ st✉❞② ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s ♥♦t ♥❡❝❡ss❛r✐❧② ✐♥✈♦❧✉t✐✈❡✳ ■♥ ✷✵✶✼✱ ●✉❛r♥✐❡r✐ ❛♥❞ ❱❡♥❞r❛♠✐♥ ♦❜t❛✐♥❡❞ ♥❡✇ r❡s✉❧ts ✐♥ t❤✐s ❝♦♥t❡①t✳ ❋✐♥❞✐♥❣ ❛♥❞ st✉❞②✐♥❣ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡s str✐❝t❧② ❧✐♥❦❡❞ ✇✐t❤ s♦❧✉t✐♦♥s ✐s ❛ ✇✐❞❡❧② ✉s❡❞ str❛t❡❣② t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ♦❢ ♦❜t❛✐♥✐♥❣ ♥❡✇ s♦❧✉t✐♦♥s✳ ❆❧t❤♦✉❣❤ ✐♥t❡r❡st✐♥❣ ❛♥❞ r❡♠❛r❦❛❜❧❡ r❡s✉❧ts ♦♥ ❝❧❛ss✐❢②✐♥❣ s♦❧✉t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♣r❡s❡♥t❡❞✱ t❤❡r❡ ❛r❡ st✐❧❧ ♠❛♥② ♦♣❡♥ r❡❧❛t❡❞ ♣r♦❜❧❡♠s✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ❝♦♥str✉❝t ♥❡✇ ❢❛♠✐❧✐❡s ♦❢ s♦❧✉t✐♦♥s st❛rt✐♥❣ ❢r♦♠ ♦t❤❡rs❄

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✹ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r ❛♥❞ ❙♦❧♦✈✐❡✈✱ ●❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ❱❛♥ ❞❡♥ ❇❡r❣❤ ❧❛✐❞ t❤❡ ❣r♦✉♥❞✇♦r❦ ❢♦r st✉❞②✐♥❣ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s✱ ♠❛✐♥❧② ✐♥ ❣r♦✉♣ t❤❡♦r② t❡r♠s✳ ▼❛♥② r❡s✉❧ts ❛r❡ ♦❜t❛✐♥❡❞ ❢♦r t❤✐s ❝❧❛ss ❜② s❡✈❡r❛❧ ❛✉t❤♦rs✳ ■♥ ✷✵✵✵✱ ▲✉✱ ❨❛♥ ❛♥❞ ❩❤✉ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❙♦❧♦✈✐❡✈ st❛rt❡❞ t♦ st✉❞② ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s ♥♦t ♥❡❝❡ss❛r✐❧② ✐♥✈♦❧✉t✐✈❡✳ ■♥ ✷✵✶✼✱ ●✉❛r♥✐❡r✐ ❛♥❞ ❱❡♥❞r❛♠✐♥ ♦❜t❛✐♥❡❞ ♥❡✇ r❡s✉❧ts ✐♥ t❤✐s ❝♦♥t❡①t✳ ❋✐♥❞✐♥❣ ❛♥❞ st✉❞②✐♥❣ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡s str✐❝t❧② ❧✐♥❦❡❞ ✇✐t❤ s♦❧✉t✐♦♥s ✐s ❛ ✇✐❞❡❧② ✉s❡❞ str❛t❡❣② t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ♦❢ ♦❜t❛✐♥✐♥❣ ♥❡✇ s♦❧✉t✐♦♥s✳ ❆❧t❤♦✉❣❤ ✐♥t❡r❡st✐♥❣ ❛♥❞ r❡♠❛r❦❛❜❧❡ r❡s✉❧ts ♦♥ ❝❧❛ss✐❢②✐♥❣ s♦❧✉t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♣r❡s❡♥t❡❞✱ t❤❡r❡ ❛r❡ st✐❧❧ ♠❛♥② ♦♣❡♥ r❡❧❛t❡❞ ♣r♦❜❧❡♠s✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ❝♦♥str✉❝t ♥❡✇ ❢❛♠✐❧✐❡s ♦❢ s♦❧✉t✐♦♥s st❛rt✐♥❣ ❢r♦♠ ♦t❤❡rs❄

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✹ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r ❛♥❞ ❙♦❧♦✈✐❡✈✱ ●❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ❱❛♥ ❞❡♥ ❇❡r❣❤ ❧❛✐❞ t❤❡ ❣r♦✉♥❞✇♦r❦ ❢♦r st✉❞②✐♥❣ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s✱ ♠❛✐♥❧② ✐♥ ❣r♦✉♣ t❤❡♦r② t❡r♠s✳ ▼❛♥② r❡s✉❧ts ❛r❡ ♦❜t❛✐♥❡❞ ❢♦r t❤✐s ❝❧❛ss ❜② s❡✈❡r❛❧ ❛✉t❤♦rs✳ ■♥ ✷✵✵✵✱ ▲✉✱ ❨❛♥ ❛♥❞ ❩❤✉ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❙♦❧♦✈✐❡✈ st❛rt❡❞ t♦ st✉❞② ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s ♥♦t ♥❡❝❡ss❛r✐❧② ✐♥✈♦❧✉t✐✈❡✳ ■♥ ✷✵✶✼✱ ●✉❛r♥✐❡r✐ ❛♥❞ ❱❡♥❞r❛♠✐♥ ♦❜t❛✐♥❡❞ ♥❡✇ r❡s✉❧ts ✐♥ t❤✐s ❝♦♥t❡①t✳ ❋✐♥❞✐♥❣ ❛♥❞ st✉❞②✐♥❣ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡s str✐❝t❧② ❧✐♥❦❡❞ ✇✐t❤ s♦❧✉t✐♦♥s ✐s ❛ ✇✐❞❡❧② ✉s❡❞ str❛t❡❣② t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ♦❢ ♦❜t❛✐♥✐♥❣ ♥❡✇ s♦❧✉t✐♦♥s✳ ❆❧t❤♦✉❣❤ ✐♥t❡r❡st✐♥❣ ❛♥❞ r❡♠❛r❦❛❜❧❡ r❡s✉❧ts ♦♥ ❝❧❛ss✐❢②✐♥❣ s♦❧✉t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♣r❡s❡♥t❡❞✱ t❤❡r❡ ❛r❡ st✐❧❧ ♠❛♥② ♦♣❡♥ r❡❧❛t❡❞ ♣r♦❜❧❡♠s✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ❝♦♥str✉❝t ♥❡✇ ❢❛♠✐❧✐❡s ♦❢ s♦❧✉t✐♦♥s st❛rt✐♥❣ ❢r♦♠ ♦t❤❡rs❄

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✹ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r ❛♥❞ ❙♦❧♦✈✐❡✈✱ ●❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ❱❛♥ ❞❡♥ ❇❡r❣❤ ❧❛✐❞ t❤❡ ❣r♦✉♥❞✇♦r❦ ❢♦r st✉❞②✐♥❣ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s✱ ♠❛✐♥❧② ✐♥ ❣r♦✉♣ t❤❡♦r② t❡r♠s✳ ▼❛♥② r❡s✉❧ts ❛r❡ ♦❜t❛✐♥❡❞ ❢♦r t❤✐s ❝❧❛ss ❜② s❡✈❡r❛❧ ❛✉t❤♦rs✳ ■♥ ✷✵✵✵✱ ▲✉✱ ❨❛♥ ❛♥❞ ❩❤✉ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❙♦❧♦✈✐❡✈ st❛rt❡❞ t♦ st✉❞② ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s ♥♦t ♥❡❝❡ss❛r✐❧② ✐♥✈♦❧✉t✐✈❡✳ ■♥ ✷✵✶✼✱ ●✉❛r♥✐❡r✐ ❛♥❞ ❱❡♥❞r❛♠✐♥ ♦❜t❛✐♥❡❞ ♥❡✇ r❡s✉❧ts ✐♥ t❤✐s ❝♦♥t❡①t✳ ❋✐♥❞✐♥❣ ❛♥❞ st✉❞②✐♥❣ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡s str✐❝t❧② ❧✐♥❦❡❞ ✇✐t❤ s♦❧✉t✐♦♥s ✐s ❛ ✇✐❞❡❧② ✉s❡❞ str❛t❡❣② t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ♦❢ ♦❜t❛✐♥✐♥❣ ♥❡✇ s♦❧✉t✐♦♥s✳ ❆❧t❤♦✉❣❤ ✐♥t❡r❡st✐♥❣ ❛♥❞ r❡♠❛r❦❛❜❧❡ r❡s✉❧ts ♦♥ ❝❧❛ss✐❢②✐♥❣ s♦❧✉t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♣r❡s❡♥t❡❞✱ t❤❡r❡ ❛r❡ st✐❧❧ ♠❛♥② ♦♣❡♥ r❡❧❛t❡❞ ♣r♦❜❧❡♠s✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ❝♦♥str✉❝t ♥❡✇ ❢❛♠✐❧✐❡s ♦❢ s♦❧✉t✐♦♥s st❛rt✐♥❣ ❢r♦♠ ♦t❤❡rs❄

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✹ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r ❛♥❞ ❙♦❧♦✈✐❡✈✱ ●❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ❱❛♥ ❞❡♥ ❇❡r❣❤ ❧❛✐❞ t❤❡ ❣r♦✉♥❞✇♦r❦ ❢♦r st✉❞②✐♥❣ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s✱ ♠❛✐♥❧② ✐♥ ❣r♦✉♣ t❤❡♦r② t❡r♠s✳ ▼❛♥② r❡s✉❧ts ❛r❡ ♦❜t❛✐♥❡❞ ❢♦r t❤✐s ❝❧❛ss ❜② s❡✈❡r❛❧ ❛✉t❤♦rs✳ ■♥ ✷✵✵✵✱ ▲✉✱ ❨❛♥ ❛♥❞ ❩❤✉ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❙♦❧♦✈✐❡✈ st❛rt❡❞ t♦ st✉❞② ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s ♥♦t ♥❡❝❡ss❛r✐❧② ✐♥✈♦❧✉t✐✈❡✳ ■♥ ✷✵✶✼✱ ●✉❛r♥✐❡r✐ ❛♥❞ ❱❡♥❞r❛♠✐♥ ♦❜t❛✐♥❡❞ ♥❡✇ r❡s✉❧ts ✐♥ t❤✐s ❝♦♥t❡①t✳ ❋✐♥❞✐♥❣ ❛♥❞ st✉❞②✐♥❣ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡s str✐❝t❧② ❧✐♥❦❡❞ ✇✐t❤ s♦❧✉t✐♦♥s ✐s ❛ ✇✐❞❡❧② ✉s❡❞ str❛t❡❣② t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥ ♦❢ ♦❜t❛✐♥✐♥❣ ♥❡✇ s♦❧✉t✐♦♥s✳ ❆❧t❤♦✉❣❤ ✐♥t❡r❡st✐♥❣ ❛♥❞ r❡♠❛r❦❛❜❧❡ r❡s✉❧ts ♦♥ ❝❧❛ss✐❢②✐♥❣ s♦❧✉t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♣r❡s❡♥t❡❞✱ t❤❡r❡ ❛r❡ st✐❧❧ ♠❛♥② ♦♣❡♥ r❡❧❛t❡❞ ♣r♦❜❧❡♠s✳ Pr♦❜❧❡♠ ❍♦✇ t♦ ❝♦♥str✉❝t ♥❡✇ ❢❛♠✐❧✐❡s ♦❢ s♦❧✉t✐♦♥s st❛rt✐♥❣ ❢r♦♠ ♦t❤❡rs❄

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✹ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r✱ ❛♥❞ ❙♦❧♦✈✐❡✈ ✐♥tr♦❞✉❝❡❞ t❤❡ ❡①t❡♥s✐♦♥s ♦❢ t✇♦ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s (X, rX) ❛♥❞ (Y , rY )✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡② ♦❜t❛✐♥ ❛ ♥❡✇ s♦❧✉t✐♦♥ ♦♥ t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ s❡ts X ❛♥❞ Y ✳

• ❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ▼❛❥✐❞ ✭✷✵✵✽✮ ✐♠♣r♦✈❡❞ t❤✐s r❡s✉❧t ❜② r❡❣✉❧❛r ❡①t❡♥s✐♦♥ ❛♥❞

t❤❡② ❢♦✉♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ r❡❣✉❧❛r ❡①t❡♥s✐♦♥s ❛♥❞ r❡❣✉❧❛r ♣❛✐rs ♦❢ ❛❝t✐♦♥s✳ ●✐✈❡♥ t✇♦ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ (X, rX) ❛♥❞ (Y , rY ) t❤❡② ✐♥tr♦❞✉❝❡ ❛♥♦t❤❡r ✇❛② t♦ ♦❜t❛✐♥ ❛ ♥❡✇ s♦❧✉t✐♦♥ ♦✈❡r X ∪ Y ✱ t❤❡ str♦♥❣ t✇✐st❡❞ ✉♥✐♦♥s✳ ❘❡❝❡♥t❧②✱ ❇❛❝❤✐❧❧❡r ❛♥❞ ❈❡❞ó ♦❜t❛✐♥ ❛ ♥❡✇ ♠❡t❤♦❞ t♦ ❝♦♥str✉❝t ✐♥✈♦❧✉t✐✈❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s

• X n, r (n)

♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥✱ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✱ ❢r♦♠ ❛ ❣✐✈❡♥ s♦❧✉t✐♦♥ (X, r)✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✺ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r✱ ❛♥❞ ❙♦❧♦✈✐❡✈ ✐♥tr♦❞✉❝❡❞ t❤❡ ❡①t❡♥s✐♦♥s ♦❢ t✇♦ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s (X, rX) ❛♥❞ (Y , rY )✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡② ♦❜t❛✐♥ ❛ ♥❡✇ s♦❧✉t✐♦♥ ♦♥ t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ s❡ts X ❛♥❞ Y ✳

• ❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ▼❛❥✐❞ ✭✷✵✵✽✮ ✐♠♣r♦✈❡❞ t❤✐s r❡s✉❧t ❜② r❡❣✉❧❛r ❡①t❡♥s✐♦♥ ❛♥❞

t❤❡② ❢♦✉♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ r❡❣✉❧❛r ❡①t❡♥s✐♦♥s ❛♥❞ r❡❣✉❧❛r ♣❛✐rs ♦❢ ❛❝t✐♦♥s✳ ●✐✈❡♥ t✇♦ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ (X, rX) ❛♥❞ (Y , rY ) t❤❡② ✐♥tr♦❞✉❝❡ ❛♥♦t❤❡r ✇❛② t♦ ♦❜t❛✐♥ ❛ ♥❡✇ s♦❧✉t✐♦♥ ♦✈❡r X ∪ Y ✱ t❤❡ str♦♥❣ t✇✐st❡❞ ✉♥✐♦♥s✳ ❘❡❝❡♥t❧②✱ ❇❛❝❤✐❧❧❡r ❛♥❞ ❈❡❞ó ♦❜t❛✐♥ ❛ ♥❡✇ ♠❡t❤♦❞ t♦ ❝♦♥str✉❝t ✐♥✈♦❧✉t✐✈❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s

• X n, r (n)

♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥✱ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✱ ❢r♦♠ ❛ ❣✐✈❡♥ s♦❧✉t✐♦♥ (X, r)✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✺ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r✱ ❛♥❞ ❙♦❧♦✈✐❡✈ ✐♥tr♦❞✉❝❡❞ t❤❡ ❡①t❡♥s✐♦♥s ♦❢ t✇♦ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s (X, rX) ❛♥❞ (Y , rY )✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡② ♦❜t❛✐♥ ❛ ♥❡✇ s♦❧✉t✐♦♥ ♦♥ t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ s❡ts X ❛♥❞ Y ✳

• ❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ▼❛❥✐❞ ✭✷✵✵✽✮ ✐♠♣r♦✈❡❞ t❤✐s r❡s✉❧t ❜② r❡❣✉❧❛r ❡①t❡♥s✐♦♥ ❛♥❞

t❤❡② ❢♦✉♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ r❡❣✉❧❛r ❡①t❡♥s✐♦♥s ❛♥❞ r❡❣✉❧❛r ♣❛✐rs ♦❢ ❛❝t✐♦♥s✳ ●✐✈❡♥ t✇♦ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ (X, rX) ❛♥❞ (Y , rY ) t❤❡② ✐♥tr♦❞✉❝❡ ❛♥♦t❤❡r ✇❛② t♦ ♦❜t❛✐♥ ❛ ♥❡✇ s♦❧✉t✐♦♥ ♦✈❡r X ∪ Y ✱ t❤❡ str♦♥❣ t✇✐st❡❞ ✉♥✐♦♥s✳ ❘❡❝❡♥t❧②✱ ❇❛❝❤✐❧❧❡r ❛♥❞ ❈❡❞ó ♦❜t❛✐♥ ❛ ♥❡✇ ♠❡t❤♦❞ t♦ ❝♦♥str✉❝t ✐♥✈♦❧✉t✐✈❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s

• X n, r (n)

♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥✱ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✱ ❢r♦♠ ❛ ❣✐✈❡♥ s♦❧✉t✐♦♥ (X, r)✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✺ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r✱ ❛♥❞ ❙♦❧♦✈✐❡✈ ✐♥tr♦❞✉❝❡❞ t❤❡ ❡①t❡♥s✐♦♥s ♦❢ t✇♦ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s (X, rX) ❛♥❞ (Y , rY )✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡② ♦❜t❛✐♥ ❛ ♥❡✇ s♦❧✉t✐♦♥ ♦♥ t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ s❡ts X ❛♥❞ Y ✳

• ❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ▼❛❥✐❞ ✭✷✵✵✽✮ ✐♠♣r♦✈❡❞ t❤✐s r❡s✉❧t ❜② r❡❣✉❧❛r ❡①t❡♥s✐♦♥ ❛♥❞

t❤❡② ❢♦✉♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ r❡❣✉❧❛r ❡①t❡♥s✐♦♥s ❛♥❞ r❡❣✉❧❛r ♣❛✐rs ♦❢ ❛❝t✐♦♥s✳ ●✐✈❡♥ t✇♦ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ (X, rX) ❛♥❞ (Y , rY ) t❤❡② ✐♥tr♦❞✉❝❡ ❛♥♦t❤❡r ✇❛② t♦ ♦❜t❛✐♥ ❛ ♥❡✇ s♦❧✉t✐♦♥ ♦✈❡r X ∪ Y ✱ t❤❡ str♦♥❣ t✇✐st❡❞ ✉♥✐♦♥s✳ ❘❡❝❡♥t❧②✱ ❇❛❝❤✐❧❧❡r ❛♥❞ ❈❡❞ó ♦❜t❛✐♥ ❛ ♥❡✇ ♠❡t❤♦❞ t♦ ❝♦♥str✉❝t ✐♥✈♦❧✉t✐✈❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s

• X n, r (n)

♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥✱ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✱ ❢r♦♠ ❛ ❣✐✈❡♥ s♦❧✉t✐♦♥ (X, r)✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✺ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❇r✐❡✢②✱ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ✭■■✮

■♥ ✶✾✾✾ ❊t✐♥❣♦❢✱ ❙❝❤❡❞❧❡r✱ ❛♥❞ ❙♦❧♦✈✐❡✈ ✐♥tr♦❞✉❝❡❞ t❤❡ ❡①t❡♥s✐♦♥s ♦❢ t✇♦ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s (X, rX) ❛♥❞ (Y , rY )✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡② ♦❜t❛✐♥ ❛ ♥❡✇ s♦❧✉t✐♦♥ ♦♥ t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ s❡ts X ❛♥❞ Y ✳

• ❛t❡✈❛✲■✈❛♥♦✈❛ ❛♥❞ ▼❛❥✐❞ ✭✷✵✵✽✮ ✐♠♣r♦✈❡❞ t❤✐s r❡s✉❧t ❜② r❡❣✉❧❛r ❡①t❡♥s✐♦♥ ❛♥❞

t❤❡② ❢♦✉♥❞ ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ r❡❣✉❧❛r ❡①t❡♥s✐♦♥s ❛♥❞ r❡❣✉❧❛r ♣❛✐rs ♦❢ ❛❝t✐♦♥s✳ ●✐✈❡♥ t✇♦ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ (X, rX) ❛♥❞ (Y , rY ) t❤❡② ✐♥tr♦❞✉❝❡ ❛♥♦t❤❡r ✇❛② t♦ ♦❜t❛✐♥ ❛ ♥❡✇ s♦❧✉t✐♦♥ ♦✈❡r X ∪ Y ✱ t❤❡ str♦♥❣ t✇✐st❡❞ ✉♥✐♦♥s✳ ❘❡❝❡♥t❧②✱ ❇❛❝❤✐❧❧❡r ❛♥❞ ❈❡❞ó ♦❜t❛✐♥ ❛ ♥❡✇ ♠❡t❤♦❞ t♦ ❝♦♥str✉❝t ✐♥✈♦❧✉t✐✈❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥s

• X n, r (n)

♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥✱ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✱ ❢r♦♠ ❛ ❣✐✈❡♥ s♦❧✉t✐♦♥ (X, r)✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✺ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❙♦❧✉t✐♦♥✿ ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥

▲❡t X ❜❡ ❛ ♥♦♥✲❡♠♣t② s❡t ❛♥❞ r : X × X → X × X ❛ ♠❛♣✳ ■❢ λx ❛♥❞ ρx✱ ❢♦r ❡✈❡r② x ∈ X ❛r❡ ♠❛♣s s✉❝❤ t❤❛t r (x, y) = (λx (y) , ρy (x)) ❢♦r ❛❧❧ x, y ∈ X t❤❡♥ (X, r) ✐s ❛ s♦❧✉t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ❤♦❧❞✿ ✶✳ λxλy = λλx (y)λρy (x)✱ ❢♦r ❛❧❧ x, y ∈ X❀ ✷✳ ρλρy (x)(z)λx (y) = λρλy (z)(x)ρz (y)✱ ❢♦r ❛❧❧ x, y, z ∈ X❀ ✸✳ ρzρy = ρρz (y)ρλy (z)✱ ❢♦r ❛❧❧ y, z ∈ X✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✻ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❙♦❧✉t✐♦♥✿ ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥

▲❡t X ❜❡ ❛ ♥♦♥✲❡♠♣t② s❡t ❛♥❞ r : X × X → X × X ❛ ♠❛♣✳ ■❢ λx ❛♥❞ ρx✱ ❢♦r ❡✈❡r② x ∈ X ❛r❡ ♠❛♣s s✉❝❤ t❤❛t r (x, y) = (λx (y) , ρy (x)) ❢♦r ❛❧❧ x, y ∈ X t❤❡♥ (X, r) ✐s ❛ s♦❧✉t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ❤♦❧❞✿ ✶✳ λxλy = λλx (y)λρy (x)✱ ❢♦r ❛❧❧ x, y ∈ X❀ ✷✳ ρλρy (x)(z)λx (y) = λρλy (z)(x)ρz (y)✱ ❢♦r ❛❧❧ x, y, z ∈ X❀ ✸✳ ρzρy = ρρz (y)ρλy (z)✱ ❢♦r ❛❧❧ y, z ∈ X✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✻ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❉❡✜♥✐t✐♦♥✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠

▲❡t (S, rs) ❛♥❞ (T, rT) ❜❡ s♦❧✉t✐♦♥s ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s✱ ♣✉t α (u) := αu✱ ❢♦r ❡✈❡r② u ∈ T ❛♥❞ β (a) := βa✱ ❢♦r ❡✈❡r② a ∈ S✳ ■❢ S, rS, T, rT, α ❛♥❞ β s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s αuαv = αλu(v)αρv (u); βaβb = βλa(b)βρb(a); ρα−✶

u

(b)α−✶ βa(u) (a) = α−✶ βρb(a)β−✶

b

(u)ρb (a) ;

ρβ−✶

a

(v)β−✶ αu(a) (u) = β−✶ αρv (u)α−✶

v

(a)ρv (u) ;

λaαu = αβa(u)λα−✶

βa(u)(a);

λuβa = βαu(a)λβ−✶

αu(a)(u);

❢♦r ❛❧❧ u, v ∈ T ❛♥❞ a, b ∈ S✱ t❤❡♥ ✇❡ ❝❛❧❧ (S, rS, T, rT, α, β) ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠ ♦❢ s♦❧✉t✐♦♥s✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✼ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❉❡✜♥✐t✐♦♥✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠

▲❡t (S, rs) ❛♥❞ (T, rT) ❜❡ s♦❧✉t✐♦♥s ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s✱ ♣✉t α (u) := αu✱ ❢♦r ❡✈❡r② u ∈ T ❛♥❞ β (a) := βa✱ ❢♦r ❡✈❡r② a ∈ S✳ ■❢ S, rS, T, rT, α ❛♥❞ β s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s αuαv = αλu(v)αρv (u); βaβb = βλa(b)βρb(a); ρα−✶

u

(b)α−✶ βa(u) (a) = α−✶ βρb(a)β−✶

b

(u)ρb (a) ;

ρβ−✶

a

(v)β−✶ αu(a) (u) = β−✶ αρv (u)α−✶

v

(a)ρv (u) ;

λaαu = αβa(u)λα−✶

βa(u)(a);

λuβa = βαu(a)λβ−✶

αu(a)(u);

❢♦r ❛❧❧ u, v ∈ T ❛♥❞ a, b ∈ S✱ t❤❡♥ ✇❡ ❝❛❧❧ (S, rS, T, rT, α, β) ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠ ♦❢ s♦❧✉t✐♦♥s✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✼ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❉❡✜♥✐t✐♦♥✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠

▲❡t (S, rs) ❛♥❞ (T, rT) ❜❡ s♦❧✉t✐♦♥s ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s✱ ♣✉t α (u) := αu✱ ❢♦r ❡✈❡r② u ∈ T ❛♥❞ β (a) := βa✱ ❢♦r ❡✈❡r② a ∈ S✳ ■❢ S, rS, T, rT, α ❛♥❞ β s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s αuαv = αλu(v)αρv (u); βaβb = βλa(b)βρb(a); ρα−✶

u

(b)α−✶ βa(u) (a) = α−✶ βρb(a)β−✶

b

(u)ρb (a) ;

ρβ−✶

a

(v)β−✶ αu(a) (u) = β−✶ αρv (u)α−✶

v

(a)ρv (u) ;

λaαu = αβa(u)λα−✶

βa(u)(a);

λuβa = βαu(a)λβ−✶

αu(a)(u);

❢♦r ❛❧❧ u, v ∈ T ❛♥❞ a, b ∈ S✱ t❤❡♥ ✇❡ ❝❛❧❧ (S, rS, T, rT, α, β) ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠ ♦❢ s♦❧✉t✐♦♥s✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✼ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❉❡✜♥✐t✐♦♥✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠

▲❡t (S, rs) ❛♥❞ (T, rT) ❜❡ s♦❧✉t✐♦♥s ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s✱ ♣✉t α (u) := αu✱ ❢♦r ❡✈❡r② u ∈ T ❛♥❞ β (a) := βa✱ ❢♦r ❡✈❡r② a ∈ S✳ ■❢ S, rS, T, rT, α ❛♥❞ β s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s αuαv = αλu(v)αρv (u); βaβb = βλa(b)βρb(a); ρα−✶

u

(b)α−✶ βa(u) (a) = α−✶ βρb(a)β−✶

b

(u)ρb (a) ;

ρβ−✶

a

(v)β−✶ αu(a) (u) = β−✶ αρv (u)α−✶

v

(a)ρv (u) ;

λaαu = αβa(u)λα−✶

βa(u)(a);

λuβa = βαu(a)λβ−✶

αu(a)(u);

❢♦r ❛❧❧ u, v ∈ T ❛♥❞ a, b ∈ S✱ t❤❡♥ ✇❡ ❝❛❧❧ (S, rS, T, rT, α, β) ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠ ♦❢ s♦❧✉t✐♦♥s✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✼ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❉❡✜♥✐t✐♦♥✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠

▲❡t (S, rs) ❛♥❞ (T, rT) ❜❡ s♦❧✉t✐♦♥s ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s✱ ♣✉t α (u) := αu✱ ❢♦r ❡✈❡r② u ∈ T ❛♥❞ β (a) := βa✱ ❢♦r ❡✈❡r② a ∈ S✳ ■❢ S, rS, T, rT, α ❛♥❞ β s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s αuαv = αλu(v)αρv (u); βaβb = βλa(b)βρb(a); ρα−✶

u

(b)α−✶ βa(u) (a) = α−✶ βρb(a)β−✶

b

(u)ρb (a) ;

ρβ−✶

a

(v)β−✶ αu(a) (u) = β−✶ αρv (u)α−✶

v

(a)ρv (u) ;

λaαu = αβa(u)λα−✶

βa(u)(a);

λuβa = βαu(a)λβ−✶

αu(a)(u);

❢♦r ❛❧❧ u, v ∈ T ❛♥❞ a, b ∈ S✱ t❤❡♥ ✇❡ ❝❛❧❧ (S, rS, T, rT, α, β) ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠ ♦❢ s♦❧✉t✐♦♥s✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✼ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❉❡✜♥✐t✐♦♥✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠

▲❡t (S, rs) ❛♥❞ (T, rT) ❜❡ s♦❧✉t✐♦♥s ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s✱ ♣✉t α (u) := αu✱ ❢♦r ❡✈❡r② u ∈ T ❛♥❞ β (a) := βa✱ ❢♦r ❡✈❡r② a ∈ S✳ ■❢ S, rS, T, rT, α ❛♥❞ β s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s αuαv = αλu(v)αρv (u); βaβb = βλa(b)βρb(a); ρα−✶

u

(b)α−✶ βa(u) (a) = α−✶ βρb(a)β−✶

b

(u)ρb (a) ;

ρβ−✶

a

(v)β−✶ αu(a) (u) = β−✶ αρv (u)α−✶

v

(a)ρv (u) ;

λaαu = αβa(u)λα−✶

βa(u)(a);

λuβa = βαu(a)λβ−✶

αu(a)(u);

❢♦r ❛❧❧ u, v ∈ T ❛♥❞ a, b ∈ S✱ t❤❡♥ ✇❡ ❝❛❧❧ (S, rS, T, rT, α, β) ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠ ♦❢ s♦❧✉t✐♦♥s✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✼ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❚❤❡♦r❡♠✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ s♦❧✉t✐♦♥s

▲❡t (S, rS, T, rT, α, β) ❜❡ ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■❢ ✇❡ s❡t λ(a,u)(b, v) :=

• αuλα−✶

u

(a)(b), βaλβ−✶

a

(u)(v)

• ρ(b,v)(a, u) :=

 α−✶

β−✶

αuλ α−✶ u (a) (b)βaλβ−✶ a (u)(v)ρα β−✶ a (u)

(b)(a), β−✶ α−✶

βaλ β−✶ a (u) (v)αuλα−✶ u (a)(b)ρβ α−✶ u (a)

(v)(u)

  , ❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ t❤❡♥ t❤❡ ♠❛♣ r : S×T × S×T → S×T × S×T ❞❡✜♥❡❞ ❜② r ((a, u) , (b, v)) :=

• λ(a,u)(b, v), ρ(b,v)(a, u)
• ,

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ ✐s ❛ s♦❧✉t✐♦♥ t❤❛t ✇❡ ❝❛❧❧ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ♦❢ rS ❛♥❞ rT ✭✈✐❛ α ❛♥❞ β✮✱ ❞❡♥♦t❡❞ ❜② rS ⊲ ⊳ rT✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✽ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❚❤❡♦r❡♠✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ s♦❧✉t✐♦♥s

▲❡t (S, rS, T, rT, α, β) ❜❡ ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■❢ ✇❡ s❡t λ(a,u)(b, v) :=

• αuλα−✶

u

(a)(b), βaλβ−✶

a

(u)(v)

• ρ(b,v)(a, u) :=

 α−✶

β−✶

αuλ α−✶ u (a) (b)βaλβ−✶ a (u)(v)ρα β−✶ a (u)

(b)(a), β−✶ α−✶

βaλ β−✶ a (u) (v)αuλα−✶ u (a)(b)ρβ α−✶ u (a)

(v)(u)

  , ❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ t❤❡♥ t❤❡ ♠❛♣ r : S×T × S×T → S×T × S×T ❞❡✜♥❡❞ ❜② r ((a, u) , (b, v)) :=

• λ(a,u)(b, v), ρ(b,v)(a, u)
• ,

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ ✐s ❛ s♦❧✉t✐♦♥ t❤❛t ✇❡ ❝❛❧❧ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ♦❢ rS ❛♥❞ rT ✭✈✐❛ α ❛♥❞ β✮✱ ❞❡♥♦t❡❞ ❜② rS ⊲ ⊳ rT✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✽ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❚❤❡♦r❡♠✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ s♦❧✉t✐♦♥s

▲❡t (S, rS, T, rT, α, β) ❜❡ ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■❢ ✇❡ s❡t λ(a,u)(b, v) :=

• αuλα−✶

u

(a)(b), βaλβ−✶

a

(u)(v)

• ρ(b,v)(a, u) :=

 α−✶

β−✶

αuλ α−✶ u (a) (b)βaλβ−✶ a (u)(v)ρα β−✶ a (u)

(b)(a), β−✶ α−✶

βaλ β−✶ a (u) (v)αuλα−✶ u (a)(b)ρβ α−✶ u (a)

(v)(u)

  , ❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ t❤❡♥ t❤❡ ♠❛♣ r : S×T × S×T → S×T × S×T ❞❡✜♥❡❞ ❜② r ((a, u) , (b, v)) :=

• λ(a,u)(b, v), ρ(b,v)(a, u)
• ,

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ ✐s ❛ s♦❧✉t✐♦♥ t❤❛t ✇❡ ❝❛❧❧ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ♦❢ rS ❛♥❞ rT ✭✈✐❛ α ❛♥❞ β✮✱ ❞❡♥♦t❡❞ ❜② rS ⊲ ⊳ rT✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✽ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❚❤❡♦r❡♠✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ s♦❧✉t✐♦♥s

▲❡t (S, rS, T, rT, α, β) ❜❡ ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■❢ ✇❡ s❡t λ(a,u)(b, v) :=

• αuλα−✶

u

(a)(b), βaλβ−✶

a

(u)(v)

• ρ(b,v)(a, u) :=

 α−✶

β−✶

αuλ α−✶ u (a) (b)βaλβ−✶ a (u)(v)ρα β−✶ a (u)

(b)(a), β−✶ α−✶

βaλ β−✶ a (u) (v)αuλα−✶ u (a)(b)ρβ α−✶ u (a)

(v)(u)

  , ❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ t❤❡♥ t❤❡ ♠❛♣ r : S×T × S×T → S×T × S×T ❞❡✜♥❡❞ ❜② r ((a, u) , (b, v)) :=

• λ(a,u)(b, v), ρ(b,v)(a, u)
• ,

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ ✐s ❛ s♦❧✉t✐♦♥ t❤❛t ✇❡ ❝❛❧❧ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ♦❢ rS ❛♥❞ rT ✭✈✐❛ α ❛♥❞ β✮✱ ❞❡♥♦t❡❞ ❜② rS ⊲ ⊳ rT✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✽ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❚❤❡♦r❡♠✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ s♦❧✉t✐♦♥s

▲❡t (S, rS, T, rT, α, β) ❜❡ ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■❢ ✇❡ s❡t λ(a,u)(b, v) :=

• αuλα−✶

u

(a)(b), βaλβ−✶

a

(u)(v)

• ρ(b,v)(a, u) :=

 α−✶

β−✶

αuλ α−✶ u (a) (b)βaλβ−✶ a (u)(v)ρα β−✶ a (u)

(b)(a), β−✶ α−✶

βaλ β−✶ a (u) (v)αuλα−✶ u (a)(b)ρβ α−✶ u (a)

(v)(u)

  , ❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ t❤❡♥ t❤❡ ♠❛♣ r : S×T × S×T → S×T × S×T ❞❡✜♥❡❞ ❜② r ((a, u) , (b, v)) :=

• λ(a,u)(b, v), ρ(b,v)(a, u)
• ,

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ ✐s ❛ s♦❧✉t✐♦♥ t❤❛t ✇❡ ❝❛❧❧ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ♦❢ rS ❛♥❞ rT ✭✈✐❛ α ❛♥❞ β✮✱ ❞❡♥♦t❡❞ ❜② rS ⊲ ⊳ rT✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✽ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❚❤❡♦r❡♠✿ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ s♦❧✉t✐♦♥s

▲❡t (S, rS, T, rT, α, β) ❜❡ ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■❢ ✇❡ s❡t λ(a,u)(b, v) :=

• αuλα−✶

u

(a)(b), βaλβ−✶

a

(u)(v)

• ρ(b,v)(a, u) :=

 α−✶

β−✶

αuλ α−✶ u (a) (b)βaλβ−✶ a (u)(v)ρα β−✶ a (u)

(b)(a), β−✶ α−✶

βaλ β−✶ a (u) (v)αuλα−✶ u (a)(b)ρβ α−✶ u (a)

(v)(u)

  , ❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ t❤❡♥ t❤❡ ♠❛♣ r : S×T × S×T → S×T × S×T ❞❡✜♥❡❞ ❜② r ((a, u) , (b, v)) :=

• λ(a,u)(b, v), ρ(b,v)(a, u)
• ,

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✱ ✐s ❛ s♦❧✉t✐♦♥ t❤❛t ✇❡ ❝❛❧❧ t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ♦❢ rS ❛♥❞ rT ✭✈✐❛ α ❛♥❞ β✮✱ ❞❡♥♦t❡❞ ❜② rS ⊲ ⊳ rT✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✽ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

P❛rt✐❝✉❧❛r ❝❛s❡

❆ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ✐♥✈♦❧✉t✐✈❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥

▲❡t X ❜❡ ❛ ♥♦♥✲❡♠♣t② s❡t ❛♥❞ r : X × X → X × X ❛ ♠❛♣✳ ■♥❞✐❝❛t❡ t❤❡ ✐♠❛❣❡ r (x, y) := (λx (y) , ρy (x)) ❢♦r ❛❧❧ x, y ∈ X✱ ✇❤❡r❡ λx, ρx : X → X ❛r❡ ♠❛♣s✳ (X, r) ✐s ❛ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ❤♦❧❞✿ ✶✳ λx ∈ Sym (X)✱ ❢♦r ❡✈❡r② x ∈ X❀ ✷✳ ρy(x) = λ−✶

λx (y) (x)✱ ❢♦r ❛❧❧ x, y ∈ X❀

✸✳ λxλλ−✶

x

(y) = λyλλ−✶

y

(x)✱ ❢♦r ❛❧❧ x, y ∈ X✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✾ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

P❛rt✐❝✉❧❛r ❝❛s❡

❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ✭■■✮

▲❡t (S, rS)✱ (T, rT) ❜❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s t❤❛t s❛t✐s❢② αuαλ−✶

u

(v) = αvαλ−✶

v

(u)

βaβλ−✶

a

(b) = βbβλ−✶

b

(a)

λaαβ−✶

a

(u) = αuλα−✶

u

(a)

λuβα−✶

u

(a) = βuλβ−✶

a

(u)

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✳ ❚❤❡♥ (S, rS, T, rT, α, β) ✐s ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐♥ t❤✐s ❝❛s❡ t❤❡ ❝♦♥❞✐t✐♦♥s ρα−✶

u

(b)α−✶ βa(u) (a) = α−✶ βρb(a)β−✶

b

(u)ρb (a)

ρβ−✶

a

(v)β−✶ αu(a) (u) = β−✶ αρv (u)α−✶

v

(a)ρv (u)

❛r❡ s❛t✐s✜❡❞✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✶✵ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

P❛rt✐❝✉❧❛r ❝❛s❡

❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ✭■■✮

▲❡t (S, rS)✱ (T, rT) ❜❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s t❤❛t s❛t✐s❢② αuαλ−✶

u

(v) = αvαλ−✶

v

(u)

βaβλ−✶

a

(b) = βbβλ−✶

b

(a)

λaαβ−✶

a

(u) = αuλα−✶

u

(a)

λuβα−✶

u

(a) = βuλβ−✶

a

(u)

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✳ ❚❤❡♥ (S, rS, T, rT, α, β) ✐s ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐♥ t❤✐s ❝❛s❡ t❤❡ ❝♦♥❞✐t✐♦♥s ρα−✶

u

(b)α−✶ βa(u) (a) = α−✶ βρb(a)β−✶

b

(u)ρb (a)

ρβ−✶

a

(v)β−✶ αu(a) (u) = β−✶ αρv (u)α−✶

v

(a)ρv (u)

❛r❡ s❛t✐s✜❡❞✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✶✵ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

P❛rt✐❝✉❧❛r ❝❛s❡

❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ✭■■✮

▲❡t (S, rS)✱ (T, rT) ❜❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s t❤❛t s❛t✐s❢② αuαλ−✶

u

(v) = αvαλ−✶

v

(u)

βaβλ−✶

a

(b) = βbβλ−✶

b

(a)

λaαβ−✶

a

(u) = αuλα−✶

u

(a)

λuβα−✶

u

(a) = βuλβ−✶

a

(u)

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✳ ❚❤❡♥ (S, rS, T, rT, α, β) ✐s ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐♥ t❤✐s ❝❛s❡ t❤❡ ❝♦♥❞✐t✐♦♥s ρα−✶

u

(b)α−✶ βa(u) (a) = α−✶ βρb(a)β−✶

b

(u)ρb (a)

ρβ−✶

a

(v)β−✶ αu(a) (u) = β−✶ αρv (u)α−✶

v

(a)ρv (u)

❛r❡ s❛t✐s✜❡❞✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✶✵ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

P❛rt✐❝✉❧❛r ❝❛s❡

❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ✭■■✮

▲❡t (S, rS)✱ (T, rT) ❜❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s t❤❛t s❛t✐s❢② αuαλ−✶

u

(v) = αvαλ−✶

v

(u)

βaβλ−✶

a

(b) = βbβλ−✶

b

(a)

λaαβ−✶

a

(u) = αuλα−✶

u

(a)

λuβα−✶

u

(a) = βuλβ−✶

a

(u)

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✳ ❚❤❡♥ (S, rS, T, rT, α, β) ✐s ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐♥ t❤✐s ❝❛s❡ t❤❡ ❝♦♥❞✐t✐♦♥s ρα−✶

u

(b)α−✶ βa(u) (a) = α−✶ βρb(a)β−✶

b

(u)ρb (a)

ρβ−✶

a

(v)β−✶ αu(a) (u) = β−✶ αρv (u)α−✶

v

(a)ρv (u)

❛r❡ s❛t✐s✜❡❞✳

■✳ ❈♦❧❛③③♦ ✭❯♥✐❙❛❧❡♥t♦✮ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ✶✵ ✴ ✶✷

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

P❛rt✐❝✉❧❛r ❝❛s❡

❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ✭■■■✮

▲❡t (S, rS)✱ (T, rT) ❜❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s t❤❛t s❛t✐s❢② αuαλ−✶

u

(v) = αvαλ−✶

v

(u)

βaβλ−✶

a

(b) = βbβλ−✶

b

(a)

λaαβ−✶

a

(u) = αuλα−✶

u

(a)

λuβα−✶

u

(a) = βuλβ−✶

a

(u)

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✳ ❚❤❡♥ (S, rS, T, rT, α, β) ✐s ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠ ❛♥❞ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ♦❢ rS ❛♥❞ rT ✐s ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ❛♥❞ ✐♥✈♦❧✉t✐✈❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❛♠❡ ❞❡✜♥✐t✐♦♥ ♦❢ λ(a,u) : S × T → S × T✱ ✐✳❡✱ λ(a,u)(b, v) :=

• αuλα−✶

u

(a)(b), βaλβ−✶

a

(u)(v)

• ,

✇❡ ❤❛✈❡ t❤❛t t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ✐s t❤❡ ♠❛♣ r : S × T × S × T → S × T × S × T ❣✐✈❡♥ ❜② r ((a, u) , (b, v)) :=

• λ(a,u)(b, v), λ−✶

λ(a,u)(b,v)(a, u)

• ,

❢♦r ❛❧❧ a, b ∈ S✱ u, v ∈ T✳

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

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

P❛rt✐❝✉❧❛r ❝❛s❡

❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ✭■■■✮

▲❡t (S, rS)✱ (T, rT) ❜❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s t❤❛t s❛t✐s❢② αuαλ−✶

u

(v) = αvαλ−✶

v

(u)

βaβλ−✶

a

(b) = βbβλ−✶

b

(a)

λaαβ−✶

a

(u) = αuλα−✶

u

(a)

λuβα−✶

u

(a) = βuλβ−✶

a

(u)

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✳ ❚❤❡♥ (S, rS, T, rT, α, β) ✐s ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠ ❛♥❞ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ♦❢ rS ❛♥❞ rT ✐s ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ❛♥❞ ✐♥✈♦❧✉t✐✈❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❛♠❡ ❞❡✜♥✐t✐♦♥ ♦❢ λ(a,u) : S × T → S × T✱ ✐✳❡✱ λ(a,u)(b, v) :=

• αuλα−✶

u

(a)(b), βaλβ−✶

a

(u)(v)

• ,

✇❡ ❤❛✈❡ t❤❛t t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ✐s t❤❡ ♠❛♣ r : S × T × S × T → S × T × S × T ❣✐✈❡♥ ❜② r ((a, u) , (b, v)) :=

• λ(a,u)(b, v), λ−✶

λ(a,u)(b,v)(a, u)

• ,

❢♦r ❛❧❧ a, b ∈ S✱ u, v ∈ T✳

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

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

P❛rt✐❝✉❧❛r ❝❛s❡

❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ♦❢ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s ✭■■■✮

▲❡t (S, rS)✱ (T, rT) ❜❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ✐♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥ ❛♥❞ α : T → Sym (S)✱ β : S → Sym (T) ♠❛♣s t❤❛t s❛t✐s❢② αuαλ−✶

u

(v) = αvαλ−✶

v

(u)

βaβλ−✶

a

(b) = βbβλ−✶

b

(a)

λaαβ−✶

a

(u) = αuλα−✶

u

(a)

λuβα−✶

u

(a) = βuλβ−✶

a

(u)

❢♦r ❛❧❧ a, b ∈ S ❛♥❞ u, v ∈ T✳ ❚❤❡♥ (S, rS, T, rT, α, β) ✐s ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠ ❛♥❞ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ♦❢ rS ❛♥❞ rT ✐s ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ ❛♥❞ ✐♥✈♦❧✉t✐✈❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s❛♠❡ ❞❡✜♥✐t✐♦♥ ♦❢ λ(a,u) : S × T → S × T✱ ✐✳❡✱ λ(a,u)(b, v) :=

• αuλα−✶

u

(a)(b), βaλβ−✶

a

(u)(v)

• ,

✇❡ ❤❛✈❡ t❤❛t t❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s♦❧✉t✐♦♥ ✐s t❤❡ ♠❛♣ r : S × T × S × T → S × T × S × T ❣✐✈❡♥ ❜② r ((a, u) , (b, v)) :=

• λ(a,u)(b, v), λ−✶

λ(a,u)(b,v)(a, u)

• ,

❢♦r ❛❧❧ a, b ∈ S✱ u, v ∈ T✳

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

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❆♥ ❡①❛♠♣❧❡

▲❡t r : S × S → S × S ❜❡ ❛♥ ✐♥✈♦❧✉t✐✈❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥✳ ■❢ α, β : S → Sym (S) ❛r❡ ❞❡✜♥❡❞ ❜② αu := λu ❛♥❞ βa := λa✱ ❢♦r ❛❧❧ a, u ∈ S✱ t❤❡♥ (S, r, S, r, α, β) ✐s ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■♥ ❢❛❝t ✐❢ s❛t✐s✜❡s αuαλ−✶

u

(v) = αvαλ−✶

v

(u)

βaβλ−✶

a

(b) = βbβλ−✶

b

(a)

λaαβ−✶

a

(u) = αuλα−✶

u

(a)

λuβα−✶

u

(a) = βuλβ−✶

a

(u),

s✐♥❝❡ (S, r) ✐s ❛♥ ✐♥✈♦❧✉t✐✈❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥✳

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

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❆♥ ❡①❛♠♣❧❡

▲❡t r : S × S → S × S ❜❡ ❛♥ ✐♥✈♦❧✉t✐✈❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥✳ ■❢ α, β : S → Sym (S) ❛r❡ ❞❡✜♥❡❞ ❜② αu := λu ❛♥❞ βa := λa✱ ❢♦r ❛❧❧ a, u ∈ S✱ t❤❡♥ (S, r, S, r, α, β) ✐s ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■♥ ❢❛❝t ✐❢ s❛t✐s✜❡s αuαλ−✶

u

(v) = αvαλ−✶

v

(u)

βaβλ−✶

a

(b) = βbβλ−✶

b

(a)

λaαβ−✶

a

(u) = αuλα−✶

u

(a)

λuβα−✶

u

(a) = βuλβ−✶

a

(u),

s✐♥❝❡ (S, r) ✐s ❛♥ ✐♥✈♦❧✉t✐✈❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥✳

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

❙♦❧✉t✐♦♥s ♦❢ t❤❡ ❨❛♥❣✲❇❛①t❡r ❡q✉❛t✐♦♥ ❚❤❡ ♠❛t❝❤❡❞ ♣r♦❞✉❝t ■♥✈♦❧✉t✐✈❡ s♦❧✉t✐♦♥s

❆♥ ❡①❛♠♣❧❡

▲❡t r : S × S → S × S ❜❡ ❛♥ ✐♥✈♦❧✉t✐✈❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥✳ ■❢ α, β : S → Sym (S) ❛r❡ ❞❡✜♥❡❞ ❜② αu := λu ❛♥❞ βa := λa✱ ❢♦r ❛❧❧ a, u ∈ S✱ t❤❡♥ (S, r, S, r, α, β) ✐s ❛ ♠❛t❝❤❡❞ ♣r♦❞✉❝t s②st❡♠✳ ■♥ ❢❛❝t ✐❢ s❛t✐s✜❡s αuαλ−✶

u

(v) = αvαλ−✶

v

(u)

βaβλ−✶

a

(b) = βbβλ−✶

b

(a)

λaαβ−✶

a

(u) = αuλα−✶

u

(a)

λuβα−✶

u

(a) = βuλβ−✶

a

(u),

s✐♥❝❡ (S, r) ✐s ❛♥ ✐♥✈♦❧✉t✐✈❡ ❧❡❢t ♥♦♥✲❞❡❣❡♥❡r❛t❡ s♦❧✉t✐♦♥✳

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

❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦