t trsrt tss strtrs - - PowerPoint PPT Presentation

❖♥ t❤❡ tr❛♥s♣♦rt ♦❢ ✜♥✐t❡♥❡ss str✉❝t✉r❡s

▲✐♦♥❡❧ ❱❛✉①✱ ♠❛✐♥❧② ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❈❤r✐st✐♥❡ ❚❛ss♦♥

■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ▲✉♠✐♥②✱ ▼❛rs❡✐❧❧❡✱ ❋r❛♥❝❡

❚❆❈▲ ✷✵✶✶✱ ▼❛rs❡✐❧❧❡ ❏✉❧② ✷✻✲✸✵ ✷✵✶✶

❙❡ts ❛♥❞ r❡❧❛t✐♦♥s

❉❡✜♥✐t✐♦♥

❚❤❡ ❝❛t❡❣♦r② Rel ♦❢ s❡ts ❛♥❞ r❡❧❛t✐♦♥s ❤❛s s❡ts ❛s ♦❜❥❡❝ts ❛♥❞ r❡❧❛t✐♦♥s ❛s ♠♦r♣❤✐s♠s✿ f ∈ Rel(A, B) ⇐ ⇒ f ⊆ A × B✳ ❘❡❧❛t✐♦♥❛❧ ❝♦♠♣♦s✐t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ (α, γ) ∈ g ◦ f ⇐ ⇒ ∃β, (α, β) ∈ f ∧ (β, γ) ∈ g.

Rel ❛s ❛ ♠♦❞❡❧ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝

◮ ❝♦♠♣❛❝t ❝❧♦s❡❞✿ ⊗ = × ❛♥❞ f⊥ = tf❀ ◮ ❝❛rt❡s✐❛♥ ❛♥❞ ❝♦❝❛rt❡s✐❛♥✿ ✐s ❛ ❜✐♣r♦❞✉❝t❀ ◮ ❡①♣♦♥❡♥t✐❛❧ str✉❝t✉r❡✿ ❣✐✈❡♥ ❜② t❤❡ ❝♦♠♦♥❛❞ ! = Mf✳

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

❆♣♣❧② t❤❡ ❝♦✲❑❧✐❡s❧✐ ❝♦♥str✉❝t✐♦♥

Rel!(A, B) = Rel(!A, B) ✇✐t❤ ❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡♥ ❜②✿ g ◦! f = n

• i=1

αi, γ

• ; ∃([β1, . . . , βn] , γ) ∈ g ∧ ∀i, (αi, βi) ∈ f
• ✐s ❝❛rt❡s✐❛♥ ❝❧♦s❡❞ ✇✐t❤ ♣r♦❞✉❝t

✳ ▼♦r❡♦✈❡r ✐t ✐s ❝♣♦✲❡♥r✐❝❤❡❞ ✭❢♦r s❡t ✐♥❝❧✉s✐♦♥✮✳

❆ ❦❡② ✐♥t✉✐t✐♦♥

▼♦r♣❤✐s♠s ✐♥ ❛r❡ t❤❡ s✉♣♣♦rt ♦❢ ♣♦✇❡r s❡r✐❡s✳

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

❆♣♣❧② t❤❡ ❝♦✲❑❧✐❡s❧✐ ❝♦♥str✉❝t✐♦♥

Rel!(A, B) = Rel(!A, B) ✇✐t❤ ❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡♥ ❜②✿ g ◦! f = n

• i=1

αi, γ

• ; ∃([β1, . . . , βn] , γ) ∈ g ∧ ∀i, (αi, βi) ∈ f
• Rel! ✐s ❝❛rt❡s✐❛♥ ❝❧♦s❡❞ ✇✐t❤ ♣r♦❞✉❝t ⊎✳

▼♦r❡♦✈❡r ✐t ✐s ❝♣♦✲❡♥r✐❝❤❡❞ ✭❢♦r s❡t ✐♥❝❧✉s✐♦♥✮✳

❆ ❦❡② ✐♥t✉✐t✐♦♥

▼♦r♣❤✐s♠s ✐♥ ❛r❡ t❤❡ s✉♣♣♦rt ♦❢ ♣♦✇❡r s❡r✐❡s✳

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

❆♣♣❧② t❤❡ ❝♦✲❑❧✐❡s❧✐ ❝♦♥str✉❝t✐♦♥

Rel!(A, B) = Rel(!A, B) ✇✐t❤ ❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡♥ ❜②✿ g ◦! f = n

• i=1

αi, γ

• ; ∃([β1, . . . , βn] , γ) ∈ g ∧ ∀i, (αi, βi) ∈ f
• Rel! ✐s ❝❛rt❡s✐❛♥ ❝❧♦s❡❞ ✇✐t❤ ♣r♦❞✉❝t ⊎✳

▼♦r❡♦✈❡r ✐t ✐s ❝♣♦✲❡♥r✐❝❤❡❞ ✭❢♦r s❡t ✐♥❝❧✉s✐♦♥✮✳

❆ ❦❡② ✐♥t✉✐t✐♦♥

▼♦r♣❤✐s♠s ✐♥ ❛r❡ t❤❡ s✉♣♣♦rt ♦❢ ♣♦✇❡r s❡r✐❡s✳

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

❆♣♣❧② t❤❡ ❝♦✲❑❧✐❡s❧✐ ❝♦♥str✉❝t✐♦♥

Rel!(A, B) = Rel(!A, B) ✇✐t❤ ❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡♥ ❜②✿ g ◦! f = n

• i=1

αi, γ

• ; ∃([β1, . . . , βn] , γ) ∈ g ∧ ∀i, (αi, βi) ∈ f
• Rel! ✐s ❝❛rt❡s✐❛♥ ❝❧♦s❡❞ ✇✐t❤ ♣r♦❞✉❝t ⊎✳

▼♦r❡♦✈❡r ✐t ✐s ❝♣♦✲❡♥r✐❝❤❡❞ ✭❢♦r s❡t ✐♥❝❧✉s✐♦♥✮✳

❆ ❦❡② ✐♥t✉✐t✐♦♥

▼♦r♣❤✐s♠s ✐♥ ❛r❡ t❤❡ s✉♣♣♦rt ♦❢ ♣♦✇❡r s❡r✐❡s✳

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

❆♣♣❧② t❤❡ ❝♦✲❑❧✐❡s❧✐ ❝♦♥str✉❝t✐♦♥

Rel!(A, B) = Rel(!A, B) ✇✐t❤ ❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡♥ ❜②✿ g ◦! f = n

• i=1

αi, γ

• ; ∃([β1, . . . , βn] , γ) ∈ g ∧ ∀i, (αi, βi) ∈ f
• Rel! ✐s ❝❛rt❡s✐❛♥ ❝❧♦s❡❞ ✇✐t❤ ♣r♦❞✉❝t ⊎✳

▼♦r❡♦✈❡r ✐t ✐s ❝♣♦✲❡♥r✐❝❤❡❞ ✭❢♦r s❡t ✐♥❝❧✉s✐♦♥✮✳

❆ ❦❡② ✐♥t✉✐t✐♦♥

▼♦r♣❤✐s♠s ✐♥ ❛r❡ t❤❡ s✉♣♣♦rt ♦❢ ♣♦✇❡r s❡r✐❡s✳

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

❆♣♣❧② t❤❡ ❝♦✲❑❧✐❡s❧✐ ❝♦♥str✉❝t✐♦♥

Rel!(A, B) = Rel(!A, B) ✇✐t❤ ❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡♥ ❜②✿ g ◦! f = n

• i=1

αi, γ

• ; ∃([β1, . . . , βn] , γ) ∈ g ∧ ∀i, (αi, βi) ∈ f
• Rel! ✐s ❝❛rt❡s✐❛♥ ❝❧♦s❡❞ ✇✐t❤ ♣r♦❞✉❝t ⊎✳

▼♦r❡♦✈❡r ✐t ✐s ❝♣♦✲❡♥r✐❝❤❡❞ ✭❢♦r s❡t ✐♥❝❧✉s✐♦♥✮✳

❆ ❦❡② ✐♥t✉✐t✐♦♥

▼♦r♣❤✐s♠s ✐♥ ❛r❡ t❤❡ s✉♣♣♦rt ♦❢ ♣♦✇❡r s❡r✐❡s✳

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

❆♣♣❧② t❤❡ ❝♦✲❑❧✐❡s❧✐ ❝♦♥str✉❝t✐♦♥

Rel!(A, B) = Rel(!A, B) ✇✐t❤ ❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡♥ ❜②✿ g ◦! f = n

• i=1

αi, γ

• ; ∃([β1, . . . , βn] , γ) ∈ g ∧ ∀i, (αi, βi) ∈ f
• Rel! ✐s ❝❛rt❡s✐❛♥ ❝❧♦s❡❞ ✇✐t❤ ♣r♦❞✉❝t ⊎✳

▼♦r❡♦✈❡r ✐t ✐s ❝♣♦✲❡♥r✐❝❤❡❞ ✭❢♦r s❡t ✐♥❝❧✉s✐♦♥✮✳

❆ ❦❡② ✐♥t✉✐t✐♦♥

▼♦r♣❤✐s♠s ✐♥ ❛r❡ t❤❡ s✉♣♣♦rt ♦❢ ♣♦✇❡r s❡r✐❡s✳

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

❆♣♣❧② t❤❡ ❝♦✲❑❧✐❡s❧✐ ❝♦♥str✉❝t✐♦♥

Rel!(A, B) = Rel(!A, B) ✇✐t❤ ❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡♥ ❜②✿ g ◦! f = n

• i=1

αi, γ

• ; ∃([β1, . . . , βn] , γ) ∈ g ∧ ∀i, (αi, βi) ∈ f
• Rel! ✐s ❝❛rt❡s✐❛♥ ❝❧♦s❡❞ ✇✐t❤ ♣r♦❞✉❝t ⊎✳

▼♦r❡♦✈❡r ✐t ✐s ❝♣♦✲❡♥r✐❝❤❡❞ ✭❢♦r s❡t ✐♥❝❧✉s✐♦♥✮✳

❆ ❦❡② ✐♥t✉✐t✐♦♥

▼♦r♣❤✐s♠s ✐♥ Rel! ❛r❡ t❤❡ s✉♣♣♦rt ♦❢ ♣♦✇❡r s❡r✐❡s✳

◗✉❛♥t✐t❛t✐✈❡ s❡♠❛♥t✐❝s

■❞❡❛ ✭●✐r❛r❞✱ ♣r❡✲▲▲✮

■♥t❡r♣r❡t ❛ t❡r♠ s ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥✿ s =

α∈ssαα

s♦ t❤❛t ❛♣♣❧✐❝❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ s tβ =

• (α,β)∈s

s(α,β)tα ✇❤❡r❡ t[α1,...,αk] = tα1 · · · tαk✳ ❲❡ ♥❡❡❞ s♦♠❡ ♥♦t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✦ ❙✉❝❤ ✐♥t✉✐t✐♦♥s ✇❡r❡ ❛t t❤❡ ❝♦r❡ ♦❢ t❤❡ ✐♥✈❡♥t✐♦♥ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝✳

❋✐♥✐t❡♥❡ss s♣❛❝❡s ✭❊❤r❤❛r❞✱ ✷✵✵✵✬s✮

■♥ ❛ t②♣❡❞ s❡tt✐♥❣✱ t❤❡ s✉♠ ✐s ❛❧✇❛②s ✜♥✐t❡✳ ▲❡❞ t♦ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ✲❝❛❧❝✉❧✉s ✭❊❤r❤❛r❞✕❘❡❣♥✐❡r✱ ✷✵✵✹✮✿ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛s ❛ ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ✳

◗✉❛♥t✐t❛t✐✈❡ s❡♠❛♥t✐❝s

■❞❡❛ ✭●✐r❛r❞✱ ♣r❡✲▲▲✮

■♥t❡r♣r❡t ❛ t❡r♠ s ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥✿ s =

α∈ssαα

s♦ t❤❛t ❛♣♣❧✐❝❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ s tβ =

• (α,β)∈s

s(α,β)tα ✇❤❡r❡ t[α1,...,αk] = tα1 · · · tαk✳ ❲❡ ♥❡❡❞ s♦♠❡ ♥♦t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✦ ❙✉❝❤ ✐♥t✉✐t✐♦♥s ✇❡r❡ ❛t t❤❡ ❝♦r❡ ♦❢ t❤❡ ✐♥✈❡♥t✐♦♥ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝✳

❋✐♥✐t❡♥❡ss s♣❛❝❡s ✭❊❤r❤❛r❞✱ ✷✵✵✵✬s✮

■♥ ❛ t②♣❡❞ s❡tt✐♥❣✱ t❤❡ s✉♠ ✐s ❛❧✇❛②s ✜♥✐t❡✳ ▲❡❞ t♦ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ✲❝❛❧❝✉❧✉s ✭❊❤r❤❛r❞✕❘❡❣♥✐❡r✱ ✷✵✵✹✮✿ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛s ❛ ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ✳

◗✉❛♥t✐t❛t✐✈❡ s❡♠❛♥t✐❝s

■❞❡❛ ✭●✐r❛r❞✱ ♣r❡✲▲▲✮

■♥t❡r♣r❡t ❛ t❡r♠ s ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥✿ s =

α∈ssαα

s♦ t❤❛t ❛♣♣❧✐❝❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ s tβ =

• (α,β)∈s

s(α,β)tα ✇❤❡r❡ t[α1,...,αk] = tα1 · · · tαk✳ ❲❡ ♥❡❡❞ s♦♠❡ ♥♦t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✦ ❙✉❝❤ ✐♥t✉✐t✐♦♥s ✇❡r❡ ❛t t❤❡ ❝♦r❡ ♦❢ t❤❡ ✐♥✈❡♥t✐♦♥ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝✳

❋✐♥✐t❡♥❡ss s♣❛❝❡s ✭❊❤r❤❛r❞✱ ✷✵✵✵✬s✮

■♥ ❛ t②♣❡❞ s❡tt✐♥❣✱ t❤❡ s✉♠ ✐s ❛❧✇❛②s ✜♥✐t❡✳ ▲❡❞ t♦ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ✲❝❛❧❝✉❧✉s ✭❊❤r❤❛r❞✕❘❡❣♥✐❡r✱ ✷✵✵✹✮✿ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛s ❛ ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ✳

◗✉❛♥t✐t❛t✐✈❡ s❡♠❛♥t✐❝s

■❞❡❛ ✭●✐r❛r❞✱ ♣r❡✲▲▲✮

■♥t❡r♣r❡t ❛ t❡r♠ s ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥✿ s =

α∈ssαα

s♦ t❤❛t ❛♣♣❧✐❝❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ s tβ =

• (α,β)∈s

s(α,β)tα ✇❤❡r❡ t[α1,...,αk] = tα1 · · · tαk✳ ❲❡ ♥❡❡❞ s♦♠❡ ♥♦t✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✦ ❙✉❝❤ ✐♥t✉✐t✐♦♥s ✇❡r❡ ❛t t❤❡ ❝♦r❡ ♦❢ t❤❡ ✐♥✈❡♥t✐♦♥ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝✳

❋✐♥✐t❡♥❡ss s♣❛❝❡s ✭❊❤r❤❛r❞✱ ✷✵✵✵✬s✮

■♥ ❛ t②♣❡❞ s❡tt✐♥❣✱ t❤❡ s✉♠ ✐s ❛❧✇❛②s ✜♥✐t❡✳ ▲❡❞ t♦ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ λ✲❝❛❧❝✉❧✉s ✭❊❤r❤❛r❞✕❘❡❣♥✐❡r✱ ✷✵✵✹✮✿ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛s ❛ ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ A ⊗ !A ⊸ !A✳

❋✐♥✐t❡♥❡ss s♣❛❝❡s

❙❤♦rt ✈❡rs✐♦♥

❚❤❡ ❝❛t❡❣♦r② Fin ♦❢ ✜♥✐t❡♥❡ss s♣❛❝❡s ✐s t❤❡ t✐❣❤t ♦rt❤♦❣♦♥❛❧✐t② ❝❛t❡❣♦r② ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❍②❧❛♥❞✕❙❝❤❛❧❦✱ ✷✵✵✸✮ ♦❜t❛✐♥❡❞ ❢r♦♠ Rel ❜② s❡tt✐♥❣✿ a ⊥A a′ ⇐ ⇒ a ∩ a′ ∈ Pf (A)

▼♦r❡ ❡①♣❧✐❝✐t❧②

❆ ✜♥✐t❡♥❡ss s♣❛❝❡ ✐s ❛ ♣❛✐r s✳t✳ ✐s ❛ s❡t ❛♥❞ ✳ ❆ ✜♥✐t❛r② r❡❧❛t✐♦♥ ✐s ❛ r❡❧❛t✐♦♥ s✳t✳✿

✐♠♣❧✐❡s ❀ ✐♠♣❧✐❡s ✳

❋✐♥✐t❡♥❡ss s♣❛❝❡s

❙❤♦rt ✈❡rs✐♦♥

❚❤❡ ❝❛t❡❣♦r② Fin ♦❢ ✜♥✐t❡♥❡ss s♣❛❝❡s ✐s t❤❡ t✐❣❤t ♦rt❤♦❣♦♥❛❧✐t② ❝❛t❡❣♦r② ✭✐♥ t❤❡ s❡♥s❡ ♦❢ ❍②❧❛♥❞✕❙❝❤❛❧❦✱ ✷✵✵✸✮ ♦❜t❛✐♥❡❞ ❢r♦♠ Rel ❜② s❡tt✐♥❣✿ a ⊥A a′ ⇐ ⇒ a ∩ a′ ∈ Pf (A)

▼♦r❡ ❡①♣❧✐❝✐t❧②

◮ ❆ ✜♥✐t❡♥❡ss s♣❛❝❡ ✐s ❛ ♣❛✐r (|A| , F (A)) s✳t✳ |A| ✐s ❛ s❡t ❛♥❞

F (A) = F (A)⊥⊥ ⊆ P (|A|)✳

◮ ❆ ✜♥✐t❛r② r❡❧❛t✐♦♥ f ∈ Fin(A, B) ✐s ❛ r❡❧❛t✐♦♥

f ∈ Rel(|A| , |B|) s✳t✳✿

◮ a ∈ F (A) ✐♠♣❧✐❡s f · a ∈ F (B)❀ ◮ b′ ∈ F

• B⊥

✐♠♣❧✐❡s tf · b′ ∈ F

• A⊥

✳

❋✐♥✐t❡♥❡ss s♣❛❝❡s ❛s ❛ ♠♦❞❡❧ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝

❙❤♦rt ✈❡rs✐♦♥

❆❧❧ t❤❡ ❝♦♥str✉❝t✐♦♥s ❢♦r ♠✉❧t✐♣❧✐❝❛t✐✈❡✱ ❛❞❞✐t✐✈❡ ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ str✉❝t✉r❡ ✇♦r❦ ♦✉t ❛s ❞❡s❝r✐❜❡❞ ❜② ❍②❧❛♥❞ ❛♥❞ ❙❝❤❛❧❦✳ ▼♦r❡♦✈❡r✱ ❛❧❧ t❤✐s str✉❝t✉r❡ ✐s ♣r❡s❡r✈❡❞ ❜② t❤❡ ❢♦r❣❡t❢✉❧ ❢✉♥❝t♦r |−| : Fin → Rel✳

■♥ ♦t❤❡r ✇♦r❞s

❚❤❡ r❡❧❛t✐♦♥❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝ ✭♦r t②♣❡❞ ✲❝❛❧❝✉❧✉s✮ ✐s ❛❧✇❛②s ✜♥✐t❛r②✳

❇✉t✳ ✳ ✳

❖♥❡ ♠✉st ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦♥str✉❝t✐♦♥s ❞♦ ♣r♦✈✐❞❡ t❤❡ ♥❡❝❡ss❛r② str✉❝t✉r❡ ✏❜② ❤❛♥❞✑✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ❛ss♦❝✐❛t✐✈✐t② ♦❢ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t

❋✐♥✐t❡♥❡ss s♣❛❝❡s ❛s ❛ ♠♦❞❡❧ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝

❙❤♦rt ✈❡rs✐♦♥

❆❧❧ t❤❡ ❝♦♥str✉❝t✐♦♥s ❢♦r ♠✉❧t✐♣❧✐❝❛t✐✈❡✱ ❛❞❞✐t✐✈❡ ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ str✉❝t✉r❡ ✇♦r❦ ♦✉t ❛s ❞❡s❝r✐❜❡❞ ❜② ❍②❧❛♥❞ ❛♥❞ ❙❝❤❛❧❦✳ ▼♦r❡♦✈❡r✱ ❛❧❧ t❤✐s str✉❝t✉r❡ ✐s ♣r❡s❡r✈❡❞ ❜② t❤❡ ❢♦r❣❡t❢✉❧ ❢✉♥❝t♦r |−| : Fin → Rel✳

■♥ ♦t❤❡r ✇♦r❞s

❚❤❡ r❡❧❛t✐♦♥❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝ ✭♦r t②♣❡❞ λ✲❝❛❧❝✉❧✉s✮ ✐s ❛❧✇❛②s ✜♥✐t❛r②✳

❇✉t✳ ✳ ✳

❖♥❡ ♠✉st ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦♥str✉❝t✐♦♥s ❞♦ ♣r♦✈✐❞❡ t❤❡ ♥❡❝❡ss❛r② str✉❝t✉r❡ ✏❜② ❤❛♥❞✑✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ❛ss♦❝✐❛t✐✈✐t② ♦❢ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t

❋✐♥✐t❡♥❡ss s♣❛❝❡s ❛s ❛ ♠♦❞❡❧ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝

❙❤♦rt ✈❡rs✐♦♥

❆❧❧ t❤❡ ❝♦♥str✉❝t✐♦♥s ❢♦r ♠✉❧t✐♣❧✐❝❛t✐✈❡✱ ❛❞❞✐t✐✈❡ ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ str✉❝t✉r❡ ✇♦r❦ ♦✉t ❛s ❞❡s❝r✐❜❡❞ ❜② ❍②❧❛♥❞ ❛♥❞ ❙❝❤❛❧❦✳ ▼♦r❡♦✈❡r✱ ❛❧❧ t❤✐s str✉❝t✉r❡ ✐s ♣r❡s❡r✈❡❞ ❜② t❤❡ ❢♦r❣❡t❢✉❧ ❢✉♥❝t♦r |−| : Fin → Rel✳

■♥ ♦t❤❡r ✇♦r❞s

❚❤❡ r❡❧❛t✐♦♥❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝ ✭♦r t②♣❡❞ λ✲❝❛❧❝✉❧✉s✮ ✐s ❛❧✇❛②s ✜♥✐t❛r②✳

❇✉t✳ ✳ ✳

❖♥❡ ♠✉st ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦♥str✉❝t✐♦♥s ❞♦ ♣r♦✈✐❞❡ t❤❡ ♥❡❝❡ss❛r② str✉❝t✉r❡ ✏❜② ❤❛♥❞✑✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ❛ss♦❝✐❛t✐✈✐t② ♦❢ ⊗ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t {a × b; a ∈ F (A) , b ∈ F (B)}⊥⊥ = {c ⊆ |A ⊗ B| ; c1 ∈ F (A) , c2 ∈ F (B)} .

❚r❛♥s♣♦rt✐♥❣ ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

▲❡t f ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |B| s✉❝❤ t❤❛t f · α ∈ F (B) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ F = {a ⊆ A; f · a ∈ F (B) } ✐s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿

❚r❛♥s♣♦rt✐♥❣ ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

▲❡t f ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |B| s✉❝❤ t❤❛t f · α ∈ F (B) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ F = {a ⊆ A; f · a ∈ F (B) } ✐s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿

❘❡♠❛r❦

❚❤✐s ♠❡❛♥s f ♠❛♣s ✜♥✐t❡ s✉❜s❡ts t♦ ✜♥✐t❛r② s✉❜s❡ts✱ ✇❤✐❝❤ ✐s ♥❡❝❡ss❛r② ❢♦r F t♦ ❝♦♥t❛✐♥ ❛❧❧ ✜♥✐t❡ s✉❜s❡ts ♦❢ A✳

❚r❛♥s♣♦rt✐♥❣ ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

▲❡t f ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |B| s✉❝❤ t❤❛t f · α ∈ F (B) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ F = {a ⊆ A; f · a ∈ F (B) } ✐s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿ F = {f \ b; b ∈ F (B) }⊥⊥ .

❚r❛♥s♣♦rt✐♥❣ ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

▲❡t f ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |B| s✉❝❤ t❤❛t f · α ∈ F (B) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ F = {a ⊆ A; f · a ∈ F (B) } ✐s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿ F = {f \ b; b ∈ F (B) }⊥⊥ .

❉❡✜♥✐t✐♦♥

f \ b =

• {a ⊆ A; f · a ⊆ b}

❚r❛♥s♣♦rt✐♥❣ ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

▲❡t f ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |B| s✉❝❤ t❤❛t f · α ∈ F (B) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ F = {a ⊆ A; f · a ∈ F (B) } ✐s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿ F = {f \ b; b ∈ F (B) }⊥⊥ .

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t❤❡ s❡t Mf (|B|) ❛♥❞ t❤❡ s✉♣♣♦rt r❡❧❛t✐♦♥ σ✳ ❚❤❡♥ σ · b = supp

• b
• ✱ f \ b = b! = Mf (b) ❛♥❞

F (!B) =

• b!; b ∈ F (B)

⊥⊥ =

• b ⊆ |!B| ; supp
• b
• ∈ F (B)
• .

❚r❛♥s♣♦rt✐♥❣ ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

▲❡t f ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |B| s✉❝❤ t❤❛t f · α ∈ F (B) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ F = {a ⊆ A; f · a ∈ F (B) } ✐s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿ F = {f \ b; b ∈ F (B) }⊥⊥ .

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳

❚❛❦❡ a ∈ {f \ b; b ∈ F (B)}⊥⊥ ❛♥❞ b′ ∈ F

• B⊥

✱ ❛♥❞ ✜♥❞ ✭✉s✐♥❣ ❆❈✮ a′ ⊆f A s✳t✳ f · a ∩ b′ ⊆ f · a′✳

✭❱❡r② s✐♠✐❧❛r t♦ t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ !A ✐♥ ❊❤r❤❛r❞✬s ♣❛♣❡r✳✮

❚r❛♥s♣♦rt✐♥❣ ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

▲❡t fi ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |Bi| s✉❝❤ t❤❛t fi · α ∈ F (Bi) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ F = {a ⊆ A; fi · a ∈ F (Bi) , ∀i ∈ I} ✐s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿ F =

i∈I f \ bi; bi ∈ F (Bi) , ∀i ∈ I

⊥⊥ .

❚r❛♥s♣♦rt✐♥❣ ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

▲❡t fi ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |Bi| s✉❝❤ t❤❛t fi · α ∈ F (Bi) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ F = {a ⊆ A; fi · a ∈ F (Bi) , ∀i ∈ I} ✐s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿ F =

i∈I f \ bi; bi ∈ F (Bi) , ∀i ∈ I

⊥⊥ .

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t❤❡ s❡t |A| × |B| ❛♥❞ t❤❡ ♣r♦❥❡❝t✐♦♥ r❡❧❛t✐♦♥s✳ ❚❤❡♥ {a × b; a ∈ F (A) , b ∈ F (B)}⊥⊥ = {c ⊆ |A ⊗ B| ; c1 ∈ F (A) , c2 ∈ F (B)} .

❚r❛♥s♣♦rt ✐s ❢✉♥❝t♦r✐❛❧

✭✇❤❡♥ ✐t ❝♦♥t❛✐♥s ✜♥✐t❡ ❞❛t❛✮

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❢✉♥❝t♦rs ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

❆ss✉♠❡ T : Rel → Rel ✐s ❛ ❢✉♥❝t♦r ♦♥ r❡❧❛t✐♦♥s✱ ❛♥❞ φ : T ⇒ 1Rel ✐s ❛♥ ❛❧♠♦st✲❢✉♥❝t✐♦♥❛❧ ❧❛① ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✳ ❆ss✉♠❡ ♠♦r❡♦✈❡r t❤❛t t❤❡r❡ ❡①✐sts ❛ s❤❛♣❡ r❡❧❛t✐♦♥ ♦♥ ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥❡s ❛ ❢✉♥❝t♦r T : Fin → Fin ✇✐t❤ ✇❡❜ T✿

◮ ❢♦r ❛❧❧ A ∈ Fin✱ |T A| = T |A| ❛♥❞ F (T A) ✐s tr❛♥s♣♦rt❡❞

❢r♦♠ F (A) ❛❧♦♥❣ φ|A|❀

◮ ❢♦r ❛❧❧ f ∈ Fin(A, B)✱ T f = Tf✳

❚r❛♥s♣♦rt ✐s ❢✉♥❝t♦r✐❛❧

✭✇❤❡♥ ✐t ❝♦♥t❛✐♥s ✜♥✐t❡ ❞❛t❛✮

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❢✉♥❝t♦rs ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

❆ss✉♠❡ T : Rel → Rel ✐s ❛ ❢✉♥❝t♦r ♦♥ r❡❧❛t✐♦♥s✱ ❛♥❞ φ : T ⇒ 1Rel ✐s ❛♥ ❛❧♠♦st✲❢✉♥❝t✐♦♥❛❧ ❧❛① ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✳ ❆ss✉♠❡ ♠♦r❡♦✈❡r t❤❛t t❤❡r❡ ❡①✐sts ❛ s❤❛♣❡ r❡❧❛t✐♦♥ ♦♥ ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥❡s ❛ ❢✉♥❝t♦r T : Fin → Fin ✇✐t❤ ✇❡❜ T✿

◮ ❢♦r ❛❧❧ A ∈ Fin✱ |T A| = T |A| ❛♥❞ F (T A) ✐s tr❛♥s♣♦rt❡❞

❢r♦♠ F (A) ❛❧♦♥❣ φ|A|❀

◮ ❢♦r ❛❧❧ f ∈ Fin(A, B)✱ T f = Tf✳

❉❡✜♥✐t✐♦♥

φ : T ⇒ U ✐s ❧❛① ♥❛t✉r❛❧ ✐❢ φB ◦ Tf ⊆ Uf ◦ φA

❊①❛♠♣❧❡

❚❤❡ s✉♣♣♦rt r❡❧❛t✐♦♥ σ : Mf ⇒ 1Rel✳

❚r❛♥s♣♦rt ✐s ❢✉♥❝t♦r✐❛❧

✭✇❤❡♥ ✐t ❝♦♥t❛✐♥s ✜♥✐t❡ ❞❛t❛✮

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❢✉♥❝t♦rs ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

❆ss✉♠❡ T : Rel → Rel ✐s ❛ ❢✉♥❝t♦r ♦♥ r❡❧❛t✐♦♥s✱ ❛♥❞ φ : T ⇒ 1Rel ✐s ❛♥ ❛❧♠♦st✲❢✉♥❝t✐♦♥❛❧ ❧❛① ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✳ ❆ss✉♠❡ ♠♦r❡♦✈❡r t❤❛t t❤❡r❡ ❡①✐sts ❛ s❤❛♣❡ r❡❧❛t✐♦♥ ♦♥ ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥❡s ❛ ❢✉♥❝t♦r T : Fin → Fin ✇✐t❤ ✇❡❜ T✿

◮ ❢♦r ❛❧❧ A ∈ Fin✱ |T A| = T |A| ❛♥❞ F (T A) ✐s tr❛♥s♣♦rt❡❞

❢r♦♠ F (A) ❛❧♦♥❣ φ|A|❀

◮ ❢♦r ❛❧❧ f ∈ Fin(A, B)✱ T f = Tf✳

❉❡✜♥✐t✐♦♥

f : A → B ✐s ❛❧♠♦st✲❢✉♥❝t✐♦♥❛❧ ✐❢ α· ∈ Pf (B) ❢♦r ❛❧❧ α ∈ A✳ ■♥ ♦t❤❡r ✇♦r❞s✿ f ♣r❡s❡r✈❡s ✜♥✐t❡ s❡ts✳

❘❡♠❛r❦

❚❤✐s ❡♥s✉r❡s t❤❡ tr❛♥s♣♦rt t❤❡♦r❡♠ ❛❧✇❛②s ❛♣♣❧✐❡s✳

❚r❛♥s♣♦rt ✐s ❢✉♥❝t♦r✐❛❧

✭✇❤❡♥ ✐t ❝♦♥t❛✐♥s ✜♥✐t❡ ❞❛t❛✮

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❢✉♥❝t♦rs ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

❆ss✉♠❡ T : Rel → Rel ✐s ❛ ❢✉♥❝t♦r ♦♥ r❡❧❛t✐♦♥s✱ ❛♥❞ φ : T ⇒ 1Rel ✐s ❛♥ ❛❧♠♦st✲❢✉♥❝t✐♦♥❛❧ ❧❛① ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✳ ❆ss✉♠❡ ♠♦r❡♦✈❡r t❤❛t t❤❡r❡ ❡①✐sts ❛ s❤❛♣❡ r❡❧❛t✐♦♥ ♦♥ ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥❡s ❛ ❢✉♥❝t♦r T : Fin → Fin ✇✐t❤ ✇❡❜ T✿

◮ ❢♦r ❛❧❧ A ∈ Fin✱ |T A| = T |A| ❛♥❞ F (T A) ✐s tr❛♥s♣♦rt❡❞

❢r♦♠ F (A) ❛❧♦♥❣ φ|A|❀

◮ ❢♦r ❛❧❧ f ∈ Fin(A, B)✱ T f = Tf✳

❘❡♠❛r❦

Pr❡s❡r✈❛t✐♦♥ ♦❢ ✐❞❡♥t✐t✐❡s ❛♥❞ ❝♦♠♣♦s✐t✐♦♥ ✐s tr✐✈✐❛❧❧② ❞❡❞✉❝❡❞ ❢r♦♠ t❤❛t ♦❢ T✳

❚r❛♥s♣♦rt ✐s ❢✉♥❝t♦r✐❛❧

✭✇❤❡♥ ✐t ❝♦♥t❛✐♥s ✜♥✐t❡ ❞❛t❛✮

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❢✉♥❝t♦rs ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

❆ss✉♠❡ T : Rel → Rel ✐s ❛ ❢✉♥❝t♦r ♦♥ r❡❧❛t✐♦♥s✱ ❛♥❞ φ : T ⇒ 1Rel ✐s ❛♥ ❛❧♠♦st✲❢✉♥❝t✐♦♥❛❧ ❧❛① ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✳ ❆ss✉♠❡ ♠♦r❡♦✈❡r t❤❛t t❤❡r❡ ❡①✐sts ❛ s❤❛♣❡ r❡❧❛t✐♦♥ ♦♥ ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥❡s ❛ ❢✉♥❝t♦r T : Fin → Fin ✇✐t❤ ✇❡❜ T✿

◮ ❢♦r ❛❧❧ A ∈ Fin✱ |T A| = T |A| ❛♥❞ F (T A) ✐s tr❛♥s♣♦rt❡❞

❢r♦♠ F (A) ❛❧♦♥❣ φ|A|❀

◮ ❢♦r ❛❧❧ f ∈ Fin(A, B)✱ T f = Tf✳

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳

■t ♦♥❧② r❡♠❛✐♥s t♦ ♣r♦✈❡ Tf ∈ Fin(T A, T B)✱ ✐✳❡✳✿

◮ a ∈ F (T A) ✐♠♣❧✐❡s Tf · a ∈ F (T B)✿ ❜② ❧❛① ♥❛t✉r❛❧✐t②❀ ◮ b ∈ F (T B)⊥ ✐♠♣❧✐❡s t(Tf) · b ∈ F (T A)⊥✿

❚r❛♥s♣♦rt ✐s ❢✉♥❝t♦r✐❛❧

✭✇❤❡♥ ✐t ❝♦♥t❛✐♥s ✜♥✐t❡ ❞❛t❛✮

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❢✉♥❝t♦rs ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

❆ss✉♠❡ T : Rel → Rel ✐s ❛ ❢✉♥❝t♦r ♦♥ r❡❧❛t✐♦♥s✱ ❛♥❞ φ : T ⇒ 1Rel ✐s ❛♥ ❛❧♠♦st✲❢✉♥❝t✐♦♥❛❧ ❧❛① ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✳ ❆ss✉♠❡ ♠♦r❡♦✈❡r t❤❛t t❤❡r❡ ❡①✐sts ❛ s❤❛♣❡ r❡❧❛t✐♦♥ ♦♥ ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥❡s ❛ ❢✉♥❝t♦r T : Fin → Fin ✇✐t❤ ✇❡❜ T✿

◮ ❢♦r ❛❧❧ A ∈ Fin✱ |T A| = T |A| ❛♥❞ F (T A) ✐s tr❛♥s♣♦rt❡❞

❢r♦♠ F (A) ❛❧♦♥❣ φ|A|❀

◮ ❢♦r ❛❧❧ f ∈ Fin(A, B)✱ T f = Tf✳

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳

■t ♦♥❧② r❡♠❛✐♥s t♦ ♣r♦✈❡ Tf ∈ Fin(T A, T B)✱ ✐✳❡✳✿

◮ a ∈ F (T A) ✐♠♣❧✐❡s Tf · a ∈ F (T B)✿ ❜② ❧❛① ♥❛t✉r❛❧✐t②❀ ◮ b ∈ F (T B)⊥ ✐♠♣❧✐❡s t(Tf) · b ∈ F (T A)⊥✿ ❄❄❄

❚r❛♥s♣♦rt ✐s ❢✉♥❝t♦r✐❛❧

✭✇❤❡♥ ✐t ❝♦♥t❛✐♥s ✜♥✐t❡ ❞❛t❛✮

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❢✉♥❝t♦rs ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

❆ss✉♠❡ T : Rel → Rel ✐s ❛ ❢✉♥❝t♦r ♦♥ r❡❧❛t✐♦♥s✱ ❛♥❞ φ : T ⇒ 1Rel ✐s ❛♥ ❛❧♠♦st✲❢✉♥❝t✐♦♥❛❧ ❧❛① ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✳ ❆ss✉♠❡ ♠♦r❡♦✈❡r t❤❛t t❤❡r❡ ❡①✐sts ❛ s❤❛♣❡ r❡❧❛t✐♦♥ ♦♥ ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥❡s ❛ ❢✉♥❝t♦r T : Fin → Fin ✇✐t❤ ✇❡❜ T✿

◮ ❢♦r ❛❧❧ A ∈ Fin✱ |T A| = T |A| ❛♥❞ F (T A) ✐s tr❛♥s♣♦rt❡❞

❢r♦♠ F (A) ❛❧♦♥❣ φ|A|❀

◮ ❢♦r ❛❧❧ f ∈ Fin(A, B)✱ T f = Tf✳

❈♦✉♥t❡r✲❡①❛♠♣❧❡ ❚❤❡ ❢✉♥❝t♦r −∞ ♦❢ str❡❛♠s✱ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♦❜✈✐♦✉s s✉♣♣♦rt r❡❧❛t✐♦♥✱ ❞♦❡s ♥♦t ♣r❡s❡r✈❡ ✜♥✐t❛r② r❡❧❛t✐♦♥s✦ ❊✳❣✳ t❤❡ t♦t❛❧ ❡♥❞♦r❡❧❛t✐♦♥ ✐s ✜♥✐t❛r② ♦♥ 2✱ ❜✉t ♥♦t ♦♥ 2∞✳

❚r❛♥s♣♦rt ✐s ❢✉♥❝t♦r✐❛❧

✭✇❤❡♥ ✐t ❝♦♥t❛✐♥s ✜♥✐t❡ ❞❛t❛✮

❚❤❡♦r❡♠ ✭❚r❛♥s♣♦rt ❢✉♥❝t♦rs ❬❚❛ss♦♥✕❱✳ ✷✵✶✶❪✮

❆ss✉♠❡ T : Rel → Rel ✐s ❛ s②♠♠❡tr✐❝ ❢✉♥❝t♦r ♦♥ r❡❧❛t✐♦♥s✱ ❛♥❞ φ : T ⇒ 1Rel ✐s ❛♥ ❛❧♠♦st✲❢✉♥❝t✐♦♥❛❧ ❧❛① ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✳ ❆ss✉♠❡ ♠♦r❡♦✈❡r t❤❛t t❤❡r❡ ❡①✐sts ❛ s❤❛♣❡ r❡❧❛t✐♦♥ ♦♥ (T, φ)✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥❡s ❛ ❢✉♥❝t♦r T : Fin → Fin ✇✐t❤ ✇❡❜ T✿

◮ ❢♦r ❛❧❧ A ∈ Fin✱ |T A| = T |A| ❛♥❞ F (T A) ✐s tr❛♥s♣♦rt❡❞

❢r♦♠ F (A) ❛❧♦♥❣ φ|A|❀

◮ ❢♦r ❛❧❧ f ∈ Fin(A, B)✱ T f = Tf✳

❉❡✜♥✐t✐♦♥

❆ s❤❛♣❡ r❡❧❛t✐♦♥ ♦♥ (T, φ) ✐s ❛♥ ❛❧♠♦st✲❢✉♥❝t✐♦♥❛❧ ❧❛① ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ µ ❢r♦♠ T t♦ ❛ ❝♦♥st❛♥t ❢✉♥❝t♦r Z s✉❝❤ t❤❛t✿ ❢♦r ❛❧❧ a ⊆ TA✱ a ✐s ✜♥✐t❡ ❛s s♦♦♥ ❛s φA · a ❛♥❞ µA · a ❛r❡✳ T ✐s s②♠♠❡tr✐❝ ✐❢ T tf = tTf✳

❲❤❛t ✐s tr❛♥s♣♦rt ❣♦♦❞ ❢♦r❄

❈♦♥str✉❝t✐♥❣ ✜♥✐t❡♥❡ss s♣❛❝❡s

❡✳❣✳✱ t❤❡ ✜♥✐t❡♥❡ss s♣❛❝❡ ♦❢ ❜✐♥❛r② tr❡❡s ✇✐t❤ ♥♦❞❡s ✐♥ |A| ❛♥❞ ❧❡❛✈❡s ✐♥ |B|✱ ✇✐t❤ ✜♥✐t❡ss str✉❝t✉r❡ ❣✐✈❡♥ ❜② ❜♦✉♥❞❡❞ ❤❡✐❣❤t✱ ✜♥✐t❛r② A✲s✉♣♣♦rt ❛♥❞ ✜♥✐t❛r② B✲s✉♣♣♦rt✳

✳ ✳ ✳ ❢✉♥❝t♦r✐❛❧❧②

✐✳❡✳ ❞❛t❛t②♣❡s✳

❈❤❛r❛❝t❡r✐③❡ t❤❡ ❧❡❛st ✜①♣♦✐♥ts ♦❢ ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❢✉♥❝t♦rs

❛♠♦♥❣ ✇❤✐❝❤ t❤♦s❡ ❢♦r ❛❧❣❡❜r❛✐❝ ❞❛t❛t②♣❡s✳ Pr♦✈✐❞❡❞ ❛ ✜♥✐t❛r② s❡♠❛♥t✐❝s ♦❢ t②♣❡❞ r❡❝✉rs✐♦♥ ❬❚❛ss♦♥✕❱✳✱ ✷✵✶✶❪

❍✐❣❤❡r ♦r❞❡r ❧✐♥❡❛r ❧♦❣✐❝ ❄

❲❤❛t ✐s tr❛♥s♣♦rt ❣♦♦❞ ❢♦r❄

❈♦♥str✉❝t✐♥❣ ✜♥✐t❡♥❡ss s♣❛❝❡s

❡✳❣✳✱ t❤❡ ✜♥✐t❡♥❡ss s♣❛❝❡ ♦❢ ❜✐♥❛r② tr❡❡s ✇✐t❤ ♥♦❞❡s ✐♥ |A| ❛♥❞ ❧❡❛✈❡s ✐♥ |B|✱ ✇✐t❤ ✜♥✐t❡ss str✉❝t✉r❡ ❣✐✈❡♥ ❜② ❜♦✉♥❞❡❞ ❤❡✐❣❤t✱ ✜♥✐t❛r② A✲s✉♣♣♦rt ❛♥❞ ✜♥✐t❛r② B✲s✉♣♣♦rt✳

✳ ✳ ✳ ❢✉♥❝t♦r✐❛❧❧②

✐✳❡✳ ❞❛t❛t②♣❡s✳

❈❤❛r❛❝t❡r✐③❡ t❤❡ ❧❡❛st ✜①♣♦✐♥ts ♦❢ ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❢✉♥❝t♦rs

❛♠♦♥❣ ✇❤✐❝❤ t❤♦s❡ ❢♦r ❛❧❣❡❜r❛✐❝ ❞❛t❛t②♣❡s✳ Pr♦✈✐❞❡❞ ❛ ✜♥✐t❛r② s❡♠❛♥t✐❝s ♦❢ t②♣❡❞ r❡❝✉rs✐♦♥ ❬❚❛ss♦♥✕❱✳✱ ✷✵✶✶❪

❍✐❣❤❡r ♦r❞❡r ❧✐♥❡❛r ❧♦❣✐❝ ❄

❲❤❛t ✐s tr❛♥s♣♦rt ❣♦♦❞ ❢♦r❄

❈♦♥str✉❝t✐♥❣ ✜♥✐t❡♥❡ss s♣❛❝❡s

❡✳❣✳✱ t❤❡ ✜♥✐t❡♥❡ss s♣❛❝❡ ♦❢ ❜✐♥❛r② tr❡❡s ✇✐t❤ ♥♦❞❡s ✐♥ |A| ❛♥❞ ❧❡❛✈❡s ✐♥ |B|✱ ✇✐t❤ ✜♥✐t❡ss str✉❝t✉r❡ ❣✐✈❡♥ ❜② ❜♦✉♥❞❡❞ ❤❡✐❣❤t✱ ✜♥✐t❛r② A✲s✉♣♣♦rt ❛♥❞ ✜♥✐t❛r② B✲s✉♣♣♦rt✳

✳ ✳ ✳ ❢✉♥❝t♦r✐❛❧❧②

✐✳❡✳ ❞❛t❛t②♣❡s✳

❈❤❛r❛❝t❡r✐③❡ t❤❡ ❧❡❛st ✜①♣♦✐♥ts ♦❢ ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❢✉♥❝t♦rs

❛♠♦♥❣ ✇❤✐❝❤ t❤♦s❡ ❢♦r ❛❧❣❡❜r❛✐❝ ❞❛t❛t②♣❡s✳ Pr♦✈✐❞❡❞ ❛ ✜♥✐t❛r② s❡♠❛♥t✐❝s ♦❢ t②♣❡❞ r❡❝✉rs✐♦♥ ❬❚❛ss♦♥✕❱✳✱ ✷✵✶✶❪

❍✐❣❤❡r ♦r❞❡r ❧✐♥❡❛r ❧♦❣✐❝ ❄

❲❤❛t ✐s tr❛♥s♣♦rt ❣♦♦❞ ❢♦r❄

❈♦♥str✉❝t✐♥❣ ✜♥✐t❡♥❡ss s♣❛❝❡s

❡✳❣✳✱ t❤❡ ✜♥✐t❡♥❡ss s♣❛❝❡ ♦❢ ❜✐♥❛r② tr❡❡s ✇✐t❤ ♥♦❞❡s ✐♥ |A| ❛♥❞ ❧❡❛✈❡s ✐♥ |B|✱ ✇✐t❤ ✜♥✐t❡ss str✉❝t✉r❡ ❣✐✈❡♥ ❜② ❜♦✉♥❞❡❞ ❤❡✐❣❤t✱ ✜♥✐t❛r② A✲s✉♣♣♦rt ❛♥❞ ✜♥✐t❛r② B✲s✉♣♣♦rt✳

✳ ✳ ✳ ❢✉♥❝t♦r✐❛❧❧②

✐✳❡✳ ❞❛t❛t②♣❡s✳

❈❤❛r❛❝t❡r✐③❡ t❤❡ ❧❡❛st ✜①♣♦✐♥ts ♦❢ ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❢✉♥❝t♦rs

❛♠♦♥❣ ✇❤✐❝❤ t❤♦s❡ ❢♦r ❛❧❣❡❜r❛✐❝ ❞❛t❛t②♣❡s✳ Pr♦✈✐❞❡❞ ❛ ✜♥✐t❛r② s❡♠❛♥t✐❝s ♦❢ t②♣❡❞ r❡❝✉rs✐♦♥ ❬❚❛ss♦♥✕❱✳✱ ✷✵✶✶❪

❍✐❣❤❡r ♦r❞❡r ❧✐♥❡❛r ❧♦❣✐❝ ❄

❲❤❛t ✐s tr❛♥s♣♦rt ❣♦♦❞ ❢♦r❄

❈♦♥str✉❝t✐♥❣ ✜♥✐t❡♥❡ss s♣❛❝❡s

❡✳❣✳✱ t❤❡ ✜♥✐t❡♥❡ss s♣❛❝❡ ♦❢ ❜✐♥❛r② tr❡❡s ✇✐t❤ ♥♦❞❡s ✐♥ |A| ❛♥❞ ❧❡❛✈❡s ✐♥ |B|✱ ✇✐t❤ ✜♥✐t❡ss str✉❝t✉r❡ ❣✐✈❡♥ ❜② ❜♦✉♥❞❡❞ ❤❡✐❣❤t✱ ✜♥✐t❛r② A✲s✉♣♣♦rt ❛♥❞ ✜♥✐t❛r② B✲s✉♣♣♦rt✳

✳ ✳ ✳ ❢✉♥❝t♦r✐❛❧❧②

✐✳❡✳ ❞❛t❛t②♣❡s✳

❈❤❛r❛❝t❡r✐③❡ t❤❡ ❧❡❛st ✜①♣♦✐♥ts ♦❢ ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❢✉♥❝t♦rs

❛♠♦♥❣ ✇❤✐❝❤ t❤♦s❡ ❢♦r ❛❧❣❡❜r❛✐❝ ❞❛t❛t②♣❡s✳ Pr♦✈✐❞❡❞ ❛ ✜♥✐t❛r② s❡♠❛♥t✐❝s ♦❢ t②♣❡❞ r❡❝✉rs✐♦♥ ❬❚❛ss♦♥✕❱✳✱ ✷✵✶✶❪

❍✐❣❤❡r ♦r❞❡r ❧✐♥❡❛r ❧♦❣✐❝ ❄

❚r❛♥s♣♦rt ♦❢ ♦t❤❡r str✉❝t✉r❡s

❈♦❤❡r❡♥❝❡ s♣❛❝❡s

▲❡t f ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |B| s✉❝❤ t❤❛t f · α ∈ C(B) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ C = {a ⊆ A; f · a ∈ C(B)} ✐s ❛ ❝♦❤❡r❡♥❝❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿ C = {f \ b; b ∈ C(B)}⊥⊥ . ❱❡r② ❡❛s②✳

❚♦t❛❧✐t② s♣❛❝❡s

❋❛✐❧ ❄❄❄

❚r❛♥s♣♦rt ♦❢ ♦t❤❡r str✉❝t✉r❡s

❈♦❤❡r❡♥❝❡ s♣❛❝❡s

▲❡t f ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |B| s✉❝❤ t❤❛t f · α ∈ C(B) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ C = {a ⊆ A; f · a ∈ C(B)} ✐s ❛ ❝♦❤❡r❡♥❝❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿ C = {f \ b; b ∈ C(B)}⊥⊥ . ❱❡r② ❡❛s②✳

❚♦t❛❧✐t② s♣❛❝❡s

❋❛✐❧ ❄❄❄

❚r❛♥s♣♦rt ♦❢ ♦t❤❡r str✉❝t✉r❡s

❈♦❤❡r❡♥❝❡ s♣❛❝❡s

▲❡t f ❜❡ ❛ r❡❧❛t✐♦♥ ❢r♦♠ A t♦ |B| s✉❝❤ t❤❛t f · α ∈ C(B) ❢♦r ❛❧❧ α ∈ A✳ ❚❤❡♥ C = {a ⊆ A; f · a ∈ C(B)} ✐s ❛ ❝♦❤❡r❡♥❝❡ ♦♥ A✳ ▼♦r❡ ♣r❡❝✐s❡❧②✿ C = {f \ b; b ∈ C(B)}⊥⊥ . ❱❡r② ❡❛s②✳

❚♦t❛❧✐t② s♣❛❝❡s

❋❛✐❧ ❄❄❄

❚r❛♥s♣♦rt ❛♥❞ ♦rt❤♦❣♦♥❛❧✐t②

❚❤❡② ♣❧❛② ❝♦♠♣❧❡♠❡♥t❛r② r♦❧❡s✿

◮ ♦rt❤♦❣♦♥❛❧✐t② ♣r♦✈✐❞❡s t❤❡ ❣❡♥❡r✐❝ str✉❝t✉r❡ ❛♥❞ ❛①✐♦♠s❀ ◮ tr❛♥s♣♦rt ♣r♦✈✐❞❡s ❛ s✐♠♣❧❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❛♥❞ ❛❧❧♦✇s t♦

♣r♦✈❡ t❤❡ ❛①✐♦♠s✳

❚♦✇❛r❞s ❛ ♠♦r❡ ❣❡♥❡r❛❧ ♥♦t✐♦♥ ♦❢ tr❛♥s♣♦rt❄

♦♥ t♦♣ ♦❢ ♦rt❤♦❣♦♥❛❧✐t②❀ r❡str✐❝t❡❞ t♦ ✇❡❜❜❡❞ ♠♦❞❡❧s ✭ ✮ ♦r ✐♥ ❛♥ ❡♥r✐❝❤❡❞ s❡tt✐♥❣✳

❚r❛♥s♣♦rt ❛♥❞ ♦rt❤♦❣♦♥❛❧✐t②

❚❤❡② ♣❧❛② ❝♦♠♣❧❡♠❡♥t❛r② r♦❧❡s✿

◮ ♦rt❤♦❣♦♥❛❧✐t② ♣r♦✈✐❞❡s t❤❡ ❣❡♥❡r✐❝ str✉❝t✉r❡ ❛♥❞ ❛①✐♦♠s❀ ◮ tr❛♥s♣♦rt ♣r♦✈✐❞❡s ❛ s✐♠♣❧❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❛♥❞ ❛❧❧♦✇s t♦

♣r♦✈❡ t❤❡ ❛①✐♦♠s✳

❚♦✇❛r❞s ❛ ♠♦r❡ ❣❡♥❡r❛❧ ♥♦t✐♦♥ ♦❢ tr❛♥s♣♦rt❄

◮ ♦♥ t♦♣ ♦❢ ♦rt❤♦❣♦♥❛❧✐t②❀ ◮ r❡str✐❝t❡❞ t♦ ✇❡❜❜❡❞ ♠♦❞❡❧s ✭Rel✮ ♦r ✐♥ ❛♥ ❡♥r✐❝❤❡❞ s❡tt✐♥❣✳

❋✐♥