The algebraic lambda-calculus

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Submitted on 29 Apr 2009

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The algebraic lambda-calculus

Lionel Vaux

To cite this version:

Lionel Vaux. The algebraic lambda-calculus. Mathematical Structures in Computer Science, Cam-

bridge University Press (CUP), 2009, accepted for publication. �hal-00379750�

❯♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ❢♦r ♣✉❜❧✐❝❛t✐♦♥ ✐♥ ▼❛t❤✳ ❙tr✉❝t✳ ✐♥ ❈♦♠♣✳ ❙❝✐❡♥❝❡

❆❧❣❡❜r❛✐❝ ▲❛♠❜❞❛✲❈❛❧❝✉❧✉s

▲ ■ ❖ ◆ ❊ ▲ ❱ ❆ ❯ ❳

▲❛❜♦r❛t♦✐r❡ ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ❧✬❯♥✐✈❡rs✐té ❞❡ ❙❛✈♦✐❡✱

❯❋❘ ❙❋❆✱ ❈❛♠♣✉s ❙❝✐❡♥t✐✜q✉❡✱ ✼✸✸✼✻ ▲❡ ❇♦✉r❣❡t✲❞✉✲▲❛❝ ❈❡❞❡①✱ ❋r❛♥❝❡

❘❡❝❡✐✈❡❞ ❉❡❝❡♠❜❡r ✷✻✱ ✷✵✵✽

❲❡ ✐♥tr♦❞✉❝❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣✉r❡ ❧❛♠❜❞❛✲❝❛❧❝✉❧✉s ❜② ❡♥❞♦✇✐♥❣ t❤❡ s❡t ♦❢ t❡r♠s

✇✐t❤ ❛ str✉❝t✉r❡ ♦❢ ✈❡❝t♦r s♣❛❝❡✱ ♦r ♠♦r❡ ❣❡♥❡r❛❧❧② ♦❢ ♠♦❞✉❧❡✱ ♦✈❡r ❛ ✜①❡❞ s❡t ♦❢ s❝❛❧❛rs✳

❚❡r♠s ❛r❡ ♠♦r❡♦✈❡r s✉❜❥❡❝t t♦ ✐❞❡♥t✐t✐❡s s✐♠✐❧❛r t♦ ✉s✉❛❧ ♣♦✐♥t✇✐s❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♥❡❛r

❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ ✈❛❧✉❡s ✐♥ ❛ ✈❡❝t♦r s♣❛❝❡✳ ❲❡ t❤❡♥ st✉❞② ❛ ♥❛t✉r❛❧

❡①t❡♥s✐♦♥ ♦❢ ❜❡t❛✲r❡❞✉❝t✐♦♥ ✐♥ t❤✐s s❡tt✐♥❣✿ ✇❡ ♣r♦✈❡ ✐t ✐s ❝♦♥✢✉❡♥t✱ t❤❡♥ ❞✐s❝✉ss

❝♦♥s✐st❡♥❝② ❛♥❞ ❝♦♥s❡r✈❛t✐✈✐t② ♦✈❡r t❤❡ ♦r❞✐♥❛r② ❧❛♠❜❞❛✲❝❛❧❝✉❧✉s✳ ❲❡ ❛❧s♦ ♣r♦✈✐❞❡

♥♦r♠❛❧✐③❛t✐♦♥ r❡s✉❧ts ❢♦r ❛ s✐♠♣❧❡ t②♣❡ s②st❡♠✳

✶✳ ■♥tr♦❞✉❝t✐♦♥

Pr❡❧✐♠✐♥❛r② ❉❡✜♥✐t✐♦♥s ❛♥❞ ◆♦t❛t✐♦♥s✳ ❘❡❝❛❧❧ t❤❛t ❛ r✐❣ ✭♦r ✏s❡♠✐r✐♥❣ ✇✐t❤ ③❡r♦ ❛♥❞

✉♥✐t②✑✮ ✐s t❤❡ s❛♠❡ t❤✐♥❣ ❛s ❛ ✉♥✐t❛❧ r✐♥❣✱ ✇✐t❤♦✉t t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t ❡✈❡r② ❡❧❡♠❡♥t

❛❞♠✐ts ❛♥ ❛❞❞✐t✐✈❡ ✐♥✈❡rs❡✳ ▲❡t R= (R,+,0,×,1) ❜❡ ❛ r✐❣✿ (R,+,0) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡

♠♦♥♦✐❞✱(R,×,1)✐s ❛ ♠♦♥♦✐❞✱×✐s ❞✐str✐❜✉t✐✈❡ ♦✈❡r+❛♥❞0✐s ❛❜s♦r❜✐♥❣ ❢♦r×✳ ❲❡ ✇r✐t❡

R^• ❢♦rR\ {0}✳ ❲❡ ❞❡♥♦t❡ ❜② ❧❡tt❡rsa, b, ct❤❡ ❡❧❡♠❡♥ts ♦❢R✱ ❛♥❞ s❛② t❤❛t R✐s ♣♦s✐t✐✈❡

✐❢✱ ❢♦r ❛❧❧a, b∈R✱a+b = 0✐♠♣❧✐❡s a= 0❛♥❞ b= 0✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ♣♦s✐t✐✈❡ r✐❣ ✐s N✱ t❤❡ s❡t ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs✱ ✇✐t❤ ✉s✉❛❧ ♦♣❡r❛t✐♦♥s✳

❆ ♠♦❞✉❧❡ ♦✈❡r r✐❣R✱ ♦rR✲♠♦❞✉❧❡✱ ✐s ❞❡✜♥❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛s ❛ ✉♥✐t❛❧ ♠♦❞✉❧❡ ♦✈❡r

❛ r✐♥❣✱ ❛❣❛✐♥ ✇✐t❤♦✉t t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t ❡✈❡r② ❡❧❡♠❡♥t ❛❞♠✐ts ❛♥ ❛❞❞✐t✐✈❡ ✐♥✈❡rs❡✳ ❋♦r

❛❧❧ s❡tX✱ t❤❡ s❡t ♦❢ ❢♦r♠❛❧ ✜♥✐t❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ♦❢X ✇✐t❤ ❝♦❡✣❝✐❡♥ts

✐♥R✐s t❤❡ ❢r❡❡R✲♠♦❞✉❧❡ ♦✈❡rX✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜②RhX i✳

▲✐♥❡❛r✐t② ✐♥ t❤❡λ✲❈❛❧❝✉❧✉s✳ ●✐r❛r❞✬s ❧✐♥❡❛r ❧♦❣✐❝ ✭●✐r✽✼✮✱ ❜② ❞❡❝♦♠♣♦s✐♥❣ ✐♥t✉✐t✐♦♥✐st✐❝

✐♠♣❧✐❝❛t✐♦♥✱ ♠❛❞❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♥❝❡♣t ♦❢ ❧✐♥❡❛r✐t② ♣r♦♠✐♥❡♥t✱ ✇❤✐❧❡ r❡❧❛t✐♥❣ ✐t

✇✐t❤ t❤❡ ✉s✉❛❧ ❛❧❣❡❜r❛✐❝ ♥♦t✐♦♥✳ ❆ ♣r♦❣r❛♠ ✐s s❛✐❞ t♦ ❜❡ ❧✐♥❡❛r ✐❢ ✐t ✉s❡s ✐ts ❛r❣✉♠❡♥t

❡①❛❝t❧② ♦♥❝❡✳ ❚❤✐s ✈❛❣✉❡ ✐❞❡❛ ❝❛♥ ❜❡ ♠❛❞❡ ♠♦r❡ ♣r❡❝✐s❡✱ ❜② ❞❡✜♥✐♥❣ ✇❤✐❝❤ s✉❜t❡r♠s ♦❢

❛ t❡r♠u❛r❡ ✐♥ ❧✐♥❡❛r ♣♦s✐t✐♦♥ ✐♥u✿

✖ ✐♥ ❛ t❡r♠ ✇❤✐❝❤ ✐s ♦♥❧② ❛ ✈❛r✐❛❜❧❡x✱ t❤❛t ♦❝❝✉rr❡♥❝❡ ♦❢ ✈❛r✐❛❜❧❡ ✐s ✐♥ ❧✐♥❡❛r ♣♦s✐t✐♦♥❀

✖ ✐♥ ❛♥ ❛❜str❛❝t✐♦♥ u = λx s✱ t❤❡ s✉❜t❡r♠s ✐♥ ❧✐♥❡❛r ♣♦s✐t✐♦♥ ✐♥ u ❛r❡ t❤♦s❡ ♦❢ t❤❡

❛❜str❛❝t❡❞ s✉❜t❡r♠ s✱ ❛♥❞u✐ts❡❧❢❀

✖ ✐♥ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ u = (s)t✱ t❤❡ s✉❜t❡r♠s ✐♥ ❧✐♥❡❛r ♣♦s✐t✐♦♥ ✐♥ u ❛r❡ t❤♦s❡ ♦❢ t❤❡

❢✉♥❝t✐♦♥ s✉❜t❡r♠s✱ ❛♥❞u✐ts❡❧❢✳

▲✳ ❱❛✉① ✷

■♥ ♣❛rt✐❝✉❧❛r✱ ❛♣♣❧✐❝❛t✐♦♥ ✐s ❧✐♥❡❛r ✐♥ t❤❡ ❢✉♥❝t✐♦♥ ❜✉t ♥♦t ✐♥ t❤❡ ❛r❣✉♠❡♥t✳ ❚❤✐s ✐s t♦

❜❡ r❡❧❛t❡❞ ✇✐t❤ ❤❡❛❞ r❡❞✉❝t✐♦♥ ❛♥❞ ♠❡♠♦r② ♠❛♥❛❣❡♠❡♥t✿ t❤♦s❡ s✉❜t❡r♠s t❤❛t ❛r❡ ✐♥

❧✐♥❡❛r ♣♦s✐t✐♦♥ ❛r❡ ❡✈❛❧✉❛t❡❞ ❡①❛❝t❧② ♦♥❝❡ ✐♥ t❤❡ ❤❡❛❞ r❡❞✉❝t✐♦♥✱ t❤❡② ❛r❡ ♥♦t ❝♦♣✐❡❞ ♥♦r

❞✐s❝❛r❞❡❞✳

❆❧❣❡❜r❛✐❝ ❧✐♥❡❛r✐t② ✐s ❣❡♥❡r❛❧❧② t❤♦✉❣❤t ♦❢ ❛s ❝♦♠♠✉t❛t✐♦♥ ✇✐t❤ s✉♠s✳ ■t ✐s ✇❡❧❧ ❦♥♦✇♥

t❤❛t t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❢r♦♠ s♦♠❡ s❡t t♦ s♦♠❡ ✜①❡❞R✲♠♦❞✉❧❡ ✐s ✐ts❡❧❢ ❛♥R✲♠♦❞✉❧❡✱

✇✐t❤ ♦♣❡r❛t✐♦♥s ♦♥ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♣♦✐♥t✇✐s❡✿ ❢♦r ✐♥st❛♥❝❡✱ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s

❞❡✜♥❡❞ ❜②(f+g)(x) =f(x) +g(x)✳ ■♥ ✭❊❤r✵✶✮ ❛♥❞ ✭❊❤r✵✺✮✱ ❊❤r❤❛r❞ ✐♥tr♦❞✉❝❡❞ ❞❡♥♦✲

t❛t✐♦♥❛❧ ♠♦❞❡❧s ♦❢ ❧✐♥❡❛r ❧♦❣✐❝ ✇❤❡r❡ ❢♦r♠✉❧❛s ❛r❡ ✐♥t❡r♣r❡t❡❞ ❛s ♣❛rt✐❝✉❧❛r ✈❡❝t♦r s♣❛❝❡s

♦r ♠♦❞✉❧❡s ❛♥❞ ♣r♦♦❢s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ λ✲t❡r♠s ❛r❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s

❞❡✜♥❡❞ ❜② ♣♦✇❡r s❡r✐❡s ♦♥ t❤❡s❡ s♣❛❝❡s✿ t❤✐s ✐s t❤❡ ❜❛s✐❝ ✐❞❡❛ ♦❢ ●✐r❛r❞✬s q✉❛♥t✐t❛t✐✈❡

s❡♠❛♥t✐❝s ✭●✐r✽✽✮✳ ❚❤✐s ♥♦t ♦♥❧② ❣✉✐❞❡❞ t❤❡ st✉❞② ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ λ✲❝❛❧❝✉❧✉s ❜②

❊❤r❤❛r❞ ❛♥❞ ❘❡❣♥✐❡r ✐♥ ✭❊❘✵✸✮✱ ❜✉t ❛❧s♦ ♦✛❡r❡❞ s❡r✐♦✉s ❣r♦✉♥❞✐♥❣ t♦ ❡♥❞♦✇ t❤❡ s❡t ♦❢

t❡r♠s ✇✐t❤ ❛ str✉❝t✉r❡ ♦❢ ✈❡❝t♦r s♣❛❝❡✱ ♦r ♦❢ R✲♠♦❞✉❧❡✱ ✇❤❡r❡ R✐s ❛ r✐❣✿ ♦♥❡ ❝❛♥ ❢♦r♠

❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❡r♠s✱ s✉❜❥❡❝t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ✐❞❡♥t✐t✐❡s✿

λx Xn i=1

aisi

!

= Xn i=1

aiλx si ✭✶✮

❛♥❞

Xn i=1

aisi

! u=

Xn i=1

ai(si)u ✭✷✮

❢♦r ❛❧❧ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ Pn

i=1aisi ♦❢ t❡r♠s✳ ❲❡ r❡❝♦✈❡r t❤❡ ❢❛❝t t❤❛t ❛♣♣❧✐❝❛t✐♦♥ ✐s

❧✐♥❡❛r ✐♥ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ♥♦t ✐♥ t❤❡ ❛r❣✉♠❡♥t✱ ✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧

♥♦t✐♦♥ ♦❢ ❧✐♥❡❛r✐t②✳

❘❡❞✉❝✐♥❣ ▲✐♥❡❛r ❈♦♠❜✐♥❛t✐♦♥s ♦❢ λ✲t❡r♠s✳ ❆♣❛rt ❢r♦♠ ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ♦♥❡ ✐♠♣♦rt❛♥t

❢❡❛t✉r❡ ♦❢ t❤❡ ❝❛❧❝✉❧✉s ♦❢ ✭❊❘✵✸✮ ✐s t❤❡ ✇❛②β✲r❡❞✉❝t✐♦♥ ✐s ❡①t❡♥❞❡❞ t♦ s✉❝❤ ❧✐♥❡❛r ❝♦♠✲

❜✐♥❛t✐♦♥s ♦❢ t❡r♠s✳ ❆♠♦♥❣ t❡r♠s✱ s♦♠❡ ❛r❡ ❝♦♥s✐❞❡r❡❞ s✐♠♣❧❡✿ t❤❡② ❝♦♥t❛✐♥ ♥♦ s✉♠ ✐♥

❧✐♥❡❛r ♣♦s✐t✐♦♥✱ s♦ t❤❛t ♥♦r ✭✶✮ ♥♦r ✭✷✮ ❛♣♣❧✐❡s❀ ❤❡♥❝❡ t❤❡② ❛r❡ ✐♥tr✐♥s✐❝❛❧❧② ♥♦t s✉♠s✳

❚❤❡s❡ ❢♦r♠ ❛ ❜❛s✐s ♦❢ t❤❡R✲♠♦❞✉❧❡ ♦❢ t❡r♠s✳ ❘❡❞✉❝t✐♦♥→✐s t❤❡♥ t❤❡ ❧❡❛st ❝♦♥t❡①t✉❛❧

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

(λx s)t→s[t/x] ✭✸✮

❛♥❞✱ ✐❢a∈R^•✐s ❛ ♥♦♥✲③❡r♦ s❝❛❧❛r✱

s→s^′ ✐♠♣❧✐❡sas+t→as^′+t . ✭✹✮

❙✐♥❝❡ ❡✈❡r② ♦r❞✐♥❛r② λ✲t❡r♠ ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s ❛ s✐♠♣❧❡ t❡r♠✱ ✭✸✮ ❡①t❡♥❞s ✉s✉❛❧ β✲ r❡❞✉❝t✐♦♥✳ ❚❤❡ r❡q✉✐r❡♠❡♥t t❤❛ts ✐s s✐♠♣❧❡ ✐♥ ✭✸✮ ❛♥❞ ✭✹✮✱ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥

a 6= 0 ✐♥ ✭✹✮✱ ❡♥s✉r❡ → ❛❝t✉❛❧❧② r❡❞✉❝❡s s♦♠❡t❤✐♥❣✱ s♦ t❤❛t r❡❞✉❝t✐♦♥ ✐s ♥♦t tr✐✈✐❛❧❧② r❡✢❡①✐✈❡✳

❆❧t❤♦✉❣❤ t❤❡ ♣r❡✈✐♦✉s ❞❡✜♥✐t✐♦♥ ♠✐❣❤t s❡❡♠ ❝♦♥t♦rt❡❞✱ ✐t ✐s t❡❝❤♥✐❝❛❧❧② ❡✣❝✐❡♥t✳ ❋♦r

✐♥st❛♥❝❡✱ ✐t ✐s ♣❛rt✐❝✉❧❛r❧② ✇❡❧❧ s✉✐t❡❞ ❢♦r ♣r♦✈✐♥❣ ❝♦♥✢✉❡♥❝❡ ✈✐❛ ✉s✉❛❧ ❚❛✐t✕▼❛rt✐♥✲▲ö❢

t❡❝❤♥✐q✉❡✿ ✐♥tr♦❞✉❝❡ ❛ ♣❛r❛❧❧❡❧ ✈❡rs✐♦♥⇒♦❢→s✉❝❤ t❤❛t→ ⊆⇒⊆ →^∗✱ ❛♥❞ ♣r♦✈❡ t❤❛t

❆❧❣❡❜r❛✐❝ ▲❛♠❜❞❛✲❈❛❧❝✉❧✉s ✸

⇒❡♥❥♦②s t❤❡ ❞✐❛♠♦♥❞ ♣r♦♣❡rt②✳ ❍❡r❡⇒✐s r❡✢❡①✐✈❡ ❛♥❞ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❜❡❤❛✈✐♦✉r ♦♥

❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❡r♠s✿

Xn i=1

aisi⇒ Xn i=1

ais^′_i ❛s s♦♦♥ ❛s✱ ❢♦r ❛❧❧i✱si⇒s^′_i ❛♥❞si ✐s s✐♠♣❧❡. ✭✺✮

❆ss✉♠✐♥❣ s ⇒ s^′ ⇒ s^′′ ❛r❡ s✐♠♣❧❡ t❡r♠s✱ ✇❡ ❤❛✈❡ s+s^′ ⇒ 2s^′ ❛♥❞ s+s^′ ⇒ s+s^′′✿ t❤❡♥ ✭✺✮ ❛❧❧♦✇s t♦ ❝❧♦s❡ t❤❛t ♣❛✐r ♦❢ r❡❞✉❝t✐♦♥s ❜②2s^′ ⇒ s^′+s^′′ ❛♥❞ s+s^′′ ⇒s^′+s^′′✳

❚❤✐s ✇♦✉❧❞ ♥♦t ❤♦❧❞ ✐❢ ✇❡ ❤❛❞ ❢♦r❝❡❞ t❤❡si✬s ✐♥ ✭✺✮ t♦ ❜❡ ❞✐st✐♥❝t s✐♠♣❧❡ t❡r♠s ✖ t❤❛t

❝♦♥❞✐t✐♦♥ ✇♦✉❧❞ ❛♠♦✉♥t t♦ r❡❞✉❝❡ ❡❛❝❤ ❡❧❡♠❡♥t ♦❢ t❤❡ ❜❛s❡ ♦❢ s✐♠♣❧❡ t❡r♠s✱ ✐♥ ♣❛r❛❧❧❡❧✱

✇❤✐❝❤ ♠❛② s❡❡♠ ❛ ♥❛t✉r❛❧ ❝❤♦✐❝❡ ❛t ✜rst✳

❈♦❧❧❛♣s❡✳ ■♥ ✭❱❛✉✵✼❛✮✱ ❤♦✇❡✈❡r✱ t❤❡ ❛✉t❤♦r ♣r♦✈❡❞ t❤❛t t❤❡ ❛❜♦✈❡ ❤✐❣❤❡r✲♦r❞❡r r❡✇r✐t✐♥❣

♦❢ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ❝♦❧❧❛♣s❡s ❛s s♦♦♥ ❛s t❤❡ r✐❣ ♦❢ s❝❛❧❛rs ❛❞♠✐ts ♥❡❣❛t✐✈❡ ❡❧❡♠❡♥ts✿

✐❢ −1 ∈ R ✭s♦ t❤❛t 1 + (−1) = 0✮✱ t❤❡♥ ❢♦r ❛❧❧ t❡r♠s s ❛♥❞ t✱ s →^∗ t✳ ❚❤✐s s❤♦✉❧❞

♥♦t ❜❡ ❛ s✉r♣r✐s❡✱ s✐♥❝❡ ✐♥ t❤❛t ❝❛s❡ t❤❡ s②st❡♠ ✐♥✈♦❧✈❡s ❜♦t❤ ♥❡❣❛t✐✈❡ ♥✉♠❜❡rs ❛♥❞

♣♦t❡♥t✐❛❧ ✐♥✜♥✐t② t❤r♦✉❣❤ ❛r❜✐tr❛r② ✜①❡❞ ♣♦✐♥ts✳ ■♥❞❡❡❞✱ t❛❦❡ Θ ❛ ✜①♣♦✐♥t ♦♣❡r❛t♦r ♦❢

t❤❡λ✲❝❛❧❝✉❧✉s✱ s✉❝❤ t❤❛t(Θ)s→^∗(s) (Θ)s❢♦r ❛❧❧λ✲t❡r♠s✳ ❲r✐t❡∞s❢♦r(Θ)λx(s+x)❀

t❤❡♥∞s→^∗s+∞s✱ ❤❡♥❝❡∞sst❛♥❞s ❢♦r ❛♥ ✐♥✜♥✐t❡ ❛♠♦✉♥t ♦❢s✳ ❲❡ ❣❡t✿

s=s+∞s− ∞s+∞t− ∞t→^∗s−s+t=t .

❆❧s♦✱ ✐❢ ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r ❢r❛❝t✐♦♥s ♦❢ s❝❛❧❛rs✱ str♦♥❣ ♥♦r♠❛❧✐③❛❜✐❧✐t② ❤♦❧❞s ♦♥❧② ❢♦r

♥♦r♠❛❧ t❡r♠s✿ ❛ss✉♠❡s→s^′ ❛♥❞R❝♦♥t❛✐♥s ❞②❛❞✐❝ r❛t✐♦♥❛❧s❀ t❤❡♥

s = 1 2s+1

2s → 1 2s+1

2s^′ → 1 4s+3

4s^′ → · · ·

❇♦t❤ t❤❡s❡ ❢❛✐❧✉r❡s ✐♥❞✐❝❛t❡ t❤❛t ♠✉❝❤ ❝❛r❡ ✐s ♥❡❡❞❡❞ ✇❤❡♥ ❞❡❛❧✐♥❣ ✇✐t❤ ❧✐♥❡❛r ❝♦♠❜✐✲

♥❛t✐♦♥s ♦❢λ✲t❡r♠s✿ t❤❡s❡ ♠❛❦❡ t❤❡ ✐❞❡♥t✐t② ♦❢ t❡r♠s ✈❡r② ✐♥tr✐❝❛t❡✱ ♠✉❝❤ ♠♦r❡ s♦ t❤❛♥

♣❧❛✐♥ α✲❡q✉✐✈❛❧❡♥❝❡✱ s♦ t❤❛t ✐ts ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ❤✐❣❤❡r✲♦r❞❡r r❡✇r✐t✐♥❣ ❜❡❝♦♠❡s tr✐❝❦②✳

❆s ❛ r❡s✉❧t✱ ❛❧t❤♦✉❣❤ t❤❡ ♣r♦❜❧❡♠ ❛❜♦✉t ♥♦r♠❛❧✐③❛❜✐❧✐t② ✇❛s ✇❡❧❧ ♥♦t❡❞ ✐♥ ✭❊❘✵✸✮✱ t❤❡

❝♦❧❧❛♣s❡ ♦❢ r❡❞✉❝t✐♦♥ ✐♥ ♣r❡s❡♥❝❡ ♦❢ ♥❡❣❛t✐✈❡ ❝♦❡✣❝✐❡♥ts ❡❧✉❞❡❞ t❤❡ ❛✉t❤♦rs ♦❢ t❤❛t ♣❛✲

♣❡r✳ ■♥ t❤❡ ♣r❡s❡♥t ❝♦♥tr✐❜✉t✐♦♥✱ ✇❡ ❣✐✈❡ ❛ s②♥t❛❝t✐❝ ❢r❛♠❡✇♦r❦ ❢♦r t❤❡ st✉❞② ♦❢ ❧✐♥❡❛r

❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❡r♠s✱ ✇❤✐❝❤ ❛✐♠s t♦ ❜❡ ♠♦r❡ r✐❣♦r♦✉s ❛♥❞ ❢♦r♠❛❧ t❤❛♥ t❤❛t ❞❡✈❡❧♦♣❡❞

✐♥ ✭❊❘✵✸✮ ♦r ✭❱❛✉✵✼❛✮✿ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ♣✉t ♠✉❝❤ ❝❛r❡ ✐♥ ❞❡✈❡❧♦♣♣✐♥❣ ❛♥ ❡①♣❧✐❝✐t ✐♠♣❧❡✲

♠❡♥t❛t✐♦♥ ♦❢ t❤❡R✲♠♦❞✉❧❡ ♦❢ t❡r♠s✳ ❆❧s♦✱ ✇❡ ❞♦ ♥♦t ❝♦♥s✐❞❡r ❞✐✛❡r❡♥t✐❛t✐♦♥ ♥♦r ❝❧❛ss✐❝❛❧

❝♦♥tr♦❧ ♦♣❡r❛t♦rs✱ ❛♥❞ ♦♥❧② ❢♦❝✉s ♦♥ t❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ♦❢ t❡r♠s ❛♥❞ t❤❡ ✐♥t❡r❛❝t✐♦♥

❜❡t✇❡❡♥ ❝♦❡✣❝✐❡♥ts ❛♥❞ r❡❞✉❝t✐♦♥✳ ❲❡ ❝❛❧❧ t❤❡ ♦❜t❛✐♥❡❞ s②st❡♠ t❤❡ ❛❧❣❡❜r❛✐❝λ✲❝❛❧❝✉❧✉s✳

❈♦♥tr✐❜✉t✐♦♥s✳ ■♥ s❡❝t✐♦♥ ✷✱ ✇❡ ❢♦r♠❛❧✐③❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ R✲♠♦❞✉❧❡ ♦❢ t❡r♠s✱ ✈❛❧✲

✐❞❛t✐♥❣ ✐❞❡♥t✐t✐❡s ✭✶✮ ❛♥❞ ✭✷✮✱ ❛♥❞ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❦❡② ♥♦t✐♦♥ ♦❢ ❝❛♥♦♥✐❝❛❧ ❢♦r♠s✳ ❲❡

❛❧s♦ ❝♦♠♣❛r❡ t❤✐s ♣r❡s❡♥t❛t✐♦♥ t♦ t❤❛t ♦❢ ✭❊❘✵✸✮✿ t❡r♠s à ❧❛ ❊❤r❤❛r❞✕❘❡❣♥✐❡r ❛r❡ ❥✉st

❝❛♥♦♥✐❝❛❧ ❢♦r♠s ♦❢ t❡r♠s ✐♥ ♦✉r s❡tt✐♥❣✳ ❚❤✐s ✐s ❛♥ ✐♠♣♦rt❛♥t ♣❛rt ♦❢ t❤❡ ♣r❡s❡♥t ✇♦r❦✱

✇❤✐❝❤ ✇❡ ❤♦♣❡ s❤❡❞s ♥❡✇ ❧✐❣❤t ♦♥ t❤❡ str✉❝t✉r❡ t❤❡R✲♠♦❞✉❧❡ ♦❢ t❡r♠s✳ ■♥ s❡❝t✐♦♥ ✸ ✇❡

❞❡✜♥❡ r❡❞✉❝t✐♦♥✱ ✉s✐♥❣ r✉❧❡ ✭✹✮ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ s✉♠✱ ❛♥❞ ❞✐s❝✉ss ❝♦♥s❡r✈❛t✐✈✐t② ✇✳r✳t✳ ♦r✲

❞✐♥❛r②β✲r❡❞✉❝t✐♦♥✳ ❙❡❝t✐♦♥ ✹ ♣r❡s❡♥ts ❛ ❈✉rr②✲st②❧❡ s✐♠♣❧❡ t②♣❡ s②st❡♠ ❢♦r t❤❡ ❛❧❣❡❜r❛✐❝

λ✲❝❛❧❝✉❧✳ ❲❡ ♣r♦✈❡ s✉❜❥❡❝t r❡❞✉❝t✐♦♥ ❤♦❧❞s ✐✛ t❤❡ r✐❣ ♦❢ s❝❛❧❛rs ✐s ♣♦s✐t✐✈❡✳ ■♥ s❡❝t✐♦♥ ✺✱

▲✳ ❱❛✉① ✹

✇❡ ❞✐s❝✉ss ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r str♦♥❣ ♥♦r♠❛❧✐③❛t✐♦♥ ♦❢ t②♣❡❞ t❡r♠s t♦ ❤♦❧❞❀ ✇❡ r❡✜♥❡

t❤❡s❡ t♦ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❡ t❤❡ ♣r♦♦❢ ♦❢ str♦♥❣ ♥♦r♠❛❧✐③❛t✐♦♥ ♦❢ ❞✐✛❡r✲

❡♥t✐❛❧ λ✲❝❛❧❝✉❧✉s ❜② ❊❤r❤❛r❞ ❛♥❞ ❘❡❣♥✐❡r ✭❊❘✵✸✮✳ ❲❡ ❝♦♥❝❧✉❞❡ ❜② ❞✐s❝✉ss✐♥❣ ♣♦ss✐❜❧❡

♦t❤❡r ❛♣♣r♦❛❝❤❡s ✐♥ s❡❝t✐♦♥ ✻✳

❆❜♦✉t Pr❡✈✐♦✉s ❲♦r❦s✳ ▼♦st ♦❢ t❤❡ r❡s✉❧ts ♦❢ t❤✐s ♣❛♣❡r ✇❡r❡ ❛❧r❡❛❞② ♣r❡s❡♥t ✐♥ ✭❱❛✉✵✼❛✮

♦r ❡✈❡♥ ✭❊❘✵✸✮✱ s♦♠❡t✐♠❡s ✐♥ ❛ ✇❡❛❦❡r ❢♦r♠✳ ■♥ t❤♦s❡ t✇♦ ♣r❡✈✐♦✉s ✇♦r❦s✱ ❤♦✇❡✈❡r✱ t❤❡

❢♦❝✉s ✇❛s ♦♥ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❡r♠s ❛♥❞ t❤❡✐r

❡✛❡❝ts ♦♥ r❡❞✉❝t✐♦♥ ✇❡r❡ ❝♦♥s✐❞❡r❡❞ ♦❢ ♠❛r❣✐♥❛❧ ✐♥t❡r❡st✳ ❆s ✇❡ st❛t❡❞ ❜❡❢♦r❡✱ t❤✐s ♠❛②

✐♥ ♣❛rt✐❝✉❧❛r ❡①♣❧❛✐♥ ✇❤② s♦♠❡ ♦❢ t❤❡ ♣r♦❜❧❡♠s ✇❡ ✐♥s✐st ♦♥ ✐♥ t❤✐s ♣❛♣❡r ✇❡r❡ ♣✉t ❛s✐❞❡

✐♥ ✭❊❘✵✸✮✳ ❚❤❡ ♠❛t❡r✐❛❧ ♦❢ s❡❝t✐♦♥s ✷ ❛♥❞ ✸ ✇❛s t❤❡ s✉❜❥❡❝t ♦❢ t❤❡ ❘❚❆✬✵✼ ❝♦♥❢❡r❡♥❝❡

❡①t❡♥❞❡❞ ❛❜str❛❝t ✭❱❛✉✵✼❜✮✳ ❆❧t❤♦✉❣❤ ❛ ✈❡r② ❜r✐❡❢ ♦✉t❧✐♥❡ ♦❢ ❛ ♣r❡❧✐♠✐♥❛r② ✈❡rs✐♦♥ ♦❢

s❡❝t✐♦♥ ✺ ✇❛s ❣✐✈❡♥ ✐♥ t❤❛t ❧❛st ♣❛♣❡r✱ t❤❡ ♥♦r♠❛❧✐③❛t✐♦♥ r❡s✉❧ts ♦❢ t❤❡ ♣r❡s❡♥t ❛rt✐❝❧❡ ❛r❡

❝♦♠♣❧❡t❡❧② ♥❡✇✱ ✐♥ t❤❛t t❤❡② str✐❝t❧② ❣❡♥❡r❛❧✐③❡ t❤♦s❡ ♦❢ ✭❱❛✉✵✼❛✮✳

✷✳ ▲✐♥❡❛r ❈♦♠❜✐♥❛t✐♦♥s ♦❢ ❚❡r♠s

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ s❡t ♦❢ t❡r♠s ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝λ✲❝❛❧❝✉❧✉s ✐♥ s❡✈❡r❛❧ st❡♣s✳

❋✐rst ✇❡ ❣✐✈❡ ❛ ❣r❛♠♠❛r ♦❢ t❡r♠s✱ ♦♥ ✇❤✐❝❤ ✇❡ ❞❡✜♥❡α✲❡q✉✐✈❛❧❡♥❝❡ ❛♥❞ s✉❜st✐t✉t✐♦♥ ❛s

✐♥ ❑r✐✈✐♥❡✬s ✭❑r✐✾✵✮✳ ❚❤❡♥ ✇❡ ❞❡✜♥❡ ❛ ♥♦t✐♦♥ ♦❢ ❛❧❣❡❜r❛✐❝ ❡q✉❛❧✐t② ♦♥ t❤❡s❡ t❡r♠s✿ t❤✐s ✐s

❣✐✈❡♥ ❜② ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ,♦♥ t❡r♠s s✉❝❤ t❤❛t t❤❡ ❛ss♦❝✐❛t❡❞ q✉♦t✐❡♥t s❡t ✐s ❛♥

R✲♠♦❞✉❧❡✱ ♠♦r❡♦✈❡r ✈❛❧✐❞❛t✐♥❣ ✐❞❡♥t✐t✐❡s ✭✶✮ ❛♥❞ ✭✷✮✳ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ t❤✐s q✉♦t✐❡♥t s❡t

❛r❡ t❤❡ ♦❜❥❡❝ts ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝λ✲❝❛❧❝✉❧✉s✳ ❲❡ t❤❡♥ ✐♥tr♦❞✉❝❡ ❝❛♥♦♥✐❝❛❧ ❢♦r♠s ♦❢ t❡r♠s ❛s

❞✐st✐♥❣✉✐s❤❡❞ ❡❧❡♠❡♥ts ♦❢,✲❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✳ ❲❡ s❤♦✇ t❤✐s ❝♦♥str✉❝t✐♦♥ ❡♥❝♦♠♣❛ss❡s t❤❡ ❛❜str❛❝t ♣r❡s❡♥t❛t✐♦♥ ❜② ❊❤r❤❛r❞ ❛♥❞ ❘❡❣♥✐❡r ✐♥ ✭❊❘✵✸✮✱ ❜❛s❡❞ ♦♥ ❛♥ ✐♥❝r❡❛s✐♥❣

s❡q✉❡♥❝❡ ♦❢ q✉♦t✐❡♥ts✳

✷✳✶✳ ❘❛✇ ❚❡r♠s

▲❡t ❜❡ ❣✐✈❡♥ ❛ ❞❡♥✉♠❡r❛❜❧❡ s❡t V ♦❢ ✈❛r✐❛❜❧❡s✳ ❲❡ ✉s❡ ❧❡tt❡rs ❛♠♦♥❣ x, y, z t♦ ❞❡♥♦t❡

✈❛r✐❛❜❧❡s✳

❉❡✜♥✐t✐♦♥ ✷✳✶✳ ❚❤❡ ❧❛♥❣✉❛❣❡ L⁰_R ♦❢ t❤❡ r❛✇ t❡r♠s ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ λ✲❝❛❧❝✉❧✉s ♦✈❡r R

✭❞❡♥♦t❡❞ ❜② ❝❛♣✐t❛❧ ❧❡tt❡rsL, M, N✮ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♠♠❛r✿

M, N, . . .::=x|λx M |(M)N |0|aM |M +N .

❉❡✜♥✐t✐♦♥ ✷✳✷✳ ❲❡ ❞❡✜♥❡ ❢r❡❡ ✈❛r✐❛❜❧❡s ♦❢ t❡r♠s ❛s ❢♦❧❧♦✇s✿

✖ ✈❛r✐❛❜❧❡x✐s ❢r❡❡ ✐♥ t❡r♠y ✐❢x=y❀

✖ ✈❛r✐❛❜❧❡x✐s ❢r❡❡ ✐♥λy M ✐❢x6=y ❛♥❞x✐s ❢r❡❡ ✐♥M❀

✖ ✈❛r✐❛❜❧❡x✐s ❢r❡❡ ✐♥(M)N ✐❢x✐s ❢r❡❡ ✐♥M ♦r ✐♥N❀

✖ ✈❛r✐❛❜❧❡x✐s ❢r❡❡ ✐♥aM ✐❢x✐s ❢r❡❡ ✐♥M❀

✖ ✈❛r✐❛❜❧❡x✐s ❢r❡❡ ✐♥ t❡r♠M+N ✐❢x✐s ❢r❡❡ ✐♥M ♦r ✐♥N✳

❆❧❣❡❜r❛✐❝ ▲❛♠❜❞❛✲❈❛❧❝✉❧✉s ✺

■♥ ♣❛rt✐❝✉❧❛r✱ ♥♦ ✈❛r✐❛❜❧❡ ♦❝❝✉rs ❢r❡❡ ✐♥ t❡r♠ 0✳ ◆♦t✐❝❡ ❤♦✇❡✈❡r t❤❛t✱ ❜② t❤❡ ♣r❡✈✐♦✉s

❞❡✜♥✐t✐♦♥✱aM ♠✐❣❤t ❤❛✈❡ ❢r❡❡ ✈❛r✐❛❜❧❡s ❡✈❡♥ ✐❢a= 0✿ ❛s ❢❛r ❛s r❛✇ t❡r♠s ❛r❡ ❝♦♥❝❡r♥❡❞✱

0M ✐s ♥♦t t❤❡ s❛♠❡ ❛s 0✳

❋r♦♠ t❤✐s ❞❡✜♥✐t✐♦♥ ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s✱ ✇❡ ❞❡r✐✈❡ α✲❡q✉✐✈❛❧❡♥❝❡ ✭❞❡♥♦t❡❞ ❜② ∼✮ ❛s ✐♥

✭❑r✐✾✵✮✳ ❲❡ ✇✐❧❧ ❛❧✇❛②s ❝♦♥s✐❞❡r r❛✇ t❡r♠s ✉♣✲t♦α✲❡q✉✐✈❛❧❡♥❝❡✳ ▼♦r❡ ❢♦r♠❛❧❧②✿

❉❡✜♥✐t✐♦♥ ✷✳✸✳ ❚❤❡ s❡t LR ♦❢ t❤❡ r❛✇ t❡r♠s ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ λ✲❝❛❧❝✉❧✉s ♦✈❡r R✐s t❤❡

q✉♦t✐❡♥t s❡tL⁰_R/∼✳

❆❣❛✐♥✱ ✇❡ ❞❡r✐✈❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ s✉❜st✐t✉t✐♦♥ ❢♦❧❧♦✇✐♥❣ t❤❛t ✐♥ ✭❑r✐✾✵✮✳ ❲❡ ✇r✐t❡M[N/x]

❢♦r t❤❡ ✭❝❛♣t✉r❡✲❛✈♦✐❞✐♥❣✮ s✉❜st✐t✉t✐♦♥ ♦❢N ❢♦rx✐♥M✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✐❢x1, . . . , xn ❛r❡

❞✐st✐♥❝t ✈❛r✐❛❜❧❡s ❛♥❞N1, . . . , Nn ❛r❡ t❡r♠s✱ ✇❡ ✇r✐t❡M[N1, . . . , Nn/x1, . . . , xn]❢♦r t❤❡

s✐♠✉❧t❛♥❡♦✉s ❝❛♣t✉r❡ ❛✈♦✐❞✐♥❣ s✉❜st✐t✉t✐♦♥ ♦❢ ❡❛❝❤Ni ❢♦r ❡❛❝❤xi ✐♥M✳ ❲❡ ♦❜t❛✐♥ t❤❡

❢♦❧❧♦✇✐♥❣ ✈❛r✐❛♥ts ♦❢ ❞❡✜♥✐t✐♦♥s ❛♥❞ ♣r♦♣❡rt✐❡s ❢r♦♠ ✭❑r✐✾✵✮✳

Pr♦♣♦s✐t✐♦♥ ✷✳✹✳ ❋♦r ❛❧❧ t❡r♠s M, N1, . . . , Nn, L1, . . . , Lp ❛♥❞ ❛❧❧ ❞✐st✐♥❝t ✈❛r✐❛❜❧❡s x1, . . . , xn, y1, . . . , yp✱

M[N1, . . . , Nn/x1, . . . , xn] [L1, . . . , Lp/y1, . . . , yp]

∼ M[N₁^′, . . . , N_n^′, L1, . . . , Lp/x1, . . . , xn, y1, . . . , yp]

✇❤❡r❡Ni^′=Ni[L1, . . . , Lp/y1, . . . , yp]✳

❉❡✜♥✐t✐♦♥ ✷✳✺✳ ❆ ❜✐♥❛r② r❡❧❛t✐♦♥ r♦♥ r❛✇ t❡r♠s ✐s s❛✐❞ t♦ ❜❡ ❝♦♥t❡①t✉❛❧ ✐❢ ✐t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿

✖xrx❀

✖λx M rλx M^′ ❛s s♦♦♥ ❛s M rM^′❀

✖(M)Nr (M^′)N^′ ❛s s♦♦♥ ❛sM rM^′ ❛♥❞N rN^′❀

✖0r0❀

✖aM raM^′ ❛s s♦♦♥ ❛sM rM^′❀

✖M +NrM^′+N^′ ❛s s♦♦♥ ❛sM rM^′ ❛♥❞NrN^′✳

❚❤✐s ♥♦t✐♦♥ ♦❢ ❝♦♥t❡①t✉❛❧ r❡❧❛t✐♦♥ ✐s t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ ❛λ✲❝♦♠♣❛t✐❜❧❡ r❡❧❛t✐♦♥ ✐♥ ✭❑r✐✾✵✮✳

■♥ ♣❛rt✐❝✉❧❛r✱ ❛ ❜✐♥❛r② r❡❧❛t✐♦♥r✐s ❝♦♥t❡①t✉❛❧ ✐✛ ✐t ✐s r❡✢❡①✐✈❡ ❛♥❞✿

✖λx M rλx M^′ ❛s s♦♦♥ ❛s M rM^′❀

✖(M)Nr (M^′)N^′ ❛s s♦♦♥ ❛sM rM^′ ❛♥❞N rN^′❀

✖aM raM^′ ❛s s♦♦♥ ❛sM rM^′❀

✖M +NrM^′+N^′ ❛s s♦♦♥ ❛sM rM^′ ❛♥❞NrN^′✳

Pr♦♣♦s✐t✐♦♥ ✷✳✻✳ ■❢ r ✐s ❛ ❝♦♥t❡①t✉❛❧ r❡❧❛t✐♦♥✱ t❤❡♥ M[N/x] r M[N^′/x] ❛s s♦♦♥ ❛s N rN^′✳

❆❣❛✐♥✱ t❤✐s r❡s✉❧t ✐s ♦♥❧② ❛♥ ♦❜✈✐♦✉s ✈❛r✐❛♥t ♦❢ t❤❛t ♦❢ ✭❑r✐✾✵✮✳

▲✳ ❱❛✉① ✻

✷✳✷✳ ❚❤❡R✲▼♦❞✉❧❡ ♦❢ ❚❡r♠s

❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❛❝t✉❛❧ ❛❧❣❡❜r❛✐❝ ❝♦♥t❡♥t ♦❢ t❤❡ ❝❛❧❝✉❧✉s ❜② ❞❡✜♥✐♥❣ ❛♥ ❡q✉✐✈❛❧❡♥❝❡

r❡❧❛t✐♦♥,❡♥❝♦♠♣❛ss✐♥❣ ✉s✉❛❧ ✐❞❡♥t✐t✐❡s ❜❡t✇❡❡♥ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s✱ t♦❣❡t❤❡r ✇✐t❤ ✭✶✮

❛♥❞ ✭✷✮✳

❉❡✜♥✐t✐♦♥ ✷✳✼✳ ❆❧❣❡❜r❛✐❝ ❡q✉❛❧✐t② , ✐s ❞❡✜♥❡❞ ♦♥ r❛✇ t❡r♠s ❛s t❤❡ ❧❡❛st ❝♦♥t❡①t✉❛❧

❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s ❤♦❧❞✿

✖ ❛①✐♦♠s ♦❢ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞✿

0+M , M ✭✻❛✮

(M+N) +L , M+ (N+L) ✭✻❜✮

M +N , N+M ✭✻❝✮

✖ ❛①✐♦♠s ♦❢ ♠♦❞✉❧❡ ♦✈❡r r✐❣R✿

a(M +N) , aM+aN ✭✼❛✮

aM +bM , (a+b)M ✭✼❜✮

a(bM) , (ab)M ✭✼❝✮

1M , M ✭✼❞✮

0M , 0 ✭✼❡✮

a0 , 0 ✭✼❢✮

✖ ❧✐♥❡❛r✐t② ✐♥ t❤❡λ✲❝❛❧❝✉❧✉s✿

λx0 , 0 ✭✽❛✮

λx(aM) , a(λx M) ✭✽❜✮

λx(M+N) , λx M+λx N ✭✽❝✮

(0)L , 0 ✭✽❞✮

(aM)L , a((M)L) ✭✽❡✮

(M +N)L , (M)L+ (N)L ✭✽❢✮

❲❡ ❝❛❧❧ ❛❧❣❡❜r❛✐❝λ✲t❡r♠s t❤❡ ❡❧❡♠❡♥ts ♦❢LR/,✱ ✐✳❡✳ t❤❡,✲❝❧❛ss❡s ♦❢ r❛✇ t❡r♠s✳ ■❢M ∈LR✱

✇❡ ✇r✐t❡M ❢♦r ✐ts,✲❝❧❛ss✳

◆♦t✐❝❡ t❤❛t ✐❞❡♥t✐t② ✭✼❢✮ ❝♦✉❧❞ ❜❡ r❡♠♦✈❡❞✱ ❛s ✐t ✐s ❞❡r✐✈❡❞ ❢r♦♠ ✭✼❡✮ ❛♥❞ ✭✼❝✮✳ ■❞❡♥t✐t✐❡s

✭✽❛✮ t❤r♦✉❣❤ ✭✽❝✮ s✉❜s✉♠❡ ✭✶✮ ❛♥❞ ✐❞❡♥t✐t✐❡s ✭✽❞✮ t❤r♦✉❣❤ ✭✽❢✮ s✉❜s✉♠❡ ✭✷✮✳ ❚❤❡♥ t❤❡

q✉♦t✐❡♥t s❡tLR/,✐s ❛♥R✲♠♦❞✉❧❡ ✈❛❧✐❞❛t✐♥❣ ✭✶✮ ❛♥❞ ✭✷✮✳

❉❡✜♥✐t✐♦♥ ✷✳✽✳ ❋♦r ❛❧❧M1, . . . , Mn∈LR✱ ✇❡ ✇r✐t❡M1+· · ·+Mn♦r ❡✈❡♥Pn

i=1Mi ❢♦r t❤❡ t❡r♠M1+ (· · ·+Mn)✭♦r0✐❢n= 0✮✳

❖♥❡ ♠✐❣❤t t❤✐♥❦ ♦❢ ❛ r❛✇ t❡r♠M ∈LR❛s ❛ ✇r✐t✐♥❣ ♦❢ ✐ts,✲❝❧❛ss✱ ✇❤✐❝❤ ✐s ❛♥ ❡❧❡♠❡♥t

♦❢ t❤❡ R✲♠♦❞✉❧❡ L_R/,✳ ❆♠♦♥❣ r❛✇ t❡r♠s✱ s♦♠❡ s❤♦✉❧❞ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❛s ❝❛♥♦♥✐❝❛❧

✇r✐t✐♥❣s✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ✇❛♥t t♦ ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t ♠❡❛♥✐♥❣❢✉❧✿ ❡✈❡r②

❆❧❣❡❜r❛✐❝ ▲❛♠❜❞❛✲❈❛❧❝✉❧✉s ✼ t❡r♠ M ∈ L_R ❝❛♥ ❜❡ ✉♥✐q✉❡❧② ✇r✐tt❡♥ ❛s M , Pn

i=1aisi ✇❤❡r❡ t❤❡ si✬s ❛r❡ ♣❛✐r✇✐s❡

❞✐st✐♥❝t ❜❛s❡ ❡❧❡♠❡♥ts ❛♥❞ t❤❡ai✬s ❛r❡ ♥♦♥ ③❡r♦✳

❆ ❣♦♦❞ ❝❛♥❞✐❞❛t❡ ❢♦r s✉❝❤ ❛ ❝❛♥♦♥✐❝❛❧ ❜❛s❡ ✐s ♦❜t❛✐♥❡❞ ❛s ❢♦❧❧♦✇s✿

✖ ❛❧❧ t❤❡ ✐❞❡♥t✐t✐❡s ✐♥ ❣r♦✉♣s ♦❢ ❡q✉❛t✐♦♥s ✭✻✮ ✭✼✮ ❛♥❞ ✭✽✮✱ ❡①❝❡♣t ✭✻❝✮✱ ❝❛♥ ❜❡ ♦r✐❡♥t❡❞

❢r♦♠ ❧❡❢t t♦ r✐❣❤t t♦ ❢♦r♠ ❛ r❡✇r✐t❡ s②st❡♠❀

✖ r❛✇ t❡r♠s ✇❤✐❝❤ ❛r❡ ♥♦r♠❛❧ ✐♥ t❤✐s r❡✇r✐t❡ s②st❡♠✱ ❛♥❞ ❛r❡ ♦❢ t❤❡ s❤❛♣❡x✱λx M ♦r (M)N✱ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❜❛s❡ ❡❧❡♠❡♥ts ✭t❤❡② ❛r❡ ♥♦t s✉♠s✮❀

✖ ❡✈❡r② M ∈ LR ❤❛s ❛ ♥♦r♠❛❧ ❢♦r♠ ✐♥ t❤✐s s②st❡♠✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ❧✐♥❡❛r

❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❜❛s❡ t❡r♠s✳

◆♦t✐❝❡ ❤♦✇❡✈❡r t❤❛t ❛ ♥♦r♠❛❧ ❢♦r♠ ✐♥ t❤✐s s②st❡♠ ♥❡❡❞ ♥♦t ❜❡ ❝❛♥♦♥✐❝❛❧✿ ❝♦♥s✐❞❡r✱ ❡✳❣✳✱

x+y+x✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ♦❢ ❝♦✉rs❡ t❤❛t ✇❡ ❧❡❢t ♦✉t ❝♦♠♠✉t❛t✐✈✐t②✿ ❛❞❞✐♥❣ ✭✻❝✮ ✇♦✉❧❞

❜r❡❛❦ t❤❡ ✈❡r② ♥♦t✐♦♥ ♦❢ ♥♦r♠❛❧ ❢♦r♠✳ ❘❡✇r✐t✐♥❣ ✉♣ t♦ ❝♦♠♠✉t❛t✐✈✐t②✱ ♦r ✉♣ t♦ ❛ss♦❝✐❛✲

t✐✈✐t② ❛♥❞ ❝♦♠♠✉t❛t✐✈✐t②✱ ✐s ❛ ♥♦t❛❜❧❡ tr❡♥❞ ✐♥ r❡✇r✐t✐♥❣ t❤❡♦r②✱ ✇✐t❤ ✇❡❧❧✲❡st❛❜❧✐s❤❡❞

❧✐tt❡r❛t✉r❡✿ ❧❡t✬s ❥✉st ❝✐t❡ ✭P❙✽✶✮✳ ❊✈❡♥ ❝❧♦s❡r t♦ ♦✉r s✉❜❥❡❝t✱ ❆rr✐❣❤✐ ❛♥❞ ❉♦✇❡❦ ♣r♦♣♦s❡❞

✐♥ ✭❆❉✵✺✮ ❛♥ ❛ss♦❝✐❛t✐✈❡✕❝♦♠♠✉t❛t✐✈❡ r❡✇r✐t❡ s②st❡♠ ✐♠♣❧❡♠❡♥t✐♥❣ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧ ♥♦✲

t✐♦♥ ♦❢ ✈❡❝t♦r s♣❛❝❡✱ ✇❤✐❝❤ ✐s ✈❡r② ❝❧♦s❡ t♦ ✇❤❛t ✇❡ ❤❛✈❡ ❥✉st ♦✉t❧✐♥❡❞✳

■♥ t❤❡ ❝✉rr❡♥t s❡tt✐♥❣✱ ❤♦✇❡✈❡r✱ ♦✉r ❢♦❝✉s ✐s ♦♥ ♣r❡❝✐s✐♥❣ t❤❡ s②♥t❛① ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝

λ✲❝❛❧❝✉❧✉s✿ ✇❡ ❛r❡ ♦♥❧② ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝❛♥♦♥✐❝❛❧ ❢♦r♠s ❛♥❞ ❜❛s❡ ❡❧❡♠❡♥ts✳

❍❡♥❝❡ ✇❡ ❞♦ ♥♦t ❢✉❧❧② r❡♣r♦❞✉❝❡ s✉❝❤ ❛ r❡✇r✐t❡✲t❤❡♦r❡t✐❝ ❞❡✈❡❧♦♣♠❡♥t✳ ❲❡ r❛t❤❡r ❡①t❡♥❞

♦✉r ♥♦t✐♦♥ ♦❢ ❡q✉❛❧✐t② ♦❢ t❡r♠s ❛ ♠✐♥✐♠❛✱ s♦ t❤❛t t❤❡ ♦r❞❡r ♦❢ s✉♠♠❛♥❞s ✐♥Pn

i=1Mi ♥♦

❧♦♥❣❡r ♠❛tt❡rs✳ ❆s ❢❛r ❛s s②♥t❛① ✐s ❝♦♥❝❡r♥❡❞✱ t❤✐s ✐s q✉✐t❡ ❜❡♥✐❣♥✳ ▼♦r❡♦✈❡r✱ t❤❡ r❡❞✉❝t✐♦♥

♦❢ t❤❡ ❛❧❣❡❜r❛✐❝λ✲❝❛❧❝✉❧✉s✱ t♦ ❜❡ ❞❡✜♥❡❞ ✐♥ s❡❝t✐♦♥ ✸✱ ✐s ✐♥tr♦❞✉❝❡❞ ❛s ❛ r❡❧❛t✐♦♥ ♦♥LR/,✿

❛ss♦❝✐❛t✐✈✐t② ❛♥❞ ❝♦♠♠✉t❛t✐✈✐t② ✇✐❧❧ ❜❡ ❞✐ss♦❧✈❡❞ ✐♥,✳

❉❡✜♥✐t✐♦♥ ✷✳✾✳ P❡r♠✉t❛t✐✈❡ ❡q✉❛❧✐t② ≡ ⊆L_R×L_R ✐s t❤❡ ❧❡❛st ❝♦♥t❡①t✉❛❧ ❡q✉✐✈❛❧❡♥❝❡

r❡❧❛t✐♦♥ s✉❝❤ t❤❛t✱Pn

i=1Mi ≡ Pn

i=1Mf(i) ❤♦❧❞s✱ ❢♦r ❛❧❧ M1, . . . , Mn ∈LR ❛♥❞ ❛❧❧ ♣❡r✲

♠✉t❛t✐♦♥f ♦❢{1, . . . , n}✳

❙✐♥❝❡ ❢r❡❡ ✈❛r✐❛❜❧❡s ♦❢ ❛ s✉♠ ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♦r❞❡r ♦❢ t❤❡ s✉♠♠❛♥❞s✱≡♣r❡s❡r✈❡s

❢r❡❡ ✈❛r✐❛❜❧❡s✳

❉❡✜♥✐t✐♦♥ ✷✳✶✵✳ ❲❡ ✇r✐t❡ΛR ❢♦r t❤❡ q✉♦t✐❡♥t s❡tL_R/≡✱ ❛♥❞ ✇❡ ❝❛❧❧ ♣❡r♠✉t❛t✐✈❡ t❡r♠s t❤❡ ❡❧❡♠❡♥ts ♦❢ΛR✳

Pr♦♣♦s✐t✐♦♥ ✷✳✶✶✳ ❙✉❜st✐t✉t✐♦♥ ✐s ✇❡❧❧ ❞❡✜♥❡❞ ♦♥ ΛR✿ ✐❢ M, M^′ ∈ LR ❛r❡ s✉❝❤ t❤❛t M ≡ M^′ ❛♥❞✱ ❢♦r ❛❧❧ i ∈ {1, . . . , n}✱ Ni, Ni′ ∈ L_R ❛r❡ s✉❝❤ t❤❛t Ni ≡ Ni^′✱ t❤❡♥

M[N1, . . . , Nn/x1, . . . , xn] ≡ M^′[N₁^′, . . . , N_n^′/x1, . . . , xn] ❢♦r ❛❧❧ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ✈❛r✐✲

❛❜❧❡sx1, . . . , xn✳

❊①❝❡♣t ✇❤❡♥ st❛t❡❞ ♦t❤❡r✇✐s❡✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥ ❢♦r ❛ r❛✇ t❡r♠ M ❛♥❞

✐ts≡❝❧❛ss✱ ❛♥❞ ✉s❡ t❤❡♠ ✐♥t❡r❝❤❛♥❣❡❛❜❧②✳ ❚❤✐s ✐s ❤❛r♠❧❡ss ✐♥ ❣❡♥❡r❛❧✿ t❤❡ ♣r♦♣❡rt✐❡s ✇❡

❝♦♥s✐❞❡r ❛r❡ ❛❧❧ ✐♥✈❛r✐❛♥t ✉♥❞❡r ≡❛♥❞ ✇❡ ❞❡✜♥❡ ❢✉♥❝t✐♦♥s ♦♥ ΛR ❜② ✐♥❞✉❝t✐♦♥ ♦♥ r❛✇

t❡r♠s✱ ❝♦♠♣❛t✐❜✐❧✐t② ✇✐t❤≡❜❡✐♥❣ ♦❜✈✐♦✉s✳

◆♦t✐❝❡ ❛❧r❡❛❞② t❤❛t ❛❧❣❡❜r❛✐❝ ❡q✉❛❧✐t② s✉❜s✉♠❡s ♣❡r♠✉t❛t✐✈❡ ❡q✉❛❧✐t② ♦♥ r❛✇ t❡r♠s✱

s♦ t❤❛t,✐s ✇❡❧❧ ❞❡✜♥❡❞ ♦♥ΛR ❛♥❞(LR/,) = (ΛR/,)✳

▲✳ ❱❛✉① ✽

✷✳✸✳ ❈❛♥♦♥✐❝❛❧ ❋♦r♠s

❲❡ ❝❛♥ ♥♦✇ ❞❡✜♥❡ ❝❛♥♦♥✐❝❛❧ ❢♦r♠s ♦❢ t❡r♠s ❛s ♣❛rt✐❝✉❧❛r ♣❡r♠✉t❛t✐✈❡ t❡r♠s s✉❝❤ t❤❛t

❡✈❡r② ❝❧❛ss ✐♥ΛR/,❝♦♥t❛✐♥s ❡①❛❝t❧② ♦♥❡ ❝❛♥♦♥✐❝❛❧ ❡❧❡♠❡♥t✳

❉❡✜♥✐t✐♦♥ ✷✳✶✷✳ ❲❡ ❞❡✜♥❡ t❤❡ s❡t CR ⊂ ΛR ♦❢ ❝❛♥♦♥✐❝❛❧ t❡r♠s ✭❞❡♥♦t❡❞ ❜② ❝❛♣✐t❛❧

❧❡tt❡rsS✱T✱U✱V✱W✮ ❛♥❞ t❤❡ s❡tB_R⊂C_R ♦❢ ❜❛s❡ t❡r♠s ✭❞❡♥♦t❡❞ ❜② s♠❛❧❧ ❧❡tt❡rss✱t✱

u✱ v✱w✮ ❜② ♠✉t✉❛❧ ✐♥❞✉❝t✐♦♥ ❛s ❢♦❧❧♦✇s✿

✖ ❛♥② ✈❛r✐❛❜❧❡x✐s ❛ ❜❛s❡ t❡r♠❀

✖ ❧❡tx∈ V ❛♥❞s❛ ❜❛s❡ t❡r♠✱ t❤❡♥λx s✐s ❛ ❜❛s❡ t❡r♠❀

✖ ❧❡ts❛ ❜❛s❡ t❡r♠ ❛♥❞T ❛ ❝❛♥♦♥✐❝❛❧ t❡r♠✱ t❤❡♥(s)T ✐s ❛ ❜❛s❡ t❡r♠❀

✖ ❧❡t a1, . . . , an ∈ R^• ❛♥❞ s1, . . . , sn ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ❜❛s❡ t❡r♠s✱ t❤❡♥ Pn

i=1aisi ✐s ❛

❝❛♥♦♥✐❝❛❧ t❡r♠✳

❚❤❡ r❡❛❞❡r s❤♦✉❧❞ ❡❛s✐❧② ❣❡t t❤❡ ✐♥t✉✐t✐♦♥ t❤❛t ❢♦r ❛❧❧ ❝❛♥♦♥✐❝❛❧ t❡r♠sS, T ∈C_R✱ S,T

✐✛S=T ✭❛ ❢♦r♠❛❧ ♣r♦♦❢ ♦❢ t❤✐s r❡s✉❧t ❢♦❧❧♦✇s✱ ❛s ❛ ❝♦r♦❧❧❛r② ♦❢ ❚❤❡♦r❡♠ ✷✳✶✼✮✳ ▼❛♣♣✐♥❣

st♦ t❤❡ ✏s✐♥❣❧❡t♦♥✑ 1s❞❡✜♥❡s ❛♥ ✐♥❥❡❝t✐♦♥ ❢r♦♠ ❜❛s❡ t❡r♠s ✐♥t♦ ❝❛♥♦♥✐❝❛❧ t❡r♠s✳

❉❡✜♥✐t✐♦♥ ✷✳✶✸✳ ❲❡ ❞❡✜♥❡ t❤❡ ❤❡✐❣❤t ♦❢ ❜❛s❡ t❡r♠s ❛♥❞ ❝❛♥♦♥✐❝❛❧ t❡r♠s ❜② ♠✉t✉❛❧

✐♥❞✉❝t✐♦♥✿

✖h(x) = 1❀

✖h(λx s) = 1 + h(s)❀

✖h((s)T) = 1 + max(h(s),h(T))❀

✖h(Pn

i=1aisi) = max1≤i≤n(h(si))✭✇❤✐❝❤ ✐s ✵ ✐✛n= 0✮✳

❉❡✜♥✐t✐♦♥ ✷✳✶✹✳ ▲❡t M = Pn

i=1aisi ∈ ΛR ❜❡ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❜❛s❡ t❡r♠s✱

♥♦t ♥❡❝❡ss❛r✐❧② ❝❛♥♦♥✐❝❛❧✳ ❋♦r ❛❧❧ ❜❛s❡ t❡r♠ s✱ ✇❡ ❝❛❧❧ ❝♦❡✣❝✐❡♥t ♦❢ s ✐♥ M t❤❡ s❝❛❧❛r P

1≤i≤n, s_i=sai✭t❤❡ s✉♠ ♦❢ t❤♦s❡ ai✬s s✉❝❤ t❤❛tsi=s✮✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜②M(s)✳ ❚❤❡♥

✇❡ ❞❡✜♥❡cansum(M)∈C_R ❜②✿

cansum(M) = Xp j=1

M(t_j)tj

✇❤❡r❡{t1, . . . , tp} ✐s t❤❡ s❡t ♦❢ t❤♦s❡si✬s ✇✐t❤ ❛ ♥♦♥✲③❡r♦ ❝♦❡✣❝✐❡♥t ✐♥M✳

❲❡ ♥♦✇ ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ ♠❛♣♣✐♥❣ t❡r♠s ✐♥ΛR t♦ t❤❡✐r ❝❛♥♦♥✐❝❛❧ ❢♦r♠s✳

❉❡✜♥✐t✐♦♥ ✷✳✶✺✳ ❈❛♥♦♥✐③❛t✐♦♥ ♦❢ t❡r♠scan: ΛR−→CR ✐s ❣✐✈❡♥ ❜②

✖can(x) = 1x❀

✖ ✐❢can(M) =Pn

i=1aisi t❤❡♥can(λx M) =Pn

i=1ai(λx si)❀

✖ ✐❢can(M) =Pn

i=1aisi ❛♥❞can(N) =T t❤❡♥can((M)N) =Pn

i=1ai(si)T❀

✖can(0) =0❀

✖ ✐❢can(M) =Pn

i=1aisi t❤❡♥can(aM) =cansum(Pn

i=1(aai)si)❀

✖ ✐❢can(M) =Pn

i=1aisi ❛♥❞can(N) =Pn+p

i=n+1aisi t❤❡♥

can(M +N) =cansum

n+pX

i=1

aisi

! .

❆❧❣❡❜r❛✐❝ ▲❛♠❜❞❛✲❈❛❧❝✉❧✉s ✾

◆♦t✐❝❡ t❤❛t ✐♥ t❤❡ ♣❡♥✉❧t✐♠❛t❡ ❝❛s❡ ✭❞❡✜♥✐t✐♦♥ ♦❢can(aM)✮ t❤❡ ♦♥❧② ❡✛❡❝t ♦❢ t❤❡ ❛♣♣❧✐✲

❝❛t✐♦♥ ♦❢cansum✐s t♦ ♣r✉♥❡ ❛❧❧ t❤❡ s✉♠♠❛♥❞s(aai)si s✉❝❤ t❤❛taai= 0✳

▲❡♠♠❛ ✷✳✶✻✳ ❈❛♥♦♥✐③❛t✐♦♥ ❡♥❥♦②s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✳

✭✐✮ ❱❛r✐❛❜❧❡s ❢r❡❡ ✐♥can(M)❛r❡ ❛❧s♦ ❢r❡❡ ✐♥M✳ ❚❤❡ ❝♦♥✈❡rs❡ ❞♦❡s ♥♦t ❤♦❧❞ ✐♥ ❣❡♥❡r❛❧✳

✭✐✐✮ ❋♦r ❛❧❧ ❜❛s❡ t❡r♠s✱can(s) = 1s✳

✭✐✐✐✮ ❋♦r ❛❧❧ ❝❛♥♦♥✐❝❛❧ t❡r♠S✱can(S) =S✳

✭✐✈✮ ❋♦r ❛❧❧ t❡r♠M ∈ΛR✱can(can(M)) =can(M)✳

✭✈✮ ❋♦r ❛❧❧M, N1, . . . , Nn∈ΛR❛♥❞ ❛❧❧ ✈❛r✐❛❜❧❡sx1, . . . , xn ♥♦t ❢r❡❡ ✐♥ ❛♥② ♦❢ t❤❡s❡ t❡r♠s✱

can(M[N1, . . . , Nn/x1, . . . , xn]) =can(can(M) [can(N1), . . . ,can(Nn)/x1, . . . , xn])✳

Pr♦♦❢✳ ❋❛❝t ✭✐✮ ✐s str❛✐❣t❤❢♦r✇❛r❞ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ❞❡✜♥✐t✐♦♥✳ ❋❛❝ts ✭✐✐✮ ❛♥❞ ✭✐✐✐✮ ❛r❡

♣r♦✈❡❞ ❜② ♠✉t✉❛❧ ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ ❜❛s❡ t❡r♠s ❛♥❞ ❝❛♥♦♥✐❝❛❧ t❡r♠s❀ ❢❛❝t

✭✐✈✮ ❢♦❧❧♦✇s ❢r♦♠ ✭✐✐✐✮✳ ❋❛❝t ✭✈✮ ✐s ♣r♦✈❡❞ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ M✿ ❛❧❧ ✐♥❞✉❝t✐✈❡ st❡♣s ❢♦❧❧♦✇

❞✐r❡❝t❧② ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ ❝❛♥♦♥✐③❛t✐♦♥ ❛♥❞ s✉❜st✐t✉t✐♦♥✳

❚❤❡♦r❡♠ ✷✳✶✼✳ ❆❧❣❡❜r❛✐❝ ❡q✉❛❧✐t② ✐s ❡q✉❛❧✐t② ♦❢ ❝❛♥♦♥✐❝❛❧ ❢♦r♠s✿ ❢♦r ❛❧❧ M, N ∈ ΛR

M ,N ✐✛can(M) =can(N)✳

Pr♦♦❢✳ ❋♦r ❛❧❧M, N ∈ΛR✱ ✇❡ ✇r✐t❡M ,^′ N ✐✛can(M) =can(N)✳ ■t s❤♦✉❧❞ ❜❡ ❝❧❡❛r t❤❛t,^′✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳ ■t ✐s ❝♦♥t❡①t✉❛❧ ❜❡❝❛✉s❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝❛♥♦♥✐③❛t✐♦♥

✐s ❜② ✐♥❞✉❝t✐♦♥ ♦♥ ♣❡r♠✉t❛t✐✈❡ t❡r♠s✳ ■t ♠♦r❡♦✈❡r ✈❛❧✐❞❛t❡s ❡q✉❛t✐♦♥s ✭✻❛✮ t❤r♦✉❣❤ ✭✽❢✮✿

❥✉st ❛♣♣❧②cant♦ ❜♦t❤ ♠❡♠❜❡rs ♦❢ ❡❛❝❤ ❡q✉❛t✐♦♥ ❛♥❞ ❝♦♥❝❧✉❞❡✳ ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ,✱

✇❡ ❣❡t , ⊆ ,^′✳ ❈♦♥✈❡rs❡❧②✱ ♦♥❡ ❝❛♥ ❡❛s✐❧② ❝❤❡❝❦ t❤❛t can(M) , M ❢♦r ❛❧❧ M ∈ ΛR✿ t❤✐s ✐s t❤❡ ✇❤♦❧❡ ♣♦✐♥t ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝❛♥♦♥✐③❛t✐♦♥✳ ❍❡♥❝❡ t❤❡ r❡✈❡rs❡ ✐♥❝❧✉s✐♦♥✿ ✐❢

M ,^′N✱ t❤❡♥M ,can(M) =can(N),N✳

❈♦r♦❧❧❛r② ✷✳✶✽✳ ❋♦r ❛❧❧S, T∈CR✱S,T ✐✛S=T✳

Pr♦♦❢✳ ❚❤✐s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠ ❛♥❞ ❢❛❝t ✭✐✐✐✮ ♦❢ ▲❡♠♠❛ ✷✳✶✻✳

❈♦r♦❧❧❛r② ✷✳✶✾✳ ❙✉❜st✐t✉t✐♦♥ ✐s ✇❡❧❧ ❞❡✜♥❡❞ ♦♥ ΛR/,✿ ✐❢ M, M^′ ∈ ΛR ❛r❡ s✉❝❤ t❤❛t M , M^′ ❛♥❞✱ ❢♦r ❛❧❧ i ∈ {1, . . . , n}✱ Ni, Ni′ ∈ ΛR ❛r❡ s✉❝❤ t❤❛t Ni , N_i^′✱ t❤❡♥

M[N1, . . . , Nn/x1, . . . , xn] , M^′[N₁^′, . . . , Nn^′/x1, . . . , xn] ❢♦r ❛❧❧ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ✈❛r✐✲

❛❜❧❡sx1, . . . , xn✳

Pr♦♦❢✳ ❋✐rst ❛♣♣❧② ❚❤❡♦r❡♠ ✷✳✶✼ t♦ t❤❡ ❤②♣♦t❤❡s❡s ❛♥❞ ❝♦♥❝❧✉s✐♦♥✿ ✇❡ ♠✉st ♣r♦✈❡

can(M[N1, . . . , Nn/x1, . . . , xn]) =can(M^′[N₁^′, . . . , Nn^′/x1, . . . , xn])

❦♥♦✇✐♥❣ t❤❛t can(M) = can(M^′) ❛♥❞✱ ❢♦r ❛❧❧ i ∈ {1, . . . , n}✱ can(Ni) = can(N_i^′)✳ ❲❡

❝♦♥❝❧✉❞❡ ❜② ❢❛❝t ✭✈✮ ♦❢ ▲❡♠♠❛ ✷✳✶✻✳

❈♦r♦❧❧❛r② ✷✳✷✵✳ ❲❡ ❝❛♥ ❞❡✜♥❡ ❛♥R✲♠♦❞✉❧❡ str✉❝t✉r❡ ♦♥CR ❛s ❢♦❧❧♦✇s✿

③❡r♦✿ 0_can=0

s✉♠✿ S⊕T =can(S+T)

❡①t❡r♥❛❧ ♣r♦❞✉❝t✿ a⊗S=can(aS)❀

▲✳ ❱❛✉① ✶✵

s♦ t❤❛tcan✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢R✲♠♦❞✉❧❡s ❢r♦♠ΛR/,t♦C_R✳

Pr♦♦❢✳ ❇② ❚❤❡♦r❡♠ ✷✳✶✼✱can✐s ✇❡❧❧ ❞❡✜♥❡❞ ♦♥ΛR/,✱ ❛♥❞ ✐s ✐♥❥❡❝t✐✈❡✳ ■t ✐s s✉r❥❡❝t✐✈❡

❜② ❢❛❝t ✭✐✐✐✮ ♦❢ ▲❡♠♠❛ ✷✳✶✻✳ ❚❤❡ R✲♠♦❞✉❧❡ str✉❝t✉r❡ ♦❢ CR t❤❡♥ ❢♦❧❧♦✇s ❢r♦♠ t❤❛t ♦❢

ΛR/,✳

❇② t❤✐s ✐s♦♠♦r♣❤✐s♠✱ ❛♥❞,❜❡✐♥❣ ❝♦♥t❡①t✉❛❧✱ t❤❡ q✉♦t✐❡♥t str✉❝t✉r❡ ♦❢ ❛❧❣❡❜r❛✐❝ t❡r♠s

✐s s✉❜s✉♠❡❞ ❜② t❤❡ ♠✉t✉❛❧❧② ✐♥❞✉❝t✐✈❡ str✉❝t✉r❡ ♦❢ ❜❛s❡ t❡r♠s ❛♥❞ ❝❛♥♦♥✐❝❛❧ t❡r♠s✳ ■❢C

✐s ❛ s❡t ♦❢ ❝❛♥♦♥✐❝❛❧ t❡r♠s✱ ✇❡ ✇r✐t❡C={S; S ∈ C}❀ t❤❡♥(ΛR/,) =CR✳ ❲❤❡♥ ✇❡ ♣r♦✈❡

♣r♦♣❡rt✐❡s ♦♥ ❛❧❣❡❜r❛✐❝ t❡r♠s✱ ✇❡ ❝❛♥ t❤✉s ✉s❡ ✐♥❞✉❝t✐♦♥ ♦♥ ❜❛s❡ t❡r♠s ❛♥❞ ❝❛♥♦♥✐❝❛❧

t❡r♠s✿ ✇❡ t❤❡♥ ❝❤❡❝❦ t❤❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦♣❡rt② ♦♥ ❛❧❣❡❜r❛✐❝ t❡r♠s ❢♦❧❧♦✇s t❤r♦✉❣❤

can✱ ✇❤✐❝❤ ✐s ✐♥ ❣❡♥❡r❛❧ ♦❜✈✐♦✉s✳ ❲❡ ✇✐❧❧ ❛❜✉s❡ t❡r♠✐♥♦❧♦❣② ❜② ❝❧❛✐♠✐♥❣ ♦✉r ♣r♦♦❢ ✐s ❜②

✐♥❞✉❝t✐♦♥ ♦♥ ❛❧❣❡❜r❛✐❝ t❡r♠s✳ ❆❧s♦✱ ✇❡ ✇✐❧❧ ♦❢t❡♥ ❞❡✜♥❡ ❢✉♥❝t✐♦♥s ♦♥ΛR/,❜② ✐♥❞✉❝t✐♦♥

♦♥ ❜❛s❡ t❡r♠s ❛♥❞ ❝❛♥♦♥✐❝❛❧ t❡r♠s✿ t❤❡ ❛❝t✉❛❧ ❢✉♥❝t✐♦♥ ✐s ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣♦s✐t✐♦♥ ✇✐t❤

can✳ ❋♦r ✐♥st❛♥❝❡✱ ✇❡ ❞❡✜♥❡ t❤❡ ❤❡✐❣❤t ♦❢ ❛❧❣❡❜r❛✐❝ t❡r♠s ❜②✿h(M) = h(can(M))✳

✷✳✹✳ ❆❜str❛❝t ♣r❡s❡♥t❛t✐♦♥

❖✉r ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡R✲♠♦❞✉❧❡ ♦❢ t❡r♠s ❞✐✛❡rs ❢r♦♠ t❤❛t ❜② ❊❤r❤❛r❞ ❛♥❞ ❘❡❣♥✐❡r ✐♥

✭❊❘✵✸✮✱ ✐♥ t❤❛t ✇❡ ✐♥tr♦❞✉❝❡ ❡①♣❧✐❝✐t❧② t✇♦ ❞✐♥st✐♥❝t ❧❡✈❡❧s ♦❢ s②♥t❛①✿ ♣❡r♠✉t❛t✐✈❡ t❡r♠s

♦♥ t❤❡ ♦♥❡ ❤❛♥❞ ✭ΛR✮ ❛♥❞ ❛❧❣❡❜r❛✐❝ t❡r♠s ✭ΛR/,✮ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✳

❖♥❡ ❝❛♥ s❡❡ t❤❡ R✲♠♦❞✉❧❡ ♦❢ ❝❛♥♦♥✐❝❛❧ t❡r♠s ❢r♦♠ ❈♦r♦❧❧❛r② ✷✳✷✵ ❛s ❛ ❝♦♥❝r❡t❡ ♣r❡✲

s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♦♥❡ ❛❞♦♣t❡❞ ❜② ❊❤r❤❛r❞ ❛♥❞ ❘❡❣♥✐❡r✿ ❞❡✜♥❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡

(Rh∆R(k)i)k≥0 ♦❢ ❢r❡❡R✲♠♦❞✉❧❡s ❣❡♥❡r❛t❡❞ ❜② s✐♠♣❧❡ t❡r♠s ♦❢ ❜♦✉♥❞❡❞ ❤❡✐❣❤t✳

❉❡✜♥✐t✐♦♥ ✷✳✷✶✳ ❲❡ ❞❡✜♥❡ t❤❡ s❡t ∆R(k) ♦❢ s✐♠♣❧❡ t❡r♠s ♦❢ ❤❡✐❣❤t ❛t ♠♦st k✱ ❜②

✐♥❞✉❝t✐♦♥ ♦♥k✿ ❧❡t∆R(0) =∅❀ ✇❡ ❞❡✜♥❡ t❤❡ ❡❧❡♠❡♥ts ♦❢∆R(k+ 1)❢r♦♠ t❤♦s❡ ♦❢∆R(k)

❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛✉s❡s✿

✖ ✐❢σ∈∆R(k)t❤❡♥σ∈∆R(k+ 1)❀

✖ ✐❢x∈ V t❤❡♥x∈∆R(k+ 1)❀

✖ ✐❢σ∈∆R(k)t❤❡♥λx σ∈∆R(k+ 1)❀

✖ ✐❢σ∈∆R(k)❛♥❞τ∈Rh∆R(k)it❤❡♥(σ)τ ∈∆R(k+ 1)✳

❚❤❡♥ ✇❡ ❞❡✜♥❡ t❤❡ s❡t ♦❢ ❛❧❧ s✐♠♣❧❡ t❡r♠s ❛s ∆R = S

k∆R(k) ❛♥❞ t❤❡ s❡t ♦❢ t❡r♠s Rh∆Ri=S

kRh∆R(k)i✳

◆♦t✐❝❡ t❤❛t✱ ❛❧t❤♦✉❣❤ ✐t ✐s ♥♦t ♠❛❞❡ ❝❧❡❛r ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ♣❛♣❡r✱ t✇♦ q✉♦t✐❡♥t ❝♦♥str✉❝✲

t✐♦♥s ❛r❡ ✐♥t❡r❧❡❛✈❡❞ ❛t ❡❛❝❤ ❤❡✐❣❤t✿ α✲❡q✉✐✈❛❧❡♥❝❡ ❛♥❞ t❤❡ ❢r❡❡ R✲♠♦❞✉❧❡ ❝♦♥str✉❝t✐♦♥✳

■♥ ♦✉r ♦♣✐♥✐♦♥✱ t❤✐s ♠❛❦❡s ❢♦r ❛ ✈❡r② ✐♥tr✐❝❛t❡ ♥♦t✐♦♥ ♦❢ ❡q✉❛❧✐t② ♦♥ t❡r♠s✱ s♦ t❤❛t t❤❡

st❛t✉s ♦❢ ♣r♦♠✐♥❡♥t ❛♥❞ ✇❡❧❧✲❡st❛❜❧✐s❤❡❞ ♥♦t✐♦♥s ✐♥ t❤❡ s❡tt✐♥❣ ♦❢ t❤❡ ♦r❞✐♥❛r②λ✲❝❛❧❝✉❧✉s

❜❡❝♦♠❡s ❧❡ss ✐♠♠❡❞✐❛t❡✿ ❢♦r ✐♥st❛♥❝❡✱ ✇❤❛t ✐s ❛ ❢r❡❡ ♦❝❝✉rr❡♥❝❡ ♦❢ ✈❛r✐❛❜❧❡ ✐♥ ❛ t❡r♠✱

❤♦✇ ❞♦ ✇❡ ❞❡✜♥❡ ♣r♦♣❡r❧② α✲❝♦♥✈❡rs✐♦♥ ♦♥ Rh∆Ri✱ ✇❤❛t ❛r❡ t❤❡ s✉❜t❡r♠s ♦❢ ❛ t❡r♠❄

❖❢ ❝♦✉rs❡✱ t❤❡s❡ q✉❡st✐♦♥s ❝❛♥ ❜❡ ❣✐✈❡♥ s❛t✐s❢❛❝t♦r② ❛♥s✇❡rs✿ ✇❡ ♦♥❧② ❝❧❛✐♠ t❤❛t t❤❡

s✐♠♣❧✐❝✐t② ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ✐s ♦♥❧② ❛♣♣❛r❡♥t✳

❆s ❡①♣❡❝t❡❞✱Rh∆_Ri ❛♥❞(ΛR/,) ❛r❡ ❛❝t✉❛❧❧② t❤❡ s❛♠❡R✲♠♦❞✉❧❡ ♦❢ ❛❧❣❡❜r❛✐❝ t❡r♠s✿

❞❡✜♥❡ BR(k) ✭r❡s♣✳ CR(k)✮ ❛s t❤❡ s❡t ♦❢ ❜❛s❡ t❡r♠s ✭r❡s♣✳ ❝❛♥♦♥✐❝❛❧ t❡r♠s✮ ♦❢ ❤❡✐❣❤t

❆❧❣❡❜r❛✐❝ ▲❛♠❜❞❛✲❈❛❧❝✉❧✉s ✶✶

❛t ♠♦st k❀ t❤❡♥✱ ❝❧❡❛r❧②✱ ∆R(k) ✐s B_R(k) ❛♥❞ Rh∆R(k)i ✐s C_R(k)✳ ❍❡♥❝❡ ∆R = B_R ❛♥❞

Rh∆_Ri=C_R = (ΛR/,)✳ ❚❤✐s ✐s ♦♥❡ ✐♠♣♦rt❛♥t ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤❡ ♣r❡s❡♥t ♣❛♣❡r✿ ❜r✐♥❣

♥❡✇ ❧✐❣❤t ♦♥ t❤❡ str✉❝t✉r❡ ♦❢Rh∆Ri✱ ❜② ❞❡❧✐❜❡r❛t❡❧② ✐♥tr♦❞✉❝✐♥❣α✲❡q✉✐✈❛❧❡♥❝❡ ❛♥❞ ♣❡r✲

♠✉t❛t✐✈❡ ❡q✉❛❧✐t② s❡♣❛r❛t❡❧② ❢r♦♠ ❡q✉❛❧✐t② ♦❢ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ✭✐✳❡✳ ❛❧❣❡❜r❛✐❝ ❡q✉❛❧✐t②✮✳

❆❧s♦✱ t❤✐s ♠❛❦❡s ♣r♦♠✐♥❡♥t t❤❡ ❢❛❝t t❤❛t t❤❡ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝λ✲❝❛❧❝✉❧✉s ✐s ❞❡✲

✜♥❡❞ ✉♣ t♦,✭s❡❡ ♥❡①t s❡❝t✐♦♥✮✳

❙♦✱ ❢r♦♠ ♥♦✇ ♦♥✱ ✇❡ ❢♦r♠❛❧❧② ✐❞❡♥t✐❢②∆R ✇✐t❤ BR ❛♥❞ Rh∆Ri✇✐t❤ CR ❜② r❡♣❧❛❝✐♥❣

❉❡✜♥✐t✐♦♥ ✷✳✷✶ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♥❡✿

❉❡✜♥✐t✐♦♥ ✷✳✷✷✳ ❲❡ ❞❡✜♥❡ s✐♠♣❧❡ t❡r♠s ❛s t❤❡,✲❝❧❛ss❡s ♦❢ ❜❛s❡ t❡r♠s✳ ❲❡ ✇r✐t❡∆R

❢♦r t❤❡ s❡t ♦❢ s✐♠♣❧❡ t❡r♠s ❛♥❞Rh∆_Ri❢♦r t❤❡ s❡t ♦❢ ❛❧❣❡❜r❛✐❝ t❡r♠s✱ ✇❤✐❝❤ ✇❡ ♠❛② ❥✉st

❝❛❧❧ t❡r♠s✳

❲❤❡♥ ✇❡ ✇r✐t❡ ❛ s✐♠♣❧❡ t❡r♠ ✭r❡s♣✳ ❛ t❡r♠✮ ❛ss✱t✱u✱v ♦rw✱ ✭r❡s♣✳S✱T✱U✱V ♦r W✮✱

✐t ✐s ✐♠♣❧✐❝✐t t❤❛ts✱t✱u✱v✱ ♦r w✐s ❛ ❜❛s❡ t❡r♠ ✭r❡s♣✳S✱ T✱U✱ V✱ ♦rW ✐s ❛ ❝❛♥♦♥✐❝❛❧

t❡r♠✮✳ ❲❤❡♥ ✇❡ ♠❛❦❡ ♥♦ s✉❝❤ ❛ss✉♠♣t✐♦♥✱ ✇❡ ✇r✐t❡L✱ M ♦r N ♦r ✉s❡ ❣r❡❡❦ ❧❡tt❡rsσ✱

τ✱ρ✳ ❲❡ ✇✐❧❧ ♦❢t❡♥ ✉s❡ t❤❡ ♥♦t❛t✐♦♥sλx σ✱(σ)τ✱aσ✱σ+τ ✇✐t❤ t❤❡ ♦❜✈✐♦✉s s❡♥s❡✿ t❤❡s❡

❛r❡ ✇❡❧❧ ❞❡✜♥❡❞ ❜② ❝♦♥t❡①t✉❛❧✐t② ♦❢,✳

❉❡✜♥✐t✐♦♥ ✷✳✷✸✳ ❋♦r ❛❧❧ S ∈ Rh∆_Ri ❛♥❞ s ∈ ∆R✱ ✇❡ ❞❡✜♥❡ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ s ✐♥ S

❜② S_(s) = S(s)✳ ❲❡ t❤❡♥ ❞❡✜♥❡ t❤❡ s✉♣♣♦rt ♦❢ S ❛s t❤❡ s❡t ♦❢ ❛❧❧ s✐♠♣❧❡ t❡r♠s ✇✐t❤ ❛

♥♦♥✲③❡r♦ ❝♦❡✣❝✐❡♥t ✐♥S✿

Supp(S) =n

s∈∆R; S_(s)6= 0o .

■❢S ✐s ❛ s❡t ♦❢ s✐♠♣❧❡ t❡r♠s✱ ✇❡ ✇r✐t❡RhSi❢♦r t❤❡ s❡t ♦❢ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❡❧❡♠❡♥ts

♦❢S✱ ✐✳❡✳

RhSi=



 Xn i=1

aisi; ∀i∈ {1, . . . , n}, si∈ S, ai∈R





♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱RhSi={σ∈Rh∆_Ri; Supp(σ)⊆ S}✳

✸✳ ❘❡❞✉❝t✐♦♥s

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❞❡✜♥❡ r❡❞✉❝t✐♦♥ ✉s✐♥❣ ✭✸✮ ❛♥❞ ✭✹✮ ❛s ❦❡② r❡❞✉❝t✐♦♥ r✉❧❡s✿ t❤✐s ❝❛♣t✉r❡s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ r❡❞✉❝t✐♦♥ ✐♥ ✭❊❘✵✸✮✱ ♠✐♥✉s ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ✐♥ t❤❡ s❡tt✐♥❣ ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝

λ✲❝❛❧❝✉❧✉s✳

✸✳✶✳ ❘❡❞✉❝t✐♦♥ ❛♥❞ ▲✐♥❡❛r ❈♦♠❜✐♥❛t✐♦♥s ♦❢ ❚❡r♠s

❲❡ ❝❛❧❧ r❡❧❛t✐♦♥ ❢r♦♠ s✐♠♣❧❡ t❡r♠s t♦ t❡r♠s ❛♥② s✉❜s❡t ♦❢ ∆R ×Rh∆_Ri✱ ❛♥❞ ✇❡ ❝❛❧❧

r❡❧❛t✐♦♥ ❢r♦♠ t❡r♠s t♦ t❡r♠s ❛♥② s✉❜s❡t ♦❢ Rh∆Ri ×Rh∆Ri✳ ●✐✈❡♥ ❛ r❡❧❛t✐♦♥ r ❢r♦♠

s✐♠♣❧❡ t❡r♠s t♦ t❡r♠s ✇❡ ❞❡✜♥❡ t✇♦ ♥❡✇ r❡❧❛t✐♦♥sr❛♥❞er❢r♦♠ t❡r♠s t♦ t❡r♠s ❜②✿

✖σrσ^′ ✐❢σ=Pn

i=1aisi ❛♥❞σ^′=Pn

i=1aiS_i^′ ✇❤❡r❡✱ ❢♦r ❛❧❧i∈ {1, . . . , n}✱si rS_i^′❀

✖σerσ^′ ✐❢σ=as+T ❛♥❞σ^′=aS^′+T ✇❤❡r❡a6= 0❛♥❞srS^′✳

▲✳ ❱❛✉① ✶✷

❈❧❡❛r❧②✱er⊆r✳ ■t ✐s ✐♠♣♦rt❛♥t t❤❛t✱ ✐♥ t❤❡ ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥s ✇❡ ❞♦ ♥♦t r❡q✉✐r❡Pn i=1aisi

♥♦ras+Tt♦ ❜❡ ❝❛♥♦♥✐❝❛❧ t❡r♠s✿er♠❛t❝❤❡s ❡q✉❛t✐♦♥ ✭✹✮✱ ✇❤✐❧❡r♠❛t❝❤❡s ✭✺✮✳ ❲❡ ✇✐❧❧ ✉s❡

t❤❡s❡ ❝♦♥str✉❝t✐♦♥s ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ ♦♥❡✲st❡♣β✲r❡❞✉❝t✐♦♥→❛♥❞ ♣❛r❛❧❧❡❧ r❡❞✉❝t✐♦♥⇒✿

✇❡ ✐♥tr♦❞✉❝❡ t❤❡s❡ ❛s r❡❧❛t✐♦♥s ❢r♦♠ s✐♠♣❧❡ t❡r♠s t♦ t❡r♠s✱ s♦ t❤❛t t❤❡ ❛❝t✉❛❧ r❡❞✉❝t✐♦♥

r❡❧❛t✐♦♥s ♦♥ t❡r♠s ❛r❡ ♦❜t❛✐♥❡❞ ❛sf→❛♥❞⇒r❡s♣❡❝t✐✈❡❧②✳

◆♦t✐❝❡ t❤❛t ✇❡ ❝❛♥♥♦t ❞❡✜♥❡ r❡❞✉❝t✐♦♥ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ t❡r♠s✿ ✐❢ t❤❡r❡ ❛r❡a, b ∈R^• s✉❝❤ t❤❛ta+b= 0t❤❡♥0=aσ+bσ❢♦r ❛❧❧σ∈Rh∆_Ri❀ ❤❡♥❝❡✱ ❜② r✉❧❡ ✭✹✮✱0♠❛② r❡❞✉❝❡✳

❋♦❧❧♦✇✐♥❣ ✭❊❘✵✸✮✱ ✇❡ r❛t❤❡r ❞❡✜♥❡ s✐♠♣❧❡ t❡r♠ r❡❞✉❝t✐♦♥→❜② ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ❞❡♣t❤

♦❢ t❤❡ ✜r❡❞ r❡❞❡①✳

❉❡✜♥✐t✐♦♥ ✸✳✶✳ ❲❡ ❞❡✜♥❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ r❡❧❛t✐♦♥s ❢r♦♠ s✐♠♣❧❡ t❡r♠s t♦ t❡r♠s

❜② t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts✳ ▲❡t→0 ❜❡ t❤❡ ❡♠♣t② r❡❧❛t✐♦♥ ∅ ⊆ ∆R×Rh∆Ri✳ ❆ss✉♠❡

→k ✐s ❞❡✜♥❡❞✳ ❚❤❡♥ ✇❡ s❡tσ→k+1σ^′ ❛s s♦♦♥ ❛s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿

✖σ=λx s❛♥❞σ=λx S^′ ✇✐t❤s→k S^′❀

✖σ= (s)T ❛♥❞σ^′ = (S^′)T ✇✐t❤s→kS^′✱ ♦r σ^′= (s)T^′ ✇✐t❤T g→k T^′❀

✖σ= (λx s)T ❛♥❞σ^′=s[T /x]✳

▲❡t→=S

k∈N→k✳ ❲❡ ❝❛❧❧ ♦♥❡✲st❡♣ r❡❞✉❝t✐♦♥ ♦r s✐♠♣❧② r❡❞✉❝t✐♦♥✱ t❤❡ r❡❧❛t✐♦♥f→✳

▲❡♠♠❛ ✸✳✷✳ ❲❡ ❤❛✈❡f→=S

k∈Ng→k✳

Pr♦♦❢✳ ❚❤✐s ✐s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ♠♦r❡ ❣❡♥❡r❛❧ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ e·✿ ✐❢(rn)✐s ❛♥

✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ r❡❧❛t✐♦♥s ❢r♦♠ s✐♠♣❧❡ t❡r♠s t♦ t❡r♠s✱ t❤❡♥(ren)✐s ❛❧s♦ ✐♥❝r❡❛s✐♥❣

✭♠♦♥♦t♦♥②✮ ❛♥❞S^

nrn=S

nren ✭ω✲❝♦♥t✐♥✉✐t②✮✳

▲❡♠♠❛ ✸✳✸✳ ■❢σ∈∆R ❛♥❞σ^′∈Rh∆_Ri✱ t❤❡♥ σ→σ^′ ✐✛ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿

✭✐✮σ=λx τ ❛♥❞σ=λx τ^′ ✇✐t❤τ →τ^′❀

✭✐✐✮σ= (τ)ρ❛♥❞σ^′ = (τ^′)ρ✇✐t❤τ→τ^′✱ ♦rσ^′ = (τ)ρ^′ ✇✐t❤ρf→ρ^′❀

✭✐✐✐✮σ= (λx τ)ρ❛♥❞σ^′ =τ[ρ/x]❀

✇❤❡r❡✱ ✐♥ ❡❛❝❤ ❝❛s❡τ∈∆R✳

Pr♦♦❢✳ ■❢ ✭✐✮ ♦r t❤❡ ✜rst ❝❛s❡ ♦❢ ✭✐✐✮ ❤♦❧❞s✱ t❤❡♥ ✐t ❤♦❧❞s ❛t s♦♠❡ ❞❡♣t❤k✱ ❤❡♥❝❡σ→k+1

σ^′✳ ■❢ t❤❡ s❡❝♦♥❞ ❝❛s❡ ♦❢ ✭✐✐✮ ❤♦❧❞s✱ t❤❡♥ ❜② ▲❡♠♠❛ ✸✳✷✱ ✇❡ ❣❡tρg→kρ^′ ❢♦r s♦♠❡k✱ ❤❡♥❝❡

σ→k+1 σ^′✳ ■❢ ✭✐✐✐✮ ❤♦❧❞s t❤❡♥σ→1 σ^′✳ ❈♦♥✈❡rs❡❧②✱ ✐❢σ→σ^′ t❤❡♥ t❤❡r❡ ✐s k s✉❝❤ t❤❛t σ→k σ^′ ❛♥❞ ♦♥❡ ♦❢ ✭✐✮ ✭✐✐✮ ♦r ✭✐✐✐✮ ❤♦❧❞s ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢→k ✭❛♥❞ ▲❡♠♠❛ ✸✳✷ ✐♥ t❤❡

s❡❝♦♥❞ ❝❛s❡ ♦❢ ✭✐✐✮✮✳

▲❡tf→^∗❜❡ t❤❡ r❡✢❡①✐✈❡ ❛♥❞ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ♦❢ f→✳

▲❡♠♠❛ ✸✳✹✳ ▲❡t σ, σ^′, τ ∈Rh∆_Ri ✇✐t❤σf→σ^′✳ ❚❤❡♥ ❢♦r ❛❧❧τ ∈Rh∆_Ri❛♥❞ ❛❧❧ a∈R

✇❡ ❤❛✈❡✿λx σf→λx σ^′✱(σ)τf→(σ^′)τ✱(τ)σf→^∗(τ)σ^′✱σ+τf→σ^′+τ ❛♥❞aσf→^∗aσ^′✳ Pr♦♦❢✳ ❲r✐t❡σ=S=bu+V ❛♥❞σ^′=S^′=bU^′+V ✇✐t❤b6= 0❛♥❞ ❛♥❞u→U^′✱ ❛♥❞

✇r✐t❡τ=T =Pn

i=1aiti✳ ❚❤❡♥ ❜② ▲❡♠♠❛ ✸✳✸✱λx u→λx U^′ ❛♥❞(u)T →(U^′)T✱ ❤❡♥❝❡

λx σ=bλx u+λx V f→bλx U^′+λx V =λx σ^′