Duality and Separation Theorems in Idempotent Semimodules
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Duality and Separation Theorems in Idempotent Semimodules Guy Cohen — Stéphane Gaubert — Jean-Pierre Quadrat. N° 4668 December 2002. ISSN 0249-6399. ISRN INRIA/RR--4668--FR+ENG. THÈME 4. apport de recherche.
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(388) hjWZUzZDxfqY3pelZªalmSh'WZrjf
(389) Y[ZwUf
(390) h'ZUnkqzjuq»h'k WagwW Z alebe-z'ZUh'czjmtQzjkqYÚhjalY[Z R hjkdhjalY[Z,abmhjWagrpf
(391) pZUz»-abr.hjWZ,rZUh R kq!QcmwDh'abkqmrªQzjkqY f[r'ZDh U hjk R » ahjWhjWZ]pBknalm*h agrZ r'cpzjZUYScY fqr ⊕ knpBZvzjf
(392) h'abkqmGf
(393) m s h'WZ,wDknm:qZUm*hjalknmf
(394) eNf ss ahjalknm¤kq@f3z'Zf
(395) eNwDkqmBr hf
(396) m*hfnr f ebZD°hk
(397) !zjabqW*h i fnwKhjalknm VWabrrZvY3abY[k s cebZ f h'Wf
(398) h Z,zjZDQZvz0fqr F QknzrWknzh0abmGh'WZtr'Z *cZve i agr0wDkqY[pebZDhjZq =±m}f[wDknY3pelZUh'Ztr'ZUY[alY[k s cebZ X » Z s 1Z kmZq»:Qkqzf
(399) ebe x, y ∈ X f
(400) m s λ ∈ K » Eqf x\y = (L ) (y) = >{λ ∈ K | xλ ≤ y} , f i E
(401) x/λ = (R ) (x) = >{y ∈ X | yλ ≤ x} . f i yfqzjfqpWzfqr'almdh'WZ s 1Z kmalh'abkqmokq@zjZvr'a s cBfih'Z s YHfqprU» xλ ≤ y ⇐⇒ λ ≤ x\y ⇐⇒ x ≤ y/λ . f< i G. f. X λ. X v. I. I. I. I. I. n×1 max. max. yRy Ô z*{1|[|[}. U max. def. X ] x. def. X ] λ. max.
(402) . r/ Z1 U **Me1[ /e^
(403) r1[! r(!. VWZ8zjZvr'a s cfihjalknmQknz'YSce f 2 i » f ? i f
(404) m s f H i u:abZUe s f x(x\y) ≤ y , (x/λ)λ ≤ x , f <(< i <f (< i (x\y)λ ≤ x\(yλ) , x(λ/µ) ≤ (xλ)/µ , <f N< w i x\(xλ) ≥ λ , (xλ)/λ ≥ x , <f N< s i x\(∧ Λ) = ∧(x\Λ) , (∧ U )/λ = ∧(U/λ) , <f (< Z i x(x\(xy)) = xy , ((xλ)/λ)λ = xλ , <f N< i x\(x(x\y)) = x\y , ((x/λ)λ)/λ = x/λ , <f N< i λ\(x\z) = (xλ)\z , (x/µ)/λ = x/(λµ) , <f N< W i (∨ U )\y = ∧(U \y) , x/(∨ Λ) = ∧(x/Λ) , Qknzf
(405) ebe x, y, z ∈ X » U ⊂ X » Λ ⊂ K » WZUzjZ (U \y) = {u\y | u ∈ U } »qf
(406) m s x/Λ = {x/λ | λ ∈ Λ} Labmf
(407) ebebuq»ial agrfabr'ZUY[abY3k s celZf
(408) m s »
(409) h'WZYHf
(410) pr f
(411) m s wDknY[Ydc-hjZq»iWZvmwDZn» :u f ?
(412) w i »-hjWXZUabz0zjZvr'a s cBfih'Z s YHfqpr0wDknY3µ,YSc-νh'Zn∈» K WagwW¤Y[Zf
(413) mrªxhjW7→fih xλ x 7→ νx (ν\x)/µ = ν\(x/µ) . f< i _-almwUZ8h'WZvz'Z,agrmk®f
(414) YSabqcalh uq» ZtrWfqeler'abY3pelu zjalh'Z ν\x/µ Qknz f < i ¤´ &- kqh'Zh'Wf
(415) h Enf agr s cf
(416) enk
(417) -hjWZ s #Z kmalh'abkqm ?- < ab m J!ahj*abmki- 1 70sihjWZegfih'h'ZUz@z'Z *calzjZvr hjWZ,fqrjrcY3ph'abkqm¤h'Wf
(418) h0h'WZ,f fqwDi h'abkqmGk
(419) qZvwDh'knzjrªkqmorjwUf
(420) egf
(421) zfrªr'f
(422) ih'agr kZr x(∧ λ) = ∧ (xλ) ^r'ZUZ J!ahj:almki # C0s \ H WabwW¤abr zjah'h'ZvmQknzªzjabqW*hªfnwKh'abkqm¤k
(423) !nZvwDh'kqzr.kqm¤rjwUfqebfqzjr"^ WZUzjZvfnrU» abm h'Wagr[pf
(424) pZUz» f Zor hjf abweg i!hji k hjWZoY[kqzj Z¤mf
(425) h'czf
(426) e0fnr'r'cY[p-h'abkqm§h'Wf
(427) h[h'Wagr[pzjkqpZUz'h u£Wkqe s r ahjW ∨ abmrh'Zf s k
(428) ∧ ¥hjWabrabrhjWZtwUfnrZ]Qknz0almBr hf
(429) mwUZ,ah'WZdcm s Zvz'ebu:almHr'ZUY[abz'abm[abrfHrZvY3abzjalm[kq@Qknz'YHf
(430) e r'ZUzjabZvrv»*knz0k
(431) YHfih'zjagwDZvrv»:kiqZvzf[wUkqY[pebZDh'Z,a s ZvY3pk
(432) hjZUm*hr'ZUY[alzjabm 2 ¸ Ï ´ ´ ¸ ¹ ´ =º agr3fwDknY3pelZUh'ZGz'abqW*h ¯±rZvY3abY[k s cebZq» ZGwvf
(433) ebe e( $[ r'ZUY[abY[k s celZ¢k
(434) X h'WZ 0 e K¯±rZvY3abY[Xk s cebZ X ahjWcm s Zvz'Kebu:alm§rZUh X »f s s alh'abkqm (x, y) 7→ h'WZ agrQkqzh'WZmfihjczf
(435) eknz s ZUz"kq f
(436) m s elZU°hªfqwDh'abkqm » Lkqz ∧ wUebfq{x,z'alh uqy}» f Zr'Wf
(437) ∧eber'kqY[ZDhjalY[Zr s ZUmk
(438) h'Z0:u (λ, x)X7→i λ x = x/λ h'WKZ×ebZD°Xhªfq→wKhjalXknmSkq(λ, Xx) → Vx/λ Wf
(439) h X agr0f[rZvY[alY[k s cebZ]Qkqebelk rªQzjkqY QkqzjYdce f <(< s i » f <N< i »f
(440) m s f <(< W i X =±mopf
(441) z'h'agwDcebfqzv» f <N< i u:abZUe s r λ∈Λ. λ∈Λ. op. op. ·. (λµ). 2 f< i. op. ·. op. op. x = x/(λµ) = (x/µ)/λ = λ op· (µ op· x) ,. Qknzf
(442) ebe λ, µ ∈ K fqm s x ∈ X » WabwW®r'Wk r W:u X YScr h.BZ8wDkqmBra s ZUzjZ s fnr"f-0 e BzfihjWZUz"h'Wfqm f [b a r'ZUY[abY3k s celZn ,=±m s ZUZ s »@wDknmr'a s Zvz'abm q fnr,fzjabqW*htfqwKhjalknm kqce s z'Z ncalzjZdhjWZ pz'knpBZvzh u[ru:Y[Y[ZDhjz'agwUfqeBhjk f < 2 i h'k3Wkne s »*hjWfih(x,abrv»:λ):u 7→f nx/λ f i » x/(λµ) = (x/λ)/µ »: c-hh'WagrªpzjkqpZUz'h u mZUZ s mkqhWkqe s QknzfHrZvY[alY[k s cebZ X kinZUz0f e"Y",/ rZvY3abzjalm K op. op. Ø y Ø°Þ.
(443)
(444) !#"$% '&#(#")* # + #", "-./01$. ZUmkqh'abmH:u. op. \. fqm s. \nf f< i. op. E. h'WZ,zjZvr'a s cfihjZ s kqpZUzfih'abkqmBr.calelhQzjkqY » Z]qZUhªQzjkqY. » f< i. op. ·. /. λ. op. x. op. \. op. ] x = (LX λ ) (x) = ⊥{y ∈ X | y/λ ≥ x} ,. = xλ ,. \q f< i. /. y=. op (RyX )] (x). = >{λ ∈ K | y/λ ≥ x} , = x\y .. nm f < \nf i abrf
(445) mGalm:qknelch'ab*alh upz'knpBZvzh uN¥h'WZdz'Zra s cf
(446) h'Z s egf kq@hjWZ,z'Zra s cf
(447) h'Z s ge f kq@hjWZtzjalnW*h fnwKhjalknm¤qk X abrhjWZ,zjalnWnh0fqwDh'abkqmok
(448) X alhjr'ZUel "VWZvz'ZUQkqzjZq» ¸ ¸ Ï ¸ e"M 01 K ^$1#"Y "./01$ X (X ) = X s op op. 2 4R6 wV 69:7P P4[:TU8947P9;
(449) Y69V R:P9; : RP w8VW4R69:U47P:
(450) 767QYV. :*PN8 *P
(451) 4 [:UT8V R:
(452) ):*8 PVWT 2 < }¸ ´ - ¸´ q ¸ !J ZDh s ZUmkqh'ZdfHwUkqY[pebZDh'Ztr'cr'ZUY[abY[k s celZdk
(453) fHwDkqY[pebZDhjZtrZvY[a¯ Y[k s cebZ X kiqZvz]f}wUkqY[pebZDhjZHa s ZvY[pBVkqh'ZUm*hdr'ZUY[abz'abm K ZwUf
(454) ebe e e # 1 , knm V h'WZ. YHfqp. PV : X → V,. PV (x) = >{v ∈ V | v ≤ x}. h'WZelZfqrh.cppBZvz.kqcm s k
(455) ∈ V | v ≤ x} ZUebkqmnrh'k,hjWZ8r'ZDhªBZwUfqcrZ V agr.wUkqY[pebZDh'Z i VW:crv» f agrªh'WZ]zjZvr'a s cBfih'Z s YHf
(456) pG{v k
(457) @h'WZtwvf
(458) mknmagwUf
(459) eabmi{ ZvwDh'abkqm "_:abmwDZ »-cr'alm ?
(460) w » P Z ®nZDhthjWfih P = P »"f
(461) m s r'abmwDZ i abrSalm
(462) { ZvwKhjalnZq»cr'ialm : Vf qf→i » X Zr'ZUZ®h'Wif
(463) h =P i : X → fV agi r r'czÇ{ ZvwDh'abqZq Z¤rjf=uh'Wf
(464) h agrdqf bw##! wb 1"Y ` ¤k
(465) f¢wUkqY[pebZDh'Z¤rcBrZvY3abY[k s cebZ V alf
(466) m:u ZvelZvY[ZUm*h v ∈ V wvf
(467) mGBZ zjalWhh'Zvm}fqr v = ∨{wλ | w ∈ W } »:Qkqzr'kqY[Z λ ∈ K ´:¸ ´ "! y"zjk
(468) { ZvwDh'knrz LkqzjYdcegf i & V $Y e")0# ,$[1$#1"Y "./_- X # . 'bw##! wb 1", 1f V. 2 V. 2 V. V. V. V. w. ? f< i
(469) r!Z 1s. V. V. w. W. PV (x) =. ∨ w(w\x) .. ZowUf
(470) m z'alh'Z P (x) = ∨ wλ »Qkqz[rknY[Z λ ∈ K LzjkqY P (x) ≤ x » Z¤nZDh »ªkqz»Z*cabif
(471) ebZUm*h'ebuq» λ ≤ w\x VWagrHrWk rShjWfih P (x) ≤ ∨ w(w\x) ¼ªc-h» wλ ≤ x abrf
(472) mGZUebZUY[ZUm*hk
(473) V » WabwW»-:u f <N< f i »abrebZvrjrªh'WBf
(474) m x "VWagr0pzjkiqZr f < ? i ∨ w(w\x) Z}YHf=u&zjZ zjahjZ f < ? i fqr P = ∨ P , WZvz'Z P s Zvmk
(475) hjZvr3hjWZ¡pzjk
(476) { ZwKh'knz[kqm h'WZ%$kqmZ s alY[ZUmBrabkqmfqe'&]rpfnwDZ wK ._:abY[alegf
(477) zQknz'YSce QkqzhjWZ,pzjk
(478) { ZvwDh'knzfqppZvf
(479) zjZ s abm f |}kqebebZUz < En q i ¸ ¸ Ï ¸ )( cBf
(480) e!wWfqzjfnwKhjZUzjal«fih'abkqmk
(481) h'WZ,pzjk
(482) { ZwKhjkqz i +* 1 V ⊂ X (# u e")0# +$[#^ $##", "-./0 # . hf bw1 1!
(483) ab 1", W s-1 P (x) = ⊥{z ∈ X | w\z ≥ w\x, ∀w ∈ W } . f< i
(484) r!Z 1s_-almwUZ w\z ≥ w\x ⇐⇒ z ≥ w(w\x) »h'WagrQknelebk rªQz'knY f < ? i V. w∈W. w. w∈W. w. w. w. w∈W. V. V. yRy Ô z*{1|[|[}. V. V. w∈W. w. w. w∈W.
(485) <. r/ Z1 U **Me1[ /e^
(486) r1[! r(!. . ´ Z[z'ZUh'czjmh'k¤h'WZ R ¯±r'ZUY[alY[k s cebZ F la m*hjz'k s cwUZ s fih xf
(487) Y[pebZ®Gf
(488) m ss abrjwDcBr'r hjWZtf
(489) ppelagwUf
(490) h'abkqmGk
(491) pz'Zv:alkncrªz'ZrcelhjrabmGh'Wagrr'ZvwDh'abkqm Labzjr0h k
(492) f
(493) ebeÇ»-knrZvz'nZhjWfih . max. ∀f, g ∈ F, f \g = inf g(u) − f (u) ,. alh'Wdh'WZwDknm:qZUm*hjalknmtWZU zjZ"hjWfih +∞−∞ = +∞ f rabmwUZ.abmSfqm*u,wUkqY[pebZDh'Zªa s ZUY[pk
(494) h'Zvm*hr'ZUY[abz'abm S » β \β = τ \τ = τ i Gr'r'cY[Zmk hjWfih abr.f,h'knpBknelknqagwUfqeqZwKh'knzr'pfqwUZf
(495) m s wUkqmr'a s ZUzh'WZ8wDkqY[pebZDhjZr'cr'ZUY[alY[k s ¯ celZ V qZvmZUzfihj Z s :u Uh'WZGrZUh W k
(496) wUkqm*h'abm:ckqcBrtebalmZf
(497) zdQcmwKhjalknmrSkiqZUz U @.elZf
(498) zjelun»h'Wagrdagrdh'WZ r'cr'ZUY[abY[k s celZ,kqe² rU w
(499) wDkqm:nZDxHQcmwDh'abkqmrkiqZvz C Lknzf
(500) m:u f ∈ F » P (f ) »Bfqr s Z1kmZ s alm e2 < » agrªh'WZtwDegfqrjragwUfqee² rU w
(501) wUkqm:qZUx®W:cele!k
(502) f Lkqz w ∈ UW » u∈U. S. S. S. S. S. 1. V. w\f = inf f (u) − w(u) = − sup w(u) − f (u) ,. WabwW[agr"mkqh'Wabm,c-h.h'WZkqppknr'ahjZ0k
(503) h'WZdLZUmBwWZUeh'zf
(504) mrQkqzjY hj WZUmou:abZUe s r u∈U. u∈U. PV (f )(·) =. ∨ w(w\f ) = ∨. f∗. kq f Zv=fqelcBfih'Z s fih w * m. ? f< i. w(·) − f ∗ (w) ,. hjWfih8abrh'korjf=uq»Bh'WZSeÇ rU w
(505) NwDknm:qZDxoW:cebekq f abrh'WZYLZUmwWZve!hjzjfqmrQkqzjY)kqh'WZLZvmwWZUe@h'zf
(506) mBr Qknz'Y kq f Labmf
(507) ebelun» nm < rjf=u-r8h'Wf
(508) h,h'WZHe² rv w
(509) !wUkqm:qZUx¡W*cele"kq f abr,h'WZHebZvfnr h,QcmBwKh'abkqm g abm F rcwW hjWfih g agrelZr'rhjWf f
(510) m!»-i kqz0Z *cfqeNh'kB» f f pkqabm*h abr'Z i 2 ´ = ´ -i- ¸ ´:¸ ´ ´:¸ ´ f DmabqZvzjrjf
(511) e_:Zvpf
(512) zfihjalknm¤VWZvkqzjZUY i * 1 V ⊂ X N1 - e")0# $e1$#1"Y "-Z^ /_ +-_1 x ∈ X sY 1 f ∀v ∈ V, v\P (x) = v\x , f <H i w∈W. ∗. w∈W. ∗. V. f <H i. x ∈ V ⇐⇒ x\PV (x) = x\x .. :_ ZvZUabm fnrf $'rjwUf
(513) egf
(514) zpzjk s cwKh &» nm f < H f i rjf=u-r!hjWfihhjWZqZvwDh'knz (x, P (x)) agr $ knzhjWkqnkqmfqe'& hjkhjWZ0rZvY[y\x alY[k s cebZ »qfqm s < H r'Wk r@h'Wf
(515) hh'WZ $W*u:pZUzjpegf
(516) mZ & rZvpf
(517) zfihjZvr x QzjkqY V »alfqm s kqmVebu®al x 6∈f V "i VWag rhjZUzjY3abmknelknqu abele5Z{ crh'{ya0kZ |s y\P abm '\(x) = y\x}
(518) r!Z 1s_-almwUZq»:u s #Z kmalh'abkqm»Bh'WZ alm < BZvelknmnr0h'k®hjWZdr'ZDh» ZtWf=qZ]hjWfih » QknzHf
(519) ebe v ∈ V Dr'abm P (x) ≤⊥x f
(520) m f s h'i WZ}Y3knmk
(521) hjkqmagwDalh u£k
(522) y 7→ v\y » Zov\P qZUh[h'(x) WZ}zj≥ZUnv\x ZUzrZ abmZ ncBf
(523) ebah un» WagwW&rWk r f < H f i =º x ∈ V »@h'WZvm P (x) = x »fqm s x\P (x) = x\x »h'zjab*agf
(524) ebebuq @.knm:qZUzr'ZUebuq»-al x\P (x) = x\x » ZtWBf=qZq»*u < i »hjWfih P (x) ≥ x(x\x) »5f
(525) m s »B:u <(< Z i »Bh'WBfih » WabwW}r'Wk r.h'Wf
(526) h P (x) ≥ x ._:fabmwDZ P (x) ≤ x » Z,Wf=nZ x = P (x) ∈f V x(x\x) = x Ã@ÄKÉqÅÏÐ À ÊÏÅÇʲÈj РʲÔdÒ=ÀÂÄKÁ²ÀÂÔÐ ÔdÅQÁæDÝ ¿=Ð"ÄKÀÂÔDÑ=¿=æDÑÄ'Ð Ã5ÐÔUÔбÊÇæDÀÂÒ
(527) ÄKÐ ÐÅÔi¿ÈʲÀÂÌN¿iÔ=ÈjÑ8Ô=Ê ÄÊÇʲ¿=ÈjÐÊÇÀÂÝ ÄKÄKÔ3ÔUæDÀÂÁÐ ÔDÈʲÖ.ÀÂÄKÒ=ÔiÀÂÈÛKÌÎ=Ã@ÄKÔ=¿Î=ÄÀÂݺÁʲ¿Èj ÊÇÈÊÇÀÂÔ=ÄKŲÈÔ8ÑdÔʲÄKÁ²ÄÌËÔ=ÈjÐÈÊÇÐÉ=Ö.Á É= Î=Ì ÍÀÂÁÇÔ Êʲ¿ÆKÐÐ ÐÅÏÉdÎ=Ì ÀÂÐÔtÈÝ Á Ö0ÄKÅÏÀÂÅÇÔ=Ð Ñ]ݱʲÃ!Ì ÍD¿ié ÈjÊÈÅÏЪʲ¿ ÐÁ²Ä.Ý ÄKʲÅÇ¿iÅÏÈjÐ Ý Ê!Ê.ÊÇ¿=ÈÀÂÌÂÁBÛKÐ äÇÒ=ÅÇÎÅÇÈÌÂбÀÂÝç V. V. V. V. V. V. V. 1. V. V. βS ⊗ τ S = βS. V. V. Rmax. −∞ + ∞ = −∞. Ø y Ø°Þ.
(528)
(529) !#"$% '&#(#")* # + #", "-./01$. <N<. ¤´ &- VWZ[rZvpf
(530) zfihjalm¤r'ZDh H = {v ∈ V | v\P (x) = v\x} abr]fGrZvY[alY[k s cebZq %=±m s ZUZ s » :u f <(< W i ahagrrhjf
(531) elZ]:uf ss ahjalknm}f
(532) m s f <N< i r'Wk r.h'Wf
(533) hahagrrhjfqelZ]:urjwUfqebfqzfnwKhjalknm ¤´ &- G0wUwDknz s alm¡h'k}hjWZ¤pz'Zv*abkqcBr,z'ZvYHf
(534) zg»alhSagrSr'c®wDabZUm*hSh'kwWZvweg < H f knmebu¢Qknz v f i zf
(535) mqabmSabm}f3qZUmZUzfih'abm[rcrZUh W kq V ´ Lknzh'WZ0r'ZUY[alY[k s cebZ alm*hjz'k s cwUZ s fih xf
(536) Y[pebZfqm s hjWZ0rcrZvY[alY[k s cebZ kq e² rU w
(537) qwUkqm:qZUxtQcmwDh'abkqmrqZvmZUzfihjZ s *udFh'WZr'cr'ZDh W kqNwUkqm*h'abm:ckqcr"elabmZvfqzQcmwDh'abkqmr.fqr s agr'wUcVrjrZ s f
(538) h xf
(539) Y[pebZ H »-h'WZSZ *cfqelalh u < H f zjZvrh'zjagwKh'Z s h'k fqr0knr'ZUzjqZ s abmoh'WZtpz'Zv*abkqcBr0z'ZvYHf
(540) z g kqNh'WZ]_:ZUpfqzjf
(541) h'abkqm®VWZvkqzjZUY+r'f f=u-rhji Wf fih.h'W5Z LZvmwWZUveh'∈zf
(542) WmrQkqzjY+k
(543) f
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L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des
Venant compléter les nombreux guides spécialisés qui informent le touriste gay sur les différents lieux et structures qu’il pourra fréquenter s’il recherche son