Complementation of Rational Sets on Scattered Linear Orderings of Finite Rank

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(1)Complementation of Rational Sets on Scattered Linear Orderings of Finite Rank Olivier Carton, Chloé Rispal. To cite this version: Olivier Carton, Chloé Rispal. Complementation of Rational Sets on Scattered Linear Orderings of Finite Rank. 6th Latin American Theoretical INformatics (LATIN’04), Apr 2004, Buenos Aires, Argentina, Argentina. pp.292-301. �hal-00619905�. HAL Id: hal-00619905 https://hal-upec-upem.archives-ouvertes.fr/hal-00619905 Submitted on 3 Oct 2011. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(10) HZRTB0X à å ä <>EgXZXFJGðã[LT=@HZBW<>=@LTBEOC!JGC8P|B/CZ=>LgX/ïN HZRTB ä <]EGXKXSJOð ä J à LjH8EOåT<>B JGC8P|=@LQEO<]XFEGL[P HZRTB ä <>EgXZX!JGð ä J à LjHKEGåT<@B0X ä EOHIHZB/CZBP&<>=>LTB/EGC'JGC8P|B CK=@L gX/í ì ÿ * fX [þ %@7ý N

(11) ý Z * Gý g >9u)a

(12) ÿ zþ þcý Z

(13) ý z R * z. b I:PAJZI>EAC éê û Ký * >9Wþ * 3>U ( ;Tý Zý þ ;[ýÿ | L

(14) ý Z Zy ý Rf[ÿoþ ný g0ý + N

(15) ý % % =- Z¡Dl EA¢.

(16) . > - 1i 354 ^ 1 d \` nl nl ¢ Z Ry^ 1 d ( T; ýZ ý¡c Z m E Z Zýy ZýowTý8ÿ þF zýÆþ ;[ý * Gý g >9 * z{ý * R * Nýý@ýh`ý [þ . LgH à =@HZ=>?GB <>æGïNHKRTBC8EOL[óÓJGðÀE&<>=>LTB/EGCJgCKP|B/CZ=>LÆ=]XHZRTBáÈzè|=>á à áL à áåQB/CJGðÀLTB/XIHZBP OE L[P *í HW=]X<>=>LTóGBPHZJÓ=H8X E à XZP|JgCIò X ä <>EgXZX/í TJGCW=>L[XIHKEOL ä BHKRTB¦JGC8P|B CK=>LgX JOð CKEGLTó&ê*EGL[P \ JOðC8EOLTó û åQB/<@JgLCKB/XZYQB ä HZ=>?GB/<@æHZJ M EGL[P \ í B ?GB/CIHKRTB <>B/XKX/ïzHZRTB ä <]EGXKX =>XÀLTJOHFBèTE ä HZ<>æHKRTBXZBHFJOðJgCKP|B/CZ=>LjXJOðxC8EOLTó Àí [JGCÀ=>L[XIHKEGL ä BgïOHZR[B JGC8P|B/CZ=>L ] => XÀ JOðC8EOLTó¥ê EOL[P¦åNB <>JGLgXHKJ \ í !RTB CKBð½JgCZBgïGâFB*âSJgCZóJGL&XZ<@=GRjHK<@æP|=òB CKB LjHF=@L[P à[ä HK=@?gB ä <]EGXKXIBX í TJgC*EOLA æ d ÆïTâFBWPTBã[LTBWHKRTB ä <]EGXK X

(17) åjæ

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(20) å Mà HÆP|JABX¦LTJOH`åNB <>JGL ÌHZJ EOL[P.HZR[BJGC8P|B CK=@L M åQB/<@JgL gXHKJ å à HP|JAB/X0LTJGHåQB/<@JgL HZ!J

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(22) í [JGCW=@LQXqH8EOL ä BHZR[BJgCKPTB CK=@L jX*JGð4CKEGLTóú EOCKBHZR[BãQLT=HKB¦JGLTBX í L HKRT=]XWY[EOYNB CïâSBCZBXqHKCZ= ä HHZJÓ<>=@LTBEOCWJGC8P|B/CZ=>L gX JOð4ãQLT=HKB¦CKEGLTó|X HZR[EOH*=>XFHKRTBXIB H

(23) % - % í æ¥BèAHZB/L[XZ=@Jg% Lï| HZR[BCKEGLTó&% JG ð4E¦âSJgCKPÓ=]X!HZRTBCKEGLTó&JGð"=@HKX <@B/L OHKREOL[PÓHKRTBC8EOLTó¥JOðE XIB H*JOð âFJGC8PTXF=]X!HZRTB à YTYQB/C*åNJ à L[PÓJGð(HZR[B0CKEGLTó|X!JOð =@HKX*B <>B á¦B LjHKX/í øBP|B LTJGHZBåAa æ & &HZRTBXIB HJGðÀEO<><(âFJGC8PTX'=>L[P|Bè|BPåAæ ä J à LjHKEGåT<>BX ä EzHZHZB CKB/P<>=>LTB/EGC JGC8P|B CK=@L gXFEOL[P¥âSBWEO<]XZJP|B/LTJOHKBWåjæ &('*) ¡CZBXIYNB ä HZ=>?GB/<@æ &,+-)ÀHZR[BWXZBH'JGð âFJGC8PTXFâ'RTJgXZB <@B/L OHKRÆ=>X!EGLÆJGC8P|B CK=>L [email protected]

(24) !/ ½CKB/XZYQB ä HK=@?gB <>æ / 4ð½JGCFXIJgá¦B =>LgHKB gB 1C 0jí !R à XÀHZRTBâSJgCKP[X JO8ð &('*)R[En?GBWECKEGLTóÆ<>JzâSB/CSHKR[EOLJgCFB à EO<HK2J 0jí 3 4 )ØTÞqÚÀ$× ) % # 'xØ # Ú 576ÌÚ4ÙQ$Û # ÚS& × %IÞI$× '*)"Ù.Ú4ÙQ(Û 'Ù[ÞZ(× + # C à 9æ B/8 CZBEGL[PøDFEOCZHZJgL é û ìRQEn?GB=>LjHZCKJAP à[ä BPCKEOHZ=>JGL[EG<(Bè|YTCKB/XKXZ=@JgL[X*EOLQP E à HKJGáÈzH8E`ð½JgC âSJgCKPTX=>L[P|B èABP åAæ ä J à LjH8EOåT<>B`X ä EzHIHKB CKB/Pø<@=>LTBEOCWJGC8P|B CK=>L gX/í !RTB æ RQEn?GBYTCZJz?gB/P HZR[EOH EÌXZBHÆJGðâFJGC8PTX¦=]X¦CKEOHZ=>JGL[EG<'=ð0EOLQP JgLT<>æ =ð0=@HÆ=>XÆCZB ä J gLT= /EOå[<@BÓBèAHZB/L[P|=>L çW<>B B L[B X HZRTB/JGCKB á5í $JgCZB'Y[CZB ä =]XIB/<@ægïzHZRTB/æRQEn?GB*P|B ã[LTB/P&EWâ'RTJG<>B*RT=>B C8EOC ä RAæJOðxC8EzHZ=>JGLQEO<TXZBHKXé ëOì í TJGC B/E ä RX à å[XZBH JGðC8EzHK=@JgL[EO<(JGYNB C8EzHZ=>JGLQX ïAHKRTB æ ä JgL[XI=]P|B/C'HZRTB ä <]EGXKX!JOð ä JgCZCKB/XZYQJgL[P|=> L CKEOHZ=>JGL[EG<<]EOL à E GB/XWEGL[PøPTBã[LTB¦HKCKEGL[XZ=HK=@JgL[XWJOð!E à HKJGáÈzH8E ä EOY|H à CK=@L ÓHKRTBÆXZEGá¦B<]EOLTö à E GB/X/í L&HZRTBð½JG<><>Jzâ'=@L XIB ä HK=@JgLïGHKRTB ä R[EOC8E ä HZB CK= /EzHK=@JgL¦JOð(C8EzHK=@JgL[EO<XIB HKXSJOð(âFJGC8PTXÀJOð ã[LT=@HZB0C8EOLTóÆ=]X!LTJGHZ=@ã[B/Pxí. : CFE ;KXM?OFPAQTKNV I <DHTJZI:S?S}O0PRQWS. !RTBC8EzHK=@JgL[EO<"XZBHKXJOðÀã[LT=@HZB¦CKEGLTó ä EGL åQB¦Jgå|HKEG=@LTBPð½CKJGá ã[LT=@HZB¦XZBHKXJGð4ã[LT=@HZBâSJgCKP[X à XI=>LWHKRTB à LT=>JGL]ïnHZR[B ä JgL ä EOHZB LQEzHZ=>JGL=@ïOHZRTB XIHKEGC?>AïnHKRTB'JGá¦BjE=HKB C8EzHZ=>JGL EOL[PHZR[B å[E ä óAâ!EOC8PTXÀJgá¦BgE=@HZB/CKEOHZ=>JGL *í BH @ EOL[P.A åNBWHqâSJ`XZBHKX!JGðâFJGC8PTX/ïAâSB0P|B ã[LTB.

(25) @ ] A @ =A @ n @ .@ @ -. d @ ` A ¢ ?2 , = , d @l d A ¢ 1 % ,X1 d l ,N1 d @ ¢ 1 3 M ,X1 ,X1 d @ ¢ 13 @ , 1 , 1 d @ ¢ . (JPTBã[LTB CKEOHZ=>JGL[EG<[XIB HKXÀJOðxâSJgCKP[X=@LQP|Bè|B/PÆåjæ¦EO<><[<>=>LTB/EGC4JGC8P|B CK=>LgX/ïnHZRTCKB B áJgCZB JGYNB CZö EzHZ=>JGLQXFEGCZBLTB/B/P|BP AHKRTBWJGC8P|=@LQEO<=HKB C8EzHK=@JgL &ïAHZRTBWå[E ä óAâFEGCKPTXÀJGC8P|=>L[EO<=@HZB C8EzHK=@JgL. EOL[P&HZRTB0=@HZB/CKEOHZ=>JGL¥ð½JGC*EG<@<<>=>LTB/EGC ä J à LgH8EOåT<>B0X ä EzHZHZB CKB/P&JGC8P|B CK=>LgX Ní .

(26).

(27). . @ - 1 354 ,X1 d l,X1 d @ ¢ @ @ - 13 @ 4 , 1 d l , 1 d @ ¢ @ A - 1354 4 1 d Nl 1 d @2=ð d øEGL[P 1 d A. @= ð d ¢ LHZRT=]X*Y[EGYQB/C/ïQâSBEOCKB0JGLT<>æ&=>LjHZB CKB/XIHZBP=@L<]EOL à EGB/X!â'R[= ä R EGCZBP|B ã[LTBP à XI=>L.]ï =@ï EGL[P *í øB&CZB ð½B C0HKRTBÆCKB/EGPTB CWHZJ é û ìÀð½JGCEYTCKB ä =>XZBÆP|B ã[LT=@HZ=>JGL JOð!JOHKRTB CCKEOHZ=>JGL[EG< JGYNB C8EzHZ=>JGLQX í ß XZBHJOðÀâFJGC8PTX JGL <>=>LTB/EGCJgCKP|B/CZ=>LjX*=]Xy OþF * R À=@ðÀ=HW=]XJgå|HKEG=@L[B/Pð½CKJGá ã[LT=@HZBXZBHKX*JOð(ãQLT=HKB0âSJgCKP[X à XI=>LHZR[B0CKEOHZ=>JGL[EG<NJgYQB/CKEOHZ=>JGL[X!P|B ã[LTBPÓEGåQJz?gBGí . : C B M}PAuKNM?KPRQV0OFQWI:KXJPAJZYWI>JZO0QT[RS. B(H kl&yl l l (åNB*E ä <]EGXKXI= ä EG<AE à HZJgáÈzHZJgLJgLã[LT=@HZB'âFJGC8PTXâ'=@HZR à X à EO<|LTJGHKEzHK=@JgL[X/í ß XxHKRTB4XZBH ÑJOðAHKCKEGL[XI=@HZ=>JGLQX=>XE'X à å[XZBH JOð & ¦ï8HKRTB4Y[EOHZR[XJOðTX àQä RWEGL0E à HKJGáÈzHKJGL EOCKB!ã[LT=@HZBGí L à[ä RT=QE à HZJgáÈzHKE[ïOE0âFJGC8P=]XÀE ä ä B/Y|HZBP=ð=@HÀ=]X"HZRTB<>EGåQB/<TJOðEWY[EOHZR gJG=>L =@L|ãQLT=HKB <>æ¦HZ=>á¦B/XÀHZRTCKJ à gR&Eg=@?gB L&XZBHFJOð(XIHKEzHKB/X/í !R[B YTCKJGåT<>B á=]X4HKR[EzHSHZRT=]X!E ä/ä B YTHZ=>L ä JGLQP|=HK=@JgLøP|JABX LTJGHB/?GB/L EO<><@Jzâ HKJ&CKB ä J GL[= /B0HZRTB ä JGL ä EzHZB/L[EzHK=@JgLJOðÀ=>L|ã[L[=HKBâFJGC8PTX/í (J ä JGYNBâ'=@HZR HKRT=]XP|= ä à <@HqæGï EXZBHJGðF<>=@á¦=@HHZC8EOLQXI=@HZ=>JGL[XW=>L ä < à PTB/P =>L 3=>X =@LjHZCKJ|P à[ä B/Pxí !RT=]X!â!EnæGïA=ðEOLÓ=>L|ã[LT=@HZB0Y[EOHZ R GJAB/XF=>L|ã[LT=@HZB/<@æ¥áÈOLAæ`HZ=>á¦B/XFHKRTCZJ à GR¥HZR[B XqH8EzHZBXFJOðE¦XZBH EGL[PÆHKR[EzH!HKRTBHZC8EOL[XZ=HK=@JgL l FBè|=]XqH8X ïAHKRTB L¥HZR[BL[BèAH*XqH8EzHZBWJGð(HZR[B Y[EzHKRÓáÈnæÆåNB Aí Zh wR@ý % BH - kl#&bl l Al !åNBHKRTBE à HZJgá¦EOHZJgLÓJGð = à CZBÆêâ'RTB CKB g%ê l û lKë ¢ :ï & - }/Rl ¢ ï U- jê ¢ ï - në ¢ í . "! "# $. % & '&(. *). ,+ &,. 0. -/.. "2. 3!. " ! # 4#. 1. 7. 6. { { ¹ A¹ bSGsq©Áw sq©/hoqm8K©ºhgilË8ilhjº ß <>=>á=@H`HZC8EOLQXI=@HZ=>JGL û ¢ ëÌ=]XÈGPTPTB/P HKJ `í LjH à =@HZ=>?GB/<@ægï4EOLÑ=@L|ãQLT=HKBY[EzHKR gJG=>L HZRTCKJ à gR ZH RTB XIHKEzHKB û =>L|ã[L[=HKB <>æ á¦EGLAæ.HK=@á¦B/XÓ<>B/EgPTX`HKJ XIHKEzHKBøë EOL[PE Y[EzHKR=@L <@BEGP|=>Lð½CZJgá XqH8EzHZB û HZJ¦XqH8EzHZBWë=>XF<]EOåNB <><@BP í =@L[EG<@<>æGïjHKRT=>X!E à HZJgá¦EOHZJgL&CKB ä J GLT= BX HZRTB0<]EOL à E gB / í 7. 5. >@?BA DC. 6 E 7"F. G H2. 2. 8 :9<;=5.

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