13
Complementation of Rational Sets on Scattered Linear Orderings Chlo´ e Rispal, Olivier Carton To cite this version: Chlo´ e Rispal, Olivier Carton. Complementation of Rational Sets on Scattered Linear Orderings. 8th International Conference on Developments in Language Theory (DLT 2004), 2004, France. Springer-Verlag, 3340, pp.381-392, 2004, LNCS. <hal-00620119> HAL Id: hal-00620119 https://hal-upec-upem.archives-ouvertes.fr/hal-00620119 Submitted on 3 Oct 2011 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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 destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Hal-Diderot

Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

  • Upload
    others

  • View
    1

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

Complementation of Rational Sets on Scattered Linear

Orderings

Chloe Rispal, Olivier Carton

To cite this version:

Chloe Rispal, Olivier Carton. Complementation of Rational Sets on Scattered Linear Orderings.8th International Conference on Developments in Language Theory (DLT 2004), 2004, France.Springer-Verlag, 3340, pp.381-392, 2004, LNCS. <hal-00620119>

HAL Id: hal-00620119

https://hal-upec-upem.archives-ouvertes.fr/hal-00620119

Submitted on 3 Oct 2011

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

brought to you by COREView metadata, citation and similar papers at core.ac.uk

provided by Hal-Diderot

Page 2: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

������������� ����������� ��������������� ������� �!�"�� $#%��&� ��!��'����(#%���(��)��)*+�� ���,�-����*�)�.�/ �0��

1325476�89;:)<7=?>A@ 4CB @EDAFHG 4 <JIK<79L 1 @EL?M 6 DANOQPSR%TVUXWZY\[^]`_badcS[fehg_%i\_�TQj�adYX_bkml^jkmn(j/lolpg_q_`U5r�sut`vXlo_q]/japi�w _qcSxyj�aSed_qcqUuz{k}|/|~�r�~

TQjadY\_bkmlJj�kmn�jl^ldg_q_���_yih_b�Q�EUXz\apj/Y\xq_`U�/�{�7�5�5�o����q�{�K�C���{���7�5�p���7���o��� U�V�AP��ZzE�)U\W3YX[o]�_badcS[feEg_%� j�ad[oc |�Uu��UK¡Xl^jxq_)¢`vXcScS[o_qvAUuz{k}|`r/�`r�£�� jad[oc3xb_yi\_b��¤`rEUuzhapj/YXxb_`U

¥ �7�7�{�J�h���7¦��\��§q�u�����7�7�X�m�¨� ©q���y�y�7�h�(�o�m�

ª�« �q§q���K�/§`� P�Y,j ¡\ad_qxb_yi\[oYX¬)¡Xj/¡u_ba3­C®(adv\¯u°_bad_±jYKi,�!j�aSedt`Y¨Uhjv\edt/²�jepj t`Y,lo[oY\k_yj�a%t/apih_bad[oYX¬/cqU¨TQz���³5´ ¤E£�µ?UAj/vhedt`²�j�epj�¶Xjy]�_)su_q_qY·[oYEeSadtEihvXxq_qi�¸�t/aZ¹!t/apihcZ[oY\kih_b�h_yi�s�¯�lo[oYX_yj�a;t/apih_bad[oYX¬/cqº¼»!¶X_qcS_�j/vhedt`²�jepj·j�ad_�j½¬`_bYX_bapjl^[o¾yj�ed[ot`Y¿t¸3j/v\kedt/²�jepj;¸�ta ÀKY\[oed_/Uu[oY\ÀKY\[fed_`U5sX[fkm[oY\ÀKY\[fed_�jYKi½_q]�_bYQeSapj/YXc�ÀXYX[fed_)¹!t/apihc c�edvKih[o_yis�¯¿®)ÁvXxp¶X[º�Ã%l^_b_qYX_`´ c�ed¶X_qtad_q²-¶Kjc�su_b_qYĬ`_bYX_bapjl^[o¾q_qiÅedtVed¶X_qcS_½¹!tapi\cqº(Æ�_¡hadt]`_)ed¶Xje±apjed[ot`YXj/l cS_bedc3t/¸�¹!tapi\c3t`Y½xqt/vXY�epj/s\l^_�cSxyj�eSed_bad_yi½lo[^Y\_yjaZt/apih_bad[oYX¬j�ad_%xqlot`cS_yiQv\YKi\_?a.xqt`²,¡\l^_b²,_qY�epjed[ot`Y�vXcS[oYX¬;j/Y½jl^¬/_qs\apj[ox�j¡X¡hadt�j/xp¶¨º

Ç ÈXɱÊuË5Ì%Í�Î�Ï{Ê5ÐḏÉ

Ñ D 2 <7=·=?9ÒQ<7DA@ 4 >A@E>{9LHÓJÔ`Õ�Ö}×.Ø 4 99/D59H= 2A6�Ù 9`FÅM 2 @�MV@hÚuM 6 ÒQ@EMb@ 6 D�ÛAD5<JM?9 Ù36 LbF5=Q@hDAFL?9/ÜhÚ 4 @EL)9Ýu>5L?9`=?=?< 6 DA= 2 @�Ih9,M 2 9�=b@EÒQ9�9Ýu>5L?9`=?=?<7Ih9,> 6�Ù 9/L/Þ�ßK<JD¨à�9�M 2 9D¼×¨M 2 <�=�Lb9/=?Ú 4 M 2 @\=á 9/9Dâ9ÝKM?9D¨Fu9/F·M 6 Ò½@hDXã½à 4 @\=?=?9/= 6Eä =pM?LbÚAàyMbÚ5Lb9/= 4 <7åh9�<JDuÛ¨D5<^Mb9 ÙZ6 LqF5=�Ó æA×Ô`ç�Ö}× á <Jè}<7DuÛAD5<JM?9Ù36 LbF5=;ÓJÔ/éA×�Ô/æEÖ}×KMbLb@hDA=dÛ¨D5<^Mb9 Ù36 LbF5=,Ó ê5×Ô�Ö�×KM?Lq@hà�9`=×KMbL?9/9/=/×u>5<�àyM?ÚAL?9`=Þ7ÞJÞ

Ñ DëÓ ìEÖ�×.@EÚuM 6 Ò½@�Mq@í@\àà9>uMb<JD5Ü 4 <JDA9/@EL?è 6 LqFu9Lb9/Fî=pM?LbÚAà�M?Ú5Lb9/= 2 @�I\9 á 99/D�<7DXM?L 6 FuÚAà�9`FÙ <^M 2 à 6 L?Lb9/=?> 6 DAFu<7D5ÜHLq@�Mb< 6 DA@ 4 9�Ýu>5Lb9/=b=p< 6 D¨=Þ�ï 2 9/=?9 4 <JDA9/@ELQ=pM?LbÚAàyMbÚ5L?9`=�<7DAà 4 ÚAF59·ÛAD5<JM?9Ù36 LbF5=/×�<7DuÛAD5<JM?9\×�MbLb@hDA=dÛ¨D5<^Mb9 ÙZ6 LqF5=�@ED¨F¿M 2 9/<JL�Ò�<7LbL 6 Lb=/Þ�ï 2 9/=?9·@EÚ5M 6 Ò½@EMb@�@ELb9½ÚA=?ÚA@ 4@EÚuM 6 Ò½@�Mq@ 6 D�ÛAD5<JM?9 ÙZ6 LqF5=×/9�ÝKM?9/DAFu9`F Ù <JM 2�4 <7ÒQ<^M!M?Lq@ED¨=p<JM?< 6 DA=/Þ�ðñØ 4 99/D59�è 4 <Jå\9.M 2 9 6 Lb9ÒÙ @\=�>AL 6 I\9/F äC6 L Ù36 LbFA= 6 Dñà 6 Ú5DXMq@ á 4 9�=?à/@�MpMb9Lb9/F 4 <7D59/@hL 6 LqFu9Lb<7D5Ü\=/Þ�:%9/à/@ 474 M 2 @EM½@hD6 LqFu9Lb<JDAÜ,<7=Z=?à/@�M?M?9Lb9/F·< ä <JM F 6 9`=±D 6 MZà 6 DXMb@h<JD·@�Fu9D¨=p9�=pÚ á 6 LqFu9/L?<7D5Ü,<�= 6 Ò 6 L?> 2 <7à%M 6�ò Þ

ó 6 L¼ÒQ@hDKã�=pM?LbÚAàyMbÚ5Lb9/=/×yM 2 9Zà 4 @h=b= 6Eä Lq@�Mb< 6 DA@ 4 =p9Mb=�<�=¼à 4J6 =p9`F�Ú5DAFu9/L(Ò½@EDKã 6 >{9Lq@�M?< 6 D¨=4 <7åh9�=?Ú á =pM?<JM?Ú5M?< 6 DA=/×A<7DKIh9Lq=?9�=pÚ á =dMb<^MbÚuM?< 6 DA=�@EDAF á 6K6h4 9/@hD 6 >¨9/Lb@EM?< 6 DA=/ÞAð�= äC6 L á 6K6\4 9`@ED6 >{9Lq@�M?< 6 D¨=×AM 2 9Qà 476 =?Ú5Lb9�Ú5DAF59L�Ú5D5< 6 Dô@hDAFH<7DXM?9Lq=?9/àyMb< 6 Dô@ELb9�@ 4 Ò 6 =pM�@ 4JÙ @�ãu=)9`@h=?ãâM 6Üh9�M`Þ ï 2 9½à 476 =?Ú5L?9�Ú5DAFu9/L,à 6 ÒQ> 4 9ÒQ9/D\Mq@�Mb< 6 Dô<�= 6hä M?9/DíÒ�ÚAà 2 Ò 6 Lb9�F5<^õ·à�Ú 4 M;M 6 >5L 6 Ih9\Þï 2 <7=)>5L 6 >¨9/LpMdã½<�=%<7ÒQ> 6 LpMq@EDXM á 6 M 2�ä L 6 ÒöM 2 9,>5Lq@hàyMb<7à/@ 4 @hDAF·M 2 9;M 2 9 6 Lb9�Mb<7à/@ 4 > 6 <7DXM 6EäIX<79 Ù Þ Ñ M ÒQ9/@ED¨=±M 2 @�MZM 2 9;à 4 @\=?= 6Eä Lb@EM?< 6 DA@ 4 =?9�Mb= äC6 L?Ò½= @ED·9÷{9`àyMb<JI\9 á 6K6h4 9/@ED�@ 4 Üh9 á Lb@AÞÑ M±<7=±ÚA=?9/F Ù%2 9/D59I\9L.= 6 ÒQ9 476 Üh<�à <�=!M?Lq@EDA= 4 @�M?9`F�<JDXM 6 @EÚuM 6 Ò½@�Mq@5Þ ó 6 L.<7DA=pMb@ED¨à�9h×\<JD á 6 M 2>5L 6K6hä = 6Eä M 2 9�Fu9`à�<�F5@ á < 4 <^Mdã 6hä M 2 9,Ò 6 DA@\Fu<�à�=?9/à 6 DAFuè 6 LbFu9/L±M 2 9 6 L?ã 6Eä M 2 9,<7D\Mb9Ü\9Lq= á ãø�ùÚ¨à 2 <�Ófú`Ö @ED¨FQM 2 9;Fu9`à�<�F5@ á < 4 <^Mdã 6Eä M 2 9�Ò 6 DA@hFu<�à�=?9/à 6 DAFuè 6 LbFu9/L�M 2 9 6 Lbã 6Eä M 2 9�<7DuÛAD5<JM?9á <JD¨@ELbã,M?Lb99 á ã�:)@ á <7D�ÓJÔ/ûEÖ}×�M 2 9)à 476 =?Ú5Lb9 Ú5DAFu9/L±à 6 Ò�> 4 9/Ò�9/DXMb@�Mb< 6 D 6hä @hÚuM 6 Ò½@�Mb@�<�=!M 2 9åh9ãâ>5L 6 >¨9/LpMdã\Þ

Page 3: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

Ñ DâÓ ìhÖ}×/M 2 9Zà 4J6 =pÚ5Lb9.Ú5D¨Fu9L!à 6 Ò�> 4 9/Ò�9/DXMb@�Mb< 6 D Ù @h= 4 9 ä M�@h=(@ED 6 >¨9/D�>5L 6 á 4 9Ò�Þ Ñ D�M 2 <7=>A@E>{9L`× Ù 9�= 6h4 Ih9,M 2 @�M;>5L 6 á 4 9Ò"<7Dô@·> 6 =?<^Mb<JI\9 Ù @�ãhÞ��í9�= 256�Ù M 2 @�M�M 2 9�à 6 Ò�> 4 9/Ò�9/DXM6Eä @,Lq@�M?< 6 D¨@ 4 =?9�M 6hä�ÙZ6 LqF5= 6 D·à 6 Ú5DXMb@ á 4 9)=bà@EMpM?9/L?9`F 4 <7D59/@hL 6 LqFu9/L?<7D5Ü\=!<�=±@ 4 = 6 Lq@�M?< 6 D¨@ 4 Þ

ï 2 9,à 4 @h=b=p<�à@ 4 ÒQ9M 256 FâM 6 Üh9M)@ED�@EÚuM 6 Ò½@�M 6 D äC6 L M 2 9,à 6 ÒQ> 4 9ÒQ9DXM 6Eä @Q=p9M 6hä Û5èD5<^Mb9 Ù36 LbFA=�@\àà9>uMb9/F á ã�@EDQ@EÚuM 6 Ò½@�M 6 D�� <�=!M 2 L 6 Ú5Ü 2 Fu9�Mb9LbÒ�<7D5<��/@EM?< 6 D�Þ Ñ M.<�=�@ 4 L?9`@hFuãD 6 Duè}M?Lb<7IX<�@ 4 M 2 @�M�M 2 9%à 6 ÒQ> 4 9/ÒQ9DXM 6hä @�Lb@EM?< 6 DA@ 4 =?9�M 6Eä <7DuÛADA<^Mb9 ÙZ6 LqF5=¼<7=!@ 4 = 6 Lq@�M?< 6 D¨@ 4 Þï 2 9,Fu9�Mb9LbÒ�<7D5<��/@EM?< 6 D�Ò�9M 256 F�à@hD5D 6 M á 9,9/@\=p< 4 ã½9�ÝKM?9/DAFu9/FVM 6 <7DuÛADA<^Mb9 Ù36 LbF5=/Þ Ñ D 2 <7==p9/Ò�<7DA@ 4 >A@h>¨9/L,Ó ú�Ö}× ø�ùÚAà 2 <�ÚA=?9/F·@hD 6 M 2 9L @E>5>AL 6 @hà 2 á @\=p9`F 6 Dâ@�à 6 D5ÜhLbÚ59/DAà�9 6 D½ÛAD5<JM?9Ù36 LbF5=Z@ED¨FV:)@hÒ½=p9/ã�� =3M 2 9 6 Lb9Ò�Þuï 2 <�=3ÒQ9M 256 F�<7= = 6 ÒQ9 2A6�Ù L?9 4 @EM?9/F·M 6Q6 ÚAL @ 4 Ü\9 á Lq@E<�à@E>5>5L 6 @\à 2 Þ�Hà��@hÚ5Ü 2 M 6 D�9�ÝKM?9/DAFu9`F�M 2 9�Fu9M?9LbÒQ<JDA<��`@�M?< 6 D�ÒQ9M 256 F�M 6 <7DuÛAD5<JM?9 Ù36 LbFA=Ó^Ô� �Ö\>AL 6 IK<7D5Ü M 2 @EM�@hDXã ø�ùÚAà 2 <X@EÚuM 6 Ò½@�M 6 D,<�=�9��XÚ5<7I�@ 4 9DXM¼M 6 @)Fu9M?9/L?ÒQ<7D5<7=pM?<�à��HÚ 4J4 9L!@EÚ5èM 6 Ò½@EM 6 D¼Þ ø�ùÚAà 2 <A>AÚA= 2 9/F ä Ú5L?M 2 9/L.M 2 <7=±ÒQ9�M 2A6 F½@hDAFQ9�ÝKMb9DAFu9`FQ<^M±M 6 MbLb@hDA=dÛ¨D5<^Mb9 Ù36 LbFA=Ó êEÖ}Þ Ñ M�<7=)M 2 9/D�Ih9/L?ã�à 6 ÒQ> 4 9�Ý Þ Ñ DîÓ EÖ}×uM 2 9�@ 4 Ü\9 á Lq@E<�à;@E>5>AL 6 @hà 2�Ù @\=%ÚA=?9/F�M 6 Üh<7Ih9�@ED5è6 M 2 9L,>5L 6K6Eä±6hä M 2 9Qà 4J6 =pÚAL?9�Ú5DAFu9/L�à 6 Ò�> 4 9/Ò�9/DXMb@�Mb< 6 D äC6 L�M?Lq@EDA=pÛAD5<JM?9 ÙZ6 LqF5=Þ Ñ D Ó ûEÖ}×Ù 9 2 @�Ih9�@ 4 Lb9/@\Fuã�>AL 6 I\9/F�M 2 9�Lb9/=?Ú 4 M äC6 L Ù36 LbFA= 6 DVà 6 ÚAD\Mq@ á 4 9�=bà@EMpM?9/L?9`F 4 <7D59`@EL 6 LqFu9L?è<JD5ÜX= 6hä ÛAD5<JM?9�Lb@hD5åu=ÞKï 2 9�Fu9�Mb9LbÒ�<7D5<��/@EM?< 6 D·ÒQ9�M 256 Fâà/@ED5D 6 M á 9�@h>5> 4 <J9`F á 9/à/@EÚA=?9�@hDXã@EÚuM 6 Ò½@�M 6 D�<7=%D 6 M)9��XÚ5<7I�@ 4 9DXM%M 6 @QFu9M?9/L?ÒQ<7D5<7=pM?<�à 6 DA9hÞ Ñ D�M 2 @�M�>A@h>¨9/L/× Ù 9,9�ÝKMb9DAFu9`FM 2 9%ÒQ9�M 256 F�ÚA=p9`F á ã ø�ùÚAà 2 <5<JD�Ófú`Ö5ÚA=?<JDAÜ;@hD½@hF5Fu<JM?< 6 D¨@ 4 <7DAFuÚAà�M?< 6 D 6 D�M 2 9%Lq@ED5å�ÞhßK<7DAà9Lb@hD5åu= 6hä à 6 Ú5DXMb@ á 4 9�=?à/@�MpMb9Lb9/F 4 <7D59/@hL 6 LbF59Lb<JD5ÜX=)Lq@ED5Ü\9 6 I\9L�@ 4J4 à 6 Ú5DXMb@ á 4 9 6 LqFu<7DA@ 4 =/×M 2 <�=�@h>5>5L 6 @hà 2 <7=�D 6 M�=?Ú5<JMb@ á 4 9 äC6 L ÙZ6 LqF5= 6 D�@ 4J4 M 2 9/=?9 6 LbFu9/L?<7D5ÜX=Þ Ñ D�M 2 <7=�>A@E>{9L`× Ù 9>5L 6 I\9%M 2 9 Ù%2A6h4 9�Lb9/=?Ú 4 M äC6 L @ 474 à 6 Ú5DXMb@ á 4 9�=?à/@�MpMb9Lb9/F 4 <7D59/@hL 6 LbFu9/L?<7D5ÜX=.ÚA=?<JDAÜ�@hDâ@ 4 Üh9èá Lb@h<7à�@E>5>AL 6 @hà 2 Þ��í9½Fu9ÛAD59Q@âÜ\9D59/Lb@ 4 <��/@EM?< 6 D 6hä =?9ÒQ<JÜ\L 6 Ú5>A=/רà/@ 474 9`F��\è�=?9ÒQ<7ÜhL 6 Ú5>A=/Þ�í9�= 256�Ù M 2 @�M/× Ù%2 9/D�ÛAD5<JM?9\×AM 2 9/=?9��\è�=?9ÒQ<7ÜhL 6 Ú5>A= @hL?9�9��XÚ5<JI�@ 4 9/D\M�M 6 @hÚuM 6 ÒQ@EMb@AÞ�¿9@ 4 = 6 = 256�Ù M 2 @�M`× á ã�@hDA@ 476 Ü\ã Ù <JM 2 M 2 9%à/@h=?9 6Eä ÛAD5<JM?9 Ù36 LbF5=/×�@�à@hD 6 DA<7à/@ 4 �hèS=p9/ÒQ<JÜ\L 6 Ú5>�×à@ 474 9/F�M 2 9�=?ãKD\Mq@hà�M?<�à��\è�=?9ÒQ<7ÜhL 6 Ú5>�×5à/@ED á 9�@\=?= 6 à�<�@�Mb9/F Ù <JM 2 @hDXã�Lq@�M?< 6 D¨@ 4 =?9�M��îÞ Ñ M2 @h=;M 2 9½>5L 6 >{9L?Mdã 6Eä á 9<7D5Ü�M 2 9·=?Ò½@ 474 9`=dM��\èS=p9/Ò�<7ÜhL 6 ÚA>HL?9`à 6 ÜhD5<��<7D5Ü��îÞ�ðöà 6 DXM?<7DXÚ5è@�M?< 6 D 6hä M 2 <7=�>A@E>{9L Ù36 Ú 4 F á 9½M 6 9ÝXMb9DAF¿M 2 9½9��XÚ5<7I�@ 4 9DAà9 á 9M Ù 9/9DÅ=pMb@EL?è ä L?9/9Q=?9�Mq=×ÛALb=pM 6 LbFu9/L 4J6 Üh<�à½@EDAFÅ@h>¨9/L?< 6 Fu<�à½=p9/Ò�<7ÜhL 6 ÚA>A=·Ó ÕhÕu×ÔìA×pÕ�ÖZ@hDAFÅ@ 4 = 6 á 9�M Ù 99/DÄLb@EM?< 6 DA@ 4=p9Mb=%@ED¨FâM 2 9,Ò 6 DA@\Fu<�à�=?9/à 6 DAF 6 LqFu9/L3M 2 9 6 L?ã\Þ

ø 6 M 2Å2 ãK> 6 M 2 9`=p9`=�M 2 @�M�M 2 9 6 LqFu9Lb<7D5Ü\=�@hL?9V=?à/@�MpMb9Lb9/FÅ@hDAFîà 6 Ú5DXMb@ á 4 9V@ELb9·L?9`@ 474 ãD59/à9/=b=?@hL?ã\Þ ø�ùÚAà 2 <3@ 4 Lb9/@\FuãH> 6 <7DXM?9/F 6 ÚuM�M 2 @�M�Lb@EM?< 6 DA@ 4 =p9Mb= 6Eä M?Lq@EDA=pÛAD5<JM?9 Ù36 LbF5= 6Eä4 9/D5ÜEM 2 ÜhLb9/@EM?9/L�M 2 @�M�� B�� M 2 9 4 9/@\=dMQD 6 D5è�à 6 Ú5DXMb@ á 4 9 6 LqFu<7DA@ 4�� @hL?9VD 6 M½à 476 =?9/FÅÚADAFu9Là 6 ÒQ> 4 9ÒQ9/D\M`Þ Ñ M�à@hD á 9½>5L 6 Ih9`F�M 2 @�M,M 2 9·=p9M 6Eä3ÙZ6 LqF5= 6 D¿=bà@EMpM?9/L?9`F 4 <7D59/@hL 6 LqFu9L?è<JD5ÜX=½<�=·D 6 MâLb@EM?< 6 DA@ 4 @h=·@Å=?Ú á =?9�M 6Eä,Ù36 LbFA= 6 D�@ 474)4 <JDA9/@EL 6 LqFu9Lb<JDAÜ\=½@ 4 M 2A6 Ú5Ü 2 <JMb=à 6 ÒQ> 4 9ÒQ9/D\M)<�= Lq@�M?< 6 D¨@ 4 Þ

G�Ú5L¼>5L 6K6Eäu6Eä M 2 93à 6 ÒQ> 4 9ÒQ9DXMq@�M?< 6 D�à 4J6 =pÚAL?9.<�=(9�÷�9/àyMb<JI\9hÞ���<JI\9D�@hD�@EÚuM 6 Ò½@�M 6 D��½×<^M Üh<7Ih9`=3@hD 6 M 2 9L @EÚuM 6 Ò½@�M 6 D ��M 2 @�M%@hàà9>uMq= ÙZ6 LqF5=±M 2 @EM @hL?9�D 6 M%@hà/à�9>5M?9/F á ã��·Þ Ñ MÜh<7Ih9/= @hD 6 M 2 9L%>5L 6K6Eä�6Eä M 2 9�Fu9`à�<�F5@ á < 4 <^Mdã 6Eä M 2 9,9��\ÚA<JI�@ 4 9D¨à�9 6Eä M 2 9`=p9�@hÚuM 6 ÒQ@EMb@�Ó ç�Ö�Þ

ï 2 <7=¼>A@E>{9L(<7= 6 LbÜ\@hD5<��9/F;@\= äC6h47476�Ù =Þ"!�9�Û¨D5<^Mb< 6 DA=�à 6 D¨à�9LbD5<7D5Ü 4 <JDA9/@EL 6 LqFu9Lb<JDAÜ\=�@hDAFLb@EM?< 6 DA@ 4 =p9Mb=�@hL?9�ÛALq=pM�L?9`à@ 4J4 9/F¿<7DÅßu9/àyMb< 6 DA=�Õ�@hDAF# 5Þ(ï 2 9D�×!ßK9/à�M?< 6 DÄìH<7DXM?L 6 F5ÚAà�9`=M 2 9�@ 4 Üh9 á Lb@h<7à,=pM?LbÚAàyMbÚ5Lb9 6Eä �\è�=?9ÒQ<7ÜhL 6 Ú5>¼Þ5ï 2 9�>5L 6K6Eä�6Eä 9��XÚ5<JI�@ 4 9/DAà�9 á 9M Ù 9/9D�ÛAD5<JM?9�hèS=p9/ÒQ<JÜ\L 6 Ú5>A=)@hDAFô@hÚuM 6 ÒQ@EMb@â<7=;=?åh9�Mqà 2 9/FH<JD á 6 M 2 Fu<JLb9/à�M?< 6 DA=�<7D¿ßK9`àyMb< 6 DA=,ç·@ED¨Fôæ5Þó <7DA@ 4J4 ãh×{M 2 9Q=?ãKDXMb@hà�M?<�à$�\èS=p9/Ò�<7ÜhL 6 ÚA>�à 6 LbL?9`=p> 6 D¨Fu<JDAܽM 6 @VLb@EM?< 6 DA@ 4 =?9�M;<7=;Fu9ÛAD59`Fô<7DßK9/à�M?< 6 DôúKÞ

Page 4: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

� � Ì.ËAÍ��î̱É��pÐ?É���!Ë Ì±ËAÍ���˨ÐpÉ���ï 2 <7=�=?9/à�M?< 6 D�L?9`à@ 474 = á @h=?<7àZFu9�Û¨D5<^Mb< 6 DA= 6 D 4 <JD59`@EL 6 LqFu9/L?<7D5Ü\= á ÚuM�M 2 9 L?9`@hFu9/L(<�=!Lb9 ä 9LbLb9/FM 6 Ó Õ5Ô�Ö äC6 L·@íà 6 ÒQ> 4 9�M?9�<7DXM?L 6 FuÚAàyMb< 6 D�Þ ��@hÚA=bF 6 L?÷ � =½à 2 @hLb@\àyMb9Lb<��`@�M?< 6 D 6hä à 6 Ú5DXMq@ á 4 9=?à/@�MpMb9Lb9/F 4 <7D59`@EL 6 LqFu9/L?<7D5Ü\=�<�=½Üh<7Ih9/Dñ@hDAF ÙZ6 LqF5=�<7DAFu9�Ýu9`F á ã 4 <7D59/@hL 6 LqFu9Lb<JDAÜ\=Q@ELb9<JDXM?L 6 FuÚAà9/F Þ� 9�M�� á 9�@�=?9�M.9��XÚ5<J>A>¨9`F Ù <^M 2 @ED 6 LqFu9L��,Þhï 2 9 6 LqFu9Lb<JDAÜ���<7=�������������< ä¨äC6 L±@EDKã��@EDAF! �<JD"��×�9<JM 2 9/L#�$�% 6 L &�'�¨Þ � 9M)( á 9)@�ÛADA<^Mb9)@ 4 > 2 @ á 9�M`Þhð+*),��.-0/"1 �3254 � 47698<JDAF59�Ýu9/F á ã�@ 4 <JDA9/@EL 6 LqFu9Lb<JDAÜ���<7=±@ ä ÚADAàyMb< 6 D ä L 6 Ò:�âM 6 (�Þ;��<�=.à@ 4J4 9/F�M 2 9<�=�>�@?5A�B 6Eä/¼Þ ó 6 L±<7DA=dMq@EDAà9 �î<�=.M 2 9 4 9/D5ÜEM 2·6Eä Lb<JÜ 2 Mpè�<JDuÛ¨D5<^Mb9 ÙZ6 LqF5= 2@CD2 B ÞJÞ7Þâ@hDAFFE,<�=±M 2 9 4 9/D5ÜEM 26Eä á <^è�<7DuÛAD5<JM?9 ÙZ6 LqF5= Þ7Þ7Þ 2HG B 2@CI2 B Þ7ÞJÞ¼ÞJLKNM OQP�RSUT#V5W$RHX�Y0RZP�SU[$\3]US#^@_^`SbaLcbdN\3]U^@e;PfRZP�SU^9P�\N]#g�[ó 6 L;@EDKã 4 <7D59`@EL 6 LqFu9Lb<JDAÜh��× Ù 9�Fu9/D 6 Mb9 á ã'iQ�ÄM 2 9 á @hàqå Ù @ELqF 4 <7D59/@hL 6 LbF59Lb<JD5Ü·M 2 @EM<7=�M 2 9½=p9M0� 9��XÚ5<J>A>¨9`F Ù <JM 2 M 2 9QLb9I\9Lq=p9 6 LqFu9Lb<JDAÜAÞ ó 6 L;<7DA=pMb@ED¨à�9h×ji � <�=�M 2 9 4 <7D59/@hL6 LqFu9Lb<JDAÜ 6hä D59/Ü\@EM?<7Ih9�<JDXM?9/Üh9/Lb=/Þ

ï 2 9�=?Ú5Òk�"lnm 6Eä M Ù36·4 <7D59/@hL 6 LbFu9/L?<7D5ÜX=%<7=�M 2 9�=?9�M0�&opm 9��XÚ5<7>5>{9/F Ù <^M 2 M 2 96 LqFu9Lb<JDAÜh� 9�ÝKM?9/DAFu<7D5ܽM 2 9 6 LbFu9/L?<7D5ÜX= 6Eä �î@EDAFqm á ã�=?9�M?M?<7D5Ür�h�s äC6 L�@EDKã$�ptu�@EDAF% +tvmîÞ ó 6 LbÒQ@ 4J4 ãh×�M 2 9'wyx{z}|47698 m 4 <7=�M 2 9H=?9�M 6Eä @ 474 >A@E<7Lq= � �~�� � =?ÚAà 2 M 2 @EM 'tum 4 9��\ÚA<J>5>{9/F Ù <JM 2 M 2 9 6 LbFu9/L?<7D5Ü�Fu9�ÛADA9/F á ã � B ~�� B � � � N ~�� N � < ä @ED¨F 6 D 4 ã�< ä� B ��� N 6 L � � B 1u� N @EDAFh B �% N <7Dpm 4�� � Þï 2 9�=pÚAÒ 6hä�4 <JDA9/@EL 6 LqFu9Lb<JDAÜ\= 2 9 4 >A=)M 6 Fu9ÛAD59,M 2 9�>5L 6 FuÚAàyMq= 6Eä!ÙZ6 LqF5=Þ � 9�M<� á 9@ 4 <7D59`@EL 6 LbF59Lb<JD5Ü�@hDAF 4 9�M � / 4 � 47698 á 9 ÙZ6 LqF5= 6Eä Lb9/=?>{9/àyMb<JI\9 4 9/D5ÜEM 2 m 4 äC6 LZ@EDKã��ft��!Þï 2 9 ÙZ6 LqFq/b1��47698 / 4 6 á Mq@E<7D59/F á ãíà 6 DAà/@�M?9/DA@�Mb< 6 D 6Eä M 2 9 Ù36 LbF5=Q/ 4 Ù <^M 2 L?9`=p>{9/à�MM 6 M 2 9 6 LqFu9/L?<7D5Ü 6 D��Ä<7= 6Eä�4 9D5ÜhM 2�� 1 |47698 m 4 Þ ó 6 L�<7DA=pMb@hDAà�9\×5< ä�äC6 L�@EDKã$�&t �)× Ù 9Fu9D 6 M?9 á ã$/ 4 1 2@�`� ×XM 2 9/DF/"1��476 �

/ 4 <�=±M 2 9 Ù36 LbF$/p1 2@�`� 6hä�4 9/D5ÜEM 2 |476 �� 4 1 � � Þ

ï 2 9)=p9��XÚ59DAà9 � / 4 � 47698 6Eä�Ù36 LbFA=(<�=�à@ 4J4 9/F�@0�����7�5�>A�,��y�=�I��AN�3,�� 6hä M 2 9 Ù36 LbF0/&1 �47698 / 4 Þ

JLK�J ��V5e{WIWI^@P�^`S�d3\3]U^`e{PFRHP�S#^@P�\3]UgH[ð 4 <7D59/@hL 6 LqFu9Lb<JDAÜ�� <�=�-5�>�;w>�H< ä�äC6 L·@hDXãq� @EDAF% <JDs�ë=?ÚAà 2 M 2 @EM<���� �×!M 2 9Lb99�Ýu<7=pMb=�@EDî9 4 9/ÒQ9DXM�� 6Eä ��=pÚ¨à 2 M 2 @EMQ�n���r�: �Þ Ñ M�<�=$w>����A�A��7�.��-¿< ä <JM�à 6 DXMb@E<7DA=,D 6Fu9DA=?9ô=?Ú á 6 LqFu9Lb<JDAÜAÞ.ï 2 9 6 LbF59Lb<JD5Ü � 6Eä DA@EM?Ú5Lq@ 4 <7DXM?9/Üh9Lq=·@hDAF�M 2 9 6 LqFu9Lb<JDAÜ�E 6EäL?9 4 @EM?<7Ih9,<7D\Mb9Ü\9Lq=)@ELb9�=?à/@�M?M?9Lb9/F�Þ� 6 Lb9,Üh9/D59Lq@ 474 ãh× 6 LbFu<7DA@ 4 =)@hL?9�=bà@EMpMb9Lb9/F 6 LqFu9Lb<7D5Ü\=/Þ�í9,Fu9/D 6 Mb9 á ã���M 2 9,=?Ú á à 4 @\=?= 6hä ÛAD5<JM?9 4 <JD59`@EL 6 LqFu9/L?<7D5Ü\=/×;� M 2 9,à 4 @h=b= 6Eä à 6 Ú5DXMq@ á 4 96 LqFu<JD¨@ 4 =.@EDAF 6hä�  M 2 9�à 4 @\=?= 6Eä à 6 Ú5DXMb@ á 4 9)=bà@EMpM?9/L?9`F 4 <7D59/@hL 6 LqFu9/L?<7D5Ü\=/ÞEï 2 9 äC6\4J476�Ù <7D5Üà 2 @ELq@hà�M?9Lb<��/@�Mb< 6 D 6Eä =bà@�M?M?9/L?9`F 6 LqFu9/L?<7D5Ü\=Z<�=)FuÚ59�M 6 ��@EÚ¨=?F 6 Lp÷!Þ¡<¢ ^`RHP�^@£¤MHK¦¥�§ ��x`w7-9,��©¨ ¥�ª5ª¬«�«¦­ ��,�x{�HA��9®7�=�r�¯���������!,��.-9�7�y���`?&�°�±w<w>�²��ANA��>�.��-f� �r����-,��Z�¯³$� �Q��®²�>�=,��@?�wQA�,µ´¶ 6@·0¸ ¶ *#B;�>�.�0A�BZ�0�>�=�Dw�w>�7w ¸ ¶ ���.������-�xZ�7A���¹D�>��³&-9�3º)����-F®>³5»ª5¼ ¸ C 1¾½�¿L~ MHÀ

Page 5: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

� ¼ ¸ ¶ 1¾½ |47698 m 4�� ��tr� oq½"��~Ii ��~²E À ����-�m 4 t ´��� ¶ ¸ � À`¼*#B;�>�.��¿b����- M ���.���.�7w��;���>AN��¹��7�¯³fA�BZ��,��.-5�>�y���@?�w<*���A�B"� �7�²,F����-F,����r�7�=�>z$�7�HA ¼Ñ D 6 LqFu9L,M 6 =?<JÒQ> 4 < ä ãíM 2 9â>5L 6K6Eä =/× Ù 9½Ú¨=p9â= 4 <JÜ 2 M 4 ãÄFu<J÷{9/L?9/DXM�<JD¨FuÚAàyMb<JI\9âà 4 @h=b=?9/=� ó 6 L@EDKã�bt��·×¨M 2 9�à 4 @h=b= � ¶ <7=�Fu9ÛAD59/F á ã���� ¶ 1 ½ |47698 m 4�� �utF� @ED¨F"m 4 t ¸ ¶ À Þï 2 9,<JD¨à 4 ÚA=p< 6 D¨= ¸ ¶�� � ¶�� ¸ ¶�� B 256\4 F äC6 L�@EDKã 6 LqFu<7DA@ 4 ÅM 2 ÚA=/×5ÚA=?<7D5Ü·ï 2 9 6 L?9/Ò-Ôh×=?à/@�MpMb9Lb9/F 4 <JD59`@EL 6 LbF59Lb<JD5ÜX=%à@hD á 9�Fu9ÛAD59`F ä L 6 Ò M 2 9�à 4 @h=b=?9/=�� ¶ á ã��   1 ´¶ 6@· � ¶ Þï 2 9!�.����� 6hä @ 4 <7D59`@EL 6 LqFu9Lb<7D5ÜF�í<7=ZM 2 9,=pÒ½@ 474 9/=pM 6 LbFu<7DA@ 4 =pÚAà 2 M 2 @EMQ� t�� ¶ Þ �¿9Fu9D 6 M?9 á ãh(���M 2 9�=p9M 6hä @ 4J4(Ù36 LbFA= 6 Ih9/L�( <7DAFu9ÝK9`F á ã�à 6 Ú5DXMb@ á 4 9�=?à/@�M?M?9Lb9/F 4 <7D59/@hL6 LqFu9Lb<JDAÜ\=/Þ

� �%��Ê5ÐḏÉ��j�r�@� Ê;�îÌ����¿Ì.˨Í��îÌ3É��pÐpÉ���!Ë Ì.˨Í���ËAÐ?É�)�ø LbÚ5ã��9/L?9Ä@EDAF 1 @hLpM 6 D 2 @�Ih9Ä<7D\MbL 6 FuÚ¨à�9/FëLq@�Mb< 6 DA@ 4 9ÝK>AL?9`=?=?< 6 DA=�@EDAF @hÚuM 6 ÒQ@EMb@ äC6 LÙ36 LbF5=�<7DAFu9ÝK9`F á ãôà 6 Ú5DXMq@ á 4 9½=?à/@�MpMb9Lb9/F 4 <7D59`@EL 6 LqFu9Lb<7D5Ü\=/Þ ï 2 9ã 2 @�Ih9�>5L 6 I\9/FHM 2 @EM@Å=p9M 6hä�ÙZ6 LqF5=Q<�=½Lq@�M?< 6 D¨@ 4 < ä @hDAF 6 D 4 ã < ä <JMâ<�=·@hà/à�9>5M?9/F á ã�@íÛAD5<JM?9í@EÚuM 6 Ò½@�M 6 D9�ÝKM?9/DAFu<7D5ÜHØ 4 9/9D59 � =�M 2 9 6 L?9/Ò�Þ¼ï 2 <�=�=p9`àyMb< 6 D¿= 256 L?M 4 ãôLb9/à/@ 474 =;Fu9ÛAD5<JM?< 6 DA= 6Eä Lb@EM?< 6 DA@ 46 >{9Lq@�M?< 6 D¨=Z@hDAF�@EÚuM 6 Ò½@�Mq@ á ÚuM)M 2 9,L?9`@hFu9/LZ<�= Lb9 ä 9LbLb9/FâM 6 Ó ìhÖ äC6 L%Ò 6 Lb9,Fu9�Mq@E< 4 =/Þ

KNM !&e{WI\NRH]Ue;d ^9_#"UP�^`[I[ \3R�]#[� 9�M!( á 9V@HÛAD5<JM?9�@ 4 > 2 @ á 9M/Þ!ï 2 9V=?9�M%$ 2'& � (�� ��6hä Lb@EM?< 6 DA@ 4 =?9�Mb= 6Eä)ÙZ6 LqF5= 6 I\9Lr(<JDAF59�Ýu9/F á ãîà 6 Ú5DXMq@ á 4 9�=bà@EMpMb9Lb9/F 4 <JDA9/@EL 6 LqFu9/L?<7D5Ü\=�<�=�M 2 9�=pÒ½@ 474 9/=pMQ=?9�M½à 6 D\Mq@E<7D5<7D5ܽ 2 À äC6 L @EDKã 2 tp(ë@EDAF·à 4J6 =p9`F½Ú5DAF59L3M 2 9 äC6h474J6�Ù <7D5Ü�Lb@EM?< 6 DA@ 4A6 >{9Lq@�Mb< 6 DA=±F59�ÛAD59`F äC6 L@EDKãâ=?Ú á =?9�Mb= � @ED¨F�( 6hä (�� á ã��

� l)( 1¾½+* � *!t��:o�( À�-,.( 1¾½ /�,0/ � /�t���~1/&t�( À ��2 1+½�34465 B / 4'�87 t!�¾~¬/ 4 t � À� � 1¾½ 4476 �

/ 4'� / 4 t�� À � G � 1+½ 4476{G �/ 48� / 4 t�� À

�:9 1¾½ 4476 ¶ / 48� ut��$~./ 4 t�� À � G 9b1+½ 4476{G ¶ / 48� �t��$~./ 4 t � À� ��(v1¾½ 447698�;=<8�> * 4�� ��t

 �?A@ ~6* 4 t � < ä �$t �¿@hDAF�* 4 tB( < ä �$tDC�E2 À Ù%2 9/L?9C�F2�1 C� ? ½ � @ ~7� � ~ � �;~ @ � À Þ

ï 2 9,D 6 Mq@�Mb< 6 DGC�ô<�=)Fu9ÛAD59`FV<7D�M 2 9,DA9�ÝKM�=p9`àyMb< 6 D�Þ K�J H&T�W RH£ue;WIeqR�]�d3\N]#^`e{PfRHP�S#^@P�\3]Ug�[ð)Dâ@hÚuM 6 ÒQ@EM 6 D 6 D 4 <7D59`@EL 6 LbFu9/L?<7D5ÜX=!<7=3@�à 4 @h=b=p<�à@ 4 ÛAD5<JM?9�@EÚuM 6 Ò½@�M 6 D Ù <JM 2 @\F5Fu<JM?< 6 DA@ 44 <7ÒQ<^M%MbLb@hDA=p<JM?< 6 D¨= 6Eä M 2 9 äC6 L?ÒJILK�MBN 6 L�N'K8MOI Ù%2 9/L?9PI <7=%@Q=?9�M 6hä =pMb@�Mb9/=/Þ

Page 6: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

� ^��j]#\3WI\3R�]+MHK<­ �ô@hÚuM 6 ÒQ@EM 6 D � 1 ��� ~.(�~��$~��Z~� �±6 D 4 <JDA9/@EL 6 LqFu9/L?<7D5Ü\=Q�±w<-5�Nº ����-®7³$� º �H��A���w>�7A ,���w�A���A��7w �� � º)�Z��A��¦��� �ZB;�9®²�>A#( �0w>�>A),���AN�²���;wy��A��3,��Zw��� ����� ( �� � o ��� ��� � ��� � o ������� ��� � � ����-h���H��AN�3��� ����-<º)�����jw7�>ANw$,¬�!wyA���A��7w��� � ����-� � ¼ï 2 9�F59�ÛAD5<JM?< 6 D 6Eä >A@EM 2 =�<�= á @h=?9/F 6 DHM 2 9�D 6 M?< 6 D 6hä à�ÚuM�M 2 @�M Ù 9�9�Ýu> 4 @E<7DHD 6�Ù � � 9M/ á 9�@ ÙZ6 LqF�<JDAF59�Ýu9/F á ã�@ED 6 LqFu9/L?<7D5Üp�ut   Þ�ï 6 @EDKãVM Ù36 è ä @\àyM 6 L?<��/@EM?< 6 D"/h1 /8* 6Eä/¼× 6 DA9�à@hDV@\=?= 6 à�<�@�M?9�@�>A@hLpMb<^Mb< 6 D 6hä �H<JDXM 6 M ÙZ6 <7DXM?9/L?I�@ 4 = � m�~ � � =pÚ¨à 2 M 2 @�M � / � 1°m@EDAF � * � 1 � ÞAßKÚAà 2 @�>A@EL?M?<JM?< 6 D�<�=%à@ 474 9/F�@"�>x{A 6hä ��Þ5ï 2 9,=?9�M C�h1¾½ � m�~ � � � m o � 1����� ntbmq~�� ��t � ~� ��!� À <�=,M 2 9â=?9�M 6hä àÚuMb= 6Eä M 2 9 6 LqFu9Lb<JDAÜ ��Þ�ï 2 9D�×�@�>A@�M 24 @ á 9 4J4 9/F�/�<�=�@ ä Ú5DAà�M?< 6 D ä L 6 Ò M 2 9�=p9M C� <7D\M 6 M 2 9�=p9M 6Eä =dMq@�M?9`=Þ!ð�=�M 2 9�=?9�M C��<7=DA@�MbÚ5Lb@ 4J4 ã�9��XÚ5<7>5>¨9`F Ù <JM 2 M 2 9 6 LbFu9/L?<7D5Ü � m B ~ � B � � � m N ~ � N � < ä @ED¨F 6 D 4 ã;< ä m B � m N ×@�>¨@�M 2�4 @ á 9 474 9`F á ãV@ ÙZ6 LqF 6Eä(4 9DAÜEM 2 �¿<7=%@ ÙZ6 LqF 6 I\9L � 6hä(4 9D5ÜhM 2 C��Þ� 9�M#"p1 � N%$ � $ 6 <8 á 9�@ ÙZ6 LqF 6Eä¼4 9DAÜEM 2 C� 6 Ih9/L � ×\M 2 9 4 <7ÒQ<^M)=?9�Mb= 6hä =dMq@�Mb9/= 6Eä "H@EM@�Ü\<JI\9D�à�ÚuM'& 6hä C�¿@hL?9;Fu9ÛAD59/F á ã��

4 <7Ò$�( "p1¾½.N0t � � � &*)#�+&�~-,.&*) )/&*)j�0&1) )j�0&�@hDAF�NQ1 N $ ) ) À4 <7Ò$�2 "p1¾½.N0t � � � & )43 &�~-,.& ) ) &��+& ) ) �0& ) @hDAF�NQ1 N $ ) ) À

� ^��j]#\3WI\3R�]°JLK65 �7A �v1 ��� ~.(�~��$~��Z~� � ®��¦���p��x{A�,�z!��A�,��h,��p�����������Q,��.-9�7�y���`?�wQ����-�=�7Aj/h1 �N2 4 � 47698 ®²�r�F*),��.-p,��<�=�7�`?�A©Bh�s,��&( ¼#­ >A@�M 2 "�,��¦�=�9®��7��/n�������±w0�F*),��.-"p1 � N $ � $ 6 <8 ,¬�¦�=�>�@?5A�BJC��,�¹��7� � wyxZ��BpA©B;��A��7,��r���Z³ � m�~ � � tDC� »798 �QA©B;�>�.�r��:9�±wyA3w-�Ut � wyxZ�²BpA�BZ��A � m:oq½;� À ~ �B? ½;� À � t C�A�BZ�7��N=<?>A@ BDC'E*FK8M�NG<?> ;DHJILK @ BNM HJILK C)t�� �>� w>� 4 <JÒ<?>A@ BDC ( "�O N=<?>A@ BDC)t�� ¼

798 �QA©B;�>�.�r��:9�±wyA3w� Fthm wyxZ�²BhA�BZ��A � m ? ½D À ~ � oq½D À � tDC��A©B;�>�N=<?>-M H�P1K @ B ;DH�P1K C E*QK�MBN=<?>A@ BDC)t�� �>� w>� NG<?>R@ BDCSO 4 <7Ò<?>A@ BDC 2 "qt�� ¼ï 2 Ú¨=×�< ä @�à�ÚuM 2 @h=,@�>5Lb9/F59/à�9`=?= 6 L 6 L�@�=pÚ¨àà�9`=?= 6 L`× ÚA=?ÚA@ 4 M?Lq@EDA=?<^Mb< 6 DA=�@hL?9QÚA=?9/F ×6 M 2 9L Ù <7=?9�M 2 9,>A@EM 2 ÚA=?9/= 4 <JÒQ<JM%M?Lq@EDA=?<JM?< 6 DA=/Þð�= C� 2 @h=¼M 2 9 4 9/@\=dM�9 4 9ÒQ9DXM � @ ~7� � @EDAF�M 2 9ZÜhLb9/@EM?9/=pM(9 4 9/ÒQ9DXM � �;~ @ � äC6 L�@hDXã 4 <7D59/@hL6 LqFu9Lb<JDAÜF��×A@Q>A@EM 2�2 @\=%@ 47Ù @�ãu= @�ÛALb=pM�@ED¨F�@ 4 @h=pM�=dMq@�Mb9hÞAð ÙZ6 LqFâ<�=<�5����� ��A��²- á ãâ@hD

@EÚuM 6 Ò½@�Mq@�< ä <^M <�=±M 2 9 4 @ á 9 4{6hä @�>¨@�M 2V4 9`@hFu<7D5Ü ä L 6 Ò @EDâ<7D5<JM?<�@ 4 =pMb@�Mb9)M 6 @�ÛADA@ 4 =pMb@EM?9hÞ�í9�Fu9D 6 M?9 á ã�TVUWNX N�M 2 9â9�Ýu<7=pM?9/DAà�9 6Eä @�>A@EM 2î4 9/@\Fu<JDAÜ ä L 6 Ò M 2 9V=pMb@EM?96TÅM 6 M 2 9=dMq@�M?9PN 6Eä!4 @ á 9 4 /¼ÞÑ M 2 @h= á 9/9Dî>5L 6 Ih9`Fô<7D Ó ìEÖ.M 2 @�MQ@EÚuM 6 Ò½@�Mq@�@EDAFÄLq@�Mb< 6 DA@ 4 9Ýu>5L?9`=?=?< 6 DA= 2 @�Ih9QM 2 9=?@hÒ�9;9ÝK>AL?9`=?=?<JI\9�> 6�Ù 9L`Þ¡<¢ ^`RHP�^@£ JLK¦¥ZY>«Q­ w>�7A�,��<*),��.-�w<����-5��:`��-&®>³&��,�x{�HA��9®7�=��w>����A�A��>�.��-$�¯���������0,��.-5�>�y���@?�w�±wQ�²��A��3,�������� �<����-",��Z�¯³$� �¦��A)�±w0�9�²��� ��A���-F®>³F��º �H��A��r��x{A�,�z$��A�� ¼[ \��¬ �^]�ËZ�!Ð?Ï�ÏN_��!Ë;��Ï{Ê`�¼ËAÐa`L��Ê5ÐdÌ±É Ì��·ËZ�(Ê5Ðp̱É����!�@� Ê;�ð =p9/Ò�<7ÜhL 6 ÚA>â<�=)@Q=p9Mcbî9��\ÚA<J>5>{9/F Ù <JM 2 @ED�@\=?= 6 à�<�@�M?<7Ih9 á <7DA@ELbã·>5L 6 FuÚAàyM`ÞAï 2 9,=?9ÒQ<^èÜhL 6 Ú5>dbH<JD Ù%2 <7à 2�2 @hF á 99/D�@\F5Fu9/F�@�D59/ÚuM?Lq@ 4 9 4 9ÒQ9/D\M�<7=!Fu9/D 6 Mb9/F á ã6b B Þ�ð�D�9 4 9/Ò�9/DXM

Page 7: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

¤ £

�¤ M�� £��� ¤��y£�� M ¤

� �{���K� �3vhedt`²�j�edt`Y�t`YQlo[oYX_yj�aZtapi\_bad[oY\¬`c±j/xqxb_q¡\ed[oYX¬�ed¶X_%cS_be�­ �� �� � µ��±º

� t bñ<7=�@ED��3-9�7z �;,�A��>�ZA�< ä � N¦1 � @hDAF�M 2 9�=?9�M 6hä <7Fu9/ÒQ> 6 M?9DXMq= 6Eä bñ<�=�F59D 6 M?9/F á ã� � b � ÞAð >A@h<JL ��� ~ � � t�b � bî<�=Q�y�¯? B{A��¯���8�9�²- � L?9`=p>{9/à�M?<7Ih9 4 ã 4 9 ä M 4 <JDAåh9/F � < ä � t�� � b �@EDAF � � 1 � � Lb9/=?>¨9`àyMb<JI\9 4 ã � � 1 � � Þ5ï Ù36 Lb<JÜ 2 M 4 <7D5åh9`FV>¨@E<7Lb= ����B ~ � B � @hDAF ��� N ~ � N � @ELb9�²,����yx5?@��A���-ô< ä M 2 9/L?9Q9ÝK<�=pMb= 2 ×��ft0b B =?ÚAà 2 M 2 @EM � B 1 2 ��× � N 1�� 2 × ��By2 1 � N @hDAF� N ��1 ��B Þ5ï 2 9;à 6 D��dÚAÜ\@hàãQL?9 4 @EM?< 6 Dâ<�=%@EDV9��XÚ5<7I�@ 4 9DAà9�Lb9 4 @�Mb< 6 D 6 DâLb<JÜ 2 M 4 <JD5å\9/FV>A@E<7Lq=Ó^Ô�ú`Ö�Þ

��KNM ���²[ ^`£u\3gHP�RHTF"U[ï 2 9H>5L 6 FuÚAàyM 6Eä =p9/ÒQ<JÜ\L 6 Ú5>A=½<�=·Ü\9D59/Lb@ 4 <��9`F M 6 Lb9/à 6 ÜhD5<��9�=?9�Mq= 6Eä,ÙZ6 LqF5=½<7DAFu9ÝK9`Fá ãHà 6 Ú5DXMq@ á 4 9�=bà@EMpMb9Lb9/F 4 <7D59/@hL 6 LbF59Lb<JD5ÜX=Þ�ð �hèS=?9ÒQ<JÜ\L 6 Ú5>�<7=,@VÜh9/D59Lq@ 4 <��/@�Mb< 6 D 6Eä @ÚA=pÚ¨@ 4 =?9ÒQ<7ÜhL 6 Ú5>¼ÞAï 2 9�>AL 6 FuÚAà�M 6Eä @·=?9��XÚ59/DAà�9�<JDAF59�Ýu9/F á ã�@EDKãV=bà@EMpMb9Lb9/F 6 LqFu9/L?<7D5Ü<7=)F59�ÛAD59`F Þ� ^��j]#\3WI\3R�] K<­ �\èS=p9/Ò�<7ÜhL 6 ÚA>&�±w��<w7�>A b°� �7x{� ���;��-�*���A�B=���., -�xZ�>A�!���b � K�M b%*#B{�3�²Bz!� �Hw����H³f*),��.-&,�����,�x{�ZA��5®>�=�¦w>����A�A��7�.��-!�¯���������<� �7�`?�A©B�,�¹D�>�'b+A�,&���'�7�=�>z$�7�HA ,¬�-b ¼7 �7,��0���Z³F�>�=�>z!�>�ZA � ,¬�'b ! �"� � 1 � ¼7 �7,��$���H³&*),��.-0/�,�¹��7� b ,¬�r��,�x{�HA��9®7� �0w7����ANA��7�²�²-&�����������r�=�7�`?�A©B�����-Q�7,��$���Z³��7�5�>�A�,��y�=� ��A��3,��"/&1 �47698 / 4 *#B;�>�.�<� t  

! � / � 1#! � $47698 ! � / 4� �

% BZ�0�=��A�A��>�<��,���-���AN�3,�� �±w��r?9�>���>�.�����=� ��A��3,���,��<�Dw�w>, �7�3��AN��¹ ��AN³ ¼ó 6 L%<7DA=pMb@hDAà�9\×KM 2 9�=?9�M�( ��9��XÚ5<7>5>{9/F Ù <^M 2 M 2 9�à 6 DAà/@�M?9/DA@�Mb< 6 DV<�=%@��hèS=?9ÒQ<JÜ\L 6 Ú5>�Þ& :`��z ���=� ª5¼ ï 2 9�=p9M b 1 ½`éZ~Ô À 9��XÚ5<7>5>¨9`F Ù <JM 2 M 2 9�>AL 6 FuÚAà�M'! Fu9ÛAD59/F äC6 L½@hDXã( t b � á ã)! � ( � 1 éQ< ä ( 2 @h=)@EM 4 9`@h=pM 6 D59 6 à/à�Ú5Lb9DAà9 6Eä M 2 9 4 9MpM?9/L�é½@ED¨F)! � ( � 1 Ô6 M 2 9L Ù <7=?9;<7=%@��\è�=?9ÒQ<7ÜhL 6 Ú5>�Þ

ó 6 L.@hDKã,M Ù36 9 4 9/Ò�9/DXMb= � @EDAF & 6Eä @ �\èS=p9/Ò�<7ÜhL 6 ÚA> � bj~*! � ×hM 2 9%Û¨D5<^Mb9%>5L 6 F5ÚAàyM+! �"�0& �<7=%ÒQ9/L?9 4 ãâF59D 6 M?9/F á ã �& Þð%wyxZ®7������w>�7zr�¯?��²,�x.�-, 6Eä @ �\è�=?9ÒQ<7ÜhL 6 Ú5>�b�<7=(@%=pÚ á =p9M 6Eä b�à 476 =?9/F�Ú5D¨Fu9L(>5L 6 FuÚ¨àyM/Þð�z$,����;B{�±wyz ,��$�H��w>�>z0�¯?5�.,�x.�ñ<7=�@EDî@h>5> 4 <7à/@�M?< 6 D Ù%2 <7à 2 >AL?9`=p9/L?I\9/=;M 2 9â>5L 6 FuÚAà�M/Þ!ð�²,��@?5�yxZ�7����� 6Eä �\èS=p9/Ò�<7ÜhL 6 ÚA>½<7=%@hDâ9��XÚ5<7I�@ 4 9DAà9�Lb9 4 @�Mb< 6 D/. =pMb@ á 4 9�Ú5D¨Fu9L%>5L 6 FuÚAà�M� Ñ ä

Page 8: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

� 4 . & 4 äC6 L(@hDKã �ft ��×/M 2 9/D ! � �47698 � 4 � . ! � �47698 & 4 � Þ�ï 2 9Z=p9M b � . <7=�@ �\è�=?9ÒQ<7ÜhL 6 Ú5>¼Þ�ð�hèS=p9/ÒQ<JÜ\L 6 Ú5>-, <�=±@ �>xZ,�A��3�>�ZA 6Eä @ �\èS=p9/Ò�<7ÜhL 6 ÚA> b¿< ä M 2 9/L?9�9�Ýu<7=pMb=3@ED 6 D\M 6 Ò 6 Lb> 2 <�=pÒä L 6 Ò9b�M 6 ,�Þ�ð �\è�=?9ÒQ<7ÜhL 6 Ú5> ,¾-���¹ �3-5�7wcbH< ä ,�<7=�M 2 9 �XÚ 6 Mb<J9/D\M 6hä @�=pÚ á è �\è�=?9ÒQ<7ÜhL 6 Ú5>6Eä b Þ

��K�J ��\N]#\3WI^)� ��[I^`£u\NgZP�R�TE"j[ð �hèS=p9/ÒQ<JÜ\L 6 Ú5> � bj~*! � <7=%=b@E<�F½M 6 á 9;ÛADA<^Mb9,< ä bî<�=ZÛAD5<JM?9\Þ��.I\9D Ù%2 9D�bî<�=ZÛAD5<JM?9\×uM 2 9ä Ú5DAàyMb< 6 D ! <�=;D 6 M,9/@\=pãHM 6 Fu9/=bà�Lb< á 9 á 9`à@hÚA=p9�M 2 9Q>5L 6 FuÚ¨àyM 6Eä @EDKãH=?9��XÚ59D¨à�9 2 @h=�M 6á 9QÜh<7Ih9/D�Þ Ñ M,M?Ú5LbDA= 6 Ú5M�M 2 @�M;M 2 9 ä Ú5DAàyMb< 6 D !�à@hD á 9·Fu9/=bà�Lb< á 9`FHÚA=?<JDAÜ�@�=?9ÒQ<JÜ\L 6 Ú5>=dMbL?ÚAà�M?Ú5Lb9 6 D b Ù <^M 2 M ÙZ6 @hF5F5<^Mb< 6 DA@ 4±ä Ú5DAàyMb< 6 DA= � à@ 4J4 9/F��î@EDAF�i�� �,ä L 6 Ò b M 6 b Þï 2 <7=3Üh<7Ih9`=±@�ÛAD5<JM?9;Fu9/=bà�Lb<J>5M?< 6 D 6Eä M 2 9 ä Ú5DAàyMb< 6 D !�ÞKï 2 9 ä Ú5DAàyMb< 6 DA=��â@EDAF�i��·@ELb9)M 2 9à 6 ÚAD\Mb9Lb>A@EL?M 6Eä!4 <7ÒQ<^M�M?Lq@EDA=?<^Mb< 6 DA= 6Eä @EÚ5M 6 Ò½@EMb@5Þ¨ï 2 <7=%ÛAD5<JM?9�Fu9`=?àL?<7>uMb< 6 DV<�= á @\=p9`F 6 DM 2 9,D59ÝKM � 9ÒQÒ½@ Ù%2 <�à 2VäC6\4J476�Ù = Fu<7L?9`àyM 4 ã ä L 6 Ò :)@EÒ½=?9ã � =%ï 2 9 6 Lb9Ò-Ó ÕEéEÖ}Þ� 9�M�/ 1 �

6 �/ @ED ��è ä @hà�M 6 Lb<��/@�Mb< 6 D�ÞAð�D 6 M 2 9L ä @hà�M 6 Lb<��/@�Mb< 6 Dh/ 1 �

6 �/ <�=�à/@ 474 9`F

@�wyx.�Z�7�©�7�9�7A�,��y�=� ��AN�3,�� < ä M 2 9Lb9�<7=½@Ä=?9��XÚ59/DAà�9 � � 6 � 6Eä <JDXMb9Üh9/Lb=Q=?ÚAà 2 M 2 @EM�/ C 1/ C�� / P�� @EDAF�/ 1°/ P�� ( � � B � / P�� äC6 L)@ 474 ��� Ô\Þ� ^`£u£ue�M�K�5 �>A�� �L(���K8M b ®��&�hz$,����;B{�±wyz ���HA�,��<º �Z��A�� ����w>�7zr�¯?5�.,�x.� ¼�� ,��F���Z³�7�5�>A�,��y�=�I��AN�3,��h/p1 �

6 �/ A�BZ�7�.�!��:9�±w�ANw0�fw�x.�Z�>�©�7�5�>A�,��y�=�I��AN�3,��h/p1 4

6 �/ ����-"�f�y�¯?DB`A

�����8�9��-=�Z����� �"� ~ � � t�b � � � b � wyxZ�²BhA�BZ��A�� � / C � 1 � ����-�� � / � 1 � �7,��0���H³¦� 3 é ¼ßKÚAà 2 @ ä @\àyM 6 Lb<��`@�M?< 6 DV<7=%à/@ 474 9`FV@"�.��z�w>�>³5���r�7�5�>A�,��y�=�I��AN�3,��A×{=p9/9;ï 2 9 6 Lb9Ò 5ÞfÕ�<7DÅÓ^Ô`ê�Ö�Þ� ^��j]#\3WI\3R�] ��K65 �7ARb:®��&��w>�7zr�¯?��²,�x.� ¼�­ ��x{���>A��3,���� �4b K8M b����.�7w��;���>AN��¹��7�¯³�i�� �b K8M b��&�±wâà 6 ÒQ>A@�Mb< á 4 9�M 6 M 2 9�L?<7Ü 2 M Ù <^M 2 b����.�7w��;���>AN��¹��7�¯³�M 6 M 2 9 4 9 ä M��"� �$����-h,��H��³� ���7,��"���H³ � & ��� b ����-'���Z³ ���ZA���?9�>� 7 A©B;�Q�7,����=,�*����`?����., �Z�7�yA��3�7w$B;,��=-5» � � & � ��� 1���& � � ����- ��� 3 � � 1 � � � �²�yw��Z�²�>A���¹D�>��³ �"�0& � G � � 1 � & � � G � ����- ��� 3 � G � 1 � G � � ¼ï 2 9�>5L 6 FuÚ¨àyM 6Eä @íÛAD5<JM?9 �\è�=?9ÒQ<7ÜhL 6 Ú5> b à@hD á 9�ÛAD5<JM?9 4 ãñFu9/=bà�Lb< á 9/F á ã ä Ú5DAà�M?< 6 DA=à 6 ÒQ>A@EM?< á 4 9�M 6 M 2 9,Lb<7Ü 2 M)@EDAFâM 6 M 2 9 4 9 ä M Ù <^M 2 b Þ¡<¢ ^`RHP�^@£ K65 �>A � bj~*! � ®��!��º �H��A�� �H��w>�>z0�¯?5�.,�x.� ¼ % B;�!®>�������y³P���., -�xZ�>A -9�3º)����-��7,�����Z³ � & ��� b ®7³ � , & 1 ! ���& � ����A�x{�.������³F�7��-5,�*jw0�!wyAN�yxZ�>A�x{�.�<,���w>�>z0�¯?5�.,�x.�u����-$A©B;���x{���>AN�3,��;w��s����-qi��°�.�7w �Z���7A���¹��7��³�-5�Nº ����-�®7³ � � 1 ! �"�>� � ����- � G � 1�! ��� G � � ���.��.�yw��Z���7A���¹D�>��³&��,�z��Z��A��3®7� ��A�,$A©B;���y�¯?DB`A�����-fA�,fA�BZ�<� �3��A *���A�B b ¼! ,��Z¹��7��w>�>��³ � �7A b�®²��� º)�Z��A�� w>�>z0�¯?5�.,�x.�q����-¦�=�>A�������-<i��q®����x{���>A��3,��;w��.�7w��;���>AN��¹��>��³�²,�z��Z��A��3®>�=�!A�,fA©B;�0�y�¯?DB`A�����-&A�,FA�BZ�0�=�3��A�*���A©B�b ¼ % B;�>� b ������®��0x{�Z� �>xZ�7��³p�7��-5,�*)�²-*���A�Bq�$wyA��yxZ�7A�x{�.�<,¬� �H��w>�>z0�¯?5�.,�x.� � bj~*! � wyxZ��B"A�BZ��A � � 1 ! ���I� � ����- � G � 1#! �"� G � � ¼ï 2 9�Û¨Lb=pM3>A@hLpM 6Eä M 2 9�M 2 9 6 Lb9Ò äC6h474J6�Ù =ZFu<7L?9`àyM 4 ã ä L 6 Ò�M 2 9�@\=?= 6 à�<�@�M?<7IK<^Mdã 6Eä M 2 9�>5L 6 FuèÚAàyM !�Þ 136 DKIh9Lq=?9 4 ãh× 4 9�M b á 9�@�ÛAD5<JM?9�=p9/ÒQ<JÜ\L 6 Ú5>î@EDAF 4 9M"� @hDAF%i�� á 9 ä Ú5DAà�M?< 6 DA=L?9`=p>{9/à�M?<7Ih9 4 ãíà 6 ÒQ>A@EM?< á 4 9âM 6 M 2 9âLb<7Ü 2 M�@EDAF¿M 6 M 2 9 4 9 ä M Ù <JM 2 b Þ(ï 2 9V>5L 6 F5ÚAàyM 6hä @Ù36 LbFp/ 1 ��� 4 � 47698 6 Ih9/L b 6hä.4 9/D5ÜEM 2 �nt   <�=�F59�ÛAD59`F á ã�<JDAF5ÚAàyMb< 6 D 6 D:�t�� äC6 L@EDKãh� tB� ¶ á ã½M 2 9 äC6\4J476�Ù <7D5Ü Ù @�ã��� 9�M0�ut � C @hDAF 4 9�M�/�t�b 8 Þ�ï 2 9Lb9�9Ýu<7=pMb=�@EDH<7DXM?9Ü\9L$# @hDAF ��B × × ��% <7D b=pÚAà 2 M 2 @�M�/"1 ��B �&% Þ �¿9�=?9�M ! � / � 1 ��B , � N �&% Þ

Page 9: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

� 9�M"� t � ¶ Ù%2 9Lb9� 3 Ô�@EDAF 4 9M!/ t b 8 Þ.ï 2 9 4 <7D59/@hL 6 LbFu9/L?<7D5Ü�� à@hD á 9Fu9/à 6 ÒQ> 6 =p9`F @\=�@¿=pÚAÒ �¾1 |

6�� m Ù%2 9Lb9��%t%� on½"��~Ii � À @ED¨F äC6 L½@ 4J4 �ft �A×m t ´��� ¶ � � Þ�ï

2 9/L?9V9�Ýu<7=pMb=�@ ä @\àyM 6 Lb<��`@�M?< 6 D�/°1 � 6�� / =?ÚAà 2 M 2 @�M äC6 LQ@ 474 �rt �A×

� / � 1�m Þ7 �h1 ½\Ô9~ � ~�# À tr� � Ù 9�=?9�M ! � / � 1#! � / B � � ! � / % � Þ7 �u1 ����ï 2 9Lb9�9Ýu<7=pMb=�@V=?Ú5>{9L ä @hà�M 6 Lb<��/@�Mb< 6 Dh/�1 4

6 �/ @EDAFí@·Lb<JÜ 2 M 4 <7D5å\9/FH>¨@E<7L

��� ~ � � t b � � � b � =?ÚAà 2 M 2 @�M � � / C � 1 � @hDAF � � / � 1 � äC6 L�@hDXã"� 3 é5Þ��í9�=p9M! � / � 1 � � � Þ7 �h1vi � �¨ßuãXÒQÒQ9�MbL?<�à@ 4J4 ã½M 6 M 2 9,>5Lb9IK< 6 ÚA=%à/@h=?9h× Ù 9�=?9�M ! � / � 1 � G � � ÞßK<JD¨à�9.M ÙZ6%4 <JD5å\9/F,>¨@E<7Lb=�@\=?= 6 à�<�@�Mb9/F Ù <^M 2 M ÙZ6%ä @hà�M 6 Lb<��/@�Mb< 6 DA= 6Eä @ Ù36 LbF;@ELb9±à 6 D��dÚ5ÜX@�Mb9/FÓ^Ô`ê�Ö�×/<JM�à/@ED á 93>AL 6 I\9/F á ã;<JDAF5ÚAàyMb< 6 D 6 D âM 2 @�M !·<�=¼Ú5DA<��XÚ59 4 ã�Fu9ÛAD59`F�@ED¨F�@h=b= 6 à<7@EM?<7Ih96 D�b �\Þ& :`��z ���=� � ¼ ï 2 9��hèS=p9/ÒQ<JÜ\L 6 Ú5>¨9db�1 ½`éZ~Ô À 6hä ��Ý5@hÒ�> 4 9VÔ�<�=�F59�ÛAD59`F á ã�M 2 9�ÛAD5<JM?9>5L 6 FuÚ¨àyM,éhé&1 é5Ôr1öÔ/é"1 éV@EDAF�ÔhÔ01 ÔQ@EDAF á ã�M 2 9Qà 6 Ò�>¨@�M?< á 4 9 ä ÚADAàyMb< 6 DA= �í@hDAFi���Fu9ÛAD59`F á ãâé � 1�é G � 1�éQ@hDAFôÔ � 1 Ô � 1 ÔhÞ

��K !&^@V9RHg�]U\��5e;aU\Nd3\©W�cÑ M�<7= Ù 9 474 åKD 6�Ù DHM 2 @�M�Lq@�Mb< 6 DA@ 4 =?9�Mb= 6hä ÛADA<^Mb9 Ù36 LbF5=;@hL?9�9�Ý5@hà�M 4 ã�M 2A6 =?9�Lb9/à 6 ÜhDA<��/9/Fá ã�ÛADA<^Mb9�=?9ÒQ<7ÜhL 6 Ú5>A=/Þ¨ï 2 <7=�Lb9/=?Ú 4 M�<7=�Ü\9D59/Lb@ 4 <��9`F äC6 L Ù36 LbF5=�<JD¨Fu9�Ýu9/F á ãHà 6 Ú5DXMq@ á 4 9=?à/@�MpMb9Lb9/F 4 <7D59`@EL 6 LqFu9/L?<7D5Ü\=/Þ� ^��j]#\3WI\3R�]��LK65 �7A^b�����- , ®���A�*), �H��w>�>z0�¯?5�.,�x.��w ¼ % B;� ����w>�7zr�¯?��²,�x.�', Lb9/à 6 ÜhD5<��9`=�fwyxZ®7w>�7A � ,¬�cb � �0����-p,��H��³&� ��A�BZ�7�²�$��:9�±wyA3w0�&z$,����ZB`�±wyz � �Db K�M , ����-h�FwyxZ®yw>�>AI� , wyxZ�²BFA©B;��A�� 1 � G B � I � ¼�­ w>�>A � %(��!�±w;L?9`à 6 Ü\D5<��/@ á 4 90� ������-!,��Z�¯³�� ��A�BZ�7�.���:9�±wyA3w<����� ���$�H��w>�>z0�¯?5�.,�x.���.���²,²?5�Z�=�>���@?F��A ¼& :`��z ���=��� ¼ ï 2 9�=p9M b 1 ½`éZ~Ô À 9��XÚ5<7>5>¨9`F Ù <JM 2 M 2 9�>AL 6 FuÚAà�M'! Fu9ÛAD59/F äC6 L½@hDXã( t�b � á ã'! � ( � 1 Ô�< ä ( t¿Ô 9 @EDAF ! � ( � 1 é 6 M 2 9L Ù <7=?9�<�=3@$�hèS=?9ÒQ<JÜ\L 6 Ú5>�Þ Ñ M <�=3@ 4 = 6Fu9�ÛADA9/F á ãQM 2 9�ÛAD5<JM?9;>5L 6 FuÚAàyM%éhé<1 é5Ô�1 Ô/é<1 é�9�M;ÔhÔ�1 Ô�@hDAF á ãQM 2 9,à 6 ÒQ>A@EM?< á 4 9ä Ú5DAàyMb< 6 DA= �î@EDAF°i�� Fu9�Û¨D59/F á ãÅé � 1"é G � 1 Ô G � 1 éH@hDAF Ô � 1 ÔhÞ !�9�ÛAD59VM 2 9Ò 6 L?> 2 <7=?Ò 6hä �\è�=?9ÒQ<7ÜhL 6 Ú5> � �5(���K8M b á ã � �N2 � 1 Ô äC6 L%@hDXã 2 tp(�ÞAï 2 9,=?9�M�( 9 <7=L?9`à 6 Ü\D5<��/@ á 4 9;=p<7DAà9Q( 9 1 � G B � ½\Ô À � Þó 6 LZ@EDKã�Û¨D5<^Mb9�@ 4 > 2 @ á 9�M�(�×'$ � & � ( � � Fu9D 6 M?9`=�M 2 9�=?9�M 6hä =?Ú á =?9�Mq= 6hä ( ��Lb9/à 6 Ü\D5<��`@ á 4 9á ãV@�ÛAD5<JM?9 �\è�=?9ÒQ<7ÜhL 6 Ú5>�Þ¡<¢ ^`RHP�^@£ ��K<­ w>�7A�,��!*),��²-Dw!����-5��:`��-q®>³h��,�x{�ZA��5®>�=�!w>����A�A��>�.��-"�������²���F,��.-9�7�y���`?�w$�±w�.��A��3,�������� ¨v��A��±w¦�.����,.?5�Z�=� �9®7�=� ¼& :`��z ���=� YH¼ ï 2 93=?9�M ��1 �32 � � �3<�=�Lb9/à 6 ÜhDA<��/9/F á ã�M 2 9��\è�=?9ÒQ<7ÜhL 6 Ú5>�b�1+½ � ~ & ~ � ~���~bé ÀÙ%256 =?9½>5L 6 F5ÚAàyM�<�=�Fu9ÛAD59`F á ã �& 1 � × & � 1���× � � 1 � ×����%1���× � � 1 � ×�� & 1 & ×� �u1 � × & � 1 & × � � 1 � × � G � 1 � ×�� � 1 & ×�� G � 1 � Ù%2 9/L?9½@EDKã 6 M 2 9/L,>5L 6 FuÚ¨àyM,<7=9��XÚA@ 4 M 6 é5Þ !�9�ÛAD5<7D5Ü�M 2 9�Ò 6 Lb> 2 <�=pÒ � �5(�� O b á ã � �N2 � 1 � @hDAF � � � � 1 & × Ù 9�Üh9M��1 � G B � � � Þ

Page 10: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

Ñ ä �"<7=�Lb9/à 6 Ü\D5<��/9/F á ã�@�Ò 6 L?> 2 <7=?Ò � ��b K8M ,�×`M 2 9)=?9�M)( � ? � <�=�@ 4 = 6 Lb9/à 6 ÜhDA<��/9/Fá ã � =?<JDAà9<(�� ? ��1 � G B � b ? I � ÞAï 2 9/L?9 äC6 Lb9h× Ù 9 6 á Mq@E<7D äC6\4J476�Ù <JDAÜ�M 2 9 6 L?9/Ò�Þ¡<¢ ^`RHP�^@£ �LK � ��AN�3,������w7�>ANw¦,¬�¦*),��.-�w�,�����,�x{�ZA��5®>�=�Qw>����A�A��>�.��-$�¯����������,��.-5�>�y���@?�w0���.��7�=,�w>�²-Fx{��-9�7�<��,�z ��� �7z$�7�HA���AN�3,�� ¼& :`��z ���=��� ¼ ï 2 9,=p9M ��1�( 2�<�= L?9`à 6 ÜhD5<��9`F á ã½M 2 9 �\è�=?9ÒQ<7ÜhL 6 Ú5> b�1 ½`é;~/Ô À Ù%256 =?9>5L 6 FuÚ¨àyM3<�=ZFu9�ÛADA9/F á ã�ÔhÔ�1 Ôh×Ké5Ô�1 Ô`é<1 éhé<1 é�@hDAF á ã�M 2 9;à 6 ÒQ>A@EM?< á 4 9 ä Ú5DAà�M?< 6 DA=é � 1 é G � 1 Ô � 1 Ô G � 1 é5Þ !�9�ÛAD59�M 2 9QÒ 6 L?> 2 <7=?Ò � �L(�� O b á ã � �32 � 1 Ô äC6 L@EDKã 2 t�(�Þ!G�DA9·Üh9Mb= � 1�� G B � Ô � @hDAF¿M 2 9Và 6 ÒQ> 4 9/ÒQ9DXM�( � ? � 1 � ( � � � ( � l(�� � (�� � G � 1 � G B � é � Þï 2 9,D59�ÝKM)=?9/àyMb< 6 D�<7=%Fu9/I 6 M?9/FâM 6 =?åh9Mbà 2 9`= 6Eä >5L 6K6hä(6Eä ï 2 9 6 L?9/ÒöìAÞ

� �%ËAÌ�� ����@���Ð�.Ë5Ì3Î����ÅÊKÌ ��Î%ÊuÌ�� ��Ê{�� 9�M � bj~ ! � á 9Q@âÛAD5<JM?9 �\èS=p9/Ò�<7ÜhL 6 ÚA>�Þ ø ãHï 2 9 6 L?9/Ò 5×{M 2 9�>AL 6 FuÚAà�M ! <�=�F59�ÛAD59`F á ãà 6 ÒQ>A@EM?< á 4 9 ä Ú5DAà�M?< 6 DA=��·@EDAFpi�� Þ � 9M � á 9�@�=?Ú á =?9�M 6Eä (���L?9`à 6 ÜhD5<��9`F á ã b ÞKï 2 9Lb99�Ýu<7=pMb=�@�Ò 6 Lb> 2 <�=pÒ 6Eä �\è�=?9ÒQ<7ÜhL 6 Ú5> � �U(��'K8M b @EDAFî@�=?Ú á =?9�M%I 6Eä b =pÚ¨à 2 M 2 @EM��1 � G B � I � ÞAßK<7DAà9�Lq@�Mb< 6 DA@ 4 =p9Mb= @ELb9�à 4J6 =p9`F½Ú5DAF59LZÛAD5<JM?9�Ú5D5< 6 D�× 6 D59�Ò½@�ã½=?Ú5>5> 6 =?9M 2 @EM I <�=�@�=?<JD5Ü 4 939 4 9ÒQ9DXM ½�T À Þ � 9�M� á 93M 2 93ÛAD5<JM?9 =?Ú á =pM?<JM?ÚuMb< 6 D Ù%2 <�à 2 @\=?= 6 à�<�@�M?9`= M 69/@hà 2 9 4 9ÒQ9DXM � 6hä b¿M 2 9�=?9�M � G B �"� ��� (�ÞußK<7DAà�9���1� � ! G B � T ��� � � ( � � � ×K<JMZ=?Úuõ·à�9/=M 6 >AL 6 I\9½M 2 @EM�M 2 9V=?9�M ! G B � T ��6hä)Ù36 LbF5= 6 Ih9/L�b Ù%256 =p9â>5L 6 FuÚ¨àyMQ<7=cT¼×�<7=�Lq@�M?< 6 D¨@ 4 Þ:%9/à/@ 474 M 2 @EM%M 2 9$��Lb99/D � = Lb9 4 @�Mb< 6 DA=%@ELb9;Fu9�ÛADA9/F ä L 6 Ò M 2 9 äC6h474J6�Ù <7D5ÜQ>5Lb9 6 LqFu9/Lb=�

����� &���� , 2 t�b B × � 1 &�2������&���� , 2 t�b B × � 1 2'&�����)&���� , 2 ~ ��t�b B × � 1 2'& �ó 6 L,@hDXã�� t%½! �~#"�~%$ À × � � & < ä @hDAF 6 D 4 ã�< ä �&��' & @hDAF &(��'#� Þ �í9½@ 4 = 6 Fu9/D 6 Mb9á ã � � '-& <J÷ �)��'-& @EDAF�D 6 M &���'�� ÞZ:)9/à@ 4J4 M 2 @�MâM 2 9ô9��XÚ5<JI�@ 4 9/DAà�9íL?9 4 @EM?< 6 D* 1+ ,"b1+"� <7= 9��XÚA@ 4 M 6 $ Ù%2 9D bî<�= ÛADA<^Mb9hÞ

ï 2 9�>5L 6K6Eä <�= á ã�<JDAF5ÚAàyMb< 6 D 6 D�M 2 9 * è�à 4 @\=?=�=dMbL?ÚAà�M?Ú5Lb9 6hä b Þ ó 6 L;@EDKã * è�à 4 @\=?=.-6Eä b ×5Fu9D 6 M?9 á ã��b0/�1+½ � t�b � � Tpt1-�~ � � � T À @EDAF ,0/�1¾½ � t b � ��Tpt2-�~ � 3 � T À�í9âF59�ÛAD59�@EDî@hÚuM 6 ÒQ@EM 6 D 6 D 4 <JD59`@EL 6 LqFu9Lb<JDAÜ\=,@hà/à�9/>uM?<7D5Ü ÙZ6 LqF5= 6 I\9L b / @hDAF

à 6 ÒQ>5Ú5M?<7D5ÜQM 2 9,>AL 6 FuÚAà�M ! 6hä <JMb= >A@EM 2 � = 4 @ á 9 4 = <JD á 6 M 2 F5<JLb9/à�M?< 6 DA=/Þ� 9�M � / 1 ��� / ~ b / ~�� / � á 9;M 2 9�@EÚuM 6 Ò½@�M 6 D�Fu9�Û¨D59/F á ã��� / 1�b B/ � b B/ �43 <�=ZM 2 9�=?9�M 6hä =pMb@�Mb9/= Ù%2 9Lb9 3 1 ½`é;~/Ô À��/ 1¾½ ��� ~65 & ~ � �87K�M ��� 5�~ & ~ �:9 � � ��t 3 ~ �:9U1 � 5rt1- � À

o°½9½ �"� ~ & ~ � � À B:; ; % K8M ��� ~ & ~ � � � ��t 3 ~ ,�Ô � � � # × � 1 Ô�×, Ô � � # × , � t�� � - � × � P � 1 �;P × � &JP 1 &JP × � 1 � P � � @EDAF &JP 1 � � & À

o°½ ��� ~ & ~ � � K8M'½ �"� ~ & ~ � � À B%; ; % � ��t 3 ~ ,�Ô � � � # × � 1 Ô�×, Ô � � # × , � t�� � - � × � P � 1 �;P × � &JP 1 &JP × & 1 � G � &JP @hDAF �;P 1 � � G � À

Page 11: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

ï 2 9 á 6K6h4 9/@hD¿à 6 ÒQ> 6 D59/D\M 6hä � / @ 474J6�Ù = 4 <JÒQ<JM,M?Lq@EDA=?<^Mb< 6 DA= 6 D 4 ãH< ä M 2 9 4 @ á 9 4�6Eä M 2 9>A@�M 2 @hF5Ò�<JMb=;@VLb@hÒ½=p9/ã\@ED ä @\àyM 6 Lb<��`@�M?< 6 Dí@h=b= 6 à�<�@�Mb9/F�M 6 @hDô<�Fu9ÒQ> 6 Mb9DXM 6hä -�Þ�ßK<7DAà9M Ù36 L?<7Ü 2 M 4 <JD5å\9/Fí>A@E<7Lq= �"��B ~ � B � @EDAF ��� N ~ � N ��6Eä @�=?@hÒ�9 * èSà 4 @h=b=,@ELb9�à 6 D��dÚAÜ\@�Mb9/Fô<^÷� B � N ×�<JM�à@hD á 9�= 256�Ù D á ã�<JD¨FuÚAàyMb< 6 D 6 DHM 2 9�Lb@hD5åVM 2 @EM � / à 6 ÒQ>5Ú5M?9/=�>5L 6 >{9L 4 ãM 2 9½>5L 6 FuÚAà�M !�Þ(ï 2 9 Ù36 LbFA= 6hä b �/ @hF5Ò�<JMpMb<JDAÜ�Lb@hÒQ=?9ãX@ED ä @\àyM 6 Lb<��`@�M?< 6 D¨=�@\=?= 6 à�<�@�Mb9/FM 6 <�Fu9ÒQ> 6 Mb9DXMb= 6Eä $QèS@ á 6 Ih9 * èSà 4 @h=b=p9`=·@ELb9�Mb@håh9/D à/@ELb9 á ãñ@Å=?Ú á =pM?<JM?ÚuMb< 6 D Ù%2 <�à 2<7=âLb@EM?< 6 DA@ 4 á ãñ<7DAFuÚ¨àyM?< 6 D¼Þ � 9M2- á 9ô@ * è�à 4 @\=?= 6Eä b @ED¨F 4 9M � á 9ôM 2 9ôLb@EM?< 6 DA@ 4=pÚ á =dMb<^MbÚuM?< 6 D�Fu9ÛAD59/F á ã��

����b0/ K8MG$ 2'& � b0/�� �� K8M

� ! G B �"� � < ä � t/,0/½ � À o���jo���� < ä � t�-

Ù%2 9Lb9 äC6 L)@EDKã � t2-H× � 1 � �� � � � � ����� � � � � ��

! G B �"��B � � ! G B ���&% �

����1 ��� � ���� ����� � ! G B � & � ! G B � � � � o ��� � ���� � ( ��� �

! G B � � � G � ! G B � & � Ñ äU� ��Fu9D 6 M?9`=ZM 2 9,=?9�M 6Eä�Ù36 LbFA=3Lb9/à 6 Ü\D5<��/9/F á ã½M 2 9,@EÚuM 6 Ò½@�M 6 D�� / Ù <JM 2 M 2 9,<JD5<JM?<�@ 4=dMq@�M?9<½ � Ô9~ � ~?é � À @hDAF�M 2 9)ÛADA@ 4 =?9�M 6hä =pMb@EM?9`=�½ �"� ~Ô5~ � � � ��t 3 À äC6 L3@EDKã � t�b / ×\<JM3à@hDá 9�>5L 6 Ih9/F�M 2 @�M äC6 L,@hDXã T�t�-�× � � ��� � 1 ! G B � T � ÚA=?<7D5ÜV@hD 6 M 2 9L�<JD¨FuÚAàyMb< 6 D 6 DHM 2 9Lb@hD5å�Þ

�%ËAÌ�� ��Î%ÊuÌ�� ��Ê{� ÊuÌ � D�@���Ð�±Ë5̱Î����ï 2 <7=�>5L 6X6hä%6Eä M 2 9�à 6 DXI\9Lq=p9Q<�=�@\F5@E>uMb9/F ä L 6 Ò Ó EÖ}Þ � 9M�� 1 ��� ~²(<~��$~��H~� � á 9â@hD@EÚuM 6 Ò½@�M 6 D 6 D 4 <7D59/@hL 6 LbFu9/L?<7D5ÜX=±@hà/à�9/>uM?<7D5Ü�@�=p9M � b( �\Þuï 2 9!�²,��ZA��7�HA 6Eä @�>A@�M 2 <7=M 2 9�=?9�M 6hä =pMb@EM?9`= 6 àà�ÚAL?Lb<JDAÜ�<7D�M 2 9,>A@�M 2 @EDAF�T UWNX! N�Fu9D 6 M?9`=)@Q>A@EM 2�4 9/@hF5<JD5Ü ä L 6 ÒTÄM 6 N 6Eä 4 @ á 9 4 /î@EDAF 6hä à 6 DXM?9/D\M IQÞ � 9�M-, 1 � ��� � á 9QM 2 9â=?9�M 6Eä @ 474 =?Ú á =?9�Mb= 6Eä� @hDAF�m 1 � � , � á 9½M 2 9·=?9�M 6hä =?Ú á =?9�Mq= 6Eä ,�Þ¼ï 2 9·=?9�M<m <�=,9��XÚ5<7>5>{9/F Ù <JM 2 M 2 9äC6h474J6�Ù <7D5ÜQ>5L 6 F5ÚAàyM)@ED¨FVÚAD5< 6 D �

; 9 1¾½ & o & 9 � & t� �× & 9 t� 9 À @hDAFh ¦l� 9 1� Qop 9� 9�M b á 9VM 2 9�=?9�M 6Eä @ 4J4 � � � Ò½@�MbL?<�à�9`= Ù%256 =p9V9DXM?Lb<79/=Q@ELb9â<7Dnm Ù <^M 2 >5L 6 FuÚAà�MFu9�ÛADA9/F á ã �

� #-, # 9 �#" @ " ) 1 �� 6%$ # " @ � , # 9� @ " ) 1 ½ & o & 9 � , Tht � × & t # " @ � × & 9 t # 9� @ " ) Àï 2 9�=?9ÒQ<7ÜhL 6 Ú5>0b <�=�Û¨D5<^Mb9�@EDAF á ãîï 2 9 6 Lb9Ò A×�<JM·=pÚ5õ½à9/=�M 6 Fu9ÛAD59�à 6 ÒQ>A@EM?< á 4 9ä Ú5DAàyMb< 6 DA= M 6 9/DAF 6�Ù @½=dMbL?ÚAà�M?Ú5Lb9 6Eä �hèS=p9/ÒQ<JÜ\L 6 Ú5>�Þ !�9ÛAD59;M 2 9 ä Ú5DAàyMb< 6 D � á ãO�

# � " @ " ) 1¾½ & oq½+N 9 À � , & 9 � & × , Tht � × & t #'&" @ � × & 9 t #(&� @ � @ED¨F & 9 K8MON 9 t�� À

Page 12: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

Ù%2 9Lb9 ! <�=·M 2 9í=?Ò½@ 474 9`=dMâ<7DXM?9/Üh9L�=pÚ¨à 2 M 2 @EM # & <7=�@ED <�Fu9ÒQ> 6 Mb9DXM�Ò½@EM?Lb<^Ý Þ3ï 2 9ä Ú5DAàyMb< 6 D%i��í<�=�Fu9ÛAD59/FÄ=?ãKÒ�ÒQ9M?Lb<7à/@ 474 ãH@hDAF¿<^M�à/@ED á 9½>5L 6 Ih9`F�M 2 @�M��Å@ED¨F�i��Ä@ELb9ä Ú5DAàyMb< 6 DA=QLb9/=?>¨9`àyM?<7Ih9 4 ãîà 6 ÒQ>A@EM?< á 4 9 6 D M 2 9�L?<7Ü 2 M½@EDAF 4 9 ä M Ù <^M 2 b%Þ Ñ MQLb9Ò½@h<JDA=�M 6Fu9�ÛADA9�@½Ò 6 Lb> 2 <�=pÒ � �{(���K8M bñLb9/à 6 Ü\D5<��/<JDAÜ�� Þ ó 6 L�9/@\à 2�4 9MpM?9/L 2 6Eä (�× Ù 9�Fu9�ÛADA9M 2 9;Ò½@�MbL?<JÝ # E 1 � �N2 � à 6 LbLb9/=?> 6 DAFu<7D5Ü�M 6 M 2 9,9/FuÜ\9/= 6Eä � 4 @ á 9 474 9/F á ã 2 �5ï 2 9;9/D\MbL?ã� N`~6N 9 � 6hä # E <�=%9��XÚA@ 4 M 6 ½9½.N`~ N 9 À9À < ä N EK8MON 9�t � 6 L @Q6 M 2 9/L Ù <�=p9\ÞAð)D�<7DAFuÚ¨àyM?< 6 D 6 DM 2 9�Lb@hD5å ÙZ6 Ú 4 FH= 256�Ù M 2 @EM äC6 L;@ 474�ÙZ6 LqFh/ut�( �X× � � / � 1 # Ù%2 9Lb9�M 2 9QÒ½@�M?Lb<JÝ #Ò�9/Ò 6 Lb<��/9/=ZM 2 9�à 6 DXMb9DXMb= 6hä >A@EM 2 = 4 @ á 9 474 9/F á ã&/E�

# " @ " ) 1+½ � � N/UWNX FN 9 À

ð ÙZ6 LqF$/�tp(�� á 9 476 D5ÜX=.M 6 � <^÷�� � / �32 @h= @ � ��~�� � D 6 Duè�9ÒQ>uMdã½9/DXM?Lbã Ù%2 9Lb9���@ED¨F �@ELb9�Lb9/=?>{9/àyMb<JI\9 4 ã½<JDA<^Mb<7@ 4 @EDAFVÛAD¨@ 4 =pMb@�Mb9/=/Þ5ï 2 ÚA= � <7= Lb9/à 6 ÜhDA<��/9/F á ã�b Þ� ��� É3Ê{��Ï{Ê5ÐpÏ�� D�@���Ð�±Ë5̱Î��� 9�M � á 9Q@âLb9/à 6 ÜhDA<��`@ á 4 9�=?Ú á =?9�M 6Eä ( �XÞ ð�Ò 6 D5Ü�@ 474 �\è�=?9ÒQ<7ÜhL 6 Ú5>¨=%L?9`à 6 ÜhD5<��<7D5Ü � ×M 2 9/L?939�Ýu<7=pMb= 6 DA9 Ù%2 <�à 2 <�=(ÒQ<7D5<7ÒQ@ 4 <7D�M 2 9 =p9/DA=p9 6Eä Fu<7IK<7=?< 6 D�Þ Ñ M�<�=�à@ 474 9/F;M 2 9Z=?ãKDXMb@hà�M?<�à�hèS=p9/ÒQ<JÜ\L 6 Ú5> 6Eä � @EDAFâ<7=ZM 2 9�ÛALq=dM%à@hD 6 D5<�à@ 4�6 á �d9`àyM%@h=b= 6 à�<�@�Mb9/FQM 6 Lb@EM?< 6 DA@ 4 =?9�Mq= 6 D4 <7D59/@hL 6 LbF59Lb<JD5ÜX=Þ ó 6 LZ@hDXã �\èS=p9/Ò�<7ÜhL 6 ÚA> � bj~ ! � @ED¨Fâ@EDKã½=?9�M I� +b ×XM 2 9�9��XÚ5<7I�@ 4 9DAà9L?9 4 @EM?< 6 D .��î<�=)Fu9�Û¨D59/F äC6 L)@hDKã � × & <JD b á ã � .�� & <J÷ äC6 L)@EDKã½<JDXMb9Üh9/L�#:�

� ��B ~ � N ~ � ~ �&% ~ &?B ~ & N ~ � ~ & % t b B ×���� B ~�� N ~ ~�� % G B t�½"��~Ii � À or�¾~! ��� % � � ��� N �"� B �& B � � & N ���� �� ( � & % � tBI

��� ! ��� % � � ��� N �"� B & & B � � & N � � � � ( � & % � t Iï 2 9â9��\ÚA<JI�@ 4 9D¨à�9âLb9 4 @�Mb< 6 D . � <�=�@íà 6 DAÜhLbÚ59DAà9 6Eä �\èS=p9/Ò�<7ÜhL 6 ÚA>�Þ Ñ ä bëÛADA<^Mb9h×(M 2 9/D@EDAFâM 2 9��XÚ 6 M?<79DXMcb � . � <�=%@ED�9�÷�9/à�M?<7Ih9 �\è�=?9ÒQ<7ÜhL 6 Ú5>�ÞÑ ä � <�=,@VL?9`à 6 Ü\D5<��/@ á 4 9�=pÚ á =p9M 6hä ( �\×�M 2 9DíM 2 9 �XÚ 6 M?<79DXM<(�� � .�� <�=�ÛAD5<JM?9½@hDAFL?9`à 6 Ü\D5<��9/= �îÞOQPDR ")R�[I\3WI\NRH]sMHK65 �7A � ®��0�!wyxZ®yw>�>A�,���( � ¼ % BZ�<w>�>A � �±wQ�.���²,²?5�Z�=� �5®>�=�0� �<����-f,��H��³� �¦A�BZ�<�²�7�=��AN�3,��/.�� �±w0�F��,��`?��yxZ�>������,¬� ����w7�>zr�¯?��.,�x.�',���º)�Z��A��¦����-5��: ¼ó 6 L�@EDKãñL?9`à 6 Ü\D5<��/@ á 4 9H=?Ú á =?9�M�� 6Eä (��K×3M 2 9 �hèS=p9/ÒQ<JÜ\L 6 Ú5>b( � � .�� <�=âà/@ 474 9`F M 2 9w�³��HA��9�7A��3��w>�7zr�¯?��²,�x.� 6hä � @ED¨F�<�=.Fu9/D 6 Mb9/F á ã�b � � � Þ Ñ M±<7=�M 2 9)=?Ò½@ 474 9`=dM �\è�=?9ÒQ<7ÜhL 6 Ú5>L?9`à 6 Ü\D5<��<7D5Ü � <7DVM 2 9�=?9DA=?9 6Eä F5<JIK<�=p< 6 D�ÞOQPDR ")R�[I\3WI\NRH]�JLK65 �7A � ®��!�&�.���²,²?5�Z�=� �5®>�=�0w>�7A�,¬��(�������-&�=�>A , ®²�!�$�H��w>�>z0�¯?5�.,�x.� ¼% BZ�7�), �.����,.?5�Z�=� �7w � � �0����-&,��H��³$� �'b � � � -���¹ �3-5�7w , ¼Ñ D¿>¨@EL?M?<�à�Ú 4 @EL`× äC6 L�@EDKãHLb9/à 6 Ü\D5<��`@ á 4 9Q=p9M � ×�M 2 9½Lb9 4 @�Mb< 6 D .���<�=;M 2 9·à 6 @hLb=?9/=pM,à 6 D5èÜhLbÚ59DAà9ô=?ÚAà 2 M 2 @�MâM 2 9 �XÚ 6 M?<79DXM"( � � .�� L?9`à 6 Ü\D5<��9/= �îÞ ó L 6 Ò ï 2 9 6 Lb9Ò ì @hDAF L 6 > 6 =?<^Mb< 6 DVÕu×u<JM äC6h474J6�Ù =3M 2 @�M M 2 9,=?ãKD\Mq@�Mb<7à �hèS=?9ÒQ<JÜ\L 6 Ú5> 6Eä @�Lq@�Mb< 6 DA@ 4 =?9�M%<�=±Û¨D5<^Mb9hÞ¡<¢ ^`RHP�^@£��K<­ w>�7A�,��!*),��²-Dw!����-5��:`��-q®>³h��,�x{�ZA��5®>�=�!w>����A�A��>�.��-"�������²���F,��.-9�7�y���`?�w$�±w�.��A��3,�������� ¨v��A3wQwy³��ZA��5�>AN�3� ����w7�>zr�¯?��.,�x.�q�±w º �Z��A�� ¼

Page 13: Complementation of Rational Sets on Scattered Linear Orderings · 2016. 12. 30. · Complementation of Rational Sets on Scattered Linear Orderings Chlo e Rispal, Olivier Carton To

�b�E�²�¼Ë;�¼É�Ï��#�£`º���º¼®�_yiht`Y¨º¿z�[oYX[fed_½jv\edt`²�j�epjVjYKiHt/apih[^YXj/locqº��������� �������������������������^U�£yr� �!^£/£#"#$K£q~/~\U£%"&"� hº

�Eº���ºE®�_yiht`YAº�³�epjaSkC¸7ad_b_3cS_bedc�t/¸A¹!t/apihc�t/Y�tapi\[oYKjlocqº('*)%+%��*�,�(�����-�������^Uu£% & .! "�/0$K£/£`£/U5�/¤/¤h£/º/hº���ºK®(_yi\t/Y½j/YXi21�ºu�!j�aSedt`YAº��3Y43¼[olo_qYEsu_bad¬,ed¶X_bt/ad_q² ¸7t/a±¹!t/apihc3t/YQxqt/vXY�epj/s\l^_%tapi\[oYKjlocqºP�Y,��lbgj/vXi\[ot��¼º`�¨v\xqx?¶\_qcS[KjYKi;�.adYKjlJiht n�º�T�t`vhapjhU�_qi\[fedt/adcqU6587��9�:)<; =�>�?����@�A��A� �9�B�7&C�'*)#+����D�E7��9�B��FdUE]�t/l^v\²,_%£#/&G/¤�t¸H58������<I������ FJ�:)K�����-�������L�������^U�¡Xj/¬`_bc.r�/#$� /~hºu³E¡\ad[oYX¬/_baSkmn(_?adlJj¬\U£%"&"�Ghº

~\º%n�ºA®(adv\¯u°_bad_�j/YXiM1�º¨�!j�aSedt`YAº;�±v\edt`²�j�epj,t`Yâl^[oY\_yja)tapi\_bad[oYX¬/cqº�P�Yâ¢\º{³h¬�jlolCU¨�)ºA�¼v\loeSayUjYKi���ºEÃ%t`lo²�j/YAUE_yi\[fedt/adcqU�N4OP�H�Q; R�STSTUUh]�t/l^v\²,_ ��£#/� �t/¸H5V�A������I������ FW�X)Y�����-�@�.�����������^U¡Xj/¬/_qc3��/& #$h��~�|EU¨�¤`¤E£`º(PSR%T ad_q¡ut/aSeZ�¤`¤h£?kd£��Eº

rEº%n�º!®¼adv\¯5°_bad_`UP1�º.�!jaSedt/Y¨U.jYKiîR�º�³(g_qY\[^¾b_bad¬`v\_qcqº »¨ad_q_âj/vhedt`²�j�epj�j/YKi¿j/v\edt/²�jepj�t/Ylo[oYX_qja,tapi\_bad[oYX¬/cqº�P�Yí»¨_badt[Z ja9\}v¿jYKi�¢`v\¶KjYX[�Ã)jad¶Ev\²½Áj�]E[ÂU�_yih[oedtadcqU_^M`�aJbW�Q; R�STS�c`U]`t`lovX²,_ �`|EUX¡Kj¬`_qc3�`�/�#$E��/h£/U¨�/¤/¤&/hº�»{vha]hvQ��_qY�ed_ba.¸�ta ��t/²,¡Xvhed_ba3³hxq[o_qY\xq_`º

hº)¢hº�d)ºA®%Áv\x?¶\[º�Æ�_qj�]½cS_qxqt/YKiEk�tapi\_ba�j�ad[fed¶X²,_bed[ox;jYKi½ÀKY\[oed_�j/vhedt`²�jepjEºfeV�HNM7��X�@�H58�g��ih��)�jkg�*��)�j�Cl�LNM7��X�@�^U@ .! & #$�"���U £#"� `¤hº

|Eº)¢hº�d)ºA®%Áv\x?¶\[º_1ZYVj�ih_qxq[ocS[ot`YV²,_bed¶XtEi½[oYVed¶\_�ad_qc�eSad[oxbed_qiVcS_qxqt/YKiEk�tapi\_ba�j�ad[fed¶X²,_bed[ox`º�P}Ym A�#�0��'*)@���n����)6g�� FF�58�g��B��oHNK� �X���#j&�&Cp�g�qf7�)�j m �6�9Cp��F � ����qf�r+WF ����� )�� � oHs-� AhT��Cl� q4U&=6tTSU¡Xj/¬/_qc%£ $K£/£`ºA³�epj/Yh¸�t/api�WZY\[^]`_badcS[fe}¯���ad_bcScqU{£#"� ���º

Ghº)¢hºud)º(®%Áv\x?¶\[CºÅ»¨apj/Y\c�ÀKYX[fed_�j/v\edt/²�jepj·ad_qxqv\adcS[ot/YXc�j/YXiH¹�_yj�]ôcS_qxqt/YKiítapi\_ba,ed¶\_qt/aS¯Ht¸tapi\[oYKjlocqº½P}Y m r�#�0�Q'*)@���-����)6g�� FFk58�g��B��oPNK� �X���#jT��Cp�g�q�oW7�)�j m �6�9Cp��F � ����qM�r+E������� )��*� ov � *��F 7&Cl� �wU&=6t�x\UX¡Kj¬`_qcZ�#$E��/hº@�ZtaSed¶4ZZt/lolJjYKi5U{£#"� �r�º

"hºy1�º �!jaSedt/Y¿jYKiH�3ºHd±[ocS¡Kj/lº���t`²,¡\lo_q²,_qY�epjed[ot`Y�t¸.apjed[ot`YXj/l�cS_bedc�t`YHcSxyjeSed_?ad_yi�l^[oY\_yjatapi\_bad[oY\¬`c.t/¸ ÀXYX[fed_Zapj/Y�]uº�P}Y258z-��'AI2; S x\Uu�¤`¤~\º

£q¤hº%w�º¼RZ[fapj/vXlfeSk�®�_yjv�{�vX[o_bayº�®�[ol^[o²,[fed_qc�ih_�l^j/Y\¬�j¬`_qc;ad_qxbt`YXYXj/[ocScdj/s\lo_qcqº|���@�A��� ������������������������^U�/&/h­m�#$6/�µ ! /&/`r#$�/~���U(£%"&G/~hº

£/£`º)z�º�Z jvXcdiht/aA} º(³h_?e�ed¶X_qtaS¯�º�P}Y~�8����C F%�A7/U@�Z_b¹��(t/a]uUA£#"`r`|Eº£y�Eº�³uº��3º�Ã%lo_q_qYX_/º�d±_q¡\ad_qcS_bYEepj�ed[ot`YÄt¸)_q]�_qY�edc�[oYÄYX_bad]`_½YX_bedc�j/YXiíÀKY\[fed_·j/v\edt/²�jepjEº P}Y

�3º 3�º ³E¶KjYXYXt/Y¨UA_yih[oedtayU(zJ�������E7���7�F�9��j���� FdU{¡Kj/¬/_qcy/0$�~\£`º���ad[oY\xq_bedt`Y�vXYX[o]�_?adcS[oe�¯V��ad_qcScqU� ad[oYXxq_?edt`Y¨UA£#"`r� Eº

£%/hº�d)ºXT�x%� jvX¬/¶Eedt/Y¨º�»¨_qc�ed[oYX¬,jYKi�¬/_qYX_?apjed[oYX¬;[oYhÀKYX[fed_ cS_%{Ev\_qYXxb_qc�s�¯�j�ÀKY\[fed_%j/vhedt`²�jedt/Y¨º'*)%+%��*�,�u�������������^U@".! r`��£ $hr�/`¤EU¼£%"& � hº

£b~\º�d)º¨T�x%�Zj/vX¬/¶�edt`Y�jYKiV³5º¨� j/¡u_?aSeyºK������)@��� J+*���7��.�����_7���7/º�T�P�»���ad_bcScqU �!j²�s\ad[^ih¬`_`UTQ�)U¨£%"�|�£`º

£yrEº%w�º�T�v\l^lo_bayº�P}YhÀKY\[oed_�cS_#{�vX_qY\xq_qc�j/YXiíÀKY\[oed_½²�jx?¶\[oYX_qcqº P}Y¿��adt�x/º!t/¸�z\t`v\aSed¶Ä�3YXYEvXj/lP�3u3�3ij�¯h²,¡¨ºoUE_yi\[fedtayU������X���*���:)�g[���@�A��*q47�)�j�58�g��B�7&C<by� F�lg�)XU\¡Kj¬`_qc�/0$X£# hU¨£#"� &/hº

£% hº�TVº��3[o]`j�e�j/YXi�w�ºh��_baSad[oY¨ºH3�YXcS_b²�sXlo_qc¼ad_qxqt/YXYKj[ocScdj/sXlo_qc±ih_Z²,t/edc(sX[fkm[oY\ÀKY\[ocqº{P�Y m A�#�*��Aj&D�X)�g�Fk�r+E�X�@��O����� ���� )@�X�MzJ)@)��@7�CHzy�8N���q����.��F�:������)����@�A��*q��r+4�����������9�:)�g/U¨¡Kj¬`_qc~�|#$Er�"hU{£#"&G`�Eº

£y|Eº)¢hº k���ºX��g_qxbvXxp¶X_beyºu3 edvKi\_%c�¯hY�epj�h[l{�vX_Zi\_qc.¡Kj�aSed[o_qc±ad_bxqt`Y\YKj/[ocScdjsXlo_qc ih_%²,t/edc�[oY\ÀXYX[ocqº¼]�t`lfkv\²,_%�/�� hUX¡Kj¬`_qcZ��"~�$�//¤&/EU�£#"�G& hº

£%Ghº%w�ºh��_baSad[oY�jYKi�¢\º k�3�ºX�¼[oY¨º{P}YhÀKYX[fed_.¹!tapi\cqº P�Y43�locS_q]E[^_?ayU\_qi\[fedt/ayU�zJ�7TjT� �k�B� m A� FFpUK�/¤/¤&/Eº£%"hº�TVº�1�º6dZjsX[oY¨º5wZ_qxq[^iXjsX[olo[oe�¯,t/¸¨cS_qxqt/YKiEk�tapi\_ba(ed¶X_qtad[o_qc�j/YXi,j/vhedt`²�jepj%t/Y�[oYhÀKYX[fed_�eSad_q_qcqº£b~\£�!^£*$�/`rEU £#"� &"Eº

�¤hº)z�º�w�ºHdZj²,cS_b¯�º�1ZYHj·¡\adt/sXlo_q² t/¸±¸�t/ad²�jl.l^t/¬`[ox`º m A�#�0�-��+f�X�@�,58��)�jT��)��E7��X�@�JF �#�0�^U//¤.! /&/�G0$6/&G/~hU�£#"���"hº

��£`º)¢hºuR�º�d3t/cS_qYXc�ed_b[^YAºn5(�:)��7�y��Aj6� *�:)6g/º!�±xyj`ih_q²,[ox%��ad_bcScqU��Z_?¹��(t/a]uU5£#"�G���º�/�Eº�TVº k���º¨³hxp¶ Ávhed¾q_qYEsu_bad¬/_bayº_1ZY½ÀXYX[fed_�²,t/YXt`[^ihc ¶Kjy]E[oYX¬�t`YXlf¯·eSad[o]E[Jjl�cSvXsX¬adt`v\¡XcqºE'*)#+%�� �_�

����)@�9A��C U�G.!o£%"`¤0$X£#"~\U £#"� �r�º