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ãã¡ã¤ãã¼ç©ï¼å¼ãæ»ããpullbackï¼ãåå¨ããåCã«å¯¾ãã¦ãã¹ãã³ã®åSpan(C)ãå®ç¾©ã§ãã¾ããç¹ã«CãéååSetã§ããå ´åãèãã¾ããSpan(Set)ã®è©±ã§ãããSpan(Set)ã®ãªãã«ãé¢ä¿åRelãåãè¾¼ããã®ã§ãSpan(Set)ã¯Relã®æ¡å¼µã ã¨ãè¨ãã¾ãã
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- Span(Set) Mat|Set|Set
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Span(Set)ã®åã«ãSPAN(Set)ãå®ç¾©ãã¾ãã大æåã ãã§æ¸ããSPAN(Set)ã¯åã«è¿ãã§ãããåã§ã¯ããã¾ããã
S = SPAN(Set) ã¨ç½®ãã¨ãSã¨ã¯æ¬¡ã®ãããªãã®ã§ãã
- |S| = |Set|
- S(A, B) ã¯ãAâXâB ã¨ããå½¢ã®ã¹ãã³ã®éåã
- (AâXâB)âS(A, B) 㨠(BâYâC)âS(B, C) ã®çµåï¼compositionï¼ã¯ãã³ã¹ãã³(XâBâY)ã®ãã¡ã¤ãã¼ç©XÃBYã使ã£ã¦ (AâXÃBYâC)âS(A, C) ã¨å®ç¾©ããã
- æç IdAâS(A, A) ããèªæãªã¹ãã³(AâAâA)ã¨ãã¦å®ç¾©ããã
S = SPAN(Set) ãåã«ãªããªãã®ã¯ã次ã®3ã¤ã®æ¡ä»¶ãæºãããªãããã§ãã
- ãã ã»ãããéåã§ããã
- çµåï¼compositionï¼ã®çµåå¾ï¼associative lawï¼ãçå¼ã¨ãã¦æç«ããã
- æçã®åä½å¾ãçå¼ã¨ãã¦æç«ããã
S = SPAN(Set) ã§ã¯ã
- S(A, B) ã¯å·¨å¤§ãªéã¾ãã§ãå°ããªéåï¼small setï¼ã¨ã¯è¨ããªãã
- çµåå¾ã¯çå¼ã¨ãã¦ã¯æç«ããªãï¼æç«ãããã¨ãä¿è¨¼ã¯ã§ããªãï¼ãup-to-isoã§ãªãæç«ããã
- åæ§ã«ãåä½å¾ãup-to-isoã§ããæç«ããªãã
ãã ã大ãéããããè¨ç®æ³åãã¦ã«ã¦ã«ãæ±ãã«ããã®ã¯äºå®ã§ããããããéåA, Bãåºå®ãã¦S(A, B)ãçºããã¨ãããã¯åã«ãªã£ã¦ãã¾ãããã®åã®å° f:(AâXâB)â(AâYâB) ã¯ãã¹ãã³ã®ããã£ã®ããã ã®åå f:XâY ã§å·¦å³ã®èã¨å¯æãªãã®ã¨å®ç¾©ããã°ããã®ã§ããç¹ã«ãA = B = 1 = åå éå ã¨ãããªããS(1, 1)ã¯éååSetã¨åååï¼ååå¤ããå¼·ãï¼ã¨ãªãã¾ããã¾ããS(A, 1)ã¯ãªã¼ãã¼åSet/Aã¨åååã§ãã
å人çã«ã¯ãS = SPAN(Set) ããã®ã¾ã¾ä½¿ã£ã¦ãããã¨æãã®ã§ãããé常ã¯ç´°å·¥ããã¦åã«ãã¾ããA, Bãã¨ã®S(A, B)ã¯åãªã®ã§ã対象ã®ããã ã®ååæ¦å¿µãå®ç¾©ã§ãã¾ããååãªå¯¾è±¡ãåä¸è¦ããã°ãçµåå¾ã¨åä½å¾ãçå¼ã§æç«ããããã«ãªãã¾ãã
- (Span(Set))(A, B) := |S(A, B)|/
ããå®ç¾©ããå ´åã(Span(Set))(A, B)ãå°ããªéåãªã®ãï¼ ããã¯ããããã§ããããä¾ãã°ã(Span(Set))(1, 1)ã¯Setã¨åä¸è¦å¯è½ã§ããSetã®ååé¡ã®éã¾ãã¯ãåºæ°ã®éã¾ãã¨åããµã¤ãºãæã¤ã®ã§ãå°ãããªããã¨åã«ã¯æããã®ã§ãããããããªãã説æãè¦ããã¨ããªãã§ãã
ã¨ã¯ããåã¯ããµã¤ãºãã¦ã«ã¦ã«ãæ°ã«ããªãã®ã§ãä½ãããªããã¨ã®ç¶æ S = SPAN(Set) ã§èãã¾ããæ°ã«ãªã人ã¯ãã ãååã§å²ã£ã¦ãã ãã -- ãµã¤ãºã¯ã¨ããããã¦ã«ã¦ã«ã¯æ¹åããã¾ãã
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ã¾ããKãå¯æåç°ã¨ãã¾ããMatKããKä¿æ°ã®è¡åã®åã¨ãã¾ããè¡åã®åã«ã¤ãã¦ã¯ãã¯ããã¦ã®åè« ãã®ç¬¬2æ©ï¼è¡åã®åãã§èª¬æãã¦ãã¾ããn, mâN ã«å¯¾ãã¦ãMatK(n, m)ã¯ãKä¿æ°ã®mè¡nåã®è¡åå ¨ä½ã®éåã§ãã
è¡åã®ä¿æ°åãåç°ããåç°åã«ä¸è¬åãã¾ããããåç°åã¯ãåç°ã¨ä¼¼ãæ§é ãæã¤åã§ãã2ã¤ã®ã¢ãã¤ãç©, ãæã£ã¦ãã¦ããããã®ããã ã«åé å¾ãup-to-isoã§æç«ãããããªåã ã¨æãã°ããã§ãããã£ã¨è©³ãããã¨ã¯ããã«ã«ãåç°åã®å®ç¾©ã確èªãã¦ã¿ãï¼ãã«ã«ãåç°ä½ç¨åã®ããã«ï¼ãã«æ¸ãã¦ããã¾ããã¯å¯¾ç§°ãªã¢ãã¤ãç©ã§ããã対称ã ã¨ä»®å®ããã»ãã話ãã¹ã ã¼ãºã§ãããã
C = (C, , O, , I) ãåç°åã¨ãã¾ããOã¯ã«å¯¾ããã¢ãã¤ãåä½ãIã¯ã«å¯¾ããã¢ãã¤ãåä½ã§ããå ·ä½ä¾ã¨ãã¦ã¯ã(Set, +, 0, Ã, 1) ã (VectR, , {0}, , R) ãªã©ãããã¾ãã
n, mâN ã«å¯¾ãã¦ã{1, ..., n}Ã{1, ..., m}â|C| ã®å½¢ã®ååã®å ¨ä½ã MATC(n, m) ã¨ãã¾ãã|C|ãå°ããéåã¨ã¯éããªãã®ã§ãMATC(n, m) ãå°ããéåã¨ã¯éãã¾ãããããã§ã大æåã§MATã¨æ¸ãã¾ããã
aâMATC(n, m) 㨠bâMATC(m, k) ã«å¯¾ãã¦ãè¡åã®ç© baâMATC(n, k) ã¯æ®éã®è¡åã¨åãããã«å®ç¾©ã§ãã¾ãã足ãç®ï¼æãç®ã¯ãCã®ã¨ã使ãã¾ããbaãa;bã¨ãæ¸ãã¾ããidnã¯ã対è§æåã«Iãããã®ä»ã¯Oã並ã¹ãæ£æ¹è¡åã¨ãã¾ãã
MATCã¯åã«ä¼¼ã¦ã¾ãããé常ã®åã§ã¯ããã¾ããããã®çç±ã¯ï¼
- MATC(n, m) ã¯å°ããªéåã¨ã¯éããªãã
- çµåå¾ã¯çå¼ã¨ãã¦æç«ããã¨ã¯éããªãã
- åä½å¾ãçå¼ã¨ãã¦æç«ããã¨ã¯éããªãã
äºæ ã¯ãSPAN(C)ãåã¨ã¯éããªãã®ã¨åæ§ã§ãã
n, mãåºå®ããã¨ãã®MATC(n, m)ã¯åã«ãªãã¾ãã|MATC(n, m)|ã¯å ã«å®ç¾©ããCä¿æ°ã®è¡åã®éã¾ãã¨ãã¦ãa, bâ|MATC(n, m)| ã«å¯¾ãã¦ãf:aâb ãCã®å°ã®è¡åã ã¨ãã¾ããã¤ã¾ããf:{1, ..., n}Ã{1, ..., m}âMor(C) ã§ããdom(f(i, j)) = a(i, j), cod(f(i, j)) = b(i, j) ãæºããã¨ãã¾ãã
対象ã®è¡åã1-ã»ã«ãå°ã®è¡åã2-ã»ã«ã¨èããã¨ãMATCã«ã¯2次å ã®åãã©ãã®æ§é ãå ¥ãã¾ãã
- 縦çµåã¯å³å¯ãªçµåå¾ãåä½å¾ãæºããã
- 横çµåï¼è¡åã®ç©ï¼ã¯up-to-isoã§çµåå¾ãåä½å¾ãæºããã
- å ¨ä½ã¨ãã¦ååã®å ¬çãæºããã
- ãããããã ã»ãããéåã¨ã¯éããªãã
è¡åæ¦å¿µãããå°ãä¸è¬åãã¦ããã¾ããããè¡åã®ã¤ã³ããã¯ã¹ã«èªç¶æ°ä»¥å¤ã許ããã¨ã«ãã¾ãã
è¡åã¯ç¸¦æ¨ª2次å ã®é ç½®ã§ãããä¿æ°ï¼æåï¼ã®ä½ç½®ã¯ {1, ..., n}Ã{1, ..., m} ã§ã¤ã³ããã¯ã¹ããã¾ããè¨ãæ¹ãæããã¨ãã¤ã³ããã¯ã¹éåã¯èªç¶æ°ã®åºéã§ãã
ã¤ã³ããã¯ã¹éåã®æ¦å¿µãæ¡å¼µããããã«ãéåã®æï¼family of setsï¼Jãèãã¾ããããã¦ãJã«å±ããA, Bã«å¯¾ã㦠MATC(A, B) ãèãããã¨ã«ãã¾ãããããã£ã¦ä½ã£ãè¡åã®ååï¼ãã ã»ãããå°ããã¨ã¯éããªãï¼ãMATJC ã¨æ¸ããã¨ã«ãã¾ããã¤ã³ããã¯ã¹éåãJå ã«åã以å¤ã¯MATCã¨å¤ããã¾ããã
SPANã¨MATã®å¯¾å¿é¢ä¿
以ä¸ã§ãåCã«å¯¾ãã¦SPAN(C)ã¨MATJCãå®ç¾©ã§ãã¾ãããSPANã¨MATã®å®ç¾©ã®ä»æ¹ã¯ã¾ã£ããéãã®ã§ãSPANã¨MATãæ¯è¼ãããã¨ã¯ä¸è¬ã«ã¯åºæ¥ã¾ãããããããC = Set ã®ã¨ããªããSPANã¨MATãæ¯è¼ã§ãã¦ãSPAN(Set)ã¨MAT|Set|Setã¯ï¼å¤§ããªãã ã»ããã許ãï¼ååã¨ãã¦åå¤ã«ãªãã¾ãã
ä»ã¾ã§ã®å®ç¾©ã«åºãã¦è¨ç®ãã¦ã¿ãã¨ãéåã«é¢ããSPANã¨MATãåãã§ãããã¨ã¯ãç´æçã«ã¯å²ã¨æããã§ããããããå³å¯ã«ç¤ºãã®ã¯é¢åã§ããã¤ã³ããã¯ã¹ã®éåãæééåã¨ã¯éããªãã®ã§ãè¡åã®ç©ã®å®ç¾©ãå¤æ´ããªãã¦ã¯ãªãã¾ãããæéåããç¡éåã¸ã®æ¡å¼µã§ãã
ç¡éåãèªç±ã«è¡ãã«ã¯ãæéçãªæ³åã ãããã¤åç°åã§ã¯ä¸è¶³ã§ãç¡éæ¼ç®ï¼âé æ¼ç®ï¼ã¨ãã¦ã®ã¢ãã¤ãç©ãå¿ è¦ã§ããéååSetã®å ´åã¯ã足ãç®ãåè«çç´åãã¤ã¾ãä½æ¥µéã§ä¸ããããã®ã§ãä½å®åæ§ã¨ãã¦ç¡éåãè¨è¿°ã§ãã¾ããéååã«éããªããç¡éç´åãå ¥ãã¦ãåé å¾ãæç«ããã®ã§ãMAT|Set|Setã®è¡åè¨ç®ã¯ãã¾ãå®ç¾©ã§ãã¾ããï¼ä½æ¥µéã¨ã¯éããªãç¡éã¢ãã¤ãç©ã¯é¢åããã§ããï¼
SPAN(Set)ã¨MAT|Set|Setã®å¯¾å¿ãå ·ä½çã«ã©ã決ãããã¨ããã¨ï¼
- |SPAN(Set)| = |MAT|Set|Set| = |Set| ãªã®ã§ã対象ã®ã¬ãã«ã§ã¯æçã§å¯¾å¿ããããã¤ã¾ãã対象é¡ã¯å ±æããã
- A, Bâ|Set| ãã¨ã«ããã é¡ã®å¯¾å¿ ΦA,B:(SPAN(Set))(A, B)âMAT|Set|Set(A, B) ãä½ãã
- x = (AâXâB) ãã¹ãã³ã§ãå·¦å³ã®èã Lx:XâA, Rx:XâB ã¨ãã¦ãΦA,B(x) := (Lx-1(a)â©Rx-1(b)| aâA, bâB) ã¨å®ç¾©ããã
è¦ããã«ãå·¦å³ã®èã®éåã«ãããã¹ãã³ã®ããã£Xãç´°ããåºåãããã§ããåºåãããåã ã®é¨åéåã¯ãAÃBã®è¦ç´ ã§ã¤ã³ããã¯ã¹ããã¾ãããªã®ã§ãéåä¿æ°ã®è¡åã¨ã¿ãªããã®ã§ãã
ãã®å®ç¾©ããã¨ã«ãä»ã®å®ç¾©ããããå¿ è¦ãªæ§è³ªã確èªãã¦ããã°ãSPAN(Set) MAT|Set|Set ãåããã¾ãããã é¡ãã¨ã«ååã§å²ãã° Span(Set) Mat|Set|Set ãå¾ãã¾ãã
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