A complete system of identities for one-letter rational expressions with multiplicities in the tropical semiring
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(84) 0®sî6N4 î ¿s ' 3 î s a î b (+* s+% C9ës+0®s.éf 1 0 9ïses+0®s a ;9 e . + /10 A ^ ®sî6'4 î s ' 3 î %s î b )(+* ¹?^abfa5gf^a\gv¹Ñk7mnÒkI_ a5aÖÀ3¸3bfacdedf]yk [^d?mnbea[^k be¬Ïm´{}]}òacié½r]y[^caOiIs ìciIs^,KÇ6]ygÑ]}dÑamndwÁ®gfk dfaagx^m´g ¹Ña ¬'l3degW¼m_ a 0(ì m´[^i a ì mn[^. i ì ¾Æk b?a_nacbe7 Á ?ô 9 ÃÅ¡½|Õ\ac[3ca gx^a-]}i^a[2gx]ygvÁ ^1m´[
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(113) gx^a-gxacbe¬®dWkn¾Ei3ac¯ beaca ç ]}["bfa{Êm´gf]}k [ts×+p õ KÇ3¹Ña,¯nag#gx^m´g iIsî ¥ì iIsfsLî '4' s¡×+pÍ÷2 Ãz.
(114) ¹?]`gxdfk ¬aDdfabf]}ad' n?« lNlî mNm¾Æk b$aK_ abeÁ »ç Ãn½ Þ i^i^]y[^¯
(115) mn{}{ëbfa{Êm´gf]}k [^d7s¡×+pÍ÷2\¾Æk b$aK_ acbwÁ çà Ç^¹Ña-]y¬®¬aci3]Êm´gfac{yÁk Ògmn]y[ o4 iIsî ìiIsfsLî bx'4 ' s×+p`Ãz2 ¹?^abfa 'i^ac[^k´gxacd?gx3a,deacbf]yacd\kn¾« lNlî"mNmȹ?^]};]}d?gx^a-dfl3¬ kn¾Emn{y{égx^a-dfabf]yacd ' ¾Æknb "ç(Ãn½ ÕWac[^a,beac{ÊmXgx]}kn[ts¡×+p`Ãz2|de^kI¹?dWgf¼m´g iIsesî Kb =o4 iIsî s¡×kpyà ÃIOp #^a-{}a¬®¬®m,¾Æk {y{}kI¹?d#[^kI¹ ]y¬®¬aci3]Êm´gfac{yÁ7¾Æbek ¬ beac{ÊmXgx]}kn[^d s¡×kp Ý qm´[^iÙs×+p`à ÃÐK½ =ag-l^d-[^kI¹1dfl3¸^¸ kndfaDgx¼m´gÏs8#$]}d m"deÁ6degfac¬ kn¾Ñ«4ÚvbxmXgx]}kn[¼mn{]yi^ac[2gf]ygx]yacd5gf¼m´g-ck [2gxmn]}[^d®s mn[3i;del^;gf¼m´g s8# sLî bL o4î . s×+pyÃIè Cp #^a[;¹Ña,cmn[
(116) ¯ ]y_na5gx^a$¾Æk {y{}kI¹?]}[^¯ {}ac¬¬®m3½ t N}| )Z\[3i^acb?gx^a ¸^bea_6]}knl^dWmndfdel^¬®¸gx]}kn[^d?mn[^i
(117) 2Á¸kngf^acdeacdcÇ ¹Ña-^mI_na s8# b .p
(118) { r]}[3caiIs 5ì iIs b ÖÇégx^a ck [^dwgmn[2g$gfacbf¬ kn
(119) ¾ ]yd3½-Õ\a[^ca7]`¾ 0íì ³ µåÇT¹Ña7¬'l^dwg ¼m_na ìc,mn[3i Qì mn[^i]yg]yd¥gf^ac[®amndwÁDgxk-i^aci^l3caknl^bÑ{yac¬¬Ïm¾Æbek ¬ bfa{Êm´gf]}k [
(120) s×+p`ÃÐè K½aKg ³ µå½ã¿a mn["gx^a[
(121) ¹?be]ygfa 0ìco4î ¹?]ygx"dek ¬®a$¸k {`Á[3k ¬®]}mn{ l^d?[^kI¹del^¸^¸k dfa$gf¼m´g#0 ìå kn¾« ÂØî Å=k´¾i^a¯ bfaa*l Ïà ½EuÔg?¾Æk {y{}kI¹?d|gx^m´g#¹|a-¼m_ a s&'¥ Obqì(so4î 4î bL sî 4î Kb ¾Æbek ¬ ¹?^]y
(122) ¹Ña,]y¬®¬aci3]Êm´gfac{yÁDi3aci^l^a s&'¥V9ïs+0®s f b s ' 3 sî =4î sfsî 4î b s¡×kpyÃIö Op )(+* z|lgW¹Ña-c{}acmnbf{`ÁÏ^mI_na sLî 4 iIsLî
(123) e =iIs b Eì o4î 4PiIsî
(124) Op o|k [3df]}i3acbf]y[^¯k [^{yÁ®gx^a a{}ac¬ac[2gfd#kn¾i^a¯ bfaaD ç ]}[gf^a ¸^bfaK_]yk l^d?]}[3ac· l¼mn{y]ygvÁ Ç6¹|a-k Ò3gxmn]}[ sLî 4 iIsLî e =iIsî Op ÕWac[^anÇ=df]y[^ca,a_nacbeÁ ]}i^a[ gf]ygvÁ
(125) Òagv¹Ñacac[tmn[2Á;¸^bfk6i^l^Kg5kn¾¥[3k bf¬®mn{}]yòcai;aÖÀ3¸3bfacdedf]yk [^dsÆgf¼m´gmn[%Òa [^knbf¬®mn{}]yòcaci
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