
ultrograph sii *-beejaganitam

ganithashaasthrathil, ultragraph sii *-aljibra ennathu ultragraphy 6-7 ll ninnu nirmmicha hilbert speysukalude shekharathil bhagika isometrikal srishtikkunna oru saarvathrika sii *-aljibrayaanu. eee sii *-aaljibrakal srishtichathu graph sii *-algibras, exel-laca algibras ennivayude klaasukal oresamayam saamaanyavalkkarikkunnathi eee vasthukkale padikkunnathinulla oru ekeekritha chattakkood nalkunnathinumaayaan. kaaranam oro graph oru ultragraphy encode cheyyaan kazhiyum, athupole thanne, oru exel-laca aaljibrakal nalkunna oro anantha graph oru ultragraphy encode cheyyaan kazhiyum.
nirvachanangal.
ultragraphukal.
oru ultrograph formula _ 1 ll oru koottam vertises formula _ 2, oru koottam edjus formula _ 3, oru sozhs mapp formula _ 4, vertex settinte nonmpty subsettukalude power settu kalakshan formula _ 6 ll moolyangal edukkunna oru rench mapp formula _ 5 enniva adangiyirikkunnu. oro attatthinteyum paridhi oru singleton aayirikkunna oru ultragraphinte prathyeka kesaanu directed graph, koodaathe ultragraphukale pothuvalkkaricha directed graphai kanakkakkaam, athil oro arikukalum orotta sheershakathil aarambhich shoonyamallaatha oru upasettilekku viral choondunnu.
udaaharanam.
oru ultragraph drishyavalkkarikkaanulla oru eluppamaarggam label cheytha oru koottam sheershakangalulla oru directed graph pariganikkuka ennathaanu, avide oro label rench maappile oru gadakathinte chithratthile oru upavibhagavumaayi poruthappedunnu. udaaharanathinu, konukalum edju labelukalumulla oru ultrograph formula _ 7, formula _ 8 srothassulla oru rench mapformula _ 9 valathuvashathulla chithramaayi drishyavalkkarikkaam.
ultrograph aaljibrakal.
oru ultrograph formula _ 1 nalkiyaal, singleton settu formula _ 13 adangiya formula _ 12-nte ettavum cheria upavibhagamaayi formula _ 11-neey njangal nirvachikkunnu, athil rench settu formula _ 14 adangiyirikkunnu, koodaathe intersectionukal, unionukal, aapekshika poorakangal ennivaykku keezhil adaykkunnu. paraspara oorthoogoonal shrenikalulla bhagika isometry formula 17-nte shekharathodoppam projection formula _ 16-nte oru shekharamaanu oru kunds-kreeger formula _ 15 famili.
ultragraph sii *-aljibra formula _ 28 ennathu oru kunds-kreeger formula _ 15 famili srishtikkunna saarvathrika sii *-aljibrayaanu. 
swathukkal.
oro graph sii *-algibra oru ultragraphinte prathyeka kesai kanakkaakkukayum formula _ 11 ennathu formula _ 2, formula _ 21 ennivayude ellah parimitha upavibhaagangaludeyum shekharamaanennu manassilakkukayum cheythukondu oro graph sii *-algibra oru ultragrag beejaganithamaayi kaanukayum cheiyunnu. oro exel-laca aaljibrakalum oru ultragrag sii *-aljibrayana formula _ 34 ennathu soochika settu formula _ 35 umm formula _ 36 leeii entrikalum ulla oru ananthamaaya chathura maatriksaanengil, oraalkk oru ultragraphyne formula _ 37, formula _ 38, formula _ 39, formula _ 40 enniva upayogichu nirvachikkaan kazhiyum. formula _ 28 ennathu exel-laca formula beejaganithathinu isomorphic aanennu kaanikkan kazhiyum.
graph sii *-aaljibrakalum exel-laca aaljibrakalum padikkaan upayogapradamaaya upakaranangalaan ultragraph sii *-aaljibrakal. mattu nettangalkoppam, oru exel-laca beejaganithathe ultragraph sii *-aljibrayaayi model cheiyunnathu, anubandha sii *-aaljibrakale padikkunnathinulla oru upakaranamaayi ultragraph upayogikkaan oraale anuvadikkunnu, athuvazhi exel-laca beejaganithathe padikkumbol matricks teknikkukalkku pakaram graph-theoritic teknikkukal upayogikkaanulla oopshan nalkunnu. oro lalithamaaya afjebra oru graph sii *-algibrayudeo exel-laca algibrayudeo isomorphic aanennu kaanikkan ultrograph sii *-aaljibrakal upayogichittundu. (poojyamallaatha) parimitha-dimensional koshyant illaatha oro aff-algibra oru exel-laca algibraikku isomorphic aanennu theliyikkaanum avaa upayogichittundu. 
graph sii *-algibras, exel-laca algibras, ultrograph sii *-algibras ennivayude klaasukalil oronninum sii *-algibras adangiyittundu, mattu rund klaasukalile eathenkilum sii *-aljibraikku isomorphic allah, moonnu klaasukalum moritta thulyathayumaayi poruthappedunnuvennu kaanichittundu.

