
samooh kaarravaai

ganit mein, ack sthaan parr ack samooh kaarravaai ack diye gaye samooh kaa ack samooh samroopata haye joe sthaan key parivartanon key samooh mein haye. isi tarah, ack ganitiya sanrachana parr ack samooh kaarravaai sanrachana key swachaalan samooh mein ack samooh kaa ack samooh samroopata haye. aisaa kahaa jaataa haye kii samooh sthaan yaa sanrachana parr "kaary" kartaa haye. yadi koee samooh kissy sanrachana parr kaary kartaa haye, tou yeh aamtaur parr usa sanrachana seey bunny vastuon parr bhee kaary karega. udaaharan key liye, euclidean samamitiyon kaa samooh euclidean sthaan parr aur usamen kheenchi gaee aakritiyon parr bhee kaary kartaa haye. udaaharan key liye, yeh sabhi trikonon key samooh parr kaary kartaa haye. isi tarah, ack polyhedron kii samroopata kaa samooh sheershon, kinaaron aur polyhedron key cheharon parr kaary kartaa haye.
vector space parr samooh kriya koo samooh kaa pratinidhitva kahaa jaataa haye. ack parimit-ayami sadish sthaan key mamle mein, yeh ack kshetra parr ayam key vyutkramaniya matricks key samooh key upsamooh key saath kaee samoohon kii pehchaan karane kii anumati detaa haye.
samamit samooh kissy bhee samooh parr tatvon key saath samooh key tatvon koo krambaddh karke kaary kartaa haye. halanki ack samooh key sabhi kramparivartanon kaa samooh aupchaarik roop seey sett parr nirbhar kartaa haye, samooh kaarravaai kii avadharana ack hii samooh koo sabhi seton key kramparivartan kaa adhyayan karane key liye ack hii samooh parr vichaar karane kii anumati detee haye.
paribhasha.
vaam samooh kaarravaai.
yadi pehchaan tatva vaalaa samooh haye, aur ack samooh haye, tou ack ("baaen") "samooh kriya" ack kaary haye.
joe nimnalikhit dou swayamsiddhiyon koo santusht kartaa haih
(aksar sankshipt key saath yaa jaba vichaar kii jaa rahee kaarravaai sandarbh seey spasht hoti haye):
sabhi key liye aur sabhi key liye.
samooh koo (baain or seey) kaary karane key liye kahaa jaataa haye. ack kriya key saath ack sett koo ack ("baaen")-"sett" kahaa jaataa haye.
inn dou swayamsiddhon seey, yeh iss prakaar haye kii kissy bhee nishchit key liye, kaary joe swayam seey maanchitrit hotaa haye vaha ack dwibhajan haye, vyutkram dwibhajan key saath sambandhit maanchitra key liye. isaliye, koee bhee vyakti samooh kii kaarravaai koo ack samooh samroopata key roop mein sabhi bijections key samamit samooh mein paribhaashit curr sakata haye.
sahee samooh kaarravaai.
isi tarah, onn kaa ack "sahee samooh kaarravaai" ack kaary haye.
joe anuroop swayamsiddhiyon koo santusht kartaa haih
sabhi key liye aur sabhi key liye.
baaen aur daaen karyon key beach kaa antar usa crum mein hotaa haye jiss parr ack utpaad kaary kartaa haye. ack baain kaarravaai key liye, pahale kaary kartaa haye, usake baad doosara. sahee kaarravaai key liye, pahale kaary karein, usake baad doosara kaary karein. sootra key kaaran, samooh key vyutkram sanchaalan key saath rachana karke ack baain kriya kaa nirmaan kiya jaa sakata haye. isake alawa, ack samooh kii sahee kaarravaai koo usake vipareet samooh kii baain kaarravaai key roop mein maanaa jaa sakata haye.
iss prakaar, samooh karyon key samanya gunon koo sthaapit karane key liye, kewal bache hue karyon parr vichaar karna paryaapt haye. halanki, aise mamle hain jahaan yeh sambhav naheen haye. udaaharan key liye, ack samooh kaa gunan samooh parr hii ack baain kriya aur ack daaeen kriya donon koo prerit kartaa haye-kramshah baain aur daaeen or gunan.
karyon key ullekhaniya guna.
maan leejiye kii sootra _ 3 ack samooh haye joe ack nirdhaarit sootra _ 4 parr kaary kartaa haye. kaarravaai koo 'yaa' kahaa jaataa haye yadi sabhi sootra _ 6 key liye sootra _ 5 kaa tatparya haye kii sootra _ 7. samatulya roop seey, sootra _ 3 seey sootra _ 4 key bijection key samooh taka kriya key anuroop rupankan injection haye.
kriya koo "(yaa" ardh-rekhiya "yaa" nishchit-bindu mukt ") kahaa jaataa haye yadi kissy sootra _ 6 key liye sootra _ 5 kaa kathan pahale seey hii usa sootra _ 7 kaa tatparya haye. dusare shabdon mein, sootra _ 3 kaa koee bhee gair-tuchchh tatva sootra _ 4 key ack bindu koo thheek naheen kartaa haye. yeh nishtha kii tulanaa mein bahut adhik majboot sampatti haye.
udaaharan key liye, kissy bhee samooh kii kriya apane aap mein baaen gunaa seey mukt haye. iss avalokan kaa tatparya haye kii kissy bhee samooh koo ack samamit samooh mein antarnihit kiya jaa sakata haye (joe samooh honey parr anant hotaa haye). ack seemit samooh apani mukhyata seey bahut chhote aakar key samooh parr vishvaspurvak kaary curr sakata haye (haalaanki aisee kaarravaai swatantra naheen hoe sakati haye). udaaharan key liye abelian 2-samooh sootra _ 15 (cardinality sootra _ 16 kaa) aakar sootra _ 17 key ack sett parr vishvaspurvak kaary kartaa haye. yeh hamesha mamla naheen hotaa haye, udaaharan key liye chakreeya samooh sootra _ 18 sootra _ 16 seey cumm aakar key sett parr vishvaspurvak kaary naheen curr sakata haye.
samanya taur parr sabse chhota sett jiss parr ack wafadar kaarravaai koo paribhaashit kiya jaa sakata haye, usee aakar key samoohon key liye bahut bhinn hoe sakata haye. udaaharan key liye, 120 aakar key teen samooh samamit samooh sootra _ 20, samtaap samooh sootra _ 21 aur chakreeya samooh sootra _ 22 hain. sabse chhote samooh jinn parr inn samoohon key liye vishvasaniya karyon koo paribhaashit kiya jaa sakata haye, way kramshah aakar 5,7 aur 16 key hain.
pargaman guna.
sootra _ 4 parr sootra _ 3 kii kriya koo "" kahaa jaataa haye yadi kissy bhee dou binduon key liye sootra _ 25 kaa ack sootra _ 26 maujood haye taaki sootra _ 27 hoe. 
yadi yeh sankraamak aur mukt donon haye tou kriya '(yaa' teevra roop seey sankraamak ', yaa') haye. iska matlab haye kii diyaa gayaa sootra 25 pargaman kii paribhasha mein tatva sootra 29 advitiya haye. yadi sootra _ 4 koo ack samooh sootra _ 3 dwara kewal pargaman parr kaary kiya jaataa haye tou ise sootra _ 3 yaa sootra _ 3-tors key liye ack pramukh sajaatiya sthaan kahaa jaataa haye. 
ack purnank sootra 34 key liye, kriya "yadi sootra 4 mein cumm seey cumm sootra 36 tatva hain, aur sootra 36-tooples sootra 38 kii kissy bhee jodee key liye jodeevaar alag pravishtiyon key saath (joe sootra 39 haye, sootra 40 jaba sootra 41 haye) ack sootra 26 maujood haye jaise kii sootra 44 key liye sootra 43. dusare shabdon mein, baar-baar pravishtiyon key binaa sootra 45 key upasamuchchay parr kaarravaai pargamansheel haye. sootra _ 46 key liye ise aksar double kahaa jaataa haye, kramshah triple, pargamansheelta. 2-sankramansheel samoohon kaa varg (yaani, ack seemit samamit samooh key upsamooh jinaki kriya 2-sankramansheel haye) aur adhik aam taur parr gunaa sankramansheel samoohon kaa adhyayan seemit samooh siddhant mein achchi tarah seey kiya jaataa haye.
ack kriya "taba hoti haye jaba sootra _ 45 mein baar-baar pravishtiyon key binaa tooples parr kaarravaai teevra roop seey sankramansheel hoti haye. 
udaaharan.
sootra _ 4 key samamit samooh kii kriya pargamansheel haye, vaastav mein sootra _ 4 kii cardinality taka kissy bhee sootra _ 36 key liye sootra _ 36-pargamansheel haye. yadi sootra _ 4 mein cardinality sootra _ 53 haye tou vaikalpik samooh kii kriya sootra _ 54-pargamansheel haye lekin sootra _ 55-pargamansheel naheen haye. 
gair-shoonya vactor key nirdhaarit sootra _ 57 parr ack vector space sootra _ 56 key samanya raikhik samooh kii kriya sankramaniya haye, lekin 2-sankramaniya naheen haye (isi tarah vishesh raikhik samooh kii kriya key liye yadi sootra _ 58 kaa ayam cumm seey cumm 2 haye). ack euclidean sthaan key orthogonal samooh kii kriya gair-shoonya vactor parr pargamansheel naheen haye, balki yeh ikai kshetra parr haye.
adim kriyaaen.
sootra _ 4 parr sootra _ 3 kii kriya koo "adim" kahaa jaataa haye yadi sootra _ 4 kaa koee vibhajan naheen haye joe sootra _ 3 key sabhi tatvon dwara sanrakshit haye, isake alawa tuchchh vibhajan (ack tukade mein vibhajan aur isake dohare, ekal mein vibhajan). 
sthalvaigyaanik guna.
maan leejiye kii sootra _ 4 ack topological sthaan haye aur sootra _ 3 kii kriya homomorphism dwara haye. 
yadi pratyek sootra _ 6 kaa ack pados sootra _ 66 haye tou kriya "bhatakana" haye, jaise kii sootra _ 68 key saath kewal kaee sootra _ 26 hain.
adhik aam taur parr, ack bindu sootra _ 6 koo sootra _ 3 kii kriya key liye ack bindu viram kahaa jaataa haye yadi ack khula upasamuchchay sootra _ 71 haye jaise kii sootra _ 68 key saath kewal kaee sootra _ 26 hain. kriya kaa "asantulan kaa kshetra" viram key sabhi binduon kaa samooh haye. samatulya roop seey yeh sabse badaa sootra _ 3-stable open subset sootra _ 75 haye jaise kii sootra _ 77 parr sootra _ 3 kii kriya bhatak rahee haye. gatishil sandarbh mein ise "bhatakane vaalaa samooh" bhee kahaa jaataa haye. 
yadi pratyek compact subset sootra 78 key liye kaee sootra _ 26 hain jaise kii sootra _ 80. yeh bhatakane seey sakhti seey majboot haye; udaaharan key liye sootra _ 83 dwara diye gaye sootra _ 82 parr sootra _ 81 kii kriya bhatak rahee haye aur mukt haye lekin thheek seey asantat naheen haye.
ack aavaran sthaan parr sthaaniya roop seey saral roop seey judey sthaan key maulik samooh key deck parivartan dwara kaarravaai bhatak rahee haye aur mukt haye. iss tarah key karyon koo nimnalikhit gunon dwara chihnit kiya jaa sakata haih pratyek sootra _ 6 kaa ack pados sootra _ 66 hotaa haye jaise kii pratyek sootra _ 87 key liye sootra _ 86. iss guna key saath karyon koo kabhi-kabhi "swatantra roop seey asantat" kahaa jaataa haye, aur sabse badaa upasamuchchay jiss parr kaarravaai swatantra roop seey asantat hoti haye, usey "mukt niyamit samooh" kahaa jaataa haye. 
sthaaniya roop seey compact space formula 4 parr samooh sootra _ 3 kii ack kriya koo "cocompact" kahaa jaataa haye yadi ack compact subset formula _ 90 maujood haye jaise kii formula _ 91. ack thheek seey asantulit kaarravaai key liye, cocompactness bhagfal space formula _ 92 kii compactness key barabar haye.
sthalvaigyaanik samoohon kii kriyaaen.
abb maan leejiye kii sootra 3 ack topological samooh haye aur sootra 4 ack topological sthaan haye jiss parr yeh homomorphism dwara kaary kartaa haye. yadi utpaad topology key liye maanchitra sootra _ 95 nirantar haye tou kaarravaai koo "nirantar" kahaa jaataa haye.
yadi sootra 97 dwara paribhaashit maanchitra sootra _ 96 uchit haye tou kaarravaai koo "kahaa jaataa haye. iska matlab haye kii diye gaye compact sett sootra _ 98 sootra _ 26 kaa sett aisaa haye kii sootra _ 100 compact haye. vishesh roop seey, yeh uchit asantulan key barabar haye jaba sootra _ 3 ack asatat samooh haye. 
yadi sootra 103 kaa ack pados sootra _ 66 maujood haye tou ise "sthaaniya roop seey mukt" kahaa jaataa haye jaise kii sabhi sootra _ 6 aur sootra _ 106 key liye sootra _ 104. 
yadi kakshiya maanchitra sootra _ 107 pratyek sootra _ 108 key liye nirantar haye tou kriya koo "dridhata seey nirantar" kahaa jaataa haye. naam seey joe pataa chalata haye, usake vipareet, yeh kriya kii nirantarta kii tulanaa mein ack kamjor guna haye.
yadi sootra _ 3 ack jhooth samooh haye aur sootra _ 4 ack avakalaniya manifold haye, tou kaarravaai key liye "chikane binduon" kaa upasthaan sootra _ 6 kaa sett haye jaise kii maanchitra sootra _ 107 chikna haye. jhooth samooh karyon kaa ack achchi tarah seey vikasit siddhant haye, i.ee. kriya joe poore sthaan parr sucharu hoti haye.
raikhik kriyaaen.
yadi sootra _ 29 ack commutative ring parr ack module parr raikhik parivartan dwara kaary kartaa haye, tou kaarravaai koo irreducible kahaa jaataa haye yadi koee uchit gair-shoonya sootra _ 29-invariant upa-module naheen hain. ise "ardh-saral" kahaa jaataa haye yadi yeh apuraniya karyon key pratyaksh yog key roop mein vighatit hoe jaataa haye.
kakshaaen aur sthirikaran.
ack samooh sootra parr vichaar karein joe ack samooh sootra parr kaary kartaa haye. sootra 4 mein ack tatva sootra 117 kaa "sootra 4 mein tatvon kaa samooh haye jisamen sootra 117 koo sootra 3 key tatvon dwara sthanantarit kiya jaa sakata haye. sootra 117 kii kaksha koo sootra 123 dwara darshaaya jaataa haih
sootra _ 124
ack samooh key paribhaashit gunon seey yeh guaranti milati haye kii sootra 3 kii kriya key tahat (bindu sootra 117 mein) sootra 4 key kakshon kaa samooh sootra 4 kaa ack vibhajan banaata haye. sambaddh samatulyata sambandh koo sootra 129 seey paribhaashit kiya jaataa haye yadi aur kewal yadi sootra 132 key saath sootra 3 mein ack sootra 29 maujood haye tou kakshaaen iss sambandh key tahat samatulya varg hain; dou tatva sootra 117 aur sootra 134 samkaksh hain yadi aur kewal tabhi jaba unaki kakshaaen samaan houn, arthaat sootra 135.
samooh kriya sankramansheel haye yadi aur kewal tabhi jaba isaki ack kaksha haye, arthaat, yadi sootra _ 138 key saath sootra _ 4 mein sootra _ 117 maujood haye tou yeh mamla haye yadi aur kewal sootra _ 4 mein sootra _ 117 key liye sootra _ 139 (usa sootra koo diye jaane parr _ 4 gair-khaali haye).
sootra 3 kii kriya key tahat sootra 4 kii sabhi kakshaon key samooh koo sootra 145 (yaa, cumm baar, sootra 92 key roop mein) key roop mein likhaa jaataa haye, aur ise 'kriya kaa' kahaa jaataa haye. jyaamitiya sthitiyon mein ise 'kahaa jaa sakata haye, jabaki beejganitiya sthitiyon mein ise' kaa sthaan 'kahaa jaa sakata haye, aur aparivartaniya (nishchit bindu) key vipareet likhit sootra _ 147, niroopit sootra _ 148: sikka parivartak ack hain jabaki aparivartak ack hain. sikka-variant shabdawali aur sanketan kaa upayog vishesh roop seey samooh samroopata aur samooh samroopata mein kiya jaataa haye, joe samaan superscript/sadasyata parampara kaa upayog karate hain.
aparivartaniya upasamuchchay.
yadi "y" "x" kaa ack upasamuchchay haye, tou sootra _ 149 nirdhaarit sootra _ 150 koo darshaata haye, upasamuchchay "y" koo "g key tahat aparivartaniya" kahaa jaataa haye yadi sootra _ 151 (joe sootra _ 152 key barabar haye). usa mamle mein, "g" bhee "y" parr kaarravaai koo "y" taka seemit karke sanchaalit kartaa haye. upasamuchchay "y" koo "g key tahat nishchit" kahaa jaataa haye yadi sootra _ 153 "g" mein sabhi "g" key liye aur "y" mein sabhi "y" key liye. pratyek upasamuchchay joe "jee" key tahat nishchit haye, vaha bhee "jee" key tahat aparivartaniya haye, lekin isake vipareet naheen.
pratyek kaksha "x" kaa ack aparivartaniya upasamuchchay haye jiss parr "g" paargamanik roop seey kaary kartaa haye. isake vipareet, "x" kaa koee bhee aparivartaniya upasamuchchay kakshaon kaa ack sangh haye. x" parr "g" kii kriya "sankramansheel" haye yadi aur kewal tabhi jaba sabhi tatva samatulya houn, jiska arth haye kii kewal ack kaksha haye.
x" kaa ack "g-invariant" tatva sootra _ 6 haye jaise kii sabhi sootra _ 156 key liye sootra _ 5 aise sabhi "x" key samooh koo sootra _ 148 seey darshaaya jaataa haye aur ise "x" key "g-invariant" kahaa jaataa haye. jaba "x" ack "g"-module haye, tou "xg" "x" mein gunank key saath "g" kaa shoonya saha-vigyaan samooh haye, aur uchch saha-vigyaan samooh "g"-invariant key functor key vyutpann kaarak hain.
nishchit bindu aur sthirikaran upsamooh.
sootra _ 158 key saath "g" mein "g" aur "x" mein "x" diye jaane parr kahaa jaataa haye kii "x" "g" kaa ack nishchit bindu haye yaa "g" "x" koo thheek kartaa haye. x" mein pratyek "x" key liye, "x" key sambandh mein "g" kaa (jise "isotropy samooh" yaa "chhota samooh" bhee kahaa jaataa haye) "g" mein sabhi tatvon kaa samooh haye joe "x" koo thheek kartaa haih
sootra 159
yeh "jee" kaa ack upsamooh haye, halanki aamtaur parr samanya naheen haye. x" parr "g" kii kriya taba mukt hoti haye jaba sabhi sthirikaarak tuchchh houn. samamit samooh key saath samroopata key kernell "aine", sootra _ 160 koo "aiks" mein sabhi "aiks" key liye sthirikaarak "gx" key pratichhedan dwara diyaa jaataa haye. yadi "aine" tuchchh haye, tou kaarravaai koo wafadar (yaa prabhaavi) kahaa jaataa haye.
maan leejiye kii "x" aur "y" "mein dou tatva hain, aur sootra _ 29 koo ack samooh tatva honey dein jaise kii sootra _ 162 tou dou sthirikaran samooh sootra _ 163 aur sootra _ 164 sootra _ 165 pramaan dwara sambandhit hainh paribhasha key anusaar, sootra _ 166 yadi aur kewal yadi sootra _ 167 iss samaantaa key donon pakshon koo sootra _ 168 lagu karane seey sootra _ 169 praapt hotaa haye, tou sootra _ 170 ack vipareet samavesh sootra _ 171 laker isi tarah seey anusaran kartaa haye aur maan leejiye sootra _ 172
uparokt kaa kahana haye kii ack hii kaksha mein tatvon key sthirikaarak ack dusare key saath sanyugmit hotey hain. iss prakaar, pratyek kaksha key liye, humm "jee" key ack upsamooh key ack sanyugma varg koo jodd sakate hain (yaani, upsamooh key sabhi sanyugmon kaa samooh). sootra _ 173 koo "h" key sanyugma varg koo darshaata haye. taba kaksha "oh" kaa prakaar sootra _ 173 hotaa haye yadi "oh" mein kuchh/kissy bhee "aiks" kaa sthirikaran sootra _ 163 sootra _ 173 seey sambandhit haye. ack adhikatam kaksha prakaar koo aksar ack pramukh kaksha prakaar kahaa jaataa haye.
aur bernside kaa lemma.
kakshaaen aur sthirikaarak nikatata seey sambandhit hain. x" mein ack nishchit "x" key liye, sootra 178 dwara diye gaye maanchitra sootra parr vichaar karein, paribhasha key anusaar iss maanchitra kaa chhavi sootra _ 17 haye kaksha sootra _ 180 dou tatvon key liye ack hii chhavi honey kii sthiti haye
sootra _ 181
dusare shabdon mein, sootra 182 "yadi aur kewal tabhi jaba" sootra 29 aur sootra 184 sthirikaran upsamooh sootra 185 key liye ack hii coaset mein sthit hain, tou "g"· "x" mein kissy bhee "y" parr "f" kaa fiber sootra 186 aise coaset mein nihit haye, aur aisaa pratyek coaset bhee ack fiber key roop mein hotaa haye. isaliye "f" stabilizer upsamooh key liye coaset key sett sootra _ 187 aur kaksha sootra _ 188 key beach ack koo prerit kartaa haye joe sootra _ 189 bhejata haye. iss parinam koo "kaksha-stabilizer pramey" key roop mein jaanaa jaataa haye.
yadi "g" parimit haye tou lagrange kii pramey key saath kaksha-sthirikaran pramey detaa haye
sootra _ 190
dusare shabdon mein, "x" kii kaksha kii lambai, isake sthirikaran key crum seey gunaa, samooh kaa crum haye. vishesh roop seey iska tatparya haye kii kaksha kii lambai samooh crum kaa ack vibhaajak haye.
yeh parinam vishesh roop seey upayogi haye kyonki ise tarkon kii ginati key liye niyojit kiya jaa sakata haye (aamtaur parr una sthitiyon mein jahaan "x" bhee seemit haye).
ack parinam joe kaksha-sthirikaran pramey seey nikatata seey sambandhit haye, vaha haye bernside kaa siddhaantah
sootra 201
jahan "x" "g" "g" dwara nirdhaarit binduon kaa samooh haye. yeh parinam mukhya roop seey taba upayog mein aataa haye jaba "jee" aur "aiks" seemit hotey hain, jaba isaki vyakhya iss prakaar kii jaa sakati haih kakshaon kii sankhya prati samooh tatva nirdhaarit binduon kii ausat sankhya key barabar hoti haye.
ack samooh "jee" koo thheek karna, seemit "jee"-sett key aupchaarik antar kaa sett ack ring banaata haye jise "jee" kaa bernside ring kahaa jaataa haye, jahaan jodd vichhed sangh key anuroop hotaa haye, aur cartasian utpaad key liye gunan hotaa haye.
samooh kriyaaen aur samooh.
samooh kriya kii dhaarana koo samooh kriya seey judey "kriya samooh" sootra _ 211 dwara kootbaddh kiya jaa sakata haye. kriya key sthirikaarak samooh key sheersh samooh hain aur kriya kii kakshaaen isake ghatak hain.
jee"-seton key beach rupankan aur samroopata.
yadi "x" aur "y" dou "g"-samuchchay hain, tou "x" seey "y" taka ack "rupankan" ack aisaa falan haye joe "g" mein sabhi "g" aur "x" mein sabhi "x" key liye haye. jee"-sett key rupankan koo "samparivartaniya maanchitra" yaa "jee-maanchitra" bhee kahaa jaataa haye.
dou rupankanon kii sanrachana phir seey ack rupankita haye. yadi ack rupankan "aff" dwi-abhikshep haye, tou iska vyutkram bhee ack rupankan haye. iss mamle mein "f" koo "samroopata" kahaa jaataa haye, aur dou "g"-samuchchay "x" aur "y" koo "samroopata" kahaa jaataa haye; sabhi vyaavhaarik uddeshyon key liye, samroopata "g"-samuchchay aprabhedya hain.
kuchh udaaharan samaroopataah
roopvaad kii iss dhaarana key saath, sabhi "jee"-seton kaa sangrah ack shreni banaata haye; yeh shreni ack grothendic topos haye (vaastav mein, ack shaastriya dhaatu vigyaan maanate hue, yeh topo booliyan bhee hoga).
prakaar aur saamaanyikaran.
humm uparokt samaan dou swayamsiddh kaa upayog karke sett parr monoids key karyon parr bhee vichaar curr sakate hain. haalaanki yeh dwi-uddeshya maanchitra aur samatulya sambandhon koo paribhaashit naheen kartaa haye. ardhsamuh kriya dekhein.
sett parr karyon key bajaay, humm ack manmaane varg kii vastuon parr samoohon aur monoids key karyon koo paribhaashit curr sakate hainh kuchh shreni key ack vastu "x" seey shuru karein, aur phir "x" parr ack kaarravaai koo "x" key endomorphism key monoid mein ack monoid homomorphism key roop mein paribhaashit karein. yadi "x" mein ack antarnihit samooh haye, tou oopar bataai gaee sabhi paribhashaon aur tathyon koo aage badhaaya jaa sakata haye. udaaharan key liye, yadi humm vector rikt sthaan kii shreni lete hain, tou humm iss tarah seey samooh pratinidhitva praapt karate hain.
humm ack samooh "jee" koo ack ekal vastu key saath ack shreni key roop mein dekh sakate hain jisamen pratyek rupankan vyutkram haye. ack (baaen) samooh kriya taba kuchh bhee naheen balki ack (sahparivaahak) fuctor haye joe "jee" seey sett kii shreni taka haye, aur ack samooh pratinidhitva "jee" seey vector rikt sthaan kii shreni taka ack fuctor haye. jee-sett key beach ack rupankan taba samooh kaarravaai functors key beach ack praakritik parivartan haye. saadrishya mein, samooh kaa ack kaary samooh seey samooh kii shreni yaa kissy anya shreni mein ack functor haye.
sthalshastriya sthaanon parr sthalshastriya samoohon key nirantar karyon key alawa, aksar jhooth samoohon key sucharu karyon, beejganitiya kismon parr beejganitiya samoohon key niyamit karyon aur yojanaon parr samooh yojanaon key karyon parr bhee vichaar kiya jaataa haye. yeye sabhi apani-apani shreni kii vastuon parr kaary karane vaali samooh vastuon key udaaharan hain.

