baind (beejganit)

ganit mein, ack baind (jise edempotent semigroup bhee kahaa jaataa haye) ack ardhsamuh haye jisamen pratyek tatva edempotent hotaa haye (dusare shabdon mein apane swayam key varg key barabar). baind kaa pahale adhyayan kiya gayaa aur unake dwara naamit kiya gayaa.
1970 key dashak kii shuruat mein biryukov, fenimore aur gerhard dwara baind kii kismon key jaali kaa swatantra roop seey varnan kiya gayaa thaa. ardh-jaali, vaam-shoonya patti, daaen-shoonya patti, aaytaakaar patti, samanya patti, baaen-niyamit patti, daaen-niyamit patti aur niyamit patti aise patton key vishisht upavarg hain joe iss jaali key nichale hisse key paas sthit hain aur joe vishesh ruchi key hain; way sankshep mein neeche varnit hain.
baind kii kismein.
baind kaa ack varg ack kism banaata haye yadi yeh upa-samoohon, samroop chhaviyon aur pratyaksh utpaad key gathan key tahat bund hotaa haye. baind kii pratyek vividhata koo ack paribhaashit pehchaan dwara paribhaashit kiya jaa sakata haye.
ardh-jaali.
ardh-jaali bilkul commutative baind hain; yaani, way sameekaran koo santusht karane vaale baind hain.
baind ack poorv crum koo prerit karate hain jise sootra _ 1 key roop mein paribhaashit kiya jaa sakata haye yadi sootra _ 2. commutavity kii aavashyakta kaa tatparya haye kii yeh poorv crum ack (ardh-jaali) aanshik crum bunn jaataa haye.
shoonya baind.
ack left-jeero baind sameekaran koo santusht karane vaalaa baind haye.
jahan seey isaki keyali table mein nirantar panktiyaan hoti hain.
samamit roop seey, ack sahee-shoonya baind ack santoshjanak haye.
taaki keyali table mein nirantar collum houn.
aaytaakaar pattiyan.
ack aaytaakaar patti ack aisee patti haye joe santusht karati haye
kissy bhee ardhsamuh mein pahali pehchaan ack kaheen bhee commutative ardhsamuh koo chihnit karane key liye paryaapt haye.
kaheen bhee commutative semigroup pahali pehchaan kaa tatparya naheen haye. 
kissy bhee lacheele magma sootra mein _ 3 isaliye pratyek tatva apane varg key saath aataa haye. isaliye kaheen bhee commutative semigroup mein harr tatva nishkriya haye iss prakaar kaheen bhee commutative semigroup vaastav mein kaheen bhee commutative baind naheen haye. 
iss prakaar kaheen bhee commutative semigroup mein
tou sootra _ 5 sootra _ 6 key saath aataa haye aur iss prakaar sootra _ 7-pahali visheshta pehchaan.
kissy bhee ardhsamuh mein pahali pehchaan kaa tatparya sootra _ 8 key baad seey astitva haye isaliye sootra _ 9 isaliye astitva (ack baind) haye. taba 
baind sootra _ 10 key baad seey kaheen bhee commutative naheen haye isaliye ack baind mein
kissy bhee ardhsamuh mein pahali pehchaan kaa arth doosara bhee hotaa haye kyonki.
ack aaytaakaar ardhsamuh key vishisht guna ack upa-pattika banaate hain joe ack aaytaakaar pattika hoti haye lekin ack aaytaakaar ardhsamuh mein aise tatva hoe sakate hain joe vishisht naheen hotey hain. ack baind mein doosari pehchaan kaa spasht roop seey pahala tatparya haye lekin isake liye suyogyata kii aavashyakta hoti haye. aise ardhsamuh maujood hain joe doosari pehchaan koo santusht karate hain lekin baind naheen hain aur pahale koo santusht naheen karate hain.
aaytaakaar pattiyon kaa ack poorna vargeekaran haye. manmaane sett diye gaye 'aur' ack setting dwara magma operation koo paribhaashit curr sakata haye
yeh operation sahayogi haye kyonki kissy bhee teen jode key liye, hamaare paas haye
yeye dou magma pahachaanen 
donon milkar saha-prabhavsheelta kaa bhee sanket dete hain. koee bhee magma joe inn dou aaytaakaar pahchaanon aur vishishtata koo santusht kartaa haye, isaliye ack aaytaakaar patti haye. isaliye koee bhee magma joe donon visheshta pahchaanon (chaar alag-alag magma pehchaan) koo santusht kartaa haye, ack baind haye aur isaliye ack aaytaakaar baind haye.
oopar paribhaashit magma operation ack aaytaakaar baind haye kyonki kissy bhee jodee key liye hamaare paas isaliye pratyek tatva edempotent haye aur pahali visheshta pehchaan dusare seey edempotence key saath aatee haye. 
lekin ack magma joe kewal pahali visheshta aur vishishtata key liye pehchaan koo santusht kartaa haye, usey sahayogi honey kii aavashyakta naheen haye, isaliye doosari visheshta kewal ack ardhsamuh mein pahale seey hii aatee haye.
koee bhee aaytaakaar patti uparokt roop mein seey ack key liye samroop haye (yaa tou sootra _ 15 khaali haye, yaa koee tatva sootra _ 16 chunein, aur phir (sootra _ 17) ack samroopata sootra _ 18 koo paribhaashit kartaa haye). baaen-shoonya aur daaen-shoonya baind aaytaakaar baind hain, aur vaastav mein pratyek aaytaakaar baind baaen-shoonya baind aur daaen-shoonya baind key pratyaksh utpaad key liye samroop haye. sabhi aaytaakaar mool crum key baind shoonya baind hotey hain, yaa tou baaen yaa daaen. ack aaytaakaar patti koo vishuddh roop seey aaytaakaar kahaa jaataa haye yadi yeh baaen-shoonya yaa daaen-shoonya patti naheen haye.
spasht bhasha mein, koee bhee kaha sakata haye kii gair-khaali aaytaakaar pattiyon kii shreni sootra _ 19 key barabar haye, jahaan sootra _ 20 gair-khaali seton key saath vastu key roop mein shreni haye aur rupankan key roop mein kaary kartaa haye. iska tatparya yeh haye kii pratyek gair-khaali aaytaakaar baind ack jodee sett seey aaney vaale ack key liye samroop haye, balki yeye sett bhee vishisht roop seey ack canonical samroopata taka nirdhaarit kiye jaate hain, aur baind key beach sabhi samroopata sett key beach karyon key jode seey aatey hain. yadi uparokt parinam mein "sett" khaali haye, tou aaytaakaar patti swatantra haye, aur isake vipareet. yahee kaaran haye kii uparokt parinam kewal khaali aaytaakaar pattiyon aur khaali seton key jode key beach ack samaantaa detaa haye.
aaytaakaar pattiyan bhee-algebra hain, jahan sett parr monad haye, sootra _ 21 vikarn maanchitra sootra _ 22 haye, aur sootra _ 23 haye.
samanya baind.
ack samanya baind ack santoshjanak baind hotaa haye.
humm yeh bhee kaha sakate hain kii ack samanya baind ack santoshjanak baind haye.
yeh vahee sameekaran haye jiska upayog madhya magma koo paribhaashit karane key liye kiya jaataa haye, isaliye ack samanya baind koo madhya baind bhee kahaa jaa sakata haye, aur samanya baind madhya magma key udaaharan hain.
baaen-niyamit baind.
ack baaen-niyamit baind ack santoshjanak baind haye.
yadi humm ack ardhsamuh lete hain aur paribhaashit karate hain kii kyaa, tou humm ack aanshik crum praapt karate hain yadi aur kewal tabhi jaba yeh ardhsamuh ack baaen-niyamit baind haye. iss prakaar baaen-niyamit baind swabhavik roop seey sthiti key adhyayan mein dikhaai dete hain.
sahee-niyamit baind.
ack sahee-niyamit baind ack santoshjanak baind haye.
koee bhee daaen-niyamit baind vipareet utpaad kaa upayog karke ack baaen-niyamit baind bunn jaataa haye. vaastav mein, baind kii pratyek vividhata kaa ack 'vipareet' sanskaran hotaa haye; yeh neeche diye gaye chitra mein pratibimb samroopata koo janm detaa haye.
niyamit baind.
ack niyamit baind ack santoshjanak baind haye.
kismon kii jaali.
jaba aanshik roop seey samavesh dwara adesh diyaa jaataa haye, tou pattiyon kii kismein swabhavik roop seey ack jaali banaati hain, jisamen dou kismon kaa millen unakaa pratichhedan hotaa haye aur dou kismon kaa judaav sabse chhoti kism hoti haye jisamen way donon hotey hain. iss jaali kii poori sanrachana gyaat haye; vishesh roop seey, yeh ganana yogya, poorna aur vitaranatmak haye. niyamit pattiyon kii 13 kismon seey yukt upa-jaali koo chitra mein dikhaaya gayaa haye. iss jaali key teen paramanu (gair-tuchchh nyuntam tatva) baaen-shoonya patton, ardh-patton aur daaen-shoonya patton kii kismein hain.
aakriti mein dikhaae gaye baind kii pratyek vividhata koo kewal ack pehchaan dwara paribhaashit kiya jaataa haye. yeh ack sanyog naheen haih vaastav mein, "pratyek" baind kii vividhata koo ack hii pehchaan dwara paribhaashit kiya jaa sakata haye.

