kriya beejganit

beejganitiya tark mein, ack kriya beejganit ack beejganitiya sanrachana haye joe ack avashisht ardh-jaali aur ack clean beejganit donon haye. yeh poorv key liye baad vaale key starr yaa reflexive transitive closure operation koo jodata haye, jabaki poorv key liye baaen aur daaen avashesh yaa nihitarth sanchaalan koo jodata haye. gatishil tark aur karyakramon key anya model tark key vipareet, jinake liye karyakram aur prastaav dou alag-alag prakaar key hotey hain, kriya beejganit donon koo ack hii prakaar mein jodata haye. ise taaron key saath antargyaanvaadi tark key ack prakaar key roop mein aur ack gair-commutative sanyojan key saath socha jaa sakata haye jisaki pehchaan sheersh tatva honey kii aavashyakta naheen haye. clean algebra key vipareet, kriya algebra ack kism banaate hain, joe aage bhee parimit roop seey swayamsiddh haye, mahatvapurna swayamsiddh "a" "a" "(" a "" "a" ") * ≤" a "haye. clean algebra (niyamit abhivyakti sameekaran) key samanuyayi siddhant key model key vipareet, kriya algebra kaa taaraa sanchaalan samikaranon key pratyek model mein prativartak sankramansheel samaapan haye. action algebra koo europeeya karyashala jelia '90 mein waghan praat dwara pesh kiya gayaa thaa. 
paribhasha.
ack kriya beejganit ("a", "Â", 0 "," 1 ",", ",", *) ack beejganitiya sanrachana haye joe ("a", ",", ", 1", ",") ward aur dialworth key arth mein ack avashisht ardh-jaali banaata haye, jabaki ("a", ", 0", ", 1", *) dextor kozen key arth mein ack clean beejganit banaata haye. arthaat, yeh donon vargon key algebra key sanyukt siddhant kaa koee bhee model haye. abb clean algebra koo quasiquations key saath swayamsiddh kiya jaataa haye, yaani dou yaa dou seey adhik samikaranon key beach nihitarth, jahan seey kriya algebra hotey hain jaba iss tarah seey seedhe swayamsiddh hotey hain. haalaanki, kriya algebra kaa laabh yeh haye kii unake paas ack samatulya swayamsiddheekaran bhee haye joe vishuddh roop seey samatulya haye. kriya algebra kii bhasha ack praakritik tareeke seey kriya jaali taka faili hui haye, arthaat ack meat operation koo shamil karke.
nimnalikhit mein humm asamaanta koo "a" ≤ "b" sameekaran "a" ′ b "=" b "key sankshipt roop key roop mein likhate hain. yeh hamein asamaantaon kaa upayog karke siddhant koo swayamsiddh karane kii anumati detaa haye, phir bhee jaba asamaantaon koo samaantaon taka vistaarit kiya jaataa haye tou unakaa vishuddh roop seey samaanugat swasiddhikaran hotaa haye.
sameekaran swayamsiddh kriya beejganit way hain joe ack avashisht ardh-jaali key liye hain, saath mein taaron key liye nimnalikhit sameekaran hain.
pahale sameekaran koo teen samikaranon mein vibhajit kiya jaa sakata haye, 1 ≤ "a" *, a "*" a "* ≤" a "*, aur" a "≤" a "*. a" koo paribhaashit karate hue jaba 1 ≤ "a" aur "a" "a" ≤ "a" koo dwiaadhaari sambandhon seey amurtata dwara, una samikaranon mein seey pahale dou "a" koo prativart aur sankramansheel honey key liye majboor karate hain jabaki teesare bull "a" "* koo" a "seey adhik yaa barabar honey key liye. agala swayamsiddh daavaa kartaa haye kii taaraa ack swar haye. antim swayamsiddh koo "a" key roop mein likhaa jaa sakata haye ("a" "a") * ≤ "a", ack roop joe preran key roop mein apani bhoomika koo adhik spasht banaata haye. yeye dou swayamsiddh ack avashisht ardh-jaali bull "a" key liye swayamsiddh key sanyojan mein "a" seey adhik yaa barabar tatvon key ardh-jaali kaa sabse cumm prativartak sankramansheel tatva honey key liye. ise "a" key prativartak sankramansheel samaapan kii paribhasha key roop mein lete hue, hamaare paas taba haye kii kissy bhee kriya beejganit key pratyek tatva "a" key liye, a "*" a "kaa prativartak sankramansheel samaapan haye.
guna.
kriya algebra key nihitarth-mukt tukade key samanuyami siddhant, una samikaranon mein joe β yaa β naheen hain, koo clean algebra key samanuyami siddhant key saath mell khaate hue dikhaaya jaa sakata haye, jise niyamit abhivyakti sameekaran key roop mein bhee jaanaa jaataa haye. usa arth mein uparokt swayamsiddh niyam niyamit abhivyaktiyon key ack seemit swayamsiddheekaran kaa gathan karate hain. redco ney 1967 mein dikhaaya kii niyamit abhivyakti samikaranon mein koee seemit swayamsiddheekaran naheen thaa, jisake liye johnn horton conway ney 1971 mein ack chhota pramaan diyaa.
 arto sailoma ney iss siddhant koo swayamsiddh karane key liye ack sameekaran diyaa, jise baad mein quasiquations kaa upayog karke ack seemit swayamsiddheekaran key roop mein sudhaar kiya gayaa, yaa samikaranon key beach nihitarth, mahatvapurna quasiquations induction key hainh yadi "x" "a" "≤" x "tou" x "" a "" * ≤ "x", aur yadi "a" "x" ≤ "x" tou "a" "*" x "≤" x ". kozen ney ack clean beejganit koo iss parimit swayamsiddheekaran key kissy bhee model key roop mein paribhaashit kiya.
conway ney dikhaaya kii niyamit abhivyaktiyon key samanuyayi siddhant ney una models koo sweekaar kiya jinamen "a" * "a" kaa prativartak sankramansheel samaapan naheen thaa, ack chaar-tatva model 0 ≤1 ≤ "a" ≤ "a" * dekar jisamen "a" "a" = "a". conway key model mein, "aeye" prativartak aur sankramansheel haye, jahan seey iska prativartak sankramansheel samaapan "aeye" honah chaahiye. haalaanki niyamit abhivyaktiyaan ise lagu naheen karati hain, jisase "a" "a" seey sakhti seey badaa hoe jaataa haye. iss tarah kaa visangat vyavhaar ack kriya beejganit mein sambhav naheen haye, joe "a" * koo sabse cumm sankramansheel prativartak tatva honey key liye majboor kartaa haye.
udaaharan.
kissy bhee highting beejganit (aur isaliye koee bhee booliyan beejganit) koo ack kriya beejganit banaaya jaataa haye jise ′ aur "a" * = 1 liya jaataa haye. yeh taaron key liye aavashyak aur paryaapt haye kyonki highting beejganit kaa sheersh tatva 1 iska ekmatra prativartak tatva haye, aur beejganit key pratyek tatva key saath-saath adhik yaa barabar bhee haye.
ack varnamala parr sabhi aupchaarik bhaashaon (seemit strings key sett) kaa sett 2σ * ack kriya beejganit banaata haye jisamen 0 khaali sett key roop mein, 1 = {ε}, Â sangh key roop mein, ′ sangh key roop mein, ′l "′" m "sabhi strings" x "key sett key roop mein jaise kii" xm "′l" (aur "m" key liye dwigunit "l"), aur "l" * "l" (clean closure) mein strings key sabhi strings key sett key roop mein.
ack samooh "x" parr sabhi dwiaadhaari sambandhon kaa sett 2 "x" 2 ack kriya beejganit banaata haye jisamen 0 khaali sambandh key roop mein, 1 pehchaan sambandh yaa samaantaa key roop mein, sangh key roop mein, sambandh sanrachana key roop mein, "r" s "sabhi jode (" x, y ") seey yukt sambandh key roop mein jaise kii" x "mein sabhi" z "key liye," ysz "kaa tatparya" xrz "(aur" s "" "r" "key liye dwigunit roop seey), aur" r * "koo" r "" key prativartak sankraman bund key roop mein paribhaashit kiya gayaa haye, jise purnank "n" ≤ 0 key liye sabhi sambandhon "r" "n" parr sangh key roop mein paribhaashit kiya gayaa haye.
dou poorvavarti udaaharan shakti samooh hain, joe sangh, pratichhedan aur poorak key samanya sett saiddhaantik sanchaalan key tahat booliyan algebra hain. yeh unhein booliyan action algebra kahane kaa auchitya haye. sambandhparak udaaharan ack sambandh beejganit kaa gathan kartaa haye joe prativartak sankramansheel samaapan key sanchaalan seey susajjit hotaa haye. dhyaan dein kii pratyek booliyan beejganit ack uchch beejganit haye aur isaliye pahale udaaharan kaa ack udaaharan honey key kaaran ack kriya beejganit haye.

