Rsr Jorl o Ali is Eiri Tolo 68: 326-334 203 IN: 2040-7459; -IN: 2040-7467 Mwll itii Oritio 203 itt: M 04 202 At: Frr 0 203 Plis: Jl 0 203 O F- -ils o -Al-Grss's Groois M K i Ais Fisl Drtt o Mttis COMAT Istitt o Iortio Tolo Aott Pist Astrt: I tis st w itro t otio o -iitio i -AG-roois wi is i t t rlitio o AG-roois W sti srl rortis o itr-rlr -AG ** -roois i trs o -lt rit two-si si itrior rli i- i- ils W ro tt ll -ils oii i itr-rlr -AG ** -roois W lso sow tt t st o -two-si ils o itr-rlr -AG ** -rooi ors siltti strtr Kwors: -AG-srooi -AG ** -rooi -AG ** -rooi itr-rlr sst - ils -AG-rooi INTRODUCTION T rl worl s lot o irt sts wi r ot sll sii Mols or rols i lost ll ils o owl li iri il si ttis sis otr si rtiiil itlli ostrt o tis t ols r r iilt to l t ossil soltios iossil Tror t lssil st tor wi is ris t is ot rorit or s rols o rtit I to's worl toris lo to l wit s rtitis or ist st tor tor o sts tor o sot ils tor o ititioisti sts tor o ro sts T tor o sot sts s litios i irt ils s s t sootss o tios tor ortios rsr Ri itrtio t T si ot o st tor ws irst i Z 965 Z 965 isss t rltiosis tw st tor roilit tor Rosl 97 iitit t ros i st tor Morso t l 2003 isss t litios o st tor i oi tot iit stt is Al-Grss's rooi AG-rooi is t rlitio o siro tor wit wi r o ss i tor o los Nsri 970 T tls o tis o-ssoiti lri strtr wr irst isor Ki Nsri 972 AG-rooi is o-ssoiti lri strtr i w tw rooi otti siro It is itrsti to ot tt AG-rooi wit rit itit os otti ooi Mst Yos 978 T ot o -siro s itro 98 s ollows: A o-t st is ll -siro i or ll A -siro is t rlitio o siro I tis st w rtri - AG ** -roois t rortis o tir -ils rli so rslts A -AG-rooi is t rlitio o AG-rooi Lt o-t sts I tr ists i writt s t is ll -AG-rooi i s tt t ollowi -lt irti lw ols or ll A -AG-rooi lso stisis t -il lw or ll w w w Not tt i -AG-rooi otis lt itit t it os AG-rooi wit lt itit A -AG-rooi is ll - AG ** -rooi i it stisis t ollowi lw or ll A - AG ** -rooi lso stisis t - w ril lw or ll w w PRELIMINARIE T ollowi iitios r ill i R 200 Corrsoi Ator: M K Drtt o Mttis COMAT Istitt o Iortio Tolo Aott Pist 326
Lt -AG-rooi o-t sst A o is ll -AG-srooi i A or ll A or i A A A A sst A o -AG-rooi is ll -lt rit il o i A A A A A is ll -two-si-il o i it is ot -lt il -rit il A sst A o -AG-rooi is ll -rli i-il o i A A A A s -AG-rooi A o -AG-rooi is ll -i-il o i A A A A sst A o -AG-rooi is ll -itrior il o i A A A sst A o -AG-rooi is ll -si-il o i A A A A sst o i st is sri s ritrr tio : [0] wr [ 0] t sl is los itrl o rl rs Now w itro t ollowi iitios Lt ssts o -AGrooi t t -rot 0 { } is i i wr otrwis A sst o -AG-rooi is ll -AG-srooi i or ll A sst o -AG-rooi is ll -lt il o i or ll A sst o -AG-rooi is ll -rit il o i or ll A sst o -AG-rooi is ll -two-si il o i it is ot -lt il -rit il o A sst o -AG-rooi is ll -rli i-il o i A sst o -AG-rooi is ll -itrior il o i or ll A sst o -AG-rooi is ll -itrior il o i -FUZZIFICATION IN -AG-GROUPOID El : Lt { 23 456789} T ollowi ltilitio tl sows tt is AG-rooi lso AG-: 2 3 4 5 6 7 8 9 4 7 3 6 8 2 9 5 2 9 2 5 7 4 8 6 3 3 6 8 3 5 9 2 4 7 4 5 9 2 4 7 6 3 8 5 3 6 8 2 5 9 7 4 6 7 4 8 3 6 9 5 2 7 8 3 6 9 2 5 7 4 8 2 5 9 4 7 3 8 6 9 4 7 6 8 3 5 2 9 Clrl is o-otti o-ssoiti s 23 32 423 423 Lt { } i i 2 2 or ll T it is s to s tt is -AGrooi lso -AG- Not tt is ootti o-ssoiti s 9 9 6 7 8 6 78 El 2: Lt {23 } i i Z Z Z or ll Z wr " " is sl strtio o itrs T Z is -AG-rooi I + + + + + 2 + 2 or ll + + + + 2 + 2 + + 2 + 2 A -AG-srooi o -AGrooi is ll -i-il o i Z Tis l is t Wi sows tt or ll or ll rliio o -AG-rooi i R 200 327 A
328 El 3: Ass tt is AG-rooi wit lt itit lt } { Di i or ll t is -AG-rooi Ts w s tt r AGrooi is -AG-rooi or } { tt is - AG-rooi is t rlitio o AG-rooi Also is -AG ** -rooi s or ll El 4: Lt AG-rooi } { Di i or ll t w ow tt is -AG-rooi Lt L lt il o AG-rooi t L L L Ts L is -lt il o Tis sows tt r -lt il o -AG-rooi is rlitio o lt il i AG-rooi or sitl iilrl ll t -ils r t rlitios o ils B i t rlitio t roo o L Tor r s s i No 200 L : Lt sst o -AGrooi t Tor : Lt -AG-rooi t t ollowi rortis ol i or ll ssts o or ll ssts o Tor 2: Lt -AG ** -rooi t t ollowi rortis ol i i or ll ssts o ii or ll ssts o Proo: i Ass tt is ritrr lt o -AG ** -rooi lt I is ot rssil s rot o two lts i t 0 Lt tr ists i s tt t si 3 w : I is ot rssil s rot o two ltsi t 0 H or ll i ii I lt o is ot rssil s rot o two lts i t st t 0 Ass tt lt tr ists i s tt t si 4 w : B i t rlitio t roo o t ollowi two ls is s s i Morso t l 2003 L 2: Lt sst o -AGrooi t t ollowi rortis ol: is -AG-srooi o i ol i is -lt rit il o i ol i is -two-si il o i ol i
L 3: Lt -AG-srooi o -AG-rooi t is -i-il o i ol i L 4: Lt -rit il -lt il o -AG-rooi t is -si il o Proo: It is s to osr t ollowi L 5: Er -si il o -AGrooi is -AG-srooi o Proo: Lt -si il o t tror H is -AG-srooi o A sst o -AG-rooi is ll -iott i L 6: I -AG-rooi r - iott -si il is -i-il o Proo: Lt -si il o t l 5 is -AG-srooi Now si 2 w A Tis ilis tt H l i-il is -i-il o L 8: I -AG-rooi o si -lt rit il o is - rli i-il o Proo: Ass tt -lt il o Lt lt Now si w Ts iilrl i t s o -rit il L 9: Lt or is -iott -si il o -AG**-rooi t or is -i-il o Proo: Clrl is -AG-srooi Now si l i-il 4 2 w : iilrl w sow tt i-il o is - L 0: T rot o two -lt rit il o -AG ** -rooi is - lt rit il o Proo: Lt two -lt ils o t si 3 w : Lt two -rit ils o t si 2 w : L 7: I -AG-rooi o si - lt rit il is -si il o Proo: It is oios Corollr : I -AG-rooi r - two- si il o is -si il o 329 -FUZZIFICATION IN INTRA-REGULAR - AG ** -GROUPOID A lt o -AG-rooi is ll itr-rlr i tr ists s tt is ll itr-rlr i r lt o is itrrlr
Not tt i itr-rlr -AG-rooi w writ El 5: Lt {2345 } AG-rooi wit t ollowi ltilitio tl: Lt i i or ll t is -AG ** - rooi s or ll Also is itrrlr s It is s to osr tt { } is -two-si il o itr-rlr -AG ** -rooi It is s to osr tt i itr-rlr -AGrooi t ollowi ols L : A sst o itr-rlr - AG-rooi is -rit il i ol i it is -lt il Proo: Ass tt is -rit il o i is itr-rlr -AG-rooi so or tr ist s tt Now lt t si w : s tt Now lt t si 3 w Ts w : { } Corollr 2: Er -two-si il o itr-rlr -AG ** -rooi is -iott Tor 4: I itr-rlr -AG ** -rooi or r -rit il r -lt il o Proo: Ass tt is itr-rlr -AG ** - rooi Lt -rit -lt il o t wi ilis tt tr i is itr-rlr so or ist s tt Now lt t si 3 5 4 w : Tror w : { } Corsl ss tt is -lt il o Now si w : Tor 3: Er -lt il o itrrlr -AG ** -rooi is -iott Proo: Ass tt is -lt il o t lrl i is itrrlr -AG-rooi so or tr ist 330 Corollr 3: I itr-rlr -AG ** -rooi or r -rit il o Tor 5: T st o -two-si ils o itr-rlr -AG** -rooi ors siltti strtr wit itit Proo: Lt l t st o -two-si ils o itr-rlr -AG ** -rooi I t lrl I is los orollr 2 orollr 3 w wr r -two-si ils o
Clrl ow si w t Also si l w : A -two-si il o -AGrooi is si to -strol irril i ol i or -two-si ils o ilis tt or T st o -two-si ils o -AGrooi is ll -totll orr r ilsio i or -two-si ils o itr or A -two-si il o -AGrooi is ll - ri il o i or -two-si o ilis tt or Tor 6: I itr-rlr -AG ** -rooi -two-si il is -strol irril i ol i it is - ri Proo: It ollows ro orollr 3 Tor 7: Er -two-si il o itrrlr -AG ** -rooi is - ri i ol i t st o -two-si ils o is - totll orr r ilsio Proo: It ollows ro orollr 3 Tor 8: For sst o itr-rlr -AG ** -rooi t ollowi sttts r ilt i is -two-si il o ii is -itrior il o Proo: i ii : Lt -two- si il o t oiosl is -itrior il o ii i : Lt -itrior il o i is itr-rlr -AGrooi so or tr ist s tt Now lt ts si 3 2 w : Also si 3 4 2 w : H is -two- si il o Tor 9: A sst o itr-rlr - AG ** -rooi is -two-si il i ol i it is -si il Proo: Lt -two-si il o t oiosl is -si il o Corsl ss tt is -si il o t si orollr 2 4 w : Tror Ts is -rit il o l is -lt il o Tor 0: For sst o itr-rlr -AG ** -rooi t ollowi oitios r ilt i is -i-il o ii is -rli i-il o Proo: i ii : Lt -i-il o t oiosl is -rli i-il o ii i : Lt -rli i-il o i is itr-rlr -AG-rooi so or tr ist s tt Now lt t si 5 4 2 3 w : Tror is i-il o 33
332 Tor : For sst o itr-rlr -AG ** -rooi t ollowi oitios r ilt i is -two- si il o ii is -i-il o Proo: : ii i Lt -two- si il o t oiosl is -i-il o : i ii Lt -i-il o i is itr-rlr -AG-rooi so or tr ist s tt Now lt t si 4 2 3 w : Now si 3 4 w : Tor 2: Lt sst o itr-rlr - AG ** -rooi t t ollowi oitios r ilt i is -i-il o ii Proo: : ii i Lt -i-il o itr-rlr -AG ** -rooi Lt A t tr ists s tt Now lt t si 3 5 4 2 w : wr Tror Now si 3 5 4 w : wr Tror: Ts As w sow tt: Lt wr Tror
333 Now si l 2 w t : i ii Ass tt is sst o itrrlr -AG ** -rooi lt t } { Now si l 2 is -i-il o Tor 3: Lt sst o itr-rlr - AG ** -rooi t t ollowi oitios r ilt i is -itrior il o ii Proo: : ii i Lt -i-il o itr-rlr -AG ** -rooi Lt A t tr ists s tt Now lt t si 3 5 4 w : Tror Now i: H it ollows tt : i ii Ass tt is sst o itrrlr -AG ** -rooi lt t: L 2: Lt sst o itr-rlr -AG ** -rooi t Proo: Lt -lt il o itrrlr -AG ** -rooi lt t tr ists s tt Lt t si 5 4 3 w : Tror: } { } { T rst o t roo ollow ro l Tor 4: For sst o itr-rlr -AG ** -rooi t ollowi oitios r ilt i is -lt il o ii is -rit il o iii is -two-si il o i is -i-il o is -rli i- il o i is -itrior il o ii is -si il o iii Proo: : iii i It ollows ro l 2 : ii iii It is oios : i ii Lt -si il o itr-rlr -AG ** -rooi lt t tr ists s tt Lt t si 3 4 w : ro o:
Now si l 2 w : Now: A { } { } Tis ilis tt tror is -itrior il o i : It ollows ro tors 8 0 i : It ollows ro tor 0 i iii : It ollows ro tor iii ii : It is oios ii i ollow ro l CONCLUION I tis st w itro -ils i - AGroos W sow tt -AG-roois stis ll t lws tt is lt irti lw il ril lws Moror w itro itr-rlr AG-roois rtri ils I or tr w will os o so w rtritios o AGroois REFERENCE Ki MA M Nsri 972 O lost siros Ali Bll Mt 2: -7 Morso JN D Mli N Kroi 2003 F iros rir-vrl Brli Gr Mst Q M Ys 978 O LA-siros Ali Bll Mt 8: 65-70 Nsri M 970 o stis i lost siros los PD Tsis Alir Msli Uirsit Alir Ii Rosl A 97 F ros J Mt Al Al 35: 52-57 MK 98 O -siros Proi o Itrtiol osi o Alr its Alitios Dr Plitio Nw Yor : 30-308 T I R 200 O M-ssts i - AG-roois Pro Pist A i 47: 33-39 Z LA 965 F sts Ior Cotrol 8: 338-353 334