A NEW INTERPRETATION OF INTERVAL-VALUED FUZZY INTERIOR IDEALS OF ORDERED SEMIGROUPS

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1 ScInLhore),7),9-37,4 ISSN 3-536; CODEN: SINTE 8 9 A NEW INTERPRETATION O INTERVAL-VALUED UZZY INTERIOR IDEALS O ORDERED SEMIGROUPS Hdy Ullh Khn, b, Nor Hnz Srmn, Asghr Khn c nd z Muhmmd Khn d Deprmen of Mhemcl Scences, culy of Scence, Unvers Teknolog Mlys 83 UTM Johor Bhru, Johor, Mlys b Deprmen of Mhemcs, Unversy of Mlknd, Chkdr, Dr L), Khyber Pkhoonkhw, Pksn c Deprmen of Mhemcs, Abdul Wl Khn Unversy Mrdn, Mrdn, Khyber Pkhoonkhw, Pksn d Deprmen of Mhemcs nd Sscs, Unversy of Sw, Sw, Khyber Pkhoonkhw, Pksn ABSTRACT The concep of nervl-vlued fuzzy se s one of he mos mporn nd useful generlson of Zdeh's fuzzy ses whch s successfully ppled by engneers nd scenss n he feld of Robocs, Conrol Theory nd Compuer Engneerng In hs pper, we nroduce new nd useful generlsons of nervl-vlued neror dels nd nervl-vlued fuzzy lef rgh) dels clled nervl-vlued, q ) -fuzzy neror dels, nervl-vlued, q ) -fuzzy lef rgh) dels, nervl-vlued, q ) -fuzzy neror dels nd nervl-vlued, q ) -fuzzy lef rgh) dels of ordered semgroups These newly defned conceps re suppored by suble exmples, nd severl fundmenl resuls re nvesged I s shown h n regulr nd semsmple ordered semgroups, boh he conceps of nervl-vlued, q ) -fuzzy neror dels nd nervl-vlued, q ) -fuzzy dels concde urher, he relon beween nervl-vlued fuzzy neror dels nd nervl-vlued fuzzy del of ype, q ) s provded Keywords Inervl-vlued fuzzy neror dels; Inervl-vlued, q ) -fuzzy neror dels; nervl-vlued, q ) -fuzzy lef rgh) dels;, q ) -fuzzy neror dels;, q ) -fuzzy lef rgh) dels Mhemcs Subjec Clssfcon 3E7; 8A7; N5 INTRODUCTION Represenon of knowledge n decson mkng process by mens of nervls s more precse rher hn by pons An nervl-vlued fuzzy subse [,] s nurl exenson of fuzzy se heory [3] nd more pplcble n rel world problems nvolvng uncernes or he frs me Bsws [4] used nervl-vlued fuzzy ses n lgebrc srucure nd gve he noon of nervl-vlued fuzzy subgroups In ddon, Shbr nd Khn [5] nroduced nervl-vlued fuzzy lef rgh, wo-sded, neror, b-) dels genered by n nervl-vlued fuzzy subse of ordered semgroups urher, Khn e l [6] ned new sor of nervlvlued fuzzy b-dels known s nervl-vlued, q) - fuzzy b-dels of ordered semgroups In ddon, Khn e l, [7,8] defned nervl-vlued, q ) -fuzzy k generlzed b-del nd nervl-vlued fuzzy dels of ype, q ) of ordered semgroups nd chrcerzed ordered k semgroups n erms of hese noons Moreover, Yn nd Zhn [9] nroduced, q ) -fuzzy mplcve, posve mplcve nd fnsc) flers nd, q ) - fuzzy mplcve, posve mplcve nd fnsc) flers of BL-lgebrs nd gve some neresng resuls urher, M e l, [] ned he concep of, q ) -fuzzy Jn-eb dels of BL-lgebrs nd dscussed severl mporn resuls In ddon, Khn e l, [] gve more generl forms of, q) -fuzzy neror dels nd, q) - fuzzy neror dels of ordered semgroups nd defned he conceps of fuzzy neror dels nd fuzzy lef resp rgh) dels of ypes, q ) nd, q ) nd chrcersed ordered semgroup by he properes of hese new noons Moreover, Khn e l, [] comprehensvely dscussed, q ) -fuzzy generlzed b-dels of ordered semgroups nd provded severl clssfcons of ordered semgroups n erms of, q ) -fuzzy generlzed b-dels Moreover, Aks nd Çgmn [3] nroduced he conceps of fuzzy subrng, fuzzy del nd fuzzy rng homomorphsm In hs pper, we exend he work of [,] nd nroduced nervl-vlued fuzzy neror dels resp dels) of ypes, q ) nd, q ), where, D[,] such h Moreover, condon s provded h when boh nervl-vlued, q ) -fuzzy neror dels nd nervl-vlued, q ) -fuzzy dels wll concde

2 3 ISSN 3-536; CODEN: SINTE 8 ScInLhore),7),9-37,4 PRELIMINARIES In hs secon, we revew some fundmenl conceps h re necessry for hs pper Throughou hs pper S wll denoe n ordered semgroup unless oherwse sed or A S, we denoe A]: { S h for some h A} If A {}, hen we wre ] nsed of {}] or A, B S, we denoe, AB : { b A, b B} A nonempy subse A of S s clled subsemgroup of S f A A A non-epmy subse A of S s clled n neror del of S f ) A A, ) SAS A nd ) f b S nd b A hen b A A non-empy subse A of S s clled lef resp rgh) del of S f ) SA A resp AS A) nd ) If b S nd b A hen b A A non-empy subse A of S s n del f s boh lef nd rgh del of S Obvously, every del of n ordered semgroup S s n neror del of S Now we recll some nervl-vlued fuzzy logc conceps INTERVAL-VALUED UZZY SET[] or non-empy se X mppng : X D[,] s clled n nervl-vlued fuzzy se of X, where D [,] denoes he fmly of ll closed subnervls of [,], nd - - [ x), x)] for ll x X, where, re fuzzy ses n X such h - x) x) for ll x X nd [ - x), x)] elemen x o he se s he grde of membershp of n INTERVAL-VALUED ORDERED UZZY POINT An nervl-vlued fuzzy subse of n ordered semgroup S of he form: D,], f y x], y) : [,], f y x], s clled n nervl-vlued fuzzy pon wh suppor x nd vlue nd s denoed by x If x) resp x) ) hen we sy h x belongs o resp x qus-concden wh) fuzzy se, wren s x resp x q ) nd wre x q f x or x q 3 LEVEL SET O AN INTERVAL-VALUED UZZY SET[] Le be n nervl-vlued fuzzy subse of X Then, he crsp se U ; ) { x X x) } for every s clled level subse of 4 INTERVAL-VALUED UZZY INTERIOR Jn-eb IDEAL[6] An nervl-vlued fuzzy subse of n ordered semgroup S s clled n nervl-vlued fuzzy neror del of S f he followng wo condons hold for ll x, y, z S : I ) xyz ) y), I ) x y x) y), I ) xy) r mn { x), y) } 3 5 INTERVAL-VALUED UZZY LET RIGHT) IDEAL[6] An nervl-vlued fuzzy subse of n ordered semgroup S s clled n nervl-vlued fuzzy lef resp rgh) del of S f he followng condons hold for ll x, y S : I 4 ) x y x) y), I ) xy) y) resp xy) x)) 5 s clled n nervl-vlued fuzzy del of S f s boh nervl-vlued fuzzy lef nd nervl-vlued fuzzy rgh del of S 3 INTERVAL-VALUED, q ) -UZZY INTERIOR IDEALS In hs secon we gve noher useful generlson of nervl-vlued fuzzy neror dels nd nervl-vlued fuzzy lef rgh) dels clled nervl-vlued, q ) - fuzzy neror del nd nervl-vlued, q ) -fuzzy lef rgh) dels In ddon, we provde severl chrcersons of ordered semgroup n erms of hese new conceps Throughou n hs pper, le, D[,] be such h or n nervl-vlued ordered fuzzy pon x nd n nervl-vlued fuzzy subse of S, we sy h x f x) x q f x) x q f x or x q x q f x nd x q x f x does no hold for {,q, q, q } 3 DEINITION An nervl-vlued fuzzy subse of S s clled n nervl-vlued, q ) -fuzzy neror del of S f: ) x y) y q x ), b) x, y s xy) q r mn{, }, s c) xy) q, for ll x,, y S nd, s D,]

3 ScInLhore),7),9-37,4 ISSN 3-536; CODEN: SINTE EXAMPLE Consder he ordered semgroup S {,,,3 } wh he followng mulplcon ble nd order relon: Tble: 3 {,),,),,), 3,3),,)} Defne n nervl-vlued fuzzy subse : S D[,] s follows: [8,9], f x =, [4,5], f x, x) = [7,8], f x =, [,], f x = 3 Then s n nervl vlued, q ) -fuzzy neror del of S 33 THEOREM 3 [,] [,] [6, 7] or n nervl-vlued fuzzy subse of S, he followng condons re equvlen; ) s n nervl-vlued, q ) -fuzzy neror del of S ) The followng condons hold for ll x,, y S : ) x y) r mx { x), } r mn { y), }), ) r mx { xy), } r mn { x), y), }, 3) r mx { xy ), } r mn { ), } Proof Le be n nervl-vlued, q ) -fuzzy neror del of S If here exs, b S such h x y nd r mx { ), } r mn { b), }, hen r mx { ), } r mn { b), } for some D,] I follows h ) b bu ) nd ), follows h nd q where, b, conrdcon nd hence ) s ccepble for ll x, y S If r mx { b), } r mn { ), b), } for some, b S, hen here exss s D,] such h r mx { b), } s r mn { ), b), } Ths mples, b bu b ) nd b ) q, gn s s s s conrdcon nd herefore we ccep h ) s vld for ll x, y S Suppose h here exs x,, y S such h Jn-eb r mx { xy ), } r mn { ), } Then r mx { xy ), } r mn { ), } for some s D,], follows h bu xy ) s s nd xy ) q, conrdcng Condon c) of Defnon 3) s Hence 3) s rue for ll x,, y S Conversely, we ssume h Condons ), ) nd 3) re ssfed for ll x,, y S Le here exs x, y S wh x y nd D,] such h y bu x nd x q Then ) y, x) nd x), showng h x) Hence r mx { x), } r mn {, } r mn { y), }, hs conrdcs ) Hence ) s vld If, s b for some, b S nd s, D,] such h b ) nd r mn{ s, } b ) q, hen r mn{ s, } ) s, ) b nd b) r mn{, s}, b) r mn{, s} Ths shows h b) nd herefore r mx { xy), } r mn{, s} r mn { x), y), }, whch conrdcs ) Hence b) s vld If here exs x,, y S nd D,] such h s bu xy ) nd xy ) q, hen ), xy) nd xy ), showng h xy ) Hence r mx { xy ), } r mn {, } r mn { ), }, conrdcs 3) nd herefore c) s vld Consequenly, s n nervl-vlued, q ) -fuzzy neror del of S 34 THEOREM or n nervl-vlued, q ) -fuzzy neror del of S, he se { S ) x x } s n neror del of S f Proof Le be n nervl-vlued, q ) -fuzzy neror del of S nd, b, hen ) nd b) Hence by b) of Defnon 3) b ) q r mn{ ), b)} e, b) r mn { ), b)} or b) r mn { ), b)} Ths shows h b

4 3 ISSN 3-536; CODEN: SINTE 8 ScInLhore),7),9-37,4 If here exs x,, y S such h, hen ) nd by c) of Defnon 3) xy ) q ) e, ) ) xy or xy ) ), showng h xy Hence s n neror del of S Drecly by Theorem 34) we hve he followng wo corollres 35 COROLLARY Le be, q ) -fuzzy neror del of S nd, hen he se { xs x) } s n neror del of S 36 COROLLARY The se { xs x) } s n neror del of S, q) -fuzzy neror del of S 37 PROPOSITION If { } s collecon of nervl-vlued I, q ) -fuzzy neror del of S, hen s n nervl-vlued, q ) -fuzzy neror del of S Proof Le I be n nervl-vlued, q ) -fuzzy neror del of S for ll I nd, b S wh b Consder r mx{ ), } {r mx{ ), }} I I {r mn{ b), }} I By Theorem 33)) r mn{ b), } I r mn{ ) b), } Nex we ke, b S nd consder r mx{ b), } {r mx{ b), }} I I {r mn{ ), b), }} I I By THeorem 33)) r mn{ ), b), } I I r mn{ ) ), ) b), } nlly, f x,, y S, hen r mx{ xy ), } {r mx{ xy), }} I I {r mn{ ), }} I By THeorem 333)) r mn{ ), } I r mn{ ) ), } I Consequenly by Theorem 33 s n nervl-vlued I I I Jn-eb, q ) -fuzzy neror del of S Now s nurl o nvesge h s n nervlvlued, q ) -fuzzy neror del of S or no for ny non-empy collecon { } of nervl-vlued I, q ) -fuzzy neror dels of S In hs regrd we consruced he followng exmple o show h s no n nervl-vlued, q ) -fuzzy neror del n generl 38 EXAMPLE We consder he ordered semgroup S {, b, c, d} defned by he followng mulplcon ble nd order relons Tble: b c d b c d d I,, b, b, c, c, d, d,, } :{ d Defne wo nervl-vlued fuzzy subses nd s [4,5], f x {, b}, x) = [,], f x { c, d}, nd [4,5], f x {, c}, x) = [,], f x { b, d} Then nd re nervl-vlued [,3], [,3] q[3, 4] ) -fuzzy neror dels of S Bu s no n nervl-vlued, q ) - [3,4] Hence I [,3] [,3] [3, 4] fuzzy neror dels of S Snce, r mx { ) bc), [,3]} r mx { ) ), d [,3]} r mx{ d) [,], d) [,], r mx [,3] [,3] nd r mn { ) b), ) c), [3,4]} r mx{ b) [4,5], b) [,]}, r mn r mx{ c) [,], c) [4,5]}, [3,4] r mn {[4,5],[4,5],[3,4]}

5 ScInLhore),7),9-37,4 ISSN 3-536; CODEN: SINTE 8 33 r mx { ) bc), [,3]} [,3] [3,4] r mn { ) b), ) c), [3,4]} 39 DEINITION Le be n nervl-vlued fuzzy subse of S Then s clled n nervl-vlued, ) -fuzzy neror del of S f for ll x,, y S nd s, D,], he followng hold: d) x y) y x ), e) x, ) s y xy r mn{, s, } f ) xy) 3 THEOREM Every nervl-vlued fuzzy neror del of n ordered semgroup S s n nervl-vlued, ) -fuzzy neror del of S Proof Le be n nervl-vlued fuzzy neror del of S nd, b S wh b such h b Then b) nd by Defnon 4 I ) ) b), follows h If here exs, b S nd s, D,] such h s nd b, hen ) s nd b) By Defnon 4 I 3 ) b) r mn { ), b)} r mn { s, }, n whch follows h b ) r mn{ s, } nlly, f here exs x,, y S nd D,] such h, hen ) nd by Defnon 4 I ) xy ) ) e, xy ) Hence s n nervl-vlued, ) -fuzzy neror del of S 3 COROLLARY Every nervl-vlued, ) -fuzzy neror del of S s n nervl-vlued, q ) -fuzzy neror del of S rom Theorem 3) nd Corollry 3) we hve he followng corollry 3 COROLLARY Ech nervl-vlued fuzzy neror del of S s n nervlvlued, q ) -fuzzy neror del of S To lnk nervl-vlued fuzzy neror del nd nervlvlued fuzzy del of ype, q ), frs we defne nervl-vlued, q ) -fuzzy lef resp rgh) del n he followng lnes 33 DEINITION An nervl-vlued fuzzy subse of S s clled n Jn-eb nervl-vlued, q ) -fuzzy lef resp rgh) del of S f he followng condons hold: g) x y) y q x ), h) y xy) q resp yx) q ), for ll x, y S nd D,] An nervl-vlued fuzzy subse of S s clled n nervl-vlued, q ) -fuzzy del of S f s boh nervl-vlued, q ) -fuzzy lef nd rgh del of S 34 THEOREM Le be n nervl-vlued fuzzy subse of S Then s n nervl-vlued, q ) -fuzzy lef resp rgh) del of S f nd only f he followng condons hold for ll x, y S : ) x y) r mx { x), } r mn { y), }), ) r mx { xy), } r mn { y), }resp r mn { x), }) Proof I s srghforwrd nd omed 35 Theorem Every nervl-vlued, q ) -fuzzy del of S s n nervl-vlued, q ) -fuzzy neror del of S Proof Le be n nervl-vlued, q ) -fuzzy del of S If, b S, hen by Theorem 34 ) r mx { b), } r mn { b), } r mn { b), r mn { ), }, } r mn { ), b), } Therefore r mx { xy), } r mn { x), y), } for ll x, y S If x,, y S such h r mx { xy ), } r mn { ), }, hen r mx { xy ), } r mn { ), } for some D,], showng h By hypohess s n nervl-vlued, q ) -fuzzy del, herefore x y)) q, bu here we observe h xy ) nd xy ) I follows h xy ) nd xy ) q, conrdcon nd hence r mx { xy ), } r mn { ), } for ll x,, y S Consequenly, s n nervl-vlued, q ) -fuzzy neror del of S The followng exmple shows h he converse of Theorem 35 s no rue n generl

6 34 ISSN 3-536; CODEN: SINTE 8 ScInLhore),7),9-37,4 36 EXAMPLE Consder he ordered semgroup S {,,,3 } nd he nervl-vlued fuzzy subse : S [,] s defned n Exmple 3) Then s n nervl-vlued [,], [,] q[6,7 ]) -fuzzy neror del of S Snce bc ) ) [8,9], hen r mx { bc) [8,9], [,]} [8, 9] r mn { b), [6, 7]} If b, hen b ) ) [8, 9], nd herefore r mx { b) [8,9], [,]} [8, 9] r mn { b), b), [6, 7]} On he oher hnd f b, hen b ) ) [4, 5] nd so r mx { b) [4,5], [,]} [4, 5] [,] r mn { b), b), [6, 7]} If b, hen b ) ) [7, 8] nd hus r mx { b) [7,8], [,]} [7, 8] [,] r mn { b), b), [6, 7]} Lsly, for, we cn see h r mx { ) [8,9], [,]} [8, 9] r mn { ) [4,5], [6, 7]} The bove dscusson shows h s n nervl-vlued, q ) -fuzzy neror del of S [,] [,] [6,7 ] However, s no n nervl-vlued [,], [,] q[6,7 ]) -fuzzy del of S, snce f nd b 3, hen b ) ) [4, 5] nd hus r mx { )3)) [4,5], [,]} [4, 5] [6, 7] r mn { ) [7,8], [6, 7]} I follows h s no n nervl-vlued, q ) -fuzzy del of S [,] [,] [6,7 ] In he followng resul we show h every nervl-vlued, q ) -fuzzy neror del of regulr ordered semgroup s n nervl-vlued, q ) -fuzzy del Jn-eb 37 THEOREM If s n nervl-vlued, q ) -fuzzy neror del of regulr ordered semgroup S, hen s n nervlvlued, q ) -fuzzy del of S Proof If, b S, hen here exss x S such h x Snce s n nervl-vlued, q ) -fuzzy neror del of S, herefore r mx { b), } r mn { x) b), } r mn { x) b), } r mn { ), } Smlrly, we cn prove h r mx { b), } r mn { b), } I follows h s n nervl-vlued, q ) -fuzzy del of S 38 PROPOSITION Every nervl-vlued, q ) -fuzzy neror del of semsmple ordered semgroup S s n nervl-vlued, q ) -fuzzy del of S Proof Le be n nervl-vlued, q ) -fuzzy neror del of S If, b S, hen here exss x, y, z S such h xyz Therefore r mx { b), } r mn { xyz ) b), } r mn { xy ) zb), } r mn { ), } Smlrly we cn prove h r mx { b), } r mn { b), } for ll, b S I follows h s n nervl-vlued, q ) -fuzzy del of S rom Theorem 35), Theorem 37) nd Proposon 38) we hve he followng corollry 39 COROLLARY Inervl-vlued, q ) -fuzzy del nd nervlvlued, q ) -fuzzy neror del concde n cse of regulr ordered semgroup nd semsmple ordered semgroup 4 INTERVAL-VALUED, q ) -UZZY INTERIOR IDEALS In hs secon, we nroduce nervl-vlued, q ) - fuzzy neror dels nd nervl-vlued, q ) -fuzzy lef rgh) dels of ordered semgroup nd chrcerse ordered semgroups by he properes of hese newly defned nervl-vlued fuzzy dels

7 ScInLhore),7),9-37,4 ISSN 3-536; CODEN: SINTE DEINITION An nervl-vlued fuzzy subse of S s clled nervlvlued, q ) -fuzzy neror del of S f for ll x,, y S nd D,] he followng condons re ssfed: ) x y) x q y ), j) xy ) x q or y q, k) xy ) q 4 EXAMPLE Consder he ordered semgroup S {,,,3 } of Exmple 3) nd defne n nervl-vlued fuzzy subse : S [,] s follows: [,3], [,], x) = [3,4], [5,6], f x =, f x, f x =, f x = 3 Then s n nervl-vlued [,], [,] q ) - fuzzy neror-del of S 43 THEOREM [67 ] A fuzzy subse of S s n nervl-vlued, q ) - fuzzy neror del of S f nd only f he followng condons hold for ll x,, y S : l) x y) r mx { x), } y) ), m) r mx { xy), } r mn { x), y) }, n) r mx { xy ), } ) Proof Le be n nervl-vlued, q ) -fuzzy neror del of S nd, b S wh b such h r mx { ), } b) Then for some D,] we hve r mx { ), } b), follows h bu b nd b q, conrdcng condon ) of Defnon 4) Hence Condon l) s vld for ll x, y S wh x y If here exs, b S such h r mx { b), } s r mn { ), b)} for some s D,], hen b ) By condon j) of s Defnon 4) q or b q h s s s ) s or ) s or b) s or b) s, conrdcon Hence r mx { xy), } r mn { x), y)} for ll x, y S Le r mx { xb), } x) for some, b, x S Then here exss D,] such h Jn-eb r mx { xb), } x) Ths mples xb ) bu x nd x q, conrdcon Hence r mx { xy ), } ) for ll x,, y S Conversely, ssume h ll he hree condons l), m) nd n) re ssfed by for ll x,, y S If, b S wh b such h, hen ) nd by l) b) r mx { ), } r mx {, }, f,, f, n whch follows h b q If xy ) for x, y S, hen xy) nd by m) r mn { x), y)} r mx { xy), } r mx {, }, f,, f I follows h x q or y q nlly, le x,, y S nd xy ) Then xy ) nd by n) ) r mx { xy), } r mx {, }, f,, f Ths shows h q Hence s n nervlvlued, q ) -fuzzy neror del of S 44 THEOREM Le us defne n nervl-vlued fuzzy subse of S by, f x A, x), f x A where A s non-empy subse of S If s n nervlvlued, q ) -fuzzy neror del of S, hen A s n neror-del of S Proof Le be n nervl-vlued, q ) -fuzzy neror del of S If, b A, hen ) b) By m) of Theorem 43), r mx { b), } r mn { ), b)} r mn {, } Ths shows b) nd hence b A

8 36 ISSN 3-536; CODEN: SINTE 8 ScInLhore),7),9-37,4 If x,, y S such h A, hen ) nd by n) of Theorem 43) r mx { xy ), } ) I follows h xy ), herefore xy A nlly, for, b S f b A, hen b) nd by l) of Theorem 43), r mx { x), } y) Ths mples h ) nd hus A Consequenly A s n neror del of S 45 DEINITION An nervl-vlued fuzzy subse of S s clled nervlvlued, q ) -fuzzy lef resp rgh) del of S, f he followng hold for ll x, y S, D,] : o) x y) x q y ), p) xy ) x q resp y q ) 46 THEOREM The followng re equvlen for ny nervl-vlued fuzzy subse of S ) s n nervl-vlued, q ) -fuzzy lef resp rgh) del of S ) or ll x, y S, ) x y) r mx { x), } y) ), ) r mx { xy), } y) resp x) ) Proof ) ) If, b S wh b such h r mx { x), } y) for some D,], hen nd b q Ths conrdcs Condon o) of Defnon 45) nd hence we ccep h ) s vld for ll x, y S wh x y Nex, suppose h r mx { b), } b) for some, b S Then here exss D,] such h r mx { b), } b), n whch follows h b ) bu b q, conrdcon nd hence ) s vld for ll x, y S ) ) If, b S wh b nd, hen ) nd by ), b) r mx { ), } r mx {, }, f,, f Jn-eb Ths mples, b q nlly, f b ) for some, b S, hen b) nd by ), b) r mx { b), } r mx {, }, f,, f I follows h b q Consequenly, s n nervl-vlued, q ) -fuzzy lef del of S Smlrly, we cn prove h s n nervl-vlued, q ) -fuzzy rgh del of S CONCLUDING REMARKS The de of usng nervls nsed of sngle numbers ply n essenl pr n he conemporry mhemcs nd severl oher ppled felds of scences lke sysem conrol heory, robocs, compuer engneerng nd uom heory The concep of nervl-vlued fuzzy se gned he enons of reserchers round he world They nvesged severl chrcerson of nervl-vlued fuzzy ses nd successfully ppled n foremenoned felds whch cn be seen n erms of reserch rcles n hghly repued journls nd well known conferences In hs regrd, we deermned new generlzon of nervl-vlued fuzzy neror dels nd nervl-vlued fuzzy lef rgh) dels by nroducng nervl-vlued, q ) -fuzzy neror dels, nervl-vlued, q ) -fuzzy lef rgh) dels, nervl-vlued, q ) -fuzzy neror dels nd nervl-vlued, q ) -fuzzy lef rgh) dels of ordered semgroups urher, exmples re lso consruced for he suppor of hese new conceps In ddon, severl clsses of ordered semgroups such s regulr ordered semgroups nd semsmple ordered semgroups re lso chrcersed by he properes of hese new noons Lsly, he lnk beween nervl-vlued fuzzy neror dels nd nervl-vlued fuzzy neror dels of ype, q ) s consruced These new nvesgons wll fll he gp presen n hose ppled felds whch re usng nervlvlued fuzzy ses UTURE WORK These new des presened n hs pper cn lso be ppled n oher lgebrc srucures lke, Rng heory, Semgroups nd Hemrngs ACKNOWLEDGMENTS The frs uhor cknowledges Unvers Teknolog Mlys for provdng prl suppor hrough Inernonl Docorl ellowshp nd Unversy of Mlknd for grnng sudy leve

9 ScInLhore),7),9-37,4 ISSN 3-536; CODEN: SINTE 8 37 REERENCES Zdeh, L A "The concep of lngusc vrble nd s pplcon o pproxme resonng-i", Informon scences, 83), ) Grn Gunness, I "uzzy Membershp Mpped ono Inervls nd Mny Vlued Qunes", Mhemcl Logc Qurerly, ), ) 3 Zdeh, L A "uzzy ses", Informon nd conrol, 83), ) 4 Bsws, R "Rosenfeld's fuzzy subgroups wh nervlvlued membershp funcons", uzzy ses nd Sysems, 63), ) 5 Shbr, M nd Khn, I A "Inervl-vlued fuzzy dels genered by n nervl-vlued fuzzy subse n ordered semgroups", Mhwre & Sof Compung, 53), ) 6 Khn, A, Jun, Y B nd Shbr, M "Ordered semgroups chrcerzed by nervl vlued, q) - fuzzy b-dels", Journl of Inellgen nd uzzy Sysems, 5), ) 7 Khn, A, Jun, Y B, Srmn, N H nd Khn, H "Inervl-vlued uzzy Generlzed B-dels of Ordered Semgroups Redefned", o be pper n World Appled Scences Journl, 7 ) 3) 8 Khn, A, Dvvz, B, Khn, H nd Srmn, N H "Generl ypes of nervl-vlued, q) -fuzzy dels n ordered semgroups", Submed o Inernonl Journl of uzzy Sysems 3) 9 Yn,Y nd Zhn, J "New ypes of fuzzy flers of BLlgebrs", Compuers & Mhemcs wh Applcons, 67), 5-5 ) M, X, Zhn, J nd Jun, Y B "Some knds of-fuzzy dels of BCI-lgebrs", Compuers & Mhemcs wh Applcons, 64), 5-5 ) Khn, M Khn, A nd Srmn, N H "Chrcerzons of ordered semgroups by, q ) -fuzzy neror dels", Lobchevsk Journl of Mhemcs, 34), ) Khn, M, Srmn, N H nd Khn, H "A novel pproch owrd fuzzy generlzed b- dels n ordered semgroups", The Scenfc World Journl, Arcle ID 75947, 9 pges hp:dxdoorg ) 3 Akş, H nd Çğmn, N "A ype of fuzzy rng", Archve for Mhemcl Logc, 463-4), ) Jn-eb

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