On cartesian product of fuzzy primary -ideals in -LAsemigroups

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1 Joural Name Orgal Research aper O caresa produc o uzzy prmary -deals -Lsemgroups aroe Yarayog Deparme o Mahemacs, Faculy o cece ad Techology, bulsogram Rajabha Uvers, hsauloe 65000, Thalad rcle hsory Receved: xx xxx 06 Revsed: xx xxxx 06 cceped: x xxxx 06 *Correspodg uhor: aroe Yarayog, Deparme o Mahemacs, Faculy o cece ad Techology, bulsogram Rajabha Uversy, hsauloe 65000, Thalad Emal: pa_yarayog@ouloo.com bsrac: The purpose o hs paper s o roduce he oo o a wealy uzzy quas-prmary -deals -L-semgroups, we sudy produc o uzzy prmary, uzzy quas-prmary, uzzy wealy compleely prmary, wealy uzzy prmary ad wealy uzzy quas-prmary -deals -Lsemgroups. ome characerzaos o wealy uzzy prmary ad wealy uzzy quas-prmary -deals are obaed. Moreover, we vesgae relaoshps bewee uzzy wealy compleely prmary ad wealy uzzy quas-prmary -deals -L-semgroups. Keywords: Fuzzy prmary, Fuzzy quas-prmary, Fuzzy wealy compleely prmary, Wealy uzzy prmary, Wealy uzzy quas-prmary Iroduco le almos semgroup (L-semgroup s a geeralzao o semgroup heory wh wde rage o usages heory o locs (Naseerudd; 970. The udameals o hs o-assocave algebrac srucure were rs dscovered by Kazm ad Naseerudd (97. groupod s called a Lsemgroup sases he le verve law: ( ab c ( cb a or all a, b, c. I s eresg o oe ha a Lsemgroup wh rgh dey becomes a commuave mood (Mushaq; 978. Ths srucure s closely relaed o a commuave semgroup. ecause o coag a rgh dey, a L-semgroup becomes a commuave mood (Mushaq; 978. le dey a L-semgroup s uque (Mushaq; 978. I les bewee a groupod ad a commuave semgroup wh wde rage o applcaos heory o locs (Naseerudd; 970. Ideals L-semgroups have bee dscussed (Mushaq ad Yousu; 988. Now we dee he coceps ha we wll used. Le be a Lsemgroup. y a L-subsemgroup o (Mushaq ad Kha; 009, we meas a o-empy subse o such ha. o-empy subse o a Lsemgroup s called a le (rgh deal o (Mushaq ad Kha; 007 (. y wo-sded deal or smply deal, we mea a o-empy subse o a L-semgrou whch s boh a le ad a rgh deal o. I 98, he oo o -semgroups was roduced by M. K. e. groupod s called a - L-semgroup sases he le verve law: ( a b c ( c b a or all a, b, c ad, (hah ad Rehma; 03. Ths srucure s also ow as a -bel- Grassma's groupod ( -G-groupod. I hs paper, we are gog o vesgae some eresg properes o recely dscovered classes, amely - L-semgroup always sases he -medal law: ( a b ( cd ( a c ( bd or all a, b, c, d ad,,, (hah ad Rehma; 03, whle a -L-semgroup wh le dey always sases -paramedal law: ( a b ( cd ( dc ( b a or all a, b, c, d ad,,, (hah ad Rehma; 03. Recely T. hah ad I. Rehma have dscussed -Ideals ad --Ideals -Lsemgroups. deal o a -L-semgroup s called prmary mples ha eher or, or all deals ad. Q. Mushaq ad M. Kha deed he drec produc o le (resp, rgh deals, prme deals, maxmal deals ad vesgae he properes o such deals (Mushaq ad Kha; 008. The udameal cocep o uzzy ses was rs 04 The uhor(s. Ths ope access arcle s dsrbued uder a Creave Commos rbuo (CC-Y 3.0 lcese.

2 Frs uhor e al, Joural Name 04, Volume Number: age Numbers roduced by Zadeh (Zadeh; Gve a se, a uzzy subse o s, by deo a arbrary mappg : [0,], where [0,] s he u erval. Kuro aed he heory o uzzy b deals semgroups (Kuro; 993. The hough o belogg ess o a uzzy po o a uzzy subse uder a aural equvalece o a uzzy subse was deed by Mural (Mural; 004. Recely, M. Kha e al. roduced he cocep o uzzy deals ad a uzzy deals o Lsemgroups hs papers (Zadeh; 965. There are may mahemacas who added several resuls o he heory uzzy -L-semgroups, see (bdullaha slama ad Naeemb; 0. I hs paper we characerze he uzzy subse -L-semgroup. We vesgae he relaoshps bewee uzzy wealy compleely prmary ad wealy uzzy quas-prmary -deals -L-semgroups. relmares Le be a -L-semgroup. oempy subse o s called a le -deal o. s called a rgh -deal o ad s called a -deal o s boh a le ad a rgh -deal o. uco rom o he u erval [0,] s a uzzy subse o. The -Lsemgroup sel s a uzzy subse o such ha ( x or all x, deoed also by. Le ad g be wo uzzy subses o. The he cluso relao g s deed ( x g( x or all x. g ad g are uzzy subses o deed by ( g( x m ( x, g( x, ( g( x max ( x, g( x or all x. The produc g (Kha, s ad Fasal; 03 s deed as ollows; sup m x, y ; z xy g z 0; z xy. s s well ow (Kha, s ad Fasal; 03. Fuzzy subse o s called a uzzy sub -L-semgroup (wealy sub -L-semgroup o ( x y m ( x, ( y ( x y m ( x, ( y, or all x, y, ad s called a uzzy le (rgh deal o ( xy ( y( ( xy ( x or all x, y, s boh uzzy le ad rgh deal o, he s called a uzzy deal o (Kha, s ad Fasal; 03. I s easy ha s a uzzy deal o ad oly ( xy max ( x, ( y or all x, y ad ay uzzy le (rgh deal o s a uzzy Lsubsemgroup o. Equvalely, We ca prove easly ha s a (le, rgh deal o ad oly he characersc uco o s a uzzy (le, rgh deal o (hah, Iayaur-Rehma ad Kha; 04. Lemma.. (Kha, s ad Fasal; 03 I s a Lsemgroup ad, g, h are uzzy subses o, he ( g h ( hg. Lemma.. (Kha, s ad Fasal; 03 I s a - L-semgroup wh le dey ad, g, h, are uzzy subses o, he. ( gh g( h$;. ( g ( h ( h ( g. Lemma.3. (Kha, s ad Fasal; 03 Le be a uzzy subse o a -L-semgroup. The he ollowg properes hold.. s a uzzy sub -L-semgroup o ad oly.. s a uzzy le -deal o ad oly. 3. s a uzzy rgh -deal o ad oly. 4. s a uzzy -deal o ad oly ad. Lemma.4. (Kha, s ad Fasal; 03 Le be a uzzy le deal o a -L-semgroup. The. ;.. Deo.5. uzzy subse o a -Lsemgroup s called uzzy quas-prmary or ay wo uzzy le -deals g ad h o such ha gh mples g or h,

3 Frs uhor e al, Joural Name 04, Volume Number: age Numbers posve eger. Deo.6. uzzy subse o a -Lsemgroup s called uzzy prmary o or ay wo uzzy -deals g ad h o such ha gh mples g or h, posve eger. I s easy o see ha every uzzy prmary -deal s uzzy quas-prmary. Deo.7. uzzy subse o a -Lsemgroup o s called uzzy wealy compleely max ( x, ( y ( x y, prmary posve eger. where x, y ad. Theorem.8. Le be a -L-semgroup. I s a uzzy wealy sub -L-semgroup o, he s uzzy wealy compleely prmary. roo. ssume ha s a uzzy wealy sub -Lsemgroup o. ce ( x y m ( x, ( y, we have x y m x y ( (, (, or some posve eger, where x, y ad. I ( x ( y, he ( x ( y, or all posve eger. The max ( x, ( y x m ( x, ( y ( x y so ha s uzzy wealy compleely prmary. I ( x ( y, posve eger, we have he same resul. Thus max ( x, ( y ( y m ( x, ( y ( x y so ha s uzzy wealy compleely prmary. Theorem.9. Le be a -L-semgroup. I s uzzy wealy compleely prmary le -deal o, he s uzzy sub -L-semgroup o. roo. uppose ha s uzzy wealy compleely prmary o. ce max ( x, ( y ( x y, we have max ( x, ( y ( x y, posve eger, where x, y ad. Thus ( x y m ( x, ( y m ( x, ( y ad hece s a uzzy sub -L-semgroup o. Theorem.0. Le be a -L-semgroup. I I are uzzy wealy compleely prmary subses, o, he subse o. I roo. uppose ha, compleely prmary subse o. s uzzy wealy compleely prmary I are uzzy wealy The ( x y max ( x, ( y, posve eger, where x, y,, ad or I. ce max ( x, ( y ( x y, I I or all I, we ge max ( x, ( y ( x y. I I I Hece s uzzy wealy compleely prmary subse o. I Theorem.. (hah, Iayaur-Rehma ad Kha; 04 Le I be a o-empy subse o a -L-semgroup ad I : [0,] be a uzzy subse o such ha ; x I I x 0; x I. The I s a le -deal (rgh -deal, -deal o ad oly I s a uzzy le -deal (resp. uzzy rgh -deal, uzzy -deal o. 3

4 Frs uhor e al, Joural Name 04, Volume Number: age Numbers Theorem.. Le I be a -deal (le, rgh - deal o a -L-semgroup, (0,]. I I s uzzy se o such ha ; x I I x 0; x I, he I s a uzzy -deal (uzzy le, uzzy rgh - deal o. Deo.3. (hah, Iayaur-Rehma ad Kha; 04 Le be a -L-semgroup, x ad [0,]. uzzy po x o s deed by he rule ha ; x y x y 0; x y. I s acceped ha x s a mappg rom o 0,, he a uzzy po o s a uzzy subse o. For ay uzzy subse o, we also deoe x by x sequel. Le deed as ollows: x be a uzzy subse o ; x 0; x. Lemma.4. Le be a subse o a -Lsemgroup ad be a uzzy se o. The he ollowg saemes are equvale. g, [0,]., [0,]. Deo.5. uzzy subse o s sad o be a wealy uzzy prmary g h mples g or h, posve eger, where ad are wo -deals o ad (0,]. Deo.6. uzzy subse o s sad o be a wealy uzzy quas-prmary g h mples g or h, posve eger, where ad are wo le -deals o ad (0,]. I s easy o see ha every wealy uzzy prmary s wealy uzzy quas-prmary. 3. Fuzzy quas-prmary -deals o -Lsemgroups The resuls o he ollowg lemmas seem o play a mpora role o sudy uzzy prmary -deals - L-semgroups; hese acs wll be used requely ad ormally we shall mae o reerece o hs lemma. Lemma 3.. Le, be ay o-empy subse o a -L-semgroup. The or ay (0,] he ollowg saemes are rue a. a 5., ad ( (. 6. I s a le -deal (rgh, -deal o, he s a uzzy le -deal (uzzy le, uzzy -deal o. roo.. I x, he ( x, ad x a b, a, b ad. Thus ( x sup m a sup( m, ( (,. I x, he ( x 0. We ow prove ha ( ( x 0. I x y z, he ( ( x 0, ad ( ( x ( x. I x y z ad y, z, he y z, so x, whch s mpossble. Thus y or z. I y, he 0. z we have y ce ( 0, m ( y, ( z 0. I z, he as he prevous case, we also have m ( y, ( z 0. Thereore, ( ( x m ( y, ( z 0.. We wll show ha ( ( x ( x, or all x. I x, he ( x. ce x ad x, we have x x, so 4

5 Frs uhor e al, Joural Name 04, Volume Number: age Numbers ha ( ( x ( x ( x. I x, he ( x 0. uppose ha x. The ( ( x ( x 0. Thus we oba ha ( ( x ( x, or all x. 3. The proo s smlar o he proo o wh suable modcao by usg he deo. 4. I x, he a ( x supa a ( x a I x, (. x x ce x, we have he 0. x a, or all a, ad so 0. a x I mples ha a ( x sup a ( x 0 ( x. a a 5. The proo s smlar o he proo o wh a slgh modcao. 6. uppose ha s a le -deal o. The ( x y ( y, or all x, y,. I y, y 0. ce s a uzzy subse o, we have ( x y 0 ( y. I y, he he y. ce s a le -deal o ad x, y,, we he have x y. Thus, ( x y ( y. Theorem 3.. Le be a deal o -L-semgroup. The s a prmary -deal o ad oly he uzzy subse s a uzzy wealy compleely prmary -deals o. roo. uppose ha s a prmary -deal o. Obvously, s a uzzy subse o. Le x, y ad. I x y, he ( x y 0 max ( x, ( y, posve eger. Le x y. The (. x y ce s a prmary -deal o, we have x or y, posve eger. Thus ( x or ( y ad so ( x y max ( x, ( y. Thereore he uzzy subse s a uzzy wealy compleely prmary -deals o. uppose ha s a uzzy wealy compleely prmary -deals o. Le x, y M be such ha x y. The ( x y. ce s a uzzy wealy compleely prmary deals o, we have ( x y max ( x, ( y, posve eger. Thus ( x or ( y ad so x or y, posve eger. Thereore s a prmary -deal o. Theorem 3.3. Le be a uzzy subses o -Lsemgroups. The s a uzzy wealy compleely prmary -deal o ad oly he level subse, Im( o s a wealy compleely prmary -deal o, or every [0,]. roo. uppose ha s a uzzy wealy compleely prmary -deal o. Le x, y, such ha x y. The ( x y. ce s a uzzy wealy compleely prmary -deal o, we have x y max x y ( (, (, posve eger. I ( x ( y, he max ( x, ( y ( x ad ( x, so x. I ( x ( y, he max ( x, ( y ( y, ad ( y, so y. uppose ha s a wealy compleely prmary -deal o, or every [0,]. Le x, y ad. The ( x y 0. ce x y, ( x y by hypohess, we have x ( x y or posve eger. Thus y, ( x y ( x ( x y or ( y ( x y ad hece max ( x, ( y ( x y. Theorem 3.4. Le be a uzzy le deal o a -Lsemgroup wh le dey. The he ollowg saemes are equvale: 5

6 Frs uhor e al, Joural Name 04, Volume Number: age Numbers. s a wealy uzzy quas-prmary o.. For ay x, y ad (0,], x( y, he x or y, posve eger. 3. For ay x, y ad (0,], x y, he x or y, posve eger. 4. I ad are le deals o such ha, he or, posve eger. roo. ( Le be a wealy uzzy quasprmary o. For ay x, y ad (0,], x ( y, he x y ( x ( y ( x (( y ( x (( y ( x (( y ( ( x (( ( y ( ( x ( y ( x( y. ce s a wealy uzzy quas-prmary -deal, we or, ge ex x posve eger. Hece x or y, posve eger. ( 3 Le x, y, (0,] ad. The x ( y ( x x y y ey (. x y Thus, by hypohess x or y, posve eger. (3 4 Le ad be le -deals o. The, by Lemma 3., we ge ad are uzzy le -deals o. uppose ha, ad y, he here exss y such ha y, or all posve eger. For ay y ad, by Lemma 3. ad hypohess, x y x y. ce y,, whch mples, ad y so x. y Lemma 3., ollows ha x. x x (4 y Deo.6, he ollowg corollary s obvous. Corollary 3.5. Le be a uzzy deal o a -Lsemgroup wh le dey. The he ollowg saemes are equvale:. s a wealy uzzy prmary -deal o.. For ay x, y ad (0,], x( y, he x or y, posve eger. 3. For ay x, y ad (0,], x y, he x or y, posve eger. 4. I ad are deals o such ha, he or, posve eger. Theorem 3.6. Le be a -L-semgroup wh le dey. I s a uzzy quas-prmary o, he ( ( a ( b max ( a, ( b, or some posve eger, where a, b. roo. uppose ha ( ( a ( b m. Le g ad h be uzzy ses o such ha ad g x 0; m; x a x a m; x b h x 0; x b The g ad h are uzzy le -deals o by heorem.. I 6

7 Frs uhor e al, Joural Name 04, Volume Number: age Numbers gh( x sup[ m g( y, h( z ] m, he here exss a u a, v b such ha u v x. u u a ad v b,, ad,,. The x ( u v a b (( ( a b (( ( b a (( ( b ( a a ( b a ( ( e b ( ( a ( b \\ m so ha gh. ce s a quas-prmary - deal, we ge g or h, posve eger. Thus or g( a g(( ee a g( a e m h b h ee b h b e m ( (( (. Theorem 3.7. Le be a -L-semgroup wh le dey ad s a uzzy -deal o. I ( x y max ( x, ( y, he s a wealy uzzy quas-prmary -deal o, posve eger, where x, y ad. roo. uppose ha x, y ( (0,] are he uzzy pos o such ha x ( y. ce ( x y ( x y x( y ad x y max x y ( (, (, we have ( x y, whch mples ha ( x or ( y, posve eger. The x or y. Corollary 3.8. Le be a -L-semgroup wh le dey. I s a uzzy wealy compleely prmary, he s wealy uzzy quas-prmary o. Theorem 3.9. Le be a -L-semgroup wh le dey. uzzy subse o a -L-semgroup s wealy uzzy quas-prmary ad oly ( x y max ( x, ( y, posve eger, where x, y ad. roo. uppose ha s a wealy uzzy quas-prmary o. ( x y max ( x, ( y, he here I exss (0, such ha Thus ( x y max ( x, ( y. x ( y ( x y ( x y, or all x, y ad. ce s a wealy uzzy quas-prmary o, we ge some posve eger, bu x or y, or x whch s mpossble. Thereore, ( x y max ( x, ( y, or all x, y ad. ad y, 4. Caresa roduc o uzzy -deals o -Lsemgroups We sar wh he ollowg heorem ha gves a relao bewee caresa produc o uzzy -deal ad uzzy -deal -L-semgroup. Our sarg pos are he ollowg deos: Le ad be wo L-semgroups. The : ( x, y x, y ad or ay ( a, b,( c, d, we dee ( a, b ( c, d : ( a c, b d, he s a - L-semgroup as well. Le : [0,] ad g : [0,] be wo uzzy subses o -Lsemgroups ad respecvely. The he produc o uzzy subses s deoed by g ad deed as g : [0,], where ( g( x, y m ( x, g( y. 7

8 Frs uhor e al, Joural Name 04, Volume Number: age Numbers Lemma 4.. I ad g are uzzy sub -Lsemgroups o ad respecvely, he g s a uzzy sub -L-semgroup o. roo. Le ( x, y,( x, y ad. The ( g(( x, y ( x, y ( g( x x, y y (, (, (, ( m ( x x, g( y y m ( x, ( x, g( y, g( y m m x g y m x g y m ( g( x, y,( g( x, y. Thereore g s a uzzy sub -L-semgroup o. Lemma 4.. I ad g are uzzy le -deals (uzzy rgh -deals, uzzy -deals o ad respecvely, he g s a uzzy le -deal (uzzy rgh -deal, uzzy -deal o. roo. Le ( x, y,( x, y ad. The ( g(( x, y ( x, y ( g( x x, y y m ( x x, g( y y m ( x, g( y ( g( x, y. Thereore g s a uzzy le -deal o. Corollary 4.3. Le,, 3,, be a uzzy subses o -L-smgroups,, 3,, respecvely. \. I,, 3,, are uzzy sub -Lsemgroups o,, 3,, respecvely, he s uzzy sub -L-semgroup o.. I,, 3,, are uzzy le -deals (uzzy rgh -deals, uzzy -deals o,, 3,, respecvely, he s uzzy le -deal (uzzy rgh -deal, uzzy -deal o. roo. Oe ca easly show by duco mehod. Lemma 4.4. Le, g be uzzy subses o -Lsmgroup wh le dey, respecvely such ha g s a uzzy sub -L-semgroup o. The or g s uzzy sub -L-subsemgroup o or respecvely. roo. We ow ha m ( e, g( e ( g( e, e ( g( x, y m ( x, g( y, or all ( x, y. The ( x ( e or g( y g( e. I ( x ( e, he ( x g( e or g( y g( e. Le ( x g( e. The ( g( x, e ( x so ha ( x y ( g( x y, e ( g(( x, e ( y, e m ( g( x, e,( g( y, e m ( x, ( y. Thereore s a uzzy sub -L-semgroup o. Now suppose ha ( x g( e s o rue or all x. I ( x g( e x, he g( y g( e, or all y. Thereore ( g( e, y g( y, or all y. mlarly g( x y ( g( e, x y ( g(( e, x ( e, y \\ m ( g( e, x,( g( e, y m g( x, g( y. Hece g s uzzy sub -L-semgroup o. 8

9 Frs uhor e al, Joural Name 04, Volume Number: age Numbers Lemma 4.5. Le, g be uzzy subses o -Lsmgroups wh le dey, respecvely such ha g be a uzzy le -deal (uzzy rgh - deal, uzzy -deal o. The or g s uzzy le -deal (uzzy rgh -deal, uzzy -deal o or respecvely. roo. We ow ha m ( e, g( e ( g( e, e ( g( x, y m ( x, g( y, or all ( x, y. The ( x ( e or g( y g( e. I ( x ( e, he ( x g( e or g( y g( e. Le ( x g( e. The ( g( x, e ( x so ha ( x y ( g( x y, e ( g(( x, e ( y, e ( g( y, e y. Thereore s a uzzy le -deal o. Now suppose ha ( x g( e s o rue or all x. I ( x g( e x, he g( y g( e, or all y. Thereore ( g( e, y g( y, or all y. mlarly g( x y ( g( e, x y ( g(( e, x( e, y ( g( e, y \\ g y. Hece g s uzzy le -deal o. Corollary 4.6. Le,, 3,, be a uzzy subses o -L-smgroups,, 3,, respecvely.. I s a uzzy sub -L-semgroup o, he or or 3 or or s a uzzy sub -L-semgroup o,, 3,, respecvely.. I s a uzzy le -deal (uzzy rgh - deal, uzzy -deal o, he or or 3 or or s a uzzy le -deal (uzzy rgh -deal, uzzy -deal o,, 3,, respecvely. roo. Oe ca easly show by duco mehod. Lemma 4.7. Le, g be uzzy subses o -Lsmgroups, respecvely ad [0,]. The ( g. roo. Le, g be uzzy subses o -L-smgroup, respecvely ad [0,]. The ( x, y g x ad y g ( x ad g( y m ( x, g( y ( g( x, y ( x, y ( g or every x, y. Hece ( g. Corollary 4.8. Le,, 3,, be a uzzy subses o -L-smgroups,, 3,, respecvely ad ad [0,]. The ( (. roo. Oe ca easly show by duco mehod. Theorem 4.9. Le ad g be wo uzzy wealy compleely prmary (uzzy prmary, quas-prmary - deals o a -L-semgroups, respecvely. The ( g s a uzzy wealy compleely prmary (uzzy prmary, quas-prmary -deal o. roo. Le ( a, b,( c, d ad. ce ad g are uzzy wealy compleely prmary - deals o, we ge ( g(( a, b ( c, d ( g( a c, b d m ( a c, g( b d (, (, (, ( (, (, (, ( m max a c max g b g d max m a g b m c g d 9

10 Frs uhor e al, Joural Name 04, Volume Number: age Numbers (, (, (, ( max m a g b m c g d max ( g( a, b,( g( c, d posve eger. Hece g s a uzzy wealy compleely prmary o. Theorem 4.0. Le, g be uzzy subses o -Lsmgroup wh le dey, respecvely such ha g s a uzzy wealy compleely prmary (uzzy prmary -deal, quas-prmary -deal o. The or g s uzzy wealy compleely prmary (uzzy prmary -deal, quas-prmary -deal o or respecvely. roo. We ow ha m ( e, g( e ( g( e, e ( g( x, y m ( x, g( y, or all ( x, y. The ( x ( e or g( y g( e. I ( x ( e, he ( x g( e or g( y g( e. Le ( x g( e. The ( g( x, e ( x. o ha ( x x ( g( x x, e ( g(( x, e ( x, e max ( g( x, e,( g( x, e max ( x, ( x posve eger, where x, x ad. Thereore s a uzzy wealy compleely prmary o. ( x g( e s o Now suppose ha rue or all x. I ( x g( e x, he g( y g( e, or all y. Thereore ( g( e, y g( y, or all y. mlarly y g e y y g( y ( (, ( g(( e, y ( e, y max ( g( e, y,( g( e, y max ( y, ( y posve eger, where y, y ad. Hece g s uzzy wealy compleely prmary o Theorem 4.. Le, be a uzzy subses o -Lsemgroups, respecvely. The g s a uzzy wealy compleely prmary -deal o ad oly he level subse ( g, Im( g o g s a wealy compleely prmary -deal o, or every [0,]. roo. ( uppose ha g s a uzzy wealy compleely prmary -deal o. Le ( x, y,( m,, such ha ( x, y ( m, ( g. The ( g(( x, y ( m, so ha ( g( x m, y. ce g s a uzzy wealy compleely prmary -deal o, we have ( g(( x, y ( m, max ( g( x, y,( g( m,, posve eger. I ( g( x, y ( g( m,, he max( g( x, y,( g( m, ( g( m,, ad ( g( m,, so ( m, ( g. I ( g( x, y ( g( m,, he max( g( x, y,( g( m, ( g( x, y, ad ( g( x, y, so ( x, y ( g. ( uppose ha ( g s a wealy compleely prmary -deal o, or every [0,]. Le ( x, y,( m, ad. The ( g(( x, y ( m, 0. ce ( x, y ( m, ( g ( g (( x, y ( m,, by hypohess, we have ( x, y ( g ( g (( x, y ( m, or 0

11 Frs uhor e al, Joural Name 04, Volume Number: age Numbers ( m, ( g ( g (( x, y ( m,, posve eger. Thus ( g( x, y ( g(( x, y ( m, or ( g( m, ( g(( x, y ( m, ad hece max( g( x, y,( g( m, ( g(( x, y ( m,. Corollary 4.. Le,, 3,, be a uzzy subses o -L-semgroups,, 3, respecvely ad ad [0,]. The wealy compleely prmary -deal o oly he level subse s a uzzy (, Im( wealy compleely prmary -deal o ad s a. roo. Oe ca easly show by duco mehod. Cocluso May ew classes o -L-semgroups have bee dscovered recely. ll hese have araced researchers o he eld o vesgae hese ewly dscovered classes deal. Ths arcle vesgaes he wealy uzzy quas-prmary -deals -L-semgroups. ome characerzaos o produc o uzzy prmary, uzzy quas-prmary, uzzy wealy compleely prmary, wealy uzzy prmary ad wealy uzzy quas-prmary -deals -L-semgroups. Fally, we oba ecessary ad suce codos o a uzzy wealy compleely prmary ad wealy uzzy quas-prmary -deals -L-semgroups cowledgeme The auhors are very graeul o he aoymous reeree or smulag commes ad mprovg preseao o he paper. Reereces bdullah,., M. slam, M. Imra ad M. Ibrar, 0. Drec produc o uosc uzzy ses Lsemgroups-II. als o Fuzzy Mahemacs ad Iormacs, (: bdullaha,., M. slama ad M. Naeemb, 0. Iuosc uzzy - -deals o -Lsemgroups. Ieraoal Joural o lgebra ad ascs, (: bdullah,. ad M. lsam, 0. O uosc uzzy prme -Ideals o -L-semgroups. J. ppl. Mah. ad Iormacs, 30(3-4: slam, M.,. bdullah ad N. Nasree, 0. Drec produc o uosc uzzy ses L-semgroups. Fuzzy es, Rough es ad Mulvalued Operaos ad pplcaos, 3(: - 9. Kazm, M.. ad M. Naseerudd, 97. O almos semgroups. The lg. ull. Mah., : - 7. Kha, M.,. s ad Fasal., 03. O uzzy- -deals o -bel-grassma's Groupods. Research Joural o ppled ceces, Egeerg ad Techology, 6(8: Kha, M.,. s ad. Lodh, 0. sudy o uzzy bel-grassma's groupods. Ieraoal Joural o he hyscal ceces, 7(4: /IJ.700 Kha, M., Y. ae Ju ad T. Mahmood, 00. Geeralzed uzzy eror deals bel Grassma's Groupods. Hdaw ublshg Corporao Ieraoal Joural o Mahemacs ad Mahemacal ceces, - 4. Kha, M., Y. ae Ju ad F. Yousaza, 05. Fuzzy deals rgh regular L-semgroups. Haceepe Joural o Mahemacs ad ascs, 44(3: Kha, M. ad Fasal, 0. O uzzy ordered bel- Grassma's Groupods. Joural o Mahemacs Research, 3(: /jmr.v3p7 Kha, M. ad M.N. Kha, 00. O uzzy bel Grassma's groupods. dvace Fuzzy Mah., 5(3: Kha, M., M.F. Iqbal ad M.N.. Kha, 00. O a uzzy deals le almos semgroups. J.Mah. Research, : /jmr.v3p03 Kha, M. ad M. Nom a slam Kha, 009. Fuzzy bel-grassma's groupods, arxv: v [mah.gr]. Kuro, N., 993. Fuzzy semprme quas deals semgroups. Iorm. c., 75(3: / ( Mordeso, J.N., 003. Fuzzy semgroups. prger- Verlag erl Hedelberg. Mural, V., 004. Fuzzy pos o equvale uzzy subses. Iorm. c., 58: /j.s

12 Frs uhor e al, Joural Name 04, Volume Number: age Numbers Mushaq, Q. ad M. Kha, 007. oe o a bel- Grassma's 3-bad. Quasgroups ad Relaed ysems, 5: Mushaq, Q. ad M. Kha, 008. Drec produc o bel Grassma's groupods. J. Ierdscp.Mah., : / Mushaq, Q. ad M. Kha, 009. Ideals le almos semgroup. arxv: v [mah.gr]. Mushaq, Q. ad.m. Yousu, 978. O Lsemgroups.978. The lg. ull. Mah., 8: Mushaq, Q. ad.m. Yousu, 988. O L-semgroup deed by a commuave verse semgroup. Mah.ech., 40:59-6. Naseerudd, M., 970. ome sudes almos semgroups ad locs. h.d. hess: lgarh Muslm Uversy: lgarh: Ida. e, M.K., 98. O -semgroups. roceedg o Ieraoal ymposum o lgebra ad Is pplcaos, Decer ublcao: New Yor, hah, T., Iayaur-Rehma ad. Kha, 04. Fuzzy -deals -G-groupods. Haceepe Joural o Mahemacs ad ascs, 43(4: hah, T. ad T. Rehma, 03. Decomposo o locally assocave -G-groupods. Nov ad J. Mah., 43(: - 8. Yousaza, F., N. Yaqoob,. Haq ad R. Mazoor, 0. oe o uosc uzzy -L-semgroups. World ppled ceces Joural, 9(: Zadeh, L.., 965. Fuzzy ses. Iorm. Corol., 8: