Applications of N-Structures to Ideal Theory of LA-Semigroup

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1 Appl. Mah. Inf. Sci. Le. 4, No. 3, (206) 97 Applied Mahemaics & Informaion Sciences Leers An Inernaional Journal hp://d.doi.org/0.8576/amisl/04030 Applicaions of N-Srucures o Ideal Theor of LA-Semigroup Qura-ul-Ain Qamar, Saleem Abdullah 2, and Muhammad Shahzad 2 Deparmen of Mahemaics, Riphah Inernaional Universi, Islamabad, Pakisan 2 Deparmen of Mahemaics, Hazara Unviersi, Mansehra, Pakisan Received: 8 Sep. 205, Revised: 30 Dec. 205, Acceped: 3 Dec. 205 Published online: Sep. 206 Absrac: In his paper, he concep of ([e],[e] [c])-ideals in an LA-semigroup is inroduced and proved fundamenal resuls o deermine he relaion beween hese noions and ideals of a LA-semigroup. Moreover, we inroduce he concep of ([e],[e] [c])-sub LA-semigroup and invesigaed relaed properies. To obain more general form [(α),(β)]-ideals and [(α),(β)]-sub LA-semigroup are inroduced and furher characerizaion is discussed. Kewords: LA-semigroup, N-srucure, Ideal, ([e],[e]v[c])-ideal Inroducion A lef almos semigroup, abbreviaed as an LA-semigroup, is an algebraic srucure bewi beween a groupoid and a commuaive semigroup. An LA-semigroup is a non-commuaive and non-associaive algebraic srucure. This srucure was inroduced b M. A. Kazim and M. Naseeruddin [4] in 972. I has been defined in [] and [2] ha a groupoid G wih lef inverive law, ha is: (ab)c = (cb)a a,b,c G is called an LA-semigroup. Naseeruddin has invesigaed some basic characerisics of his srucure in his hesis. He has generalized some imporan resuls of semigroup heor. Moreover, he has esablished he relaionships beween LA-semigroups and quasi groups, semigroups, loops, monoids and groups. Kazim and Naseeruddin in heir paper on almos semigroup [4] have shown ha G is medial. Tha is, (ab)(cd) =(ac)(bd). Laer, his srucure has furher invesigaed b Q.Mushaq and ohers and man useful resuls have been added o heor of LA-semigroups [7, 8, 9]. In 965, L. A. Zadeh generalized he Crisp Se Theor and inroduced he fundamenal concep of a fuzz se [3]. Since hen his concep has been applied o various algebraic srucures. Tha is wh; here is a rapid increase in research and lieraure on Fuzz Se Theor and is applicaions. Man areas, for eample, arificial inelligence, compuer science, conrol engineering, eper, roboics, auoma heor, finie sae machine and graph heor ec deals wih he sud of Fuzz Se Theor [3]. Ideals of LA-semigroups were defined b Mushaq and Khan in his paper [0]. Furher, Khan and Ahmad characerized he ideals of LA-semigroup in [5]. In 200, Khan and Khan inroduced fuzz ideals in LA-semigroups [6] and proved some imporan resuls. So far, no negaive informaion was involved. All he research relied on posiive informaion onl. To cope wih his problem, in 2009, Jun e al [] inroduced a new funcion which is called a negaive valued funcion and hen inroduced N-srucure and discussed N-algebras and N-ideals in BCK/BCI algebras. Laer on, Y. B. Jun and M. S. Kang [2] inroduced he noion of N-ideal of BE-algebra inroduced b, Kim and Kim [3] and invesigaed several characerizaions of N-ideals. For furher generalizaion of N-ideals, he proposed a definiion of a poin N-srucure which is emploed or condiionall emploed in an N-srucure. Using hese noions, he inroduced he concep of ([e],[e] [c])-ideals and invesigaed hese ideals on BE-algebras. The aim of his paper is o make developmen in fuzz echnolog and o promoe research in he specified field. Our goal is o eplain new mahemaical echniques and developmen in he field of fuzz algebra and LA-semigroup, which would be of grea imporance in Corresponding auhor saleemabdullah8@ahoo.com

2 98 Q. Ain e al.: Applicaions of N-srucures... fuure. This paper can be a bridge passing from he heor of N-fuzz algebra o he heor of LA-semigroup. In his paper, we have used he idea of Y. B. Jun and M. S. Kang [2] and inroduced ([e],[e] [c])-ideals in LA-semigroup. We have proved some fundamenal resuls ha deermine he relaion beween he newl inroduced noions and ideals of an LA-semigroup. Moreover, we have inroduced he noion of ([e],[e] [c])-sub LA-semigroup and invesigaed heir relaionship wih sub LA-semigroup. For he generalizaion of ([e],[e] [c])-ideals and ([e],[e] [c])-sub LA-semigroup, we have inroduced he concep of ([α],[β])-ideals and ([α],[β])-sub LA-semigroup and proved some fundamenal resuls. 2 Preliminaries Definiion.Le I be a non-emp subse of S. Then I is called a lef (resp. righ) ideal in LA-semigroup if for I, S= I (resp. I). Definiion 2.Le I be a non-emp subse of an LA-semigroup S. Then I is called a sub-la-semigroup if for all, I = I. Definiion 3.An N-srucure (S, f) is called a lef (resp. righ) N-ideal of S if for all, S, f() f() (resp. f() f()). Definiion 4.Le (X, f) be an N-srucure. Then, a poin N-srucure be defined as following f()={.0 if θ if = () where θ [,0). In his case, f is denoed b θ and we call (X, θ ) a poin N-srucure. We sa ha a poin N-srucure (X, θ ) is emploed in an N-srucure (X, f), denoed b (X, θ )[e](x, f) (or briefl θ [e] f ), if f() θ. A poin N-srucure (X, θ ) is said o be condiionall emploed in an N-srucure (X, f), denoed b (X, θ )[c](x, f) (or briefl θ [c] f ), if f()+θ+ <0. To sa ha (X, θ )[e] [c](x, f) (or briefl θ [e] [c] f ), we mean (X, θ )[e](x, f) or (X, θ )[c](x, f) (or briefl, θ [e] f or θ [c] f ). To sa ha θ α f, we mean α f does no hold for α {[e],[e] [c]}. θ 3 ([e],[e] [c])-ideals Definiion 5.An N-srucure (S, f) is called an ([e],[e] [c])-ideal in LA-semigroup if i saisfies:. 2. [e] f = [e] [c] f [e] f = [e] [c], for all, S and [,0). Eample.Le S = {,2,3} be an LA-semigroup defined b following Cale s able An N-srucure(S, f) is defined b f()={. 0.7 if =,2 0.3 if =3 Thus, b rouine (S, f) is an ([e],[e] [c])-ideal in LA-semigroup S. Eample 2.Le S = {, 2, 3, 4} be an LA-semigroup defined b following Cale s able N-srucure(S, f) is defined b 0.2 if = 0.3 if =3 f()={. 0.7 if =2 0.8 if =4 Thus, b rouine calculaion(s, f) is an([e],[e] [c])-ideal in LA-semigroup. Theorem.For an N-srucure (S, f), he following condiions are equivalen..(s, f) is an([e],[e] [c])-ideal. 2. (, S)( f() ma{ f(), 0.5} and f() ma{ f(), 0.5}) Proof.Le (S, f) is an ([e],[e] [c])-ideal of S. For all, S le f() > ma{ f(), 0.5} =. If f() = [e] f and [e] f. Bu if > 0.5 hen f() + + > = 0 = [e] [c] f so [e] f = [e] [c] f. This is a conradicion. Hence our supposiion is wrong. So f() ma{ f(), 0.5}. Similarl, we can show his for f() ma{ f(), 0.5}. Conversel, suppose ha (S, f) saisfies he condiion f() ma{ f(), 0.5}. Le, S and [,0) such ha [e] f. Then f(). Suppose ha [e] f i.e. f() >. Now eiher f() > 0.5 or f() 0.5. If f() > 0.5 hen f() ma{ f(), 0.5} = f() which is no rue. If f() 0.5 = f()+ + < 2 f()+ 2ma{ f(), 0.5}+=0 i.e. [c] f. Thus [e] [c] f. Similarl, we can show his for [e] [c] f. Thus(S, f) is an([e],[e] [c])-ideal of S. Theorem 2.For an N-srucure (S, f), he following condiions are equivalen.

3 Appl. Mah. Inf. Sci. Le. 4, No. 3, (206) / 99.(S, f) is an([e],[e] [c])-ideal. 2.C( f,) is an ideal of S. Proof.Le (S, f) is an ([e],[e] [c])-ideal and le [ 0.5,0) be such ha C( f,) Φ. B Theorem f() ma{ f(), 0.5} and f() ma{ f(), 0.5} for an, C( f, ). I follows ha f() ma{, 0.5} = = C( f,) and f() ma{, 0.5} = = C( f,). Therefore, C( f,) is an ideal of S. Conversel, suppose ha C( f,) is an ideal of S. If, S such ha f() > ma{, 0.5} hen f() > ma{, 0.5} for some [ 0.5,0). Thus C( f,) bu / C( f,) which is a conradicion. Thus f() ma{ f(), 0.5}, S. Similarl we can show his for f(). B previous heorem, we can sa ha (S, f) is an([e],[e] [c])-ideal of S. Theorem 3.Le S be an LA-semigroup. If (S, f) is an ([e],[e] [c])-ideal of S such ha f() > 0.5 S hen (S, f) is an N-ideal of S. Proof.Assume ha (S, f) is an ([e],[e] [c])-ideal of S such ha f() > 0.5 S. B Theorem f() ma{ f(), 0.5}. Since f() > 0.5 so f() f(). Similarl, we can show his for f() f(). Therefore,(S, f) is an N-ideal of S. Theorem 4.If (S, f) is an ([e],[e] [c])-ideal of an LA-semigroup, hen Q( f, ) is an ideal of S. Where Q( f,) :={ X [c] f} [, 0.5]. Proof.Le (S, f) is an ([e],[e] [c])-ideal of an LA-semigroup S. Suppose ha Q( f, ) Φ for all [, 0.5] hen here eis Q( f,) such ha f() + + < 0 = f() <. B heorem f() ma{ f(), 0.5} < ma{, 0.5} < = f()+ + <0= Q( f,). Similarl, we can show for Q( f,). Therefore, Q( f,) is an ideal of S. Theorem 5.Le S be an LA-semigroup. Then an N-srucure(S, f) is an([e],[e] [c])-ideal of S if and onl if [ f] is an ideal of S where [ f] := C( f,) Q( f,) [,0). Proof.Assume ha (S, f) is an([e],[e] [c])-ideal of S and le [,0) be such ha [ f] Φ. Then here eis [ f] such ha f() or f()++ <0. If f() hen f() ma{ f(), 0.5} ma{, 0.5} = = [ f] and if f()++<0= f()< hen f() ma{ f(), 0.5} ma{, 0.5}= = [ f]. So [ f]. Similarl we can show for [ f]. Thus [ f] is an ideal of S. Conversel, le (S, f) be an ideal of S. Suppose ha f() > ma{ f(), 0.5} for some, S. Taking := ma{ f(), 0.5}. Since [ f] is an ideal of S, so [ f] = f() or f() <. Bu he inequali 3 induces ha / C( f,). So our supposiion is wrong and f() ma{ f(), 0.5}. Similarl, i can be shown for f() ma{ f(), 0.5}. Thus (S, f) is an ([e],[e] [c])-ideal. 4 ([e],[e] [c])-sub LA-semigroup Definiion 6.Le (S, f) be an N-srucure. Then (S, f) is called ([e],[e] [c])-sub LA-semigroup if he following condiion holds. [e] f, 2 [e] f = ma{, 2 } [e] [c] f for all, S and, 2 [,0). Theorem 6.For an N-saure (S, f),he following are equivalen.(s, f) is an([e],[e] [c])-sub LA-semigroup. 2.(, S)( f() ma{ f(), f(), 0.5}). Proof.Assume ha (S, f) is an ([e],[e] [c])-sub LA-semigroup. Le f() > ma{ f(), f(), 0.5}, S be such ha := ma{ f(), f(), 0.5}. Now eiher ma{ f(), f()} > 0.5 or ma{ f(), f()} 0.5. If ma{ f(), f()} > 0.5 = f() and f() = [e] f and [e] f since f() > = [e] f. And if ma{ f(), f()} 0.5 = f() + + > = 0 = [c] f so [e] [c] f which is a conradicion. Hence f() ma{ f(), f(), 0.5}. Conversel, Suppose ha N-srucure (S, f) saisfies he given condiion f() ma{ f(), f(), 0.5}. Le, S and [,0) such ha [e] f, 2 [e] f or f(), f() 2. Then f() ma{ f(), f(), 0.5} ma{, 2, 0.5}. Suppose ha f() > ma{, 2 } or ma{, 2 }[e] f. If ma{, 2 } > 0.5 hen f() ma{ f(), f()} which is conradicion so his is no possible. Bu if ma{, 2 } 0.5 hen f() + ma{, 2 } + < f() + f() + = 2 f() + < 2ma{ f(), f(), 0.5} + < 2( 0.5)+ = 0 = f() + ma{, 2 }+ < 0 i.e. ma{, 2 } [c] f. Thus ma{, 2 }[e] [c] f and herefore (S, f) is an([e],[e] [c])-sub LA-semigroup. Theorem 7.For an N-srucure (S, f) he following are equivalen.(s, f) is an([e],[e] [c])-sub LA-semigroup. 2.C( f, ) is a sub LA-semigroup. Proof.Assume ha (S, f) is an ([e],[e] [c])-sub LA-semigroup and C( f,) 0. Le, C( f,), b Theorem 6 f() ma{ f(), f(), 0.5} < ma{,, 0.5} = = f() and [ 0.5,0) =, C( f,). Thus C( f,) is a sub LA-semigroup. Conversel, suppose ha C( f, ) is a sub LA-semigroup of S. Le f() > ma{ f(), f(), 0.5}

4 00 Q. Ain e al.: Applicaions of N-srucures... for, S hen f() > ma{ f(), f(), 0.5} for some [ 0.5,0). Thus, C( f,) bu / C( f,) which is a conradicion. Thus f() ma{ f(), f(), 0.5}. Thus (S, f) is an ([e],[e] [c])-sub LA-semigroup. Theorem 8.If(S, f) is an([e],[e] [c])-sub LA-semigroup, hen Q( f,) is a sub LA-semigroup, where Q( f,) :={ X [c] f} [, 0.5]. Proof.Le (S, f) is an ([e],[e] [c])-sub LA-semigroup. Suppose Q( f,) 0 for all [, 0.5]. Then here eis, Q( f,) = f() + + < 0 and f()+ + < 0 or f() < and f() <. Then b Theorem 6 f() ma{ f(), f(), 0.5} < ma{,, 0.5}= = f()++ <0= Q( f,). Thus Q( f, ) is a sub LA-semigroup. Theorem 9.Le S be a sub LA-semigroup. Then an N-srucure (S, f) is an ([e],[e] [c])-sub LA-semigroup if and onl if [ f] is a sub LA-semigroup for all [,0) where [ f] := C( f,) Q( f,). Proof.Assume ha (S, f) is an ([e],[e] [c])-sub LA-semigroup. Then b Theorems 7 and 8, C( f,) and Q( f,) are sub LA-semigroups. Therefore, [ f] is a sub LA-semigroup. Conversel, suppose ha [ f] is a sub LA-semigroup. Suppose ha f() > ma{ f(), f(), 0.5} for some, S. Taking := ma{ f(), f(), 0.5}. Since[ f ] is a sub LA-semigroup. So for, [ f ] = [ f ] = f() or f() <. Bu / C( f,). Thus / [ f ] which is a conradicion. Hence our supposiion is wrong. Therefore f() ma{ f(), f(), 0.5}. Thus [ f ] is an ([e],[e] [c])-sub LA-semigroup. 5 ([α],[β])-ideal and([α],[β])-sub LA-semigroup Definiion 7.Le (S, f) be an N-srucure. Then (S, f) is called a righ (resp. lef) ([α],[β])-ideal in an LA-semigroup if following condiions hold: [β] f (resp. [β] f ) where f : S [,0] and for all, S, [,0). Definiion 8.An N-srucure (S, f) is called an ([α],[β])- sub LA-semigroup if i saisfies following implicaion. [α] f,, 2 [,0) 2 ma{, 2 }[β] f for all, S and Theorem 0.Le (S, f) be a non-emp ([α],[β])-sub LAsemigroup. Then(S, f) ={ S: f()<0} is a sub LAsemigroup. Proof.Le (S, f) be a non-emp ([α],[β])-sub LA-semigroup. We have o show ha (S, f) is a sub LA-semigroup. i.e. for all, (S, f) = (S, f). Le, (S, f) i.e. f() < 0 and f()<0. Suppose ha f()=0 and α {[e],[c],[e] [c]}. (i) For α {[e],[e] [c]} we have [α] f, 2 [α] f. As f() = 0 > ma{, 2 } = f() > ma{, 2 } = ma{, 2 } [e] f. So f() + ma{, 2 } + 0 (clearl)= ma{, 2 }[c] f. So ma{, 2 } [β] f for ever β {[e],[c],[e] [c],[e] [c]}. (ii) For α = [c] we have [α] f, f() + ( ) + < 0, f() + ( ) + < 0. As f() = 0 > so f() + ( ) + = = 0 = [β] f. Thus ma{, 2 } [β] f for ever β {[e],[c],[e] [c],[e] [c]}. Hence our supposiion is wrong. So f() < 0 = (S, f). Thus (S, f) is a sub LA-semigroup. Theorem.Le(S, f) be a non-emp([α],[β])-lef ideal of S. Then(S, f) ={ S : f()<0} is a lef ideal of S. Proof.Le (S, f) be a non-emp ([α],[β])-lef ideal. We have o show ha (S, f) is a lef ideal. i.e. for (S, f) = (S, f). Le (S, f) i.e. f() < 0. Suppose ha f()=0 and α {[e],[c],[e] [c]}. (i) For α {[e],[e] [c]} we have 2 [α] f. As f() = 0 > = f() > = Thus f()++ 0 = β {[e],[c],[e] [c],[e] [c]}. ma{, 2 } [e] f. [β] f for ever (ii) for α = [c] we have f() + ( ) + < 0. As f() = 0 > = f()+( )+=0+0= 0= [β] f. Thus [β] f for ever β {[e],[c],[e] [c],[e] [c]}. Hence our supposiion is wrong. So f()<0= (S, f). Thus (S, f) is a lef ideal of S. Theorem 2.Le (S, f) be a non-emp ([α],[β])-righ ideal of S. Then (S, f) = { S: f()<0} is a righ ideal of S. Theorem 3.Le (S, f) be a lef zero LA-semigroup. Le (S, f) be a non-emp ([c],[c])-sub LA-semigroup. Then (S, f) is a consan on(s, f). Proof.Le be an elemen of S such ha f() = { f() : S}. Since f() < 0 so (S, f). Suppose ha (S, f) s.. = f() f() = hen >. Choose, 2 [,0) s.. > > > 2. Then [c] f and 2 ma{, 2 }[c] f. Because S is a lef zero LA-semigroup. This is a conradicion. Thus f() = f(e) (S, f). Therefore,(S, f) is a consan on(s, f). Theorem 4.Le (S, f) be an N-srucure in an LA-semigroup. Then (S, f) is a non-emp ([c],[c])-sub LA-semigroup iff here eis a sub LA-semigroup H of S such ha

5 Appl. Mah. Inf. Sci. Le. 4, No. 3, (206) / 0 { [,0) if S f()= 0 oherwise Proof.Le (S, f) be a non-emp ([c],[c])-sub LA-semigroup. B heorem 0, (S, f) is a sub LA-semigroup. So i is clear ha f() < 0 for all S. And b heorem 3, { f() if (S, f) f() := 0 oherwise where f() [,0). Conversel, le H be a sub LA-semigroup which saisfies f()={. [,0) if S0 oherwise Assume ha [c] f and 2 [c] f for some, 2 [,0). Then f()+ + <0 and f()+ + <0= f() 0 and f() 0. Thus, H and so H. So f()+ma{, 2 }+<0= is a {[c],[c]}-sub LA-semigroup. ma{, 2 } [c] f. Thus(S, f) Theorem 5.Le H be a sub LA-semigroup of S and(s, f) be an N-srucure in LA-semigroup S such ha (i) f()=0 S\H (ii) f() 0.5 H Then(S, f) is an([α],[e] [c]) sub LA-semigroup. Proof.Le H be a sub LA-semigroup of S and (S, f) be an N-srucure in LA-semigroup S, such ha (i) f()=0 S\H (ii) f() 0.5 H Le, S and, 2 [,0) such ha [α] f and 2 [α] f. Case I: For [α] = [e] i.e. [e] f and 2 [e] f = f() and f() 2. For / H or / H i.e. f() = 0 or f() = 0 = > 0 or 2 > 0 which is no rue. So, H = H = f() 0.5. If ma{, 2 } < 0.5 hen f() + ma{, 2 } + < ma{, 2 } = 0 = [c] f. Bu if ma{, 2 } 0.5 = ma{, 2 } 0.5 f() = f() ma{, 2 } = ma{, 2 } [e] f. Therefore ma{, 2 } [e] [c] f. Case II: For [α] = [c] i.e. and [c] f 2 f()+ + < 0 and f()+ 2 + < 0. If / H or / H = / H and f() = 0 or f() = 0 = < or 2 < which is no rue. So, H = H. Thus f() 0.5. Now if ma{, 2 } 0.5 hen ma{, 2 } 0.5 f() = f() ma{, 2 } = ma{, 2 }[e] f. Bu if ma{, 2 }< 0.5= f()+ma{, 2 }+< =0= ma{, 2 } ma{, 2 } [e] [c] f. Case III: For α [e] [c], his is obvious from Case I and Case II. Therefore,(S, f) is an ([α],[e] [c])-sub LAsemigroup. Theorem 6.Le L be a lef ideal of S and (S, f) be an N- srucure such ha (i) f()=0 S\L (ii) f() 0.5 L Then(S, f) is a ([α],[e] [c])-lef ideal. Proof.Le L be a lef ideal of S such ha (i) f()=0 S\L and(ii) f() 0.5 L Le, S and [,0) such ha [α] f and S. Case I: For [α] = [e] i.e. [e] f = f(). For L and S = L. Thus f() 0.5. If < 0.5= f()+ + < = 0= [c] f. Bu if 0.5= 0.5 f()= f() = [e] f. Thus [e] [c] f. Case II: For[α]=[c] i.e. f()+ + <0. For L and S = L. Thus f() 0.5. If 0.5 hen 0.5 f()= f() = [e] f. Bu if < 0.5= f()++< = 0= [e] [c] f. Case III: For α [e] [c], This is obvious from Case I and Case II. Therefore,(S, f) is an([α],[e] [c])-lef ideal of S. Theorem 7.Le L be an (resp. righ) ideal of S and(s, f) be an N-srucure such ha (i) f()=0 S\L (ii) f() 0.5 L Then(S, f) is a ([α],[e] [c])-(resp. righ) ideal. Proof.Sraighforward Theorem 8.For an subse A of S, le χ A denoes he N- characerisic funcion of S, defined as χ A : S {,0} { if A χ A ()= 0 if / A Then, χ A is an([e],[e] [c])-sub LA-semigroup if and onl if A is a sub LA-semigroup. Proof.Assume ha χ A is an ([e],[e] [c])-sub LA-semigroup. Le, A = [e]χ A and [e]χ A = ma{, } [e] [c]χ A. Thus eiher [e]χ A or [c]χ A = χ A () or χ A ()+( )+ < 0. If χ A () = A. Bu if χ A ()+( )+ < 0= χ A ()<0= χ A ()= = A. Thus A is a sub LA-semigroup of S. Conversel, Suppose ha A is a sub LA-semigroup of S. (i) Le, A = A, so χ A () =, χ A () =, χ A () =. So χ A ()=ma{χ A (), χ A ()} ma{χ A (), χ A (), 0.5}

6 02 Q. Ain e al.: Applicaions of N-srucures... (ii) Le eiher / A or / A = χ A () = 0 or χ A () = 0. So χ A () ma{χ A (), χ A ()} ma{χ A (), χ A (), 0.5} (iii) Le, / A = χ A () = 0,χ A () = 0. Thus χ A () ma{χ A (), χ A ()}=ma{χ A (), χ A (), 0.5} Therefore, χ A is an ([e],[e] [c])-sub LA-semigroup. Theorem 9.For an subse A of S, Le χ A denoes he N- characerisic funcion of S, defined as χ A : S {,0} χ A ()={. if A0 if / A Then, χ A is an ([e],[e] [c])-lef(resp. righ) ideal iff A is a lef(resp. righ) ideal. Proof.Assume ha χ A is an ([e],[e] [c])-lef ideal. Le A and H. Then χ A = = [e]χ A = [e] [c]χ A. Thus eiher [e]χ A or [c]χ A = χ A () or χ A ()+( )+ < 0. If χ A () = A. Bu if χ A ()+( )+ < 0= χ A ()<0= χ A ()= = A. Thus A is a lef ideal of S. Conversel, Suppose ha A is a lef ideal of S. (i) Le A= A and so χ A ()=, χ A ()=. Thus χ A ()= χ A () ma{χ A (), 0.5}. (ii) Le / A= χ A () = 0. Thus χ A () χ A () = ma{χ A (), 0.5}. Therefore, χ A is an ([e],[e] [c])-lef ideal. Similarl, we can show his for righ ideal. References [] Jun, Y. B., Lee, K. J. and Song, S. Z. on N-ideals of BCK/BCI-algebras, J. Chungcheong Mah. Soc. 22,47-437,2009. [2] Jun, Y. B.and Kang, M. S. on ideal heor of BE-algebras, Hacceepe Journal of Mah. and Sa. 4(4), ,202. [3] Kim, H. S. and Kim, Y. H. on BE-algebras, Sci. Mah. Japan. 68, ,2008. [4] Kazim, M. A. and Naseeruddin. M. on almos semigroups, Alig. Bull. Mah., 2, -7, 972. [5] Khan, M. and Ahmad, N. Characerizaion of lef almos semigroups b heir ideals, 2 (3), 6-73, 200. [6] Khan, M. and Khan, M. N. A. on fuzz abel Grassmann s groupoids, Advanced in Fuzz Mahemaics, 5(3), , 200. [7] Mushaq, Q. and Kamran, M. S. on LA-semigroups wih weak associaive law. Scienific Khber,, 69 7, 989. [8] Mushaq, Q. and Yousuf. S. M. on LA-semigroups. The Alig. Bull. Mah., 8, 65 70, 978. [9] Mushaq, Q. and Yousuf. S. M. on LA-semigroup defined b a commuaive inverse semigroup. Mah. Bech., 40, 59 62, 988. [0] Mushaq, Q. and Khan, M. on Ideals in lef almos semigroups. Proceedings of 4h Inernaional Pure Mahemaics Conference, 65 77, [] Mushaq, Q. and Inam, M. on lef almos semigroup b a free algebra, Quasigroups and Relaed Ssems 6, 69-76, [2] Naseeruddin, M. some sudies on almos semigroups and flocks, Ph.D Thesis, The Aligarh Muslim Universi India, 970. [3] Zadeh, L. A. on Fuzz ses, Inform and conrol, 8, , 965. Qura-ul-Ain Qamar has compleed her M.Sc from Inernaional Islamic Universi, Islamabad and M.Phil degree from Riphah Inernaional Universi, Islamabad, Pakisan.She is doing job as a Senior School Teacher (SST) in Governmen High School. She is working on N-srucures and Fuzz Algebras Saleem Abdullah is an assisan Professor, Deparmen of Mahemaics, Hazara Universi, Mansehra, KP, Pakisan. He compleed M.Sc from Deparmen of Mahemaics, Hazara Universi, Mansehra, KP, Pakisan, and M.Phil and Ph.D from Quaid-i-Azam Universi, Islamabad, Pakisan. He delivered differen lecures in inernaional conferences i.e Turke, China, UAE, Oman, Russia ec. He has published more 60 research papers in Inernaional repued Journals (ISI, Scopus ec). His research ineres is in Fuzz logic, Fuzz Algebra, Hper srucure heor, Logical Algebras, Fuzz decision making, Fuzz clusering ec. Muhammad Shahzad joined he Hazara Universi in 2006 as a lecurer in Mahemaics. Aferward, he was engaged in differen deparmenal aciviies, e.g. inernal eam coordinaor, member of differen commiees. In 2008, he wen o he UK for he higher qualificaion (PhD) focusing his research on Model Reducion Techniques. In 20, he rejoined he deparmen as an assisan professor. In 203 he was seleced as head of deparmen of mahemaics. His research ineress lie in Model Reducion Techniques, Phsical Chemisr, and Numerical Analsis. He has published more han 20 papers in differen repued inernaional journals.