Songklanakarin Journal of Science and Technology SJST R1 Akram. N-Fuzzy BiΓ -Ternary Semigroups

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1 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm N-Fuzzy Bi -Ternry emigroups Journl: ongklnkrin Journl of cience nd Technology Mnuscript ID JT-0-0.R Mnuscript Type: Originl Article Dte ubmitted by the Author: -Apr-0 Complete List of Authors: Akrm, Muhmmd; University Tun Hussein Onn Mlysi, Mthemtics nd ttistic; University of Gujrt, Mthemtics Jcob, Kvikumr; University Tun Hussein Onn Mlysi, Mthemtics nd ttistic Khmis, Azme; University Tun Hussein Onn Mlysi, Mthemtics nd ttistic Iqbl, Zffr; University of Gujrt, Mthemtics hmsidh, Nor; University Tun Hussein Onn Mlysi, Mthemtics nd ttistic Keyword: Bi -ternry smigroup, N-fuzzy set, N-fuzzy bi -ternry subsmigroup, N-fuzzy bi -idel.

2 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm N-Fuzzy Bi -Ternry emigroups Muhmmd Akrm ; ; Jcob Kvikumr, Azme Bin Khmis, Zffr Iqbl nd A.H. Nor hmsidh Deprtment of Mthemtics nd ttistics, Fculty of cience, Technology nd Humn Development, Universiti Tun Hussein Onn Mlysi, Btu Pht, Mlysi Deprtment of Mthemtics, University of Gujrt, Gujrt, Pkistn Corresponding uthor emil: Abstrct The notion of N-fuzzy sets (NF) hs been pplied to newly defined lgebric structure bi-ternry semigroup. The notions of N-fuzzy bi-ternry subsemigroups nd N-fuzzy bi-left (right, lterl, qusi, nd bi) idels hve been defined nd relted properties hve been investigted. The chrcteriztion of bi-ternry semigroup under these idels hve been estblished. 00 AM Clssifictions: 0A, 0N0, 0M Keywords: Bi -ternry semigroup, N-fuzzy set, N-fuzzy bi -ternry subsemigroup, N- fuzzy bi-idel.

3 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of Introduction Zdeh () introduced the concept of fuzzy set. The fuzzy set theories developed by Zdeh nd others hve been found mny pplictions in the domin of mthemtics nd elsewhere. Rosenfeld () strted the study of fuzzy lgebric structures nd introduced fuzzy subgroup (subgroupoid) nd fuzzy (left, right) idel in his pioneering pper. The concept of intuitionistic fuzzy set ws introduced by Atnssov () s generliztion of fuzzy set. Bisws (), introduced the concept of intuitionistic fuzzy subgroupoids. Kim nd Jun (00) pplied the concept of intuitionistic fuzzy sets to the idel theory of semigroup nd defined severl idels of semigroup. Mny other uthors pplied the concept of fuzzy set nd intuitionistic fuzzy set to the lgebric structures. A crisp set A in universe X cn be defined in the form of its chrcteristics function : X {0,}. o fr most of the generliztions of the crisp set hve been A conducted on the unit intervl [ 0,] nd they re consistent with the symmetry observtion. the generliztion of the crisp set to fuzzy set nd fuzzy set to intuitionistic fuzzy set relied on spreding positive informtion. Jun et l. (00) first time used the negtive mening of informtion nd introduced new function which is clled negtivevlued function (or negtive fuzzy set, briefly, N -fuzzy set) nd constructed N - structures. Khn et l. (00) used the ide of N -fuzzy sets to chrcterized ordered semigroups by their N -fuzzy idels. Akrm et l. (0) worked on the N -fuzzy idels of -AG-groupoids. Recently Akrm et l. (0) proposed new lgebric structure clled bi-ternry

4 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm semigroup s generliztion of -semigroup nd ternry semigroup. They introduced the notions of bi-ternry subsemigroup, bi-left (right, lterl) idel, bi-qusi idel nd bi-bi-idels for this structure nd discussed the reltionship between these substructures. They lso defined the regulr bi-ternry semigroup nd chrcterized it by these idels. In this pper, we hve pplied the concept of N -fuzzy set to the idel theory of bi - ternry semigroup. The notions of N -fuzzy bi -ternry subsemigroup, N -fuzzy bi - left (right, lterl, qusi nd bi) idels hve been defined nd the reltionship between them hve been investigted. The chrcteriztions of bi-ternry semigroup by these idels hve been discussed here.. Preliminry Concepts Definition. (Akrm et l., 0 ) Let T = { x, y, z,...} nd = { α, β, γ,...} be two nonempty sets. Then T is clled bi-ternry semigroup if it stisfies (i) xα yβz T (ii) ( xα yβz) γuδv = xα( yβzγu) δv= xαyβ ( zγuδv), for ll x, y, z, u, v nd α, β, γ, δ. Exmple. (Akrm et l., 0) Let T = { n+, n N} nd = { n+, n N}. Define the mpping T T T T s x α yβz= x+ α+ y+ β + z. Then T is bi-ternry semigroup.

5 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of Exmple. ( Akrm et l., 0) Let T = Z nd Z. Define xγ yδz, for x, y, z T nd γ, δ s the usul multipliction of integers. Then T is bi-ternry semigroup but not -semigroup. Exmple. (Akrm et l., 0) Let T = ir, where, i = nd R is the set of rel numbers. If R nd xα yβz is defined s the usul multipliction of complex numbers T is bi-ternry semigroup but not -semigroup. Definition. (Akrm et l., 0) A nonempty subset A of bi-ternry semigroup T is clled bi -ternry subsemigroup of T if A AA A. Exmple. (Akrm et l., 0) Let T = N = {,,,...} nd = { n+, n N}. Define x α yβz= x+ α+ y+ β + z. Then T is bi-ternry semigroup. Let A= { n, n N} be nonempty subset of T. Then A is bi-ternry subsemigroup of T. Definition. (Akrm et l., 0) A nonempty subset A of bi-ternry semigroup T is clled bi-left idel of T if T TA A. Definition. (Akrm et l., 0) A nonempty subset A of bi-ternry semigroup T is clled bi-right idel of T if A TT A.

6 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Definition. (Akrm et l., 0) A nonempty subset A of bi-ternry semigroup T is clled bi -lterl idel of T if T AT A. Definition.0 (Akrm et l., 0) A nonempty subset A of bi-ternry semigroup T is clled bi-idel of T if it is bi-left, bi-right nd bi-lterl idel of T. Definition. (Akrm et l., 0) Let T be bi-ternry semigroup. A nonempty subset Q of T is clled bi-qusi-idel of T if QTT TQT TTQ Q QTT TTQTT TTQ Q. Definition. (Akrm et l., 0) Let T be bi-ternry semigroup. A bi-ternry subsemigroup B of T is clled bi -bi-idel of T if B TBTB B. Definition. (Akrm et l., 0) Let T = { n, n N}, = { α, β, γ,...} nd A= { n, n N}. Define, x α yβz= x+ y+ z, for x, y, z T nd α, β. Then T is bi-ternry semigroup nd A is bi-left idel of T but neither bi-right nor bi-lterl idel of T. If we define, nd xα yβz= x+ y+ z nd xα yβz= x+ y+ z respectively, then A is bi -right nd bi -lterl idel of T. Exmple. In the bove exmple if we define, x α yβz= x+ y+ z, then A is bi-left, bi-right nd bi-lterl idel of T. Hence A is bi -idel of T. Proposition. (Akrm et l., 0) Let T be bi-ternry semigroup nd

7 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of φ X T. Then (i) (ii) (iii) T TX is bi-left idel of T. X TT is bi-right idel of T. TXT TTXT T is bi-lterl idel of T.. N-fuzzy bi -ternry subsemigroup nd N-fuzzy bi -idels Definition. (Khn, 00) A negtive fuzzy set (briefly, N -fuzzy set or NF) in nonempty set X is function : X [, 0]. Here we re using fuzzy function. " " for the negtive Jun et l. ( Jun et l., 00) used the term negtive-vlued function nd N -function for negtive fuzzy set nd N -fuzzy set. Definition. Let be n NF in X nd t [,0]. Then the set N( ; t) = { x X ( x) t} is clled n N -level subset of. Definition. Let nd be two NFs in X. If for ll x X, ( x ) ( x) then is clled n N -fuzzy subset (NF) of nd is written s. We sy tht = if nd only if nd. Definition. Let nd be two NFs in X. Then their union nd intersection is

8 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm lso n N -fuzzy set in X, defined s, for ll x X, ( ) = { x, min{ ( x), ( x)}} = ( ( x) ( x)), nd ( ) = { x, mx{ ( x), ( x)}} = ( ( x) ( x)). Definition. Let X = {, b, c, d} be nonempty set. Define : X [,0] s, ( ) = 0., ( b) = 0., ( c) = 0., ( d) = 0.. Then = { <, 0.>, < b, 0.>, < c, 0.>, < d, 0.> }, obviously, is n N -fuzzy set in X. Now, let = { <, >, < b, 0.>, < c, 0.>, < d, 0. } nd > = { <,.>, < b, 0.>, < c, 0.>, < d, 0.> }. Then nd re N -fuzzy sets in X. Esily we cn verify tht. If, we tke = { <, 0.>, < b, 0.>, < c, 0.>, < d, 0.> } = { <, 0.>, < b, 0.>, < c, 0.>, < d, 0.> }. Then nd re N -fuzzy sets in X nd = { <, 0.>, < b, 0.>, < c, 0.>, < d, 0.> } = { <, 0.>, < b, 0.>, < c, 0.>, < d, 0.> }. Obviously, nd re N -fuzzy sets in X. In wht follows, let T denotes bi-ternry semigroup unless otherwise specified. Definition. Let be nonempty subset of T. Then the N - chrcteristic function of

9 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of is function χ defined s, for ny x T, χ, if x ( x) = 0, if x. We denotes the N -chrcteristic function of T by χ. Definition. Let, nd be the three NFs in T. Then their product is given s,, o ( o where for ny x T, o o )( = = α β x x b c ) { ( ) ( b) ( c)}, if 0, x= αbβc, for, b, c T, α, β otherwise. Definition. Let be n NF in T. Then is clled n N -fuzzy bi-ternry subsemigroup of T if ( x αyβz) mx{ ( x), ( y), ( z)}for ll x, y, z T, α, β. Definition. Let be n NF in T. Then is clled n N -fuzzy bi-left idel of T if ( x αyβz) ( z), for ll x, y, z T, α, β. Definition.0 Let be n NF in T. Then is clled n N -fuzzy bi-right idel oft if ( x αyβz) ( x), for ll x, y, z T, α, β. Definition. Let be n NF in T. Then is clled n N -fuzzy bi-lterl idel oft if ( x αyβz) ( y), for ll x, y, z T, α, β.

10 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Definition. Let be n NF in T. Then is clled n N -fuzzy bi-idel oft if it is bi-left, bi-right nd bi-lterl idel of T. + Exmple. Let T = Z nd = Z. ThenT is bi-ternry semigroup but not semigroup under the usul multipliction of integers i.e. for x, y, z T, α, β, ( xα yβz) = xαyβz. Now define, : T [,0] s, for x T, 0., if x is even ( x) = 0., otherwise. Then is n N -fuzzy set in T. By simple clcultions we cn verify tht is n N -fuzzy bi -ternry subsemigroup of T. Exmple. Let T = {, b, c} nd = {α }. ThenT is bi-ternry semigroup under the opertion defined in the following tble, α b c b b c c b c. Define, : T [,0] such tht = { <, 0.>, < b, 0.>, < c, 0.> }. Then is n N -fuzzy set in T which is n N -fuzzy bi -ternry subsemigroup of T. Further we cn verify tht A is not n N -fuzzy bi -left (bi -right, bi -lterl) idel of T. If we tke = { <, 0.>, < b, 0.>, < c, 0.> }, then is n N -fuzzy bi-left, bi -right nd bi-lterl idel of T, hence N -fuzzy bi -idel of T. Obviously it is

11 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge 0 of n N -fuzzy bi -ternry subsemigroup of T. Exmple. Let T = { n, n N} nd = { α, β, γ,...}. Define x α yβz= x+ y+ z, for x, y, z T, α, β, then T is bi-ternry semigroup. Now define, : T [,0] s 0., if ( x) = 0., x= n for some n N Then is n N -fuzzy bi-left idel of T. otherwise. Now, for,, T, α, β, ( α ) β = () + () + = 0= (). Thus ( α β ) = (()) = 0. nd ( ) = 0. implies tht ( α β ) (). Hence is not n N -fuzzy bi -right idel of T. imilrly, we cn verify tht is not n N -fuzzy bi-lterl idel of T. If we define xα yβz= x+ y+ z nd x αyβz = x+ y+ z, then is n N -fuzzy bi-right idel oft nd N -fuzzy bi-lterl idel of T, respectively. Further more if x αyβz = x+ y+ z, then is n N -fuzzy biidel of T. From Exmple. &., we cn write the following remrk. Remrk. In bi-ternry semigroup T, (i) An N -fuzzy bi-left (right, lterl ) idel of T is n N -fuzzy bi-ternry subsemigroup of T but the converse is not true. 0

12 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm (ii) An N -fuzzy bi-left idel of T my not be n N -fuzzy bi-right (lterl) idel of T nd vice vers. Lemm. LetT be bi-ternry semigroup then, (i) The intersection of ny collection of N -fuzzy bi-ternry subsemigroups oft is n N -fuzzy bi -ternry subsemigroup of T. (ii) The intersection of ny collection of N -fuzzy bi-left (right, lterl) idels of T is n N -fuzzy bi -left (right, lterl) idel of T. Proof. (i) Let { i, i I} be collection of N -fuzzy bi-ternry subsemigroups of T, then ( xαyβz) mx{ ( x), ( y), ( z)}, for ll i I, x, y, z T, α, β. Now, for i i i i ll for i I, x, y, z T, α, β. ( i i I )(xαyβz) = (( xαyβz)) (mx{ ( x), ( y), ( z)}) Hence i I i i I i = mx{ ( x), ( y), ( z)} = mx{( )( x),( )( y),( )( z)}. i I i i I is n N -fuzzy bi-ternry subsemigroup oft. i I i (ii) Proof is similr s (i). i i I Proposition. Let be n NF in T then (i) is n N -fuzzy bi-ternry subsemigroup oft if nd only if, i i i i I i i I i i I i

13 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of o. o (ii) is n N -fuzzy bi-left idel oft if nd only if, χ o χ. o (iii) is n N -fuzzy bi-lterl idel oft if nd only if, χ o χ. o (iv) is n N -fuzzy bi -right idel oft if nd only If, o χ χ. o Proof. (i) Let be n N -fuzzy bi -ternry subsemigroup oft nd x T. Cse. If x αbβc, forα, β,, b, c T then ( o )( x ) = 0 ( )( x). Cse. If x= αbβc, forα, β nd, b, c T, then o ( o o )( x) min {mx( ( ), ( b), ( c))} min ( αbβc) ( x), x T. = x= αbβc This implies tht o. o x= αbβc Conversely, we suppose tht o. Let, α, β,, b, c T nd x= αbβc then, o ( αbβc ) = ( x) ( o o )( x) = min {mx( ( u), ( v), ( w))} mx( ( ), ( b), ( c)). x= uδvθw Hence is n N -fuzzy bi -ternry subsemigroup of T. Proof of imilrly, we cn prove (ii), (iii) nd (iv). Lemm. Let be n NF in T. Then (i) χ o χ o is n N -fuzzy bi -left idel of T. (ii) o χ o χ is n N -fuzzy bi -right idel of T. (iii) χ o o χ is n N -fuzzy bi -lterl idel of T.

14 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Proof. (i) Let L = χ o χ o. Then χ o χ o L χ o χ o χ o χ o χ o χ o = L. Hence L = χ o χ o is n N - = fuzzy bi -left idel of T. imilrly, we cn prove (ii) nd (iii). Theorem.0 Let be n NF in T. Then is n N -fuzzy bi-ternry subsemigroup of T if nd only if N ( ; t) is bi-ternry subsemigroup of T, for ll t [,0]. Proof. Let be n N -fuzzy bi -ternry subsemigroup of T. Let x, y, z N( ; t), where t [,0] then ( x) t, ( y) t nd ( z) t. Now for α, β ( xαyβz) mx{ ( x), ( y), ( z)} t. This implies tht x αyβz N( ; t), for ll x, y, z N ( ; t) nd α, β. Hence N ( ; t) is bi -ternry subsemigroup of T. Conversely, we suppose tht N ( ; t) is bi -ternry subsemigroup of T, for ll t [,0]. Let x, y, z T such tht ( x ) = tx, ( y ) = t y nd ( z ) = tz with t, t, t [,0] then x N ; t ), y N ; t ) nd z N ; t ). We my ssume tht t x x y y z z ( x A ( y ( z t t then N ; t ) N ( ; t ) N( ; t ), which implies tht x y, z N( ; t ). ( x y z, z ince, N ; t ) is bi -ternry subsemigroup of T implies tht x αyβz N ; t ), for ( z α, β. Then ( xαyβz) t = mx( t, t, t ) = mx( ( x), ( y), ( z) ), z x y z ( z for ll x, y, z T nd α, β. Hence is n N -fuzzy bi-ternry subsemigroup of T.

15 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of Theorem. Let be n NF in T. Then is n N -fuzzy bi-left (right, lterl) idel of T if nd only if N ( ; t) is bi-left (right, lterl) idel of T, for ll t [,0]. Proof. trightforwrd. Theorem. A nonempty subset of T is bi-ternry subsemigroup of T if nd only if χ is n N -fuzzy bi -ternry subsemigroup of T. Proof. Let be bi-ternry subsemigroup of T then. Let x, y, z T, α, β then we hve following cses. Cse. If x, y, z then xα yβz nd hence χ ( x) = χ ( y) = χ ( z) = χ ( xαyβz) = implies tht χ ( xαyβz) = mx{ χ ( x), χ ( y), χ ( z)}. Cse. If either x or y or z then either χ ( x) = 0 or χ ( y) = 0 or χ ( z) = 0. This implies tht, mx{ χ ( x), χ ( y), χ ( z)} = 0 but χ ( xαyβz) 0 implies tht χ ( xαyβz) mx{ χ ( x), χ ( y), χ ( z)}, for ll x, y, z T, α, β. Cse. When ny two of Cse. When ll x, y, z re not in. x, y, z re not in. Above both cses gives the sme results s in Cse. Hence χ is n N -fuzzy biternry subsemigroup of T. Conversely, we suppose tht χ is n N -fuzzy bi -ternry subsemigroup of T. Let

16 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm x, y, z nd α, β then xα yβz. By definition of χ, χ ( x) = χ ( y) = χ ( z) = implies tht mx{ χ ( x), χ ( y), χ ( z)} =. ince χ is n N -fuzzy bi -ternry subsemigroup of T, χ ( xαyβz) mx{ χ ( x), χ ( y), χ ( z)} = implies tht χ ( xαyβz) but by definition χ ( xαyβz), which implies tht χ ( xαyβz) =. This gives tht xα yβz implies tht. Hence is bi-ternry subsemigroup of T. Theorem. A nonempty subset of T is bi-left (right, lterl) idel of T if nd only if χ is n N -fuzzy bi -left (right, lterl) idel of T. Proof. trightforwrd. Definition. Let be nonempty subset of T nd, b [,0] with b. Define N -fuzzy set C in T s for ll x T, C if x ( x) = b if x. Lemm. A nonempty subset of T is bi-ternry subsemigroup (left idel, right idel, lterl idel) of T if nd only if idel, right idel, lterl idel) of T. C is n N -fuzzy bi-ternry subsemigroup (left Proof. We prove this result for bi-right idels. Let be bi- right idel of T nd x, y, z T. If x then xα yβz implies tht C ( x) = = C ( xαyβz). If x then

17 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of C ( x) = b C ( xαyβz). Hence C is n N -fuzzy bi -right idel of T. Conversely, we suppose tht C is n N -fuzzy bi -right idel of T. Let x then C ( x) =. For y, z T nd α, β, C ( xαyβz) C ( x) = but C ( xα yβz) implies tht C ( xα yβz) = implies tht xα yβz TT. Hence is biright idel of T. The result for other cses is similr.. N-fuzzy bi-qusi idels nd N-fuzzy bi-bi-idels Definition. Let be n NF in T. Then is clled n N -fuzzy bi-qusi idel of T if for ll x T, ( x) mx{( o ( x) mx{( o Alterntively, o o χo χo χ o χ o χ)( x),( χ o χ)( x),( χ o χ χ o χ χ o o χ o o χo o χ)( x),( χ o o χ χ o χ o χ o χ o χ o χ)( x),( χ o χ χ o )( x)}, χ o χ o. )( x)}. Proposition. Every N -fuzzy bi-qusi idel of T is n N -fuzzy bi-ternry subsemigroup of T. Proof. Let be n N -fuzzy bi -qusi idel of T. ince, o χ o χ o o, χ o o χ o o nd χ o χ o o o implies tht o χ o χ χ o o χ χ o χ o o o. Also, since is N -fuzzy

18 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm bi-qusi idel of T, so o χ o χ χ o o χ χ o χ o. This implies tht o o. Hence by Proposition., is n N -fuzzy bi-ternry subsemigroup of T. Lemm. Let { i, i I} be collection of N -fuzzy bi -qusi idels of T then is lso n N -fuzzy bi -qusi idel of T. Proof. trightforwrd. Lemm. Every N -fuzzy bi-left (right, lterl) idel of T is n N -fuzzy bi-qusi idel of T. Proof. trightforwrd. i I Theorem. Let be n NF in T. Then is n N -fuzzy bi-qusi idel of T if nd only if N ( ; t) is bi-qusi idel of T, for ll t [,0]. Proof. We suppose tht is n N -fuzzy bi-qusi idel of T then Let o χ o χ χ o o χ χ o χ o. m N( ; t)tt then m= nαyβz for n N( ; t), y, z T nd α, β. ince n N( ; t) implies tht ( n) t. Now, ( o χ o χ)( m) min {mx( ( n), χ( y), χ( z)} mx( t,, ) t = x= nαyβz i

19 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of imilrly, ( χ o o χ)( m t nd ( χ o χ o )( m t implies tht ) ) mx{( o χ o χ)( m),( χ o o χ)( m),( χ o χ o )( m)} t i.e. ( o χ o χ χ o o χ χ o χ o )( m) t, nd by supposition, ( m ) ( o χ o χ χ o o χ χ o χ o )( m) implies tht ( m) t. This gives tht m = nαyβz N( ; t) implies tht N ( ; t) TT N ( ; t). o, N ( ; t) TT TN ( ; t) T TTN ( ; t) N ( ; t) nd N ( ; t) TT TTN ( ; t) TT TTN ( ; t) N ( ; t). Hence N ( ; t) is bi -qusi idel of T. Conversely, we suppose tht N ( ; t) is bi-qusi idel of T, for ll t [,0]. We hve to prove tht is n N -fuzzy bi -qusi idel of T. On contrry, we suppose tht there exist some m T such tht ( o χ o χ χ o o χ χ o χ o )( m ) < ( m), i.e. ( o χ o χ)( m ) ( χ o o χ)( m) ( χ o χ o )( m) < ( m). Now, choose t [,0] such tht ince, implies tht ( o χ o χ χ o o χ χ o χ o )( m ) t < ( m), ( o χ o χ)( m t implies tht m N( ; t)tt nd ( χ o o χ)( m t This gives tht ) ) m TN (; t) T nd ( χ o χ o )( m t implies tht m TTN( ; t). ) m N( ; t) TT TN ( ; t) T TTN ( ; t) N( ; t) m N( ; t) ( m) t, which is contrdiction. o ( o χo χ χo o χ χo χo )( m) ( m), m T.

20 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm i.e. ( o χ o χ χ o o χ χ o χ o ). imilrly, we cn verify tht o χ o χ χ o χ o o χ o χ χ o χ o. Hence is n N -fuzzy bi-qusi idel of T. Theorem. A nonempty subset of T is bi-qusi idel of T if nd only if χ is n N -fuzzy bi -qusi idel of T. Proof. Let be bi-qusi idel of T. For m T either then χ ( m) = nd ( χ o o )( m) implies tht T T ( χ o χ χ)( m) χ ( m) implies tht If o m or m. If ( χ o χ o χ )( m) χ o χ o χ )( m) χ o χ o χ )( m) χ ( m) i.e. ( χ o χ o χ χ o χ o χ χ o χ o χ )( m) χ ( m). m then either m m= αbβc or m αbβc, for, b, c T nd α, β. When m αbβc, for, b, c T then ( χ o χ o χ )( m) = 0. Also χ ( m) = 0 implies tht ( χ o χ χ)( m) = χ ( m). When m= αbβc, for, b, c T then mximum two of o, b, c my contined in otherwise if, b, c then m= α bβc TT TT TT (ince is bi-qusi idel of T ) m, which is contrdiction. We hve following cses, (i) m= αbβc nd, b, c. In this cse ( χ o χ o χ )( m) = ( χ o χ o χ )( αbβc) = min{mx( χ ( ), χ( b), χ( c)) = mx{( 0,, )} = 0.

21 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge 0 of This implies tht ( χ o χ o χ )( m) = 0. imilrly, ( χ o χ o χ )( m) = 0 nd ( χ o χ o )( m) = 0. ince χ ( m) = 0, which implies tht (ii) χ ( χ o χ o χ χ o χ o χ χ o χ o χ )( m) = χ ( m). m= αbβc nd exctly one of, b, c contined in. Let then χ ( m) = 0 nd χ ( ) =. Also, b nd c ( χ o χ o χ )( m) = ( χ o χ o χ )( αbβc) = min{mx( χ ( ), χ( b), χ( c)) = mx{(,, )} =. imilrly, ( χ o χ o χ )( m) = 0 nd ( χ o χ o )( m) = 0 implies tht (iii) χ ( χ o χ o χ )( m) ( χ o χ o χ )( m) ( χ o χ o χ )( m) = 0= χ ( m). m= αbβc nd exctly two of, b, c contined in. Let, b nd c then χ ( m) = 0 nd χ ( ) =, χ ( b) =. In this cse ( χ o χ o )( m) =, T ( o χ o χ)( m) = nd ( χ o χ o )( m) = 0, which implies tht T χ ( χ o χ o χ )( m) ( χ o χ o χ )( m) ( χ o χ o χ )( m) = χ ( m). Hence, for ll m T, ( χ o χo χ)( m) ( χo χ o χ)( m) ( χo χo χ )( m) χ ( m). This implies tht χ o χ o χ χ o χ o χ χ o χ o χ χ. imilrly, we cn verify tht χ o χ o χ χ o χ o χ o χ o χ χ o χ o χ χ. Hence, χ is n N -fuzzy bi-qusi idel of T. Conversely, we suppose tht χ is n N -fuzzy bi-qusi idel of T. Let x TT TT TT. Then x TT, x TT nd x TT implies 0

22 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm tht x= α bβ c = α b β c = α b β, where c b, c, b, c,, c, b T nd α β, α, β, α, β. Then,, ( χ o χ o χ)( x) = ( χ o χ o χ)( α bβ c) =, ( χo χ o χ)( x) = ( χo χ o χ)( αbβ c) =, ( χo χo χ )( x) = ( χo χo χ )( αbβ c) =., This implies tht ( ( χ o χ o χ χ o χ o χ χ o χ o χ )( x) =. ince, χ is n N -fuzzy bi-qusi idel of T, ( χ o χ o χ χ o χ o χ χ o χ o χ )( x) χ ( x). This implies tht ( x) but χ ( x) implies tht χ ( x) = implies tht χ x tht is TT TT TT. Likewise, we cn verify tht TT TTTT TT. Hence, is bi-qusi idel of T. Lemm. A nonempty subset of T is bi-qusi-idel of T if nd only if N -fuzzy bi -qusi idel of T. Proof. trightforwrd. The following Exmples. &., shows tht the converses of Proposition. nd Lemm. re not true in generl. C is n Exmple. Let T = {, b, c} nd = {α }. Then T is bi-ternry semigroup long with the opertion defined in the tble below.

23 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of α b c Let A= { b, c} then A is bi-ternry subsemigroup of T but not bi-qusi idel of T. Now, define : T [,0] s ( ) = 0., ( b) = ( c) = 0.. Then T if N (, t) = { b, c} if Φ if b b c c b c. t [ 0., 0] t [ 0., 0.) t [, 0.). Obviously, N (, t) is bi-ternry subsemigroup of T but not bi-qusi idel of T, for ll t [,0]. Then by Proposition. nd Theorem., is n N -fuzzy bi - ternry subsemigroup of T but not n N -fuzzy bi -qusi idel of T. Exmple. In bove exmple if we tke, A = {, c} then A is bi -qusi idel of T. Further more A is bi-left idel of T but neither bi-right idel nor bi-lterl idel of T. Now, define : T [,0] s ( ) = ( c) = 0., ( b) = 0.. Then T if N (, t) = {, c} if Φ if t [ 0., 0] t [ 0., 0.) t [, 0.). Obviously, N (, t) is bi-qusi idel of T, for ll t [,0] but neither bi-right idel nor bi-lterl idel of T for t [ 0., 0.). Hence by Theorem., is n N -fuzzy bi-qusi idel of T nd by Theorem., is neither n N -fuzzy bi-right idel nor n N -fuzzy bi -lterl idel of T. imilrly, we cn construct exmples of

24 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm N -fuzzy bi -qusi idel of T, which re not N -fuzzy bi -left idel of T. Definition.0 Let be n NF in T. Then is clled n N -fuzzy bi-bi-idel of T if (i) is n N -fuzzy bi -ternry subsemigroup of T. (ii) For ll x, y, z T, α, β, η, δ, ( xαuβyηvδz) mx( ( x), ( y), ( z)). B B B Proposition. Let be n NF in T. Then is n N -fuzzy bi-bi-idel of T if nd only if (i) o nd o (ii) o χ o o χ o. Proof. We suppose tht is n N -fuzzy bi-bi-idel of T then it is bi-ternry subsemigroup of T nd by Proposition., condition (i) holds. Now for (ii), let m T. If m xαyβz, for x, y, z T nd α, β then ( o χ o o χ o )( m ) = 0 ( m). If m= xαyβz nd x = uδvθw, for u, v, w T nd δ, θ then ( o χ o o χ o )( m) = min {mx{( o χ o = = = m= xαyβz )( x), χ( y), ( z)}} min {mx{ min {mx( ( u), χ( v), ( w))}, χ( y), ( z)}} m= xα yβz x= uδvθw min {mx{ min {mx( ( u),, ( w))},, ( z)}} m= xαyβz x= uδvθw min { min {mx{mx( ( u), ( w))}, ( z)}} m= xα yβz x= uδvθw

25 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of = = min { min {mx( ( u), ( w), ( z)}} m= xα yβz x= uδvθw min m= uδvθwαyβz {mx( ( u), ( w))}, ( z)}} min ( uδvθwαyβz) = (m ). m= uδvθwαyβz This implies tht o χ o o χ o. Conversely, we suppose tht (i) nd (ii) holds for ny N -fuzzy subset of T. Let m= xαuβyδvθz, for x, y, z, u, v T, α, β, δ, θ. As by (ii), o χ o o χ o implies tht ( x αuβyδvθz) = ( m) ( o χ o o χ o )( m) = min {mx{( o χ o m= xαuβyδvθz )( xαuβy), χ( v), ( z)}} mx{ min {mx( ( x), χ( u), ( y))}, χ( v), ( z)} n= xαuβy min {mx{mx( ( x),, ( y))},, ( z)} = n = xαuβy mx( ( x ), ( y), ( z)). Hence is n N -fuzzy bi-bi-idel of T. Lemm. Let nd be two NFs in T. Then o χ o is lso n N -fuzzy bi -bi-idel of T. Proof. trightforwrd.

26 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Theorem. Let, nd be three NFs in T. Then o o is lso n N -fuzzy bi -bi-idel of T if ny one of, or is either n N -fuzzy bi-left idel or n N -fuzzy bi -right idel or n N -fuzzy bi -lterl idel of T. Proof. trightforwrd. Lemm. Every N -fuzzy bi -qusi idel of T is n N -fuzzy bi -bi-idel of T. Proof. trightforwrd. Lemm. Every N -fuzzy bi -left (right, lterl) idel of T is n N -fuzzy bi -biidel of T. Proof. trightforwrd. Lemm. Let { i, i I} be collection of N -fuzzy bi -bi-idels of T then lso n N -fuzzy bi -bi-idel of T. Proof. trightforwrd. is Theorem. Let be n NF in T. Then is n N -fuzzy bi-bi-idel of T if nd only if N ( ; t) is bi-bi-idel of T, for ll t [,0]. i I i

27 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of Proof. We suppose tht is n N -fuzzy bi-bi-idel of T nd m N( ; t) TN ( ; t) TN ( ; t). Then m= nαxβoδyθq for n, o, q N( ; t), x, y T nd α, β, δ, θ. ince n, o, q N ( ; t) implies tht ( n), ( o), ( q) t. Now, since is N -fuzzy bi-bi-idel of T so ( m) = ( nαxβoδyθq) mx( ( n ), ( o), ( q)) mx( t, t, t) = t implies tht ( m) t. This implies tht m N( ; t) tht is N ( ; t) TN ( ; t) TN ( ; t) N( ; t). Hence N ( ; t) is bi-bi-idel of T. Conversely, we suppose tht N ( ; t) is bi-bi-idel of T, for ll t [,0]. Let x, y, z T such tht ( x ) = tx, ( y) = t y nd ( z ) = tz with t x, t y, tz [,0]. Then x N ; t ), y N ( ; t ) nd z N ; t ). We my ssume tht tx t y tz nd ( x y ( z then N ; t ) N ( ; t ) N ; t ). This implies tht x y, z N( ; t ). ince N ; t ) ( x y ( z, z ( z is bi-bi-idel of T then for u, v T, α β, η, θ, x αuβyδvθz N( ; t ), so we hve, z x αuβyδvθz) t = mx( t, t, t ) = mx( ( x ), ( y), ( z)). ( z x y z Above holds for ll x, y, z, u, v T nd α, β, η, θ. Hence is n N -fuzzy bi-biidel of T. Theorem. A nonempty subset of T is bi-bi-idel of T if nd only if χ is n N -fuzzy bi -bi-idel of T.

28 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Proof. We suppose tht is bi-bi-idel of T then it is bi -ternry subsemigroup of T nd by Theorem., χ is n N -fuzzy bi -ternry subsemigroup of T. Also TT. Now for ny x, y, z, u, v T, α, β, η, θ, xαuβyηvθz T. We hve following cses, (i) If x, y, z then xα uβyηvθz TT implies tht χ ( x) = χ ( y) = χ ( z) = = χ ( xαuβyηvθz). Hence χ ( xαuβyηvθz) = = mx( χ ( x), χ ( y), χ ( z)). (ii) If either x or y or z then either χ ( x) = 0 or χ ( y) = 0 or χ ( z) = 0 implies tht mx( χ ( x), χ ( y), χ ( z)) = 0. But χ ( xαuβyηvθz) 0. This implies tht χ ( xαuβyηvθz) mx( χ ( x), χ ( y), χ ( z)). (iii) If ny two of (iv) If x A nd x, y, z re not in. It is sme like (ii). y A nd z A. It is lso sme like (ii). Hence χ is n N -fuzzy bi-bi-idel of T. Conversely, we suppose tht χ is n N -fuzzy bi-bi-idel of T. For ny t TT there exists x, y, z, u, v T nd α, β, η, θ such tht t= xαuβyηvθz. Then χ ( x) = χ ( y) = χ ( z) = implies tht mx( χ ( x), χ ( y), χ ( z)) =. ince χ is n N -fuzzy bi-bi-idel of T implies tht χ ( xαuβyηvθz) mx( χ ( x), χ ( y), χ ( z)) =. But by definition χ ( xαuβyηvθz). This gives tht χ ( xαuβyηvθz) = implies tht t= xα uβyηvθz. Which shows tht TT. Hence is bi-bi-idel of T.

29 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge of Lemm. A nonempty subset of T is bi-bi-idel of T if nd only if N -fuzzy bi -bi-idel of T. Proof. trightforwrd. C is n Exmple.0 Let T = {, b, c} nd = {α }. Then T is bi-ternry semigroup long with the opertion defined in the bellow tble. α b c Define, : T [,0] such tht = { <, 0.>, < b, 0.>, < c, 0.> }. Then is n N -fuzzy bi -bi-idel of T. Also, ( c αcαb) = ( c) = 0. 0.= ( b) nd ( cαbαc) = ( c) = 0. 0.= ( b). This implies tht is neither N -fuzzy bi -left nor N -fuzzy bi -lterl idel of T. + Exmple. Let T = Z nd = Z. Then T is bi -ternry semigroup but not - semigroup under the usul multipliction of integers i.e. for x, y, z T, α, β, ( xα yβz) = xαyβz. Define, : T [,0] s, for x T, 0., if x is even ( x) = 0., otherwise. b b c c b c.

30 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Then is n N -fuzzy set in T. By simple clcultions we cn verify tht is n N - fuzzy bi-bi-idel of T s well s N -fuzzy bi-idel T. Exmple. Let where b c T = 0 0 d,, b, c, d, e Z 0 nd = Z 0 0 e Z 0 is the set of ll non-positive integers. Then T is bi-ternry semigroup under the usul nd sclr multipliction of mtrices. Now consider 0 B = 0 0 m m, 0 m Z0. Then B is bi-bi-idel of T but not bi-qusi idel of T, s we cn see below, for s B, x, y, z T nd α, β, where s = 0 0, x= 0 0, y= 0 0 0, z= , nd 0 α =, β =, then sαxβy = z BTT, xαsβy = z TBT nd xαyβs = z TTB. This implies tht z BTT TBT TTB but z B implies tht B TT TBT TTB B. Hence, B is not bi-bi-idel of T. Then by Theorem., χ is not n N -fuzzy bi -qusi idel of T but by Theorem., χ is n N -fuzzy bi -bi-idel of T. If, we define

31 ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Pge 0 of C B 0. if x B ( x) = 0. if x B. Then by Lemm., C B (x) is not n N -fuzzy bi-qusi idel of T but by Lemm., C B (x) is n N -fuzzy bi-bi-idel of T. References Akrm, M., Kvikumr, J. nd Khmis, A. 0. On N --idels in-ag**- Groupoids. Anlele Universittii Orde Fscicol Mtemtic. Tom XXI, Issue No., -. Akrm, M., Kvikumr, J. nd Khmis, A. 0. Chrcteriztion of bi-ternry semigroups by their idels. Itlin Journl of Pure nd Applied Mthemtics. N-(), -. Atnssov, K.. Intuitionistic fuzzy sets, VII, ITKR's ession, ofi, June (Deposed in Centrl cience nd Technicl, Librry of Bulgrin Acdemy of cience, /) (in Bulg.). Bisws, R.. Intuitionistic fuzzy subgroupoids. Mthemticl Forum x, -. Jun, Y.B., Lee, K.J. nd ong,.z. 00. N -idels of BCK/BCI-lgebrs. Journl Chungcheong Mthemticl ociety., -. Khn, A., Jun, Y.B. nd hbir, M. 00. N -fuzzy idels in ordered semigroups. Interntionl Journl of Mthemtics nd Mthemticl ciences. 00, Article ID, pges. Khn, A., Jun, Y.B. nd hbir, M. 00. N -fuzzy qusi-idels in ordered semigroups. 0

32 Pge of ongklnkrin Journl of cience nd Technology JT-0-0.R Akrm Qusigroups nd Relted ystems., -. Kim, K.H. nd Jun, Y.B. 00. Intuitionistic fuzzy idels of semigroups. Indin Journl of Pure nd Applied Mthemtics. (), -. Rosenfeld, A.. Fuzzy groups. Journl of Mthemticl Anlysis nd Applictions., -. Zdeh, L.A.. Fuzzy sets. Informtion & Control., -.