A New Generalization of Fuzzy Ideals of Ternary Semigroups

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1 Appl Math Inf Sci 9, No 3, Applied Mathematics & Information Sciences An International Journal A New Generalization of Fuzzy Ideals of Ternary Semigroups Noor Rehman 1,2, and Muhammad Shabir 1 1 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan 2 Department of Basic Sciences, Riphah International University, Islamabad, Pakistan Received: 29 Aug 2014, Revised: 30 Nov 2014, Accepted: 31 Nov 2014 Published online: 1 May 2015 Abstract: Our aim in this paper is to introduce and study a new sort of fuzzy ternary subsemigroup left ideal, right ideal, lateral ideal, quasi-ideal, bi-ideal of a ternary semigroup, called interval-valued α, β-fuzzy ternary subsemigroup left ideal, right ideal, lateral ideal, quasi-ideal, bi-ideal We also characterize regular ternary semigroups in terms of these interval-valued fuzzy ideals Keywords: Ternary semigroup, fuzzy point, interval valued, q-fuzzy ideals, interval valued, q-fuzzy bi-ideals 1 Introduction The study of ternary algebraic systems was initiated by Lehmer [1] in 1932 He investigated certain algebraic systems which turn out to be ternary groups Banach [2], showed by an example that a ternary semigroup does not necessarily reduced to an ordinary semigroup Sioson [3], studied ternary semigroups with a special reference of ideals and radicals He also introduced the notion of regular ternary semigroup and characterized them by using the notion of quasi-ideals Dixit and Dewan studied quasi-ideals and bi-ideals of ternary semigroup in his paper [4] The algebraic structures play a prominent role in mathematics with wide range of applications in variety of disciplines such as computer science, information science, control engineering, pattern recognition etc The theory of fuzzy sets was first developed by Zadeh [5] and has been applied to many branches in mathematics Rosenfeld started the fuzzification of algebraic structures [6] Kuroki is responsible for much of ideal theory of fuzzy semigroups see [7, 8, 9, 10, 11] Later, in 1975 Zadeh made an extension of the concept of a fuzzy set by an interval-valued fuzzy set, that is, a fuzzy set with an interval-valued membership function [12] The interval-valued fuzzy subgroups were first defined and studied by Biswas [13] which are the fuzzy subgroups of the same nature of the the fuzzy subgroups defined by Rosenfeld A new type of fuzzy subgroup, that is,, q-fuzzy subgroup was introduced in an earlier paper of Bhakat and Das [14] by using the combined notions of belongingness and quasi-coincidence of fuzzy point and fuzzy set, which was introduced by Pu and Liu [15] In fact, the, q-fuzzy subgroup is a significant generalization of Rosenfeld s fuzzy subgroup The present authors in [16], defined and discussed α, β-fuzzy ideals in ternary semigroups For further studies on this topic the reader is referred to [17,18,19, 20] Akram et al in [21] introduced the notion of interval-valued α, β-fuzzy k-algebras Zhan et al put forth the idea of interval-valued, q-fuzzy fuzzy filters in pseudo BL-algebras [22] Ma et al discussed interval-valued fuzzy ideals of pseudo MV-agebras see [23] Recently Shabir and Mahmood studied interval-valuedα, β-fuzzy ideals of hemirings In this paper we initiate the study of interaval-valued, q-fuzzy ternary subsemigroups left ideals, right ideals, interval-valued, q-fuzzy quasi-ideals and interval-valued, q-fuzzy bi-ideals in teranary semigroups and several related properties are studied We also characterize regular ternary semigroups in terms of these ideals 2 Preliminaries A ternary semigroup is an algebraic structures,[ ] with one ternary operation satisfying the associative law: Corresponding author noorrehman82@yahoocom

2 1624 N Rehman, M Shabir: A New Generalization of Fuzzy Ideals of [[x 1 x 2 x 3 ]x 4 x 5 ] [x 1 [x 2 x 3 x 4 ]x 5 ] [x 1 x 2 [x 3 x 4 x 5 ]] for all x 1,x 2,x 3,x 4,x 5 S For the sake of convenience, [x 1 x 2 x 3 ] will be written as x 1 x 2 x 3 and consider the ternary operation as multiplication A non-empty subset A of a ternary semigroup S is called a ternary subsemigroup of S if AAA A By a left right, lateral ideal of a ternary semigroup S we mean a non-empty subset A of S such that SSA A ASS A,SAS A If a non-empty subset A of S is a left and right ideal of S, then it is called a two sided ideal of S If a non-empty subset A of a ternary semigroup S is a left, right and lateral ideal of S, then it is called an ideal of S A non-empty subset A of a ternary semigroup S is called a quasi-ideal of S if ASS SAS SSA A and ASS SSASS SSA A A is called a bi-ideal of S if it is a ternary subsemigroup of S and ASASA A A non-empty subset A of a ternary semigroup S is called a generalized bi-ideal of S if ASASA A It is clear that every left right, lateral ideal of S is a quasi-ideal, every quasi-ideal is a bi-ideal and every bi-ideal is a generalized bi-ideal of S An element a of a ternary semigroup S is called regular if there exists an element x S such that axa a A ternary semigroup S is called regular if every element of S is regular 21 Example Let Z be the set of all negative integers Then with usual ternary multiplication,z forms a ternary semigroup 22 Example The set of all odd permutations of a non-empty set X, under ternary composition forms a ternary semigroup 23 Example Let S{ i,0,i Then S is a ternary semigroup under the ternary multiplication of complex numbers 24 Theorem [3] A ternary semigroup S is regular if and only if R M L RML for every right ideal R, every lateral ideal M and every left ideal L of S 25 Theorem [24] The following conditions on a ternary semigroup S are 1 S is regular; 2 BSBSB B for every bi-ideal B of S; 3 QSQSQ Q for every quasi-ideal Q of S 26 Theorem [24] The following conditions on a ternary semigroup S are 1 S is regular; 2 R L RSL for every right ideal R and every left ideal L of S By an interval number ã we mean a closed subinterval [a,a + ] where 0 a a + 1 The set of all interval numbers is denoted by D[0,1] The interval [a,a] can be identified by the number a [0, 1] For the interval numbers ã i [a i,a+ i ] and b i [b i,b + i ] D[0,1], i Λ, we define ã i b i rmax{ã i, b i [maxa i,b i,maxa + i,b+ i ], ã i b i r min{ã i, b i [mina i,b i,mina+ i,b+ i ], rinf ã i [ i Λ a i, i Λ a + ] i, rsupãi [ i Λ a i, i Λ a + ] i Define on D[0,1] an order relation by 1 ã 1 ã 2 a 1 a 2 and a+ 1 a+ 2 2 ã 1 ã 2 a 1 a 2 and a+ 1 a+ 2 3 ã 1 < ã 2 ã 1 ã 2 and ã 1 ã 2 4 kã i [ka i,ka+ i ], whenever 0 k 1 It is clear that the set D[0, 1],,, forms a complete bounded lattice with r min, r max, 0 [0,0] as the bottom element and 1 [1,1] as the top element For any two interval-valued fuzzy subsets λ and µ of S, λ µ means that, for all x S, λx µx A fuzzy subset λ of a universe S is a function from S into the unit closed interval[0,1], that is, λ : S [0,1] Let S be a set A mapping λ : S D[0,1] is called an interval-valued fuzzy set of X For three interval-valued fuzzy subsets λ, µ and ν of a ternary semigroup S we define the multiplication of these interval-valued fuzzy subsets of S by: axyz { λx µy νz λ µ ν a if there exist x,y,z S such that axyz, [0, 0] otherwise for all a S The symbols λ µ and λ µ will mean the following interval-valued fuzzy subsets of S: λ µ x λx µx λ µ x λx µx for all x S If A S, then the interval-valued characteristic function of A is a function C A of S defined by: C A x{ [1,1] if x A [0,0] if x / A

3 Appl Math Inf Sci 9, No 3, / wwwnaturalspublishingcom/journalsasp 1625 Let λ be an interval valued fuzzy subset of a ternary semigroup S Then the crisp set { U λ ; t x S: λx t, where t D[0,1], is called the level subset of λ 27 Definition An interval-valued fuzzy subset λ of a ternary semigroup S is called an interval-valued fuzzy ternary subsemigroup of S if λxyzr min { λx, λy, λz for all x, y, z S 28 Definition S is called an interval-valued fuzzy left right, lateral ideal of S if λxyz λz λx, λy for all x,y,z S 29 Definition An interval-valued fuzzy subset ternary subsemigroup λ of a ternary semigroup S is called an interval-valued fuzzy generalized bi-ideal bi-ideal of S if λxuyvz r min { λx, λy, λz for all x, y, z, u, v S 3 Interval-valued α, β-fuzzy ideals Throughout this paper S will denote a ternary semigroup and α, β {, q, q, q unless otherwise specified The concept of quasi-coincidence of a fuzzy point can be extended to the concept of quasi-coincidence of an interval-valued fuzzy set An interval-valued fuzzy set λ of a ternary semigroup S of the form λy { t [0,0] if yx, [0,0] if y x, is called an interval-valued fuzzy point with support x and interval-value t and is denoted by x t An interval-valued fuzzy point x t is said to belongs to resp be quasi-coincident with an interval-valued fuzzy set λ, written as x t λ resp x t q λ if λx tresp λx + t > [1,1] If x t λ or x t q λ, then we write x t q λ If x t λ and x t q λ, then we write x t q λ The symbol q means the statement q does not hold We also emphasis that ã [a,a + ] must satisfy the following properties: [a,a + ]<[05,05] or[05,05] [a,a + ] for all x S 31 Definition S is called an interval-valued α, β-fuzzy ternary subsemigroup of S, if: T1 x t α λ, y r α λ and z s α λ impliesxyz r min{ t, r, s β λ for all x, y, z S, where t, r, s D[0,1] Let λ be an interval-valued fuzzy set of S such that λx [05,05], for all x S Suppose that x S and [0,0] < t [1,1] be such that x t q λ Then λx t and λx + t > [1,1] It follows that [1,1] < λx+ t λx+ λx 2 λx, which implies λx > [05,05] This means that {x t x t q λ /0 Therefore, the case α q in above definition is omitted 32 Definition S is called an interval-valuedα, β-fuzzy left resp right, lateral ideal of S, where α q, if it satisfy: T2 z t α λ and x,y S implies xyz t β λ resp zxy t β λ, xzy t β λ 33 Definition S is called an interval-valuedα,β-fuzzy two sided ideal of S if it is both an interval-valued α,β-fuzzy left ideal and interval-valued α, β-fuzzy right ideal of S An interval-valued fuzzy subset of a ternary semigroup S is called an interval-valuedα,β-fuzzy ideal of S if it is an interval-valued α, β-fuzzy left ideal, interval-valued α, β-fuzzy right ideal and interval-valued α, β-fuzzy lateral ideal of S 34 Definition S is called an interval-valued α,β-fuzzy bi-ideal of S, where α q, if it satisfies the conditions T1 and T3, where, T3 For all u, v, x, y, z S and [0,0] < t 4, t 5, t 6 D[0,1], x t4 α λ, y t5 α λ and z t6 α λ implies xuyvz r min{ t 4, t 5, t 6 β λ 35 Definition S is called an interval-valuedα, β-fuzzy generalized biideal of S, where α q, if it satisfies the condition T3

4 1626 N Rehman, M Shabir: A New Generalization of Fuzzy Ideals of 36 Lemma S is an interval-valued fuzzy ternary subsemigroup of S, if and only if λ is an interval-valued, -fuzzy ternary subsemigroup of S ProofLet λ be an interval-valued fuzzy ternary subsemigroup of S Let x, y, z S and [0,0] < t 1, t 2, t 3 D[0,1] be such that x t1 λ, y t2 λ and z t3 λ Then λx t 1, λy t 2, λz t 3 Since λ is an interval-valued fuzzy ternary subsemigroup of S, so λxyz rmin { λx, λy, λz rmin { t 1, t 2, t 3 Hencexyz rmin{ t 1, t 2, t 3 λ Conversely, assume that λ is an interval-valued, -fuzzy ternary subsemigroup of S Suppose on the contrary that there exist x, y, z S such that λxyz < λx λy λz Let [0,0] < t [1,1] be such that λxyz < t λx λy λz Then x t λ, y t λ and z t λ but xyz t λ, which contradicts our hypothesis Hence λxyz λx λy λz 37 Lemma S is an interval-valued fuzzy left right, lateral ideal of S if and only if λ is an interval-valued, -fuzzy left right, lateral ideal of S ProofThe proof is similar to the proof of Lemma 36 In a similar fashion we can prove lemma 38 Lemma the following S is an interval-valued fuzzy generalized bi-ideal bi-ideal of S if and only if λ is an interval-valued, -fuzzy generalized bi-ideal bi-ideal of S 39 Theorem Let λ be a non-zero interval-valued α,β-fuzzy ternary subsemigroup resp left ideal, right ideal, lateral ideal of a ternary { semigroup S Then the set λ0 x S: λx>[0,0] is a crisp ternary subsemigroup resp left ideal, right ideal, lateral ideal of S ProofLet x, y, z λ 0 Then λx > [0,0], λy > [0,0], λz > [0,0] Suppose λxyz [0,0] If α {, q, then x λx α λ, y λy α λ, z λz α λ but λxyz [0,0] < r min { λx, λy, λz and λxyz+rmin { λx, λy, λz [0,0] + [1,1] [1,1] This implies that xyz rmin { λx, λy, λz β λ for every β {,q, q, q, which is a contradiction Hence λxyz>[0,0], that is, xyz λ 0 Also x [1,1] q λ,y [1,1] q λ,z [1,1] q λ but xyz [1,1] β λ for every β {, q, q, q Hence λxyz > [0,0] xyz λ 0 This implies that λ0 is a ternary subsemigroup of S 310 Theorem Let λ be a non-zero interval-valued α,β-fuzzy generalized bi-ideal bi-ideal of a ternary semigroup S Then the set λ { 0 x S: λx>[0,0] is a crisp generalized bi-ideal bi-ideal of S ProofThe proof is similar to the proof of Theorem Theorem Let L be a non-empty subset of a ternary semigroup S and α {, q, q Then L is a left resp right, lateral ideal of S if and only if the interval-valued fuzzy subset λ of S defined by: λx { [05,05] if x L [0,0] if x S L is an interval-valued α, q-fuzzy left resp right, lateral ideal of S ProofSuppose L is a left ideal of S 1 Let x, y, z S and [0,0] < t [1,1] be such that z t λ Then λz t and so z L This implies that xyz L if t [05, 05], then λxyz [05,05] t implies λxyz t, that is, xyz t λ If t > [05,05], then λxyz+ t > [05,05]+[05,05] [1,1] This implies that xyz t q λ Therefore, xyz t q λ λ is an interval-valued, q-fuzzy left ideal of S 2 Let x, y, z S and [0,0] < t [1,1] be such that z t q λ Then λz+ t > [1,1], implies z L This implies that xyz L if t [05, 05], then λxyz [05,05] t This implies that xyz t λ If t > [05, 05], then λxyz+ t > [05,05]+[05,05] [1,1] This implies that λxyz + t > [1,1], that is, xyz t q λ

5 Appl Math Inf Sci 9, No 3, / wwwnaturalspublishingcom/journalsasp 1627 xyz t q λ This shows that λ is an interval-valued q, q-fuzzy left ideal of S 3 Let x, y, z S and [0,0] < t [1,1] be such that z t q λ This implies that z t λ or z t q λ Then λz t or λz+ t > [1,1] if λz t, then z L This implies that xyz L If λz+ t >[1,1], then z L This implies that xyz L Analogous to 1 and 2 we obtain xyz t q λ Hence λ is an interval-valued q, q-fuzzy left ideal of S Conversely, assume that λ is an interval-valued α, q-fuzzy left ideal of S Then L λ 0 by Theorem 39, L is a left ideal of S In a similar fashion we can prove the following theorems 312 Theorem Let A be a non-empty subset of a ternary semigroup S and α {, q, q Then A is a ternary subsemigroup biideal, generalized bi-ideal of S if and only if the intervalvalued fuzzy subset λ of S defined by: λx { [05,05] if x A [0,0] if x S A is an interval-valued α, q-fuzzy ternary subsemigroup bi-ideal, generalized bi-ideal of S 313 Proposition Every interval-valued q, q-fuzzy ternary subsemigroup left ideal, right ideal, lateral ideal of a ternary semigroup S is an interval-valued, q-fuzzy ternary subsemigroup left ideal, right ideal, lateral ideal of S 314 Remark Every interval-valued α, β-fuzzy ternary subsemigroup left ideal, right ideal, lateral ideal, bi-ideal, generalized bi-ideal of a ternary semigroup S is an interval-valued α, q-fuzzy left ideal, right ideal, lateral ideal, bi-ideal, generalized bi-ideal of S 315 Remark Every interval-valued α, q-fuzzy ternary subsemigroup left ideal, right ideal, lateral ideal, bi-ideal, generalized bi-ideal of a ternary semigroup S is an interval-valued, q-fuzzy ternary subsemigroup left ideal, right ideal, lateral ideal, bi-ideal, generalized bi-ideal of S The above remarks show the importance of interval-valued, q-fuzzy ternary subsemigroup left ideal, right ideal, lateral ideal, bi-ideal, generalized bi-ideal in interval-valued α, β-fuzzy ternary subsemigroup left ideal, right ideal, lateral ideal, bi-ideal, generalized bi-ideal of a ternary semigroup S Therefore, in next section we study interval-valued, q-fuzzy ideals of a ternary semigroup S 4 Interval-valued, q-fuzzy ideals Recall that an interval-valued fuzzy subset λ of a ternary semigroup S is called an interval-valued, q-fuzzy left right, lateral ideal of S if z t λ and x,y S implies xyz t q λ zxy t q λ, xzy t q λ for all [0,0]< t [1,1] 41 Theorem Let λ be an interval-valued fuzzy subset of a ternary semigroup S Then λ is an interval-valued, q-fuzzy left right, lateral ideal of S if and only if λxyzr min { λz, [05,05] λxyz r min λxyzr min { λy, [05,05] { λx, [05,05], ProofLet λ be an interval-valued, q-fuzzy left ideal of S Assume on the contrary that there exist x,y,z S such that λxyz < r min { λz, [05,05] Choose [0, 0] < t [1, 1] such that λxyz < t r min { λz, [05,05] Then z t λ but xyz t q λ, which is a contradiction Hence λxyzr min { λz, [05,05] Conversely, assume that λxyz r min { λz, [05, 05] Let z t λ Then λz t Now, λxyz r min { λz, [05,05] r min { t, [05,05] If t [05,05], then λxyz t so xyz t λ If t > [05,05], then λxyz > [05,05] λxyz+ t > [05,05]+[05,05] [1,1] This implies that xyz t q λ Hence xyz t q λ Therefore, λ is an interval-valued, q-fuzzy left ideal of S 42 Corollary Let λ be an interval-valued fuzzy subset of a ternary semigroup S Then λ is an interval-valued

6 1628 N Rehman, M Shabir: A New Generalization of Fuzzy Ideals of, q-fuzzy two sided ideal of S if and only if λxyz r min { λz, [05,05] and λxyzr min { λx, [05,05] 43 Theorem The intersection of interval-valued, q-fuzzy left right, lateral ideals of a ternary semigroup S is an interval-valued, q-fuzzy left right, lateral ideal of S ProofLet { λi be a family of interval-valued i Λ, q-fuzzy left ideals of a ternary semigroup S and x, y, z S Since each λ i is an interval-valued, q-fuzzy left ideal of S so λ i xyz λ i z for each i Λ i Λ λi xyz i Λ λi xyz i Λ λi z i Λ λi z i Λ λi z i Λ λi xyz i Λ λi z Hence i Λ λi is an interval-valued, q-fuzzy left ideal of S 44 Theorem S is an interval-valued, q-fuzzy ternary subsemigroup of S if and only if λxyzr min { λx, λy, λz, [05,05] ProofThe proof follows from the Theorem Remark Every, -fuzzy ternary subsemigroup of a ternary semigroup S is an interval-valued, q-fuzzy ternary subsemigroup of S but the converse need not be true 46 Example Let S { i, 0, i Then S is a ternary semigroup under the ternary multiplication of complex numbers Define an interval-valued fuzzy subset λ of S by: λi [06,08], λ i [07,075], λ0[02,035] Then it is simple to verify that λ is an interval-valued, q-fuzzy ternary subsemigroup, but not an interval-valued, -fuzzy ternary subsemigroup of S 47 Theorem An interval-valued fuzzy subset λ of a ternary semigroup S is an interval-valued, q-fuzzy generalized bi-ideal of S if and only if λxuyvzr min { λx, λy, λz, [05,05] for all u, v, x, y, z S ProofAssume that λ is an interval-valued, q-fuzzy generalized bi-ideal of S Suppose on the contrary that there exist u, v, x, y, z S such that λxuyvz < r min { λx, λy, λz, [05,05] Choose[0,0]< t [1,1] such that λxuyvz < t r min { λx, λy, λz, [05,05] Then x t λ, y t λ and z t λ but xuyvz r min{ t, t, t q λ xuyvz t q λ, which is a contradiction Hence, λxuyvzr min { λx, λy, λz,[05,05] Conversely, assume that λxuyvz r min { λx, λy, λz,[05,05] for all u, v, x, y, z S Let x t1 λ, y t2 λ and z t3 λ for all [0,0] < t 1 [1,1], [0,0] < t 2 [1,1], [0,0] < t 3 [1,1] Then λx t 1, λy t 2, λz t 3 Hence, λxuyvz r min { λx, λy, λz, [05,05] r min { t 1, t 2, t 3, [05,05] If r min { t 1, t 2, t 3 [05, 05], then λxuyvz r min { t 1, t 2, t 3 This implies that xuyvz r min{ t 1, t 2, t 3 λ If r min { t 1, t 2, t 3 > [05,05], then λxuyvz+ r min { t 1, t 2, t 3 > [05,05]+[05,05] [1,1], so xuyvz rmin{ t 1, t 2, t 3 q λ xuyvz rmin{ t 1, t 2, t 3 q λ Therefore λ is an interval-valued, q-fuzzy generalized bi-ideal of S 48 Theorem S is an interval-valued, q-fuzzy bi-ideal of S if and only if it satisfies: 1 λxyzr min { λx, λy, λz, [05,05] ; 2 λxuyvzr min { λx, λy, λz, [05,05] for all u, v, x, y, z S

7 Appl Math Inf Sci 9, No 3, / wwwnaturalspublishingcom/journalsasp 1629 ProofThe proof follows from Theorem 44 and Theorem 47 Now, we characterize the interval-valued, q-fuzzy left right, lateral ideal of S by their level subsets 49 Theorem Let λ be an interval-valued fuzzy subset of a ternary semigroup S Then λ is an interval-valued, q-fuzzy left right, lateral ideal of S if and only if U λ ; t /0 is a left right, lateral ideal of S for all [0,0]< t [1,1] ProofSuppose that λ is an interval-valued, q-fuzzy left ideal of S Let x, y S and z U λ; t for some [0, 0] < t [05, 05] Then λz t Since λxyz r min { λz, [05,05] r min { t,[05, 05] t This implies that λxyz t This implies that xyz U λ ; t Hence U λ ; t is a left ideal of S Conversely, assume that U λ ; t /0 is a left ideal of S for all [0,0] < t [05,05] We show that λ is an interval-valued, q-fuzzy left ideal of S Suppose on the contrary that there exist x, y, z S such that λxyz < r min { λz, [05,05] Choose [0, 0] < t [05, 05] such that λxyz< t r min { λz, [05,05] Then z U λ; t but xyz / U λ ; t, which is a contradiction λxyz r min { λz, [05,05] Hence λ is an interval-valued, q-fuzzy left ideal of S 410 Definition S is an interval-valued, q-fuzzy quasi-ideal of S if and only if it satisfies: 1 λxr min λ S S x, S λ S S S λ x,[05,05] 2 λ S S x, λxr min S S λ S S x, ; S S λ x,[05,05] x, ; where S is the interval-valued fuzzy subset of S mapping every element of S on[1, 1] 411 Proposition Every interval-valued, q-fuzzy quasi-ideal of a ternary semigroup S is an interval-valued, q-fuzzy ternary subsemigroup of S ProofStraightforward 412 Theorem Let λ be an interval-valued, q-fuzzy quasi-ideal of { a ternary semigroup S Then the set λ0 x S: λx>[0,0] is a quasi-ideal of S ProofTo show that λ 0 is a quasi-ideal of S we show that SS λ 0 S λ 0 S λ 0 SS λ 0 and SS λ 0 SS λ 0 SS λ 0 SS λ 0 Let a SS λ 0 S λ 0 S λ 0 SS Then a SS λ 0, a S λ 0 S and a λ 0 SS This implies that there exist x, y, z λ 0 and s 1, s 2, s 3, t 1, t 2, t 3 S such that as 1 t 1 x, as 2 yt 2, a zs 3 t 3 Now, λ S S a, S λ S a, λar min S S λ a,[05,05] Since S S λa apqr { S p S q λr S s 1 S t 1 λx [1,1] [1,1] λx λx Similarly, S λ S a λy λ S S a λz λ S S a, S λ S a, λa r min S S λ a,[05,05] r min { λz, λy, λx,[05,05] > [0,0] since > [0,0], λz>[0,0] λx>[0,0], λy and This implies that a λ 0 SS λ 0 S λ 0 S λ0 SS λ 0 Next we show SS λ 0 SS λ 0 SS λ 0 SS λ 0 Let a SS λ 0 SS λ 0 SS λ 0 SS Then a SS λ 0 and a SS λ 0 SS and a λ 0 SS a s 1 t 1 x, a zs 3 t 3, as 1 t 2 ys 4 t 4 for x,y,z λ 0 and s 1,s 2,s 3,s 4, S

8 1630 N Rehman, M Shabir: A New Generalization of Fuzzy Ideals of Now, λ S S a, λ0 ar min S S λ S S a, S S λ a,[05,05] and by above arguments S S λ a λx, λ S S a λz and S S λ S S a { arst S S λ r S s S t { rmnk S m S n arst λk S s S, t S S λ S S a { S p S q amnkst λr S s S t S s 1 S s 2 λy S s 3 S s 4 [1,1] [1,1] λy [1,1] [1,1] λy S S λ a, λa r min S S λ S S a, λ S S a, [05,05] { λx, λy, λz, [05,05] > [0,0] since λx>[0,0], λy > [0,0], λz>[0,0] a λ 0 and hence SS λ 0 SS λ 0 SS λ 0 SS λ 0 Therefore, λ 0 is a quasi-ideal of S 413 Lemma A non-empty subset Q of a ternary semigroup S is a quasiideal of S if and only if the interval-valued characteristic function C Q, of Q, is an interval-valued, q-fuzzy quasi-ideal of S ProofLet Q be a quasi-ideal of S and C Q the interval-valued characteristic function of Q and x S If x / Q, then x / SSQ or x / QSS or x / SQS If x / SSQ, then S S C Q x[0,0] so, S S CQ x, S CQ S x, r min C Q S S x,[05,05] [0,0] C Q x Similarly, for other cases If x Q, then C Q x [1,1] S S C Q r min C Q x, S C Q S S S x,[05,05] Similarly, S S C Q x, C Q xr min S S C Q S S x, C Q S S x,[05,05] x, C Q is an interval-valued, q-fuzzy quasi-ideal of S Conversely, assume that C Q is an interval-valued, q-fuzzy quasi-ideal of S We show that Q is a quasi-ideal of S Let a SSQ SQS QSS Then a SSQ and a SQS and a QSS there exist x,y,z Q and s 1,s 2,s 3,t 1,t 2,t 3 S such that as 1 t 1 x and as 2 yt 2 and azs 3 t 3 Now, C Q S S a apqr { C Q p S q S r So C Q S S CQ a [1,1] and Hence S S C Q C Q a r min C Q C Q z S s 3 S t 3 [1,1] [1,1] [1,1][1,1] S S a [1, 1] Similarly, S S C Q a [1,1] a, S C Q S S S a,[05,05] a, [05,05] C Q a[05,05] This implies that C Q a[1,1] Hence a Q so SSQ SQS QSS Q Next, let a SSQ SSQSS QSS Then a SSQ and a SSQSS and a QSS there exist s 1,s 2,s 3,s 4,t 1,t 2,t 3,t 4 S,x,y,z Q such that a s 1 t 1 x, a zs 3 t 3 and a s 2 t 2 ys 4 t 4 For a s 1 t 1 x, and a zs 3 t 3 C Q [1,1] as discussed above Consider the case if as 2 t 2 ys 4 t 4 : S S C Q S S a { arst S S C Q r S s S t { rlmn S l S m C Q n arst S s S t

9 Appl Math Inf Sci 9, No 3, / wwwnaturalspublishingcom/journalsasp 1631 S S C Q S S a { almnst S l S m CQ n S s S t S s 2 S t 2 C Q y S s 4 S t 4 [1,1] [1,1] [1,1] [1,1] [1,1] [1,1] This implies that S S CQ S S a[1,1] S S CQ a, C Q a r min S S C Q S S a, C Q S S a,[05,05] [05,05] C Q a [05,05] This implies that C Q a [1,1], and so a Q Thus SSQ SSQSS QSS Q Hence Q is a quasi-ideal of S By using similar arguments as in the proof of the above lemma we can prove the following lemmas 414 Lemma The interval-valued characteristic function C L, of L, is an interval-valued, q-fuzzy left ideal of S if and only if L is a left ideal of S 415 Lemma The interval-valued characteristic function C B, of B, is an interval-valued, q-fuzzy bi-ideal of S if and only if B is a bi-ideal of S 416 Theorem Every interval-valued, q-fuzzy left right, lateral ideal of a ternary semigroup S is an interval-valued, q-fuzzy quasi-ideal of S ProofLet λ be an interval-valued, q-fuzzy left ideal of S and a S Then, S S λa axyz { S x S y λz S S λ a λa 1 Hence λa S S λ a S S λ a, S λ S a, r min λ S S a,05 So S S λ a, S λ S a, λar min λ S S a,05 Again from1 λa S S λ a S S λ rmin a, S S λ S S a, λ S S a,[05,05] Hence S S λ a, S S λ S S a, λarmin λ S S a,[05,05] Therefore, λ is an interval-valued, q-fuzzy quasi-ideal of S 417 Theorem Every interval-valued, q-fuzzy quasi-ideal of a ternary semigroup S is an interval-valued, q-fuzzy bi-ideal of S ProofLet λ be an interval-valued, q-fuzzy quasi-ideal of S Then by Proposition 411, λ is an interval-valued, q-fuzzy ternary subsemigroup of S Now, axyz λz This implies that S S λ a axyz λz axyz λxyz λa λxuyvz λ S S xuyvz S S λ S S S S λ xuyvz xuyvz

10 1632 N Rehman, M Shabir: A New Generalization of Fuzzy Ideals of xuyvzabc { λa S b S c rs1 s 2 s 3 S s1 S s 2 xuyvzrst λs 3 S s S t { xuyvzlmn S l S m λn { λx S uyv S z S x S u λy S v S z { S x S uyv λz λx λy λz λxuyvzrmin { λx, λy, λz,[05,05] Hence λ is an interval-valued, q-fuzzy bi-ideal of S The converse of the above theorem is not true in general 418 Example Let S a b c 0 0 d : a,b,c,d Z 0 and 0 0 e B 0 x y : x,y Z , where Z 0 is the set of non-positive integers Then S is a ternary semigroup with respect to ternary matrix multiplication and B is a bi-ideal of S Let λ be an interval-valued fuzzy subset of S defined by: [07,0,8] if x 0 p q, p,q Z λx 0, [02, 04] otherwise Then λ is an interval-valued, q-fuzzy bi-ideal of S, but not an interval-valued, q-fuzzy quasi-ideal of S 419 Theorem Every interval-valued, q-fuzzy ideal of a ternary semigroup S is an interval-valued, q-fuzzy bi-ideal of S ProofThe proof follows from Theorem 416 and Theorem 417 The converse of Theorem 419 is not true in general 420 Example Let S a b c : a,b,c, p,q,r Z 0 and p q r B m n : m,n Z , where Z 0 is the set of non-positive integers Then S is a ternary semigroup with respect to ternary matrix multiplication and B is a bi-ideal of S Let λ be an interval-valued fuzzy subset of S defined by: [06,07] if x m n ;m,n Z λx [03, 04] otherwise Then λ is an interval-valued, q-fuzzy bi-ideal of S, but not an interval-valued, q-fuzzy ideal of S 5 Lower and upper parts of interval-valued, q-fuzzy ideals 51 Definition Let λ be an interval-valued fuzzy subset of a ternary semigroup S Define the fuzzy subsets λ + and λ of S as follows: λ+ x λx [05,05] and λ x λx 52 Lemma Let λ, µ and ν be interval-valued fuzzy subsets of a ternary semigroup S Then the following hold: 1 λ λ µ µ ; 2 λ λ µ µ ; 3 λ λ µ ν µ ν ProofThe proofs of1 and2 are obvious 3 Let a S If a is not expressible as a bcd for some b,c,d S, then λ µ ν a0 and λ µ ν a λ µ ν a 00 Since a is not expressible as a bcd so λ µ ν a 0 in this case

11 Appl Math Inf Sci 9, No 3, / wwwnaturalspublishingcom/journalsasp 1633 λ µ ν λ µ ν If a is expressible as axyz, then λ µ ν a λ µ ν a axyz λx λy λz λx λy axyz λz { λ axyz x µ y ν z λ µ ν a If a is expressible as abcd, then +a λ µ ν λ µ ν a [05,05] axyz { λx µy νz Therefore, [05,05] λx [05,05] axyz µy [05,05] νz [05,05] { λ+ axyz x µ + y ν + z λ+ µ + ν + a λ µ ν + λ+ µ + ν + Therefore, 53 Lemma λ µ ν λ µ ν Let λ, µ and ν be interval-valued fuzzy subsets of a ternary semigroup S Then the following hold: + 1 λ λ+ µ µ + ; + 2 λ λ+ µ µ + ; + 3 λ λ+ µ ν µ + ν + If every element x S is expressible as x abc for + some a,b,c S, then λ λ+ µ ν µ + ν + ProofThe proofs of1 and2 are obvious 3 Let a S If a is not expressible as a bcd for some b,c,d S, then λ µ ν a0 and λ µ ν +a λ µ ν a [05,05] 0 [05,05][05,05] λ+ and µ + ν + a0 So, λ µ ν + λ+ µ + ν + 54 Definition Let A be a non-empty subset of a ternary semigroup S Then C A and { C A + are defined as: C A [05,05] if x A x [0,0] if x / A and { C A + [1,1] if x A x [05,05] if x / A 55 Lemma The lower part of the interval-valued characteristic function, that is, C L is an interval-valued, q-fuzzy left resp right, lateral ideal of a ternary semigroup S if and only if L is a left resp right, lateral ideal of S ProofLet L be a left ideal of S Then by Theorem 311, C L is an interval-valued, q-fuzzy left ideal of S Conversely, assume that C L is an interval-valued, q-fuzzy left ideal of S Let z L Then C L z [05,05], so z [05,05] C L Since C L is an interval-valued, q-fuzzy left ideal of S, so xyz [05,05] q C L This implies that xyz [05,05] C L or xyz [05,05] q C L C L xyz [05,05] or C L xyz + [05,05] > [1,1] Now C L xyz + [05,05] > [1,1] is impossible C L xyz [05,05] which implies that C L xyz[05,05] This implies that xyz L Therefore L is a left ideal of S Similarly, it can be seen that the lower part of the interval-valued characteristic function C R resp C M is

12 1634 N Rehman, M Shabir: A New Generalization of Fuzzy Ideals of an interval-valued, q-fuzzy right resp lateral ideal of S if and only if R resp M is right resp lateral ideal of S the lower part of the interval-valued characteristic function C I is an interval-valued, q-fuzzy two sided ideal of S if and only if I is two sided ideal of S 56 Lemma Let Q be a non-empty subset of a ternary semigroup S Then Q is a quasi-ideal of S if and only if the lower part of the interval-valued characteristic function, that is, C Q is an interval-valued, q-fuzzy quasi-ideal of S 57 Lemma Let A, B and C be non-empty subsets of a ternary semigroup S Then the following hold: 1 C A C B C A B ; 2 C A C B C A B ; 3 C A C B C C C ABC 58 Proposition Let λ be an interval-valued, q-fuzzy left resp right, lateral ideal of a ternary semigroup S Then λ is an interval-valued fuzzy left resp right, lateral ideal of S ProofLet λ be an interval-valued, q-fuzzy left ideal of S Then for all a, b, c S, we have λabc λc λabc λc This implies that λ abc λ c Hence λ is an interval-valued fuzzy left ideal of S We now characterize regular ternary semigroups by the properties of lower parts of interval-valued, q-fuzzy ideals, interval-valued, q-fuzzy left right, lateral ideals, interval-valued, q-fuzzy quasi-ideals, interval-valued, q-fuzzy bi-ideals and interval-valued, q-fuzzy generalized bi-ideals 59 Theorem For a ternary semigroup S, the following conditions are 1 S is regular; 2 λ µ ν λ µ ν for every intervalvalued, q-fuzzy right ideal λ, every interval-valued, q-fuzzy lateral ideal µ and every interval-valued, q-fuzzy left ideal ν of S Proof1 2: Let λ be an interval-valued, q-fuzzy right ideal, µ an interval-valued, q-fuzzy lateral ideal and ν an interval-valued, q-fuzzy left ideal of S and a S Then a λ µ ν λ µ ν a axyz λx µy νz axyz λxyz µxyz νxyz λa µa νa λ µ ν a λ µ ν λ µ ν λ µ ν Since S is regular, so for any a S there exists x S such that aaxaaxaxa Now a λ µ ν λ µ ν a apqr λp µq νr λa µxax νa λa µa νa λ µ ν a λ µ ν a λ µ ν λ µ ν λ µ ν λ µ ν Hence, 2 1: Let R,M and L be the right, lateral and left ideals of S, respectively Then, by Lemma 55, lower part of the interval-valued characteristic functions C R, C M, C L are interval-valued, q-fuzzy right ideal, interval-valued, q-fuzzy lateral ideal and interval-valued, q-fuzzy left ideal of S, respectively by hypothesis C R C M C L C R C M C L C R M L C RML R M L RML Hence by Theorem 24, S is regular

13 Appl Math Inf Sci 9, No 3, / wwwnaturalspublishingcom/journalsasp Theorem For a ternary semigroup S, the following assertions are 1 S is regular; 2 λ µ λ S µ for every interval-valued, q-fuzzy right ideal λ and every interval-valued, q-fuzzy left ideal µ of S Proof1 2: Let λ be an interval-valued, q-fuzzy right ideal and µ an interval-valued, q-fuzzy left ideal of S and a S Then, a λ S µ λ S µ a axyz λx S y µz Hence axyz λx µz axyz λxyz µxyz a λa µa 05 λ µ λ S µ λ µ Since S is regular so for any a S there exists x S such that aaxaaxaxa Then, a λ S µ λ S µ a apqr λp S q µr Therefore, λa S xax µa λa µa 05 λ µ a λ S µ λ µ interval-valued, q-fuzzy left ideal of S, respectively by hypothesis C R C L C R S C L C R L C RSL R L RSL Hence by Theorem 26, S is regular 511 Theorem For a ternary semigroup S, the following conditions are 1 S is regular; 2 λ λ S λ S λ for every interval-valued, q-fuzzy generalized bi-ideal λ of S; 3 λ λ S λ S λ for every interval-valued, q-fuzzy bi-ideal λ of S; 4 λ λ S λ S λ for every interval-valued, q-fuzzy quasi-ideal λ of S Proof1 2: Let λ be an interval-valued, q-fuzzy generalized bi-ideal of S and a S Since S is regular so there exists x S such that aaxaaxaxa Now, a λ S λ S λ λ S λ S λ arst { λ S λ r S s λt { λ S λ a S x λa [05, 05] { λ S λ a λa { arst λr S s λt λt λ µ λ S µ 2 1: Let R and L be the right and left ideal of S, respectively Then, by Lemma 55, lower part of the interval-valued characteristic functions C R and C L are interval-valued, q-fuzzy right ideal and λa S x λa λa λ a λ S λ S λ λ

14 1636 N Rehman, M Shabir: A New Generalization of Fuzzy Ideals of Since λ is an interval-valued, q-fuzzy generalized bi-ideal of S, so a λ S λ S λ λ S λ S λ a axuyvz { λx S u λy S v λz axuyvz { λx λy λz axuyvz λxuyvz λa λ a λ S λ S λ λ λ S λ S λ λ Hence 2 3 4: are obvious 4 1: Let A be a quasi-ideal of S Then by Lemma 413, C A is an interval-valued, q-fuzzy quasi-ideal of S Thus by hypothesis C A C A S C A S C A C A C S C A C S C A C ASASA This implies that A ASASA Hence it follows by Theorem 25, S is regular By using similar arguments as in the proof of the Theorem 511, we can prove the following theorem 512 Theorem For a ternary semigroup S, the following conditions are 1 S is regular; 2 λ λ S λ for every interval-valued, q-fuzzy quasi-ideal λ of S; 3 λ λ S λ for every interval-valued, q-fuzzy bi-ideal λ of S 513 Theorem For a ternary semigroup S, the following statements are 1 S is regular; for 2 λ µ λ S µ every interval-valued, q-fuzzy quasi-ideal λ and every interval-valued, q-fuzzy left ideal µ of S; 3 λ µ λ S µ for every interval-valued, q-fuzzy bi-ideal λ and every interval-valued, q-fuzzy left ideal µ of S; for 4 λ µ λ S µ every interval-valued, q-fuzzy generalized bi-ideal λ and every interval-valued, q-fuzzy left ideal µ of S Proof1 4: Let λ be an interval-valued, q-fuzzy generalized bi-ideal and µ an interval-valued, q-fuzzy left ideal of S Since S is regular, so for every a S there exists x S such that aaxa a λ S µ λ S µ a apqr { λp S q µr λa S x µa λa [1,1] µa λa µa λ µ a a λ µ So, λ S µ λ µ 4 3 2: are obvious 2 1: Let λ be an interval-valued, q-fuzzy right ideal and µ an interval-valued, q-fuzzy left ideal of S Since every interval-valued, q-fuzzy right ideal is an interval-valued, q-fuzzy quasi-ideal of S So, λ µ λ S µ Let a S and consider, a λ S µ λ S µ a axyz λx S y µz axyz λx µz axyz λxyz µxyz λa µa a λ µ a λ µ So λ µ λ S µ λ µ λ S µ Therefore, by Theorem 510, S is regular

15 Appl Math Inf Sci 9, No 3, / wwwnaturalspublishingcom/journalsasp Theorem References For a ternary semigroup S, the following statements are 1 S is regular; 2 λ S λ µ µ for every interval-valued, q-fuzzy quasi-ideal λ and every interval-valued, q-fuzzy right ideal µ of S; 3 λ S λ µ µ for every interval-valued, q-fuzzy bi-ideal λ and every interval-valued, q-fuzzy right ideal µ of S Proof1 3: Let λ be an interval-valued, q-fuzzy bi-ideal and µ an interval-valued, q-fuzzy right ideal of S Since S is regular, so for any element a S there exists x S such that a axa Now, µ S λ a So 3 2: Obvious 2 1: Let λ be an interval-valued, q-fuzzy left ideal and µ an interval-valued, q-fuzzy right ideal of S and a S Now µ S λ a µ S λ a axyz µx S y λz [1] D H Lehmer, A ternary analogue of Abelian groups, Amer Jr of Math, 59, [2] J Los, On extending of models I, Fundamenta Mathematicae, 42, [3] F M Sioson, Ideal theory in ternary semigroups, Math Japon 10, [4] V N Dixit and S Dewan, A note on quasi and bi-ideals in ternary semigroups, Int J Math Math Sci, 18, [5] L A Zadeh, Fuzzy sets, Information and Control, 8, [6] A Rosenfeld, Fuzzy subgroups J Math Anal Appl, 35, [7] N Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems, 5, [8] N Kuroki, Fuzzy bi-ideals in semigroups, Comput Math Univ St Paul, 28, [9] N Kuroki, On fuzzy semigroups, Inform Sci, 53, [10] N Kuroki, Fuzzy generalized bi-ideal in Semigroups, µ S λ Inform Sci, 66, a [11] N Kuroki, Fuzzy semiprime quasi ideals in semigroups, axpqr µp S Inform Sci, 75, q λr [12] L A Zadeh, The concept of linguistic variable and its application to approximate reasoningi, Inform Sci, 8, µa S [13] R Biswas, Rosenfeld s fuzzy subgroups with intervalvalued membership function, Fuzzy Sets and Systems 63, 87- x λa µa λa [14] S K Bhakat and P Das,, q-fuzzy subgroups, Fuzzy a Sets and Systems, 80, µ λ a µ λ [15] P M Pu and Y M Liu, Fuzzy topology I, Neighborhood structure of a fuzzy point and Moor-Smith convergence, J λ S λ Math Anal Appl, 76, [16] N Rehman and M Shabir, Characterizations of ternary µ µ semigroups by α, β-fuzzy ideals World Applied Sciences Journal 1811, [17] S K Bhakat, q-level subsets, Fuzzy Sets and Systems 103, [18] S K Bhakat,, q-fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and Systems 112, [19] S K Bhakat and P Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems 51, [20] S K Bhakat and P Das, Fuzzy subrings and ideals redefined, Fuzzy Sets and Systems 81, [21] M Akram, K H Dar and K P Shum, Interval-valued α, β-fuzzy k-algebras, Applied Soft Computing 11, axyz µx λz [22] J Zhan, W A Dudek and Y B Jun, Interval-valued, q-fuzzy filter of pseudo BL-algebras, Soft Comput axyz µxyz λxyz 13, a [23] X Ma, J Zhan and Y B Jun, Interval-valued, q- µa λa λ µ λ µ µ S λ Hence by Theorem 510, S is regular fuzzy filter of pseudo MV-algebras, International Journal of Fuzzy Systems 10, [24] M Shabir and M Bano, Prime bi-ideals in ternary semigroups, Quasigroups and Related Systems 16,