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On Intuitionistic Fuzzy Magnified Translation 1 1 in Semigroups 0 2 n Sujit Kumar Sardar1 a J 8 1 Manasi Mandal2 ] M and G . Samit Kumar Majumder3 h Department of Mathematics, Jadavpur t a University, Kolkata-700032, INDIA m [email protected] [ 2manasi [email protected] 1 − [email protected] v 9 9 6 3 Abstract . 1 The notion of intuitionistic fuzzy sets was introduced by Atanassov as a gen- 0 eralizationof the notion of fuzzy sets. S.K Sardar and S.K. Majumder unified the 1 idea of fuzzy translation and fuzzy multiplication of Vasantha Kandasamy to in- 1 : troduce the conceptof fuzzy magnifiedtranslationin groupsand semigroups. The v purpose of this paper is to intuitionistically fuzzify(by using Atanassov’s idea) the i X conceptoffuzzymagnifiedtranslationinsemigroups. Hereamongotherresultswe r obtain some characterizationtheorems of regular, intra-regular,left(right) regular a semigroups in terms of intuitionistic fuzzy magnified translation. AMS Mathematics Subject Classification(2000): 08A72, 20M12,3F55 Key Words and Phrases: Semigroup, (Left, right) regular semigroup, Intra- regularsemigroup,Archimedeansemigroup,Intuitionisticfuzzy magnifiedtransla- tion, Intuitionistic fuzzy subsemigroup, Intuitionistic fuzzy bi-ideal, Intuitionistic fuzzy(1,2)-ideal,Intuitionisticfuzzyleft(right)ideal,Intuitionisticfuzzysemiprime ideal. 1 Introduction A semigroup is an algebraic structure consisting of a non-empty set S together with an associative binary operation[3]. The formal study of semigroups began in the early 20th century. Semigroups are important in many areas of mathematics, for example, 1 coding and language theory, automata theory, combinatorics and mathematical analy- sis. The concept of fuzzy sets was introduced by Lofti Zadeh[16] in his classic paper in 1965. Azirel Rosenfeld[11] used the idea of fuzzy set to introduce the notions of fuzzy subgroups. Nobuaki Kuroki[7, 8, 9] is the pioneer of fuzzy ideal theory of semigroups. The idea of fuzzy subsemigroup was also introduced by Kuroki[7, 9]. In [8], Kuroki characterized several classes of semigroups in terms of fuzzy left, fuzzy right and fuzzy bi-ideals. Others who worked on fuzzy semigroup theory, such as X.Y. Xie[14, 15], Y.B. Jun[4, 5], are mentioned in the bibliography. The notion of intuitionistic fuzzy sets was introduced by Atanassov[1, 2] as a generalization of the notion of fuzzy sets. In this paper we introduce the notion of intuitionistic fuzzy magnified translation in semigroups and observe some of its important properties. Here we characterize regular, intra-regular and left(right) regular semigroups in terms of intuitionistic fuzzy magnified translation. Finally we also observe that intuitionistic fuzzy translation and intuitionistic fuzzy multiplication are the particular cases of intuitionistic fuzzy magnified translation. 2 Preliminaries In this section we discuss some elementary definitions that we use in the sequel. If (X,∗) is a mathematical system such that ∀a,b,c ∈ X, (a∗b)∗c = a∗(b∗c), then ∗ is called associative and (X,∗) is called a semigroup[10]. Throughout the paper unless otherwise stated S will denote a semigroup. A non-empty subset A of a semigroup S is called a subsemigroup[8] of S if AA⊆ A. A subsemigroup A of S is called a bi-ideal[8] of S if ASA⊆ A. A subsemigroup A of S is called an (1,2)-ideal[8] of S if ASAA⊆ A. A left (right) ideal[10] of a semigroup S is a non-empty subset I of S such that SI ⊆ I (IS ⊆ I). If I is both a left and a right ideal of a semigroup S, then we say that I is an ideal[10] of S. A fuzzy subset[16] in S is a function µ :S −→ [0,1]. Let µ be a fuzzy subset of a set X and α ∈ [0,1−sup{µ(x) : x ∈ X}]. A mapping µT : X → [0,1] is called a fuzzy translation[6] of µ if µT(x) = µ(x)+α for all x ∈ X. α α Let µ be a fuzzy subset of a set X and β ∈ [0,1]. A mapping µM : X → [0,1] is β called a fuzzy multiplication[6] of µ if µM(x) = β·µ(x) for all x ∈ X. β Let µ be a fuzzy subset of a set X, α ∈ [0,1−sup{µ(x) : x ∈ X}] and β ∈ [0,1]. A mapping µC :X → [0,1] is called a fuzzy magnified translation[12] of µ if µC (x) = βα βα β ·µ(x)+α for all x ∈ X. A non-empty fuzzy subset µ of a semigroup S is called a fuzzy subsemigroup[10] of S if µ(xy)≥ min{µ(x),µ(y)}∀x,y ∈ S. 2 A fuzzy subsemigroup µ of a semigroup S is called a fuzzy bi-ideal[10] of S if µ(xyz) ≥ min{µ(x),µ(z)}∀x,y,z ∈ S. A fuzzy subsemigroup µ of a semigroup S is called a fuzzy (1,2)-ideal[10] of S if µ(xω(yz)) > min{µ(x),µ(y),µ(z)}∀x,ω,y,z ∈ S. A non-empty fuzzy subset µ of a semigroup S is called a fuzzy left(right) ideal[10] of S if µ(xy) ≥ µ(y)(resp. µ(xy)≥ µ(x)) ∀x,y ∈ S. A non-empty fuzzy subset µ of a semigroup S is called a fuzzy two-sided ideal or a fuzzy ideal[10] of S if it is both a fuzzy left and a fuzzy right ideal of S. A fuzzy ideal µ of a semigroup S is called a fuzzy semiprime ideal[7] of S if 2 µ(x) ≥ µ(x )∀x ∈S. Atanassov introduced in [1, 2] the concept of intuitionistic fuzzy sets defined on a non-empty set X as objects having the form A = {< x,µ (x),ν (x) >:x ∈X}, A A where the functions µ : X → [0,1] and ν : X → [0,1] denote the degree of member- A A shipandthedegreeofnon-membershipofeach element x ∈ X tothesetArespectively, and 0 ≤ µ (x)+ν (x) ≤ 1 for all x ∈ X. A A Such defined objects are studied by many authors and have many interesting ap- plications in mathematics. Let A and B be two intuitionistic fuzzy subsets of a set X. Then the following expressions are defined in [1, 2]. (i) A⊆ B if and only if µ (x) ≤ µ (x) and ν (x) ≥ ν (x), A B A B (ii) A =B if and only if A ⊆ B and B ⊆ A, (iii) AC = {< x,ν (x),µ (x)>: x ∈ X}, A A (iv) A∩B = {< x,min{µ (x),µ (x)},max{ν (x),ν (x)} >:x ∈X}, A B A B (v) A∪B = {< x,max{µ (x),µ (x)},min{ν (x),ν (x)} >:x ∈ X}. A B A B From the definition it follows that A∩B is the same as µ ∩µ and ν ∪ν . Also A B A B A∪B is the same as µ ∪µ and ν ∩ν . A B A B For thesakeofsimplicity, weshallusethesymbolA= (µ ,ν )fortheintuitionistic A A fuzzy subset A = {< x,µ (x),ν (x) >: x ∈ X}. A A A non-empty intuitionistic fuzzy subset A = (µ ,ν ) of a semigroup S is called an A A intuitionistic fuzzy subsemigroup of S if (i) µ (xy) ≥ min{µ (x),µ (y)}∀x,y ∈S, (ii) A A A ν (xy) ≤ max{ν (x),ν (y)}∀x,y ∈ S. A A A An intuitionistic fuzzy subsemigroup A = (µ ,ν ) of a semigroup S is called an A A intuitionistic fuzzy bi-ideal of S if (i) µ (xyz) ≥ min{µ (x),µ (z)}∀x,y,z ∈ S, (ii) A A A ν (xyz) ≤ max{ν (x),ν (z)}∀x,y,z ∈ S. A A A An intuitionistic fuzzy subsemigroup A = (µ ,ν ) of a semigroup S is called an in- A A tuitionistic fuzzy (1,2)-ideal of S if (i) µ (xω(yz)) > min{µ (x),µ (y),µ (z)}∀x,ω,y A A A A ,z ∈ S, (ii) ν (xω(yz)) 6 max{ν (x),ν (y),ν (z)}∀x,ω,y,z ∈ S. A A A A 3 A non-empty intuitionistic fuzzy subset A = (µ ,ν ) of a semigroup S is called an A A intuitionistic fuzzy left(right) ideal of S if (i) µ (xy) ≥ µ(y)(resp. µ (xy) ≥ µ (x)) A A A ∀x,y ∈ S, (ii) ν (xy) ≤ ν (y)(resp. ν (xy)≤ ν (x)) ∀x,y ∈S. A A A A A non-empty intuitionistic fuzzy subset A = (µ ,ν ) of a semigroup S is called an A A intuitionistic fuzzy two-sided ideal or an intuitionistic fuzzy ideal of S if it is both an intuitionistic fuzzy left and an intuitionistic fuzzy right ideal of S. Anintuitionistic fuzzyidealA =(µ ,ν )ofasemigroupS iscalled anintuitionistic A A 2 2 fuzzy semiprime ideal of S if (i) µ (x) ≥ µ (x )∀x ∈ S, (ii) ν (x) ≤ ν (x )∀x ∈ S. A A A A 3 Main Results Definition 3.1. Let A = (µ ,ν ) be an intuitionistic fuzzy subset of a set X, α ∈ A A [0,inf{ν (x) : x ∈ X}]. An object having the form AT = ((µ )T,(ν )T) is called an A α A α A α intuitionistic fuzzy translation of A if (µ )T(x) = µ (x)+α and (ν )T(x) = ν (x)−α A α A A α A for all x ∈ X. Definition 3.2. Let A = (µ ,ν ) be an intuitionistic fuzzy subset of a set X, β ∈ A A [0,1]. An object having the form AM = ((µ )M,(ν )M) is called an intuitionistic fuzzy β A β A β multiplication of A if (µ )M(x) = β·µ (x) and (ν )M(x) = β·ν (x) for all x ∈ X. A β A A β A Definition 3.3. Let A = (µ ,ν ) be an intuitionistic fuzzy subset of a set X, α ∈ A A [0,inf{β · ν (x) : x ∈ X}], where β ∈ [0,1]. An object having the form AC = A βα ((µ )C ,(ν )C )iscalledanintuitionisticfuzzymagnifiedtranslationofAif(µ )C (x)= A βα A βα A βα β ·µ (x)+α and (ν )C (x) = β ·ν (x)−α for all x ∈ X. A A βα A Example 1. Let X = {1,ω,ω2}. Let A= (µ ,ν ) be an intuitionistic fuzzy subset of A A X, defined as follows 0.3 if x = 1 0.4 if x = 1 µ (x) = 0.1 if x = ω and ν (x) = 0.25 if x = ω . A A  2  2 0.5 if x = ω 0.3 if x = ω   Let α = 0.04 andβ = 0.2. Then the intuitionisticfuzzy magnified translation of A is given by 0.1 if x = 1 0.04 if x = 1 (µ )C (x) = 0.06 if x = ω and (ν )C (x) = 0.01 if x = ω . A βα  2 A βα  2 0.14 if x = ω 0.02 if x = ω   In what follows unless otherwise mentioned A = (µA,νA) denotes a non-empty intuitionistic fuzzy subset of S and AC denotes the intuitionistic fuzzy magnified trans- βα lation of A where β ∈ (0,1],α ∈ [0,inf{β·ν (x) :x ∈ Supp(ν )}]. It can be noted here A A that AC is also non-empty. βα Theorem 3.4. The intuitionistic fuzzy magnified translation AC = ((µ )C ,(ν )C ) βα A βα A βα of A= (µ ,ν ) is an intuitionistic fuzzy subsemigroup of S if and only if A = (µ ,ν ) A A A A is an intuitionistic fuzzy subsemigroup of S. 4 Proof. Let A= (µ ,ν )bean intuitionitic fuzzy subsemigroupof asemigroup S.Then A A A is a non-empty intuitionistic fuzzy subset of S(β > 0). Hence AC is also non-empty. βα Now for x,y ∈ S, (µ )C (xy)= β ·µ (xy)+α A βα A ≥ β ·min{µ (x),µ (y)}+α(since A is an intuitionistic fuzzy A A subsemigroup of S) = min{β ·µ (x)+α,β ·µ (y)+α} A A = min{(µ )C (x),(µ )C (y)} A βα A βα and (ν )C (xy) =β ·ν (xy)−α A βα A ≤β ·max{ν (x),ν (y)}−α(since A is an intuitionistic fuzzy A A subsemigroup of S) = max{β ·ν (x)−α,β ·ν (y)−α} A A =max{(ν )C (x),(ν )C (y)}. A βα A βα Hence AC = ((µ )C ,(ν )C ) is an intuitionistic fuzzy subsemigroup of S. βα A βα A βα Conversely, let AC = ((µ )C ,(ν )C ) be an intuitionistic fuzzy subsemigroup of βα A βα A βα S. Then AC and hence A is a non-empty fuzzy subset of S. Now for all x,y ∈ S, βα (µ )C (xy) ≥ min{(µ )C (x),(µ )C (y)} A βα A βα A βα i.e., β.µ (xy)+α ≥ min{β.µ (x)+α,β.µ (y)+α} A A A i.e., β.µ (xy)+α ≥ β.min{µ (x),µ (y)}+α A A A i.e., µ (xy) ≥ min{µ (x),µ (y)} A A A and (ν )C (xy) ≤ max{(ν )C (x),(ν )C (y)} A βα A βα A βα i.e., β.µ (xy)−α ≤ max{β.ν (x)−α,β.ν (y)−α} A A A i.e., β.ν (xy)−α ≤ β.max{ν (x),ν (y)}−α A A A i.e., ν (xy) ≤ max{ν (x),ν (y)} A A A Hence A= (µ ,ν ) is an intuitionistic fuzzy subsemigroup of S. A A Theorem 3.5. The intuitionistic fuzzy magnified translation AC = ((µ )C ,(ν )C ) βα A βα A βα of A =(µ ,ν ) is an intuitionistic fuzzy bi-ideal of S if and only if A= (µ ,ν ) is an A A A A intuitionistic fuzzy bi-ideal of S. Proof. Let A = (µ ,ν ) be an intuitionistic fuzzy bi-ideal of a semigroup S. Then A A A is an intuitionistic fuzzy subsemigroup of S and hence, by Theorem 3.4, AC is an βα 5 intuitionistic fuzzy subsemigroup of S. Now for x,ω,y ∈ S, (µ )C (xωy) =β ·µ (xωy)+α≥ β ·min{µ (x),µ (y)}+α A βα A A A (since A is an intuitionistic fuzzy bi-ideal of S) = min{β ·µ (x)+α,β ·µ (y)+α} A A = min{(µ )C (x),(µ )C (y)}. A βα A βα and (ν )C (xωy) = β·ν (xωy)−α ≤β ·max{ν (x),ν (y)}−α A βα A A A (since A is an intuitionistic fuzzy bi-ideal of S) =max{β ·ν (x)−α,β ·ν (y)−α} A A =max{(ν )C (x),(ν )C (y)}. A βα A βα Hence AC = ((µ )C ,(ν )C ) is an intuitionistic fuzzy bi-ideal of S. βα A βα A βα Conversely, let AC = ((µ )C ,(ν )C ) be an intuitionistic fuzzy bi-ideal of S. βα A βα A βα Then AC = ((µ )C ,(ν )C ) is an intuitionistic fuzzy subsemigroup of S, and hence βα A βα A βα by Theorem 3.4, A= (µ ,ν ) is an intuitionistic fuzzy subsemigroup of S. Now for all A A x,ω,y ∈ S, (µ )C (xωy) ≥ min{(µ )C (x),(µ )C (y)} A βα A βα A βα i.e., β.µ (xωy)+α ≥ min{β.µ (x)+α,β.µ (y)+α} A A A i.e., β.µ (xωy)+α ≥ β.min{µ (x),µ (y)}+α A A A i.e., µ (xωy) ≥ min{µ (x),µ (y)} A A A and (ν )C (xωy) ≤ max{(ν )C (x),(ν )C (y)} A βα A βα A βα i.e., β.ν (xωy)−α ≤ max{β.ν (x)−α,β.ν (y)−α} A A A i.e., β.ν (xωy)−α ≤ β.max{ν (x),ν (y)}−α A A A i.e., ν (xωy) ≤ max{ν (x),ν (y)} A A A Hence A= (µ ,ν ) is an intuitionistic fuzzy bi-ideal of S. A A Theorem 3.6. Let A = (µ ,ν ) be a non-empty intuitionistic fuzzy subset of a semi- A A group S. Then A = (µ ,ν ) is an intuitionistic fuzzy bi-ideal of S if and only if the A A intuitionistic fuzzy magnified translation AC = ((µ )C ,(ν )C ) of A is a constant βα A βα A βα function, provided S is a group with identity e. 6 Proof. Let A= (µ ,ν ) be an intuitionistic fuzzy bi-ideal of S. Then A A (µ )C (x) = β ·µ (x)+α= β ·µ (exe)+α(since e is the identity of S) A βα A A ≥ β ·min{µ (e),µ (e)}+α(since A is an intuitionistic fuzzy A A bi-ideal of S) = β·µ (e)+α = (µ )C (e) A A βα Again (µ )C (e) = (µ )C (ee) = β ·µ (ee)+α(since e is the identity of S) A βα A βα A = β·µ ((xx−1)(x−1x))+α A = β·µ (x(x−1x−1)x)+α A ≥ β·min{µ (x),µ (x)}+α(since A is an intuitionistic fuzzy A A bi-ideal of S) = β·µ (x)+α = (µ )C (x). A A βα Thus(µ )C (x) =(µ )C (e)∀x ∈S.Similarlywecanshowthat(ν )C (x) = (ν )C (e) A βα A βα A βα A βα ∀x ∈ S. Hence AC is a constant function. βα Conversely, supposeAC is a constant function. Then AC is an intuitionistic fuzzy βα βα bi-ideal of S[7]. Hence by Theorem 3.5, A is an intuitionistic fuzzy bi-ideal of S. Note 3.7. For the converse to be true, S need not be a group. Theorem 3.8. The intuitionistic fuzzy magnified translation AC = ((µ )C ,(ν )C ) βα A βα A βα of A = (µ ,ν ) is an intuitionistic fuzzy (1,2)-ideal of S if and only if A = (µ ,ν ) A A A A is an intuitionistic fuzzy (1,2)-ideal of S. Proof. Let A = (µ ,ν ) be an intuitionistic fuzzy (1,2)-ideal of S. Then A is an A A intuitionistic fuzzy subsemigroup of S. Hence, by Theorem 3.4, AC is an intuitionistic βα fuzzy subsemigroup of S. Now for x,y,z,ω ∈ S, (µ )C (xω(yz)) = β·µ (xω(yz))+α A βα A ≥ β·min{µ (x),µ (y),µ (z)}+α A A A (since A is an intuitionistic fuzzy (1,2)-ideal of S) = min{β·µ (x)+α,β ·µ (y)+α,β ·µ (z)+α} A A A = min{(µ )C (x),(µ )C (y),(µ )C (z)}. A βα A βα A βα and (ν )C (xω(yz)) = β ·ν (xω(yz))−α A βα A ≤ β ·max{ν (x),ν (y),ν (z)}−α A A A (since A is an intuitionistic fuzzy (1,2)-ideal of S) = max{β ·ν (x)−α,β ·ν (y)−α,β ·ν (z)−α} A A A = max{(ν )C (x),(ν )C (y),(ν )C (z)}. A βα A βα A βα 7 Hence AC = ((µ )C ,(ν )C ) is an intuitionistic fuzzy (1,2)-ideal of S. βα A βα A βα Conversely, let AC = ((µ )C ,(ν )C ) be an intuitionistic fuzzy (1,2)-ideal of S. βα A βα A βα Then AC = ((µ )C ,(ν )C ) is an intuitionistic fuzzy subsemigroup of S, and hence βα A βα A βα by Theorem 3.4, A= (µ ,ν ) is an intuitionistic fuzzy subsemigroup of S. Now for all A A x,ω,y,z ∈ S, (µ )C (xω(yz)) ≥ min{(µ )C (x),(µ )C (y),(µ )C (z)} A βα A βα A βα A βα i.e., β.µ (xω(yz))+α ≥ min{β.µ (x)+α,β.µ (y)+α,β.µ (z)+α} A A A A i.e., β.µ (xω(yz))+α ≥ β.min{µ (x),µ (y),µ (z)}+α A A A A i.e., µ (xωy) ≥ min{µ (x),µ (y),µ (z)} A A A A and (ν )C (xω(yz)) ≤ max{(ν )C (x),(ν )C (y),(ν )C (z)} A βα A βα A βα A βα i.e., β.ν (xω(yz))−α ≤ max{β.ν (x)−α,β.ν (y)−α,β.ν (z)−α} A A A A i.e., β.ν (xω(yz))−α ≤ β.max{ν (x),ν (y),ν (z)}−α A A A A i.e., ν (xω(yz)) ≤ max{ν (x),ν (y),ν (z)} A A A A Hence A= (µ ,ν ) is an intuitionistic fuzzy (1,2)-ideal of S. A A Theorem 3.9. The intuitionistic fuzzy magnified translation AC = ((µ )C ,(ν )C ) βα A βα A βα of A = (µ ,ν ) is an intuitionistic fuzzy left ideal(intuitionistic fuzzy right ideal, in- A A tuitionistic fuzzy ideal) of S if and only if A = (µ ,ν ) is an intuitionstic fuzzy left A A ideal(resp. intuitionistic fuzzy right ideal, intuitionstic fuzzy ideal) of S. Proof. Let A = (µ ,ν ) be an intuitionistic fuzzy left ideal of S. Then A = (µ ,ν ) A A A A is a non-empty intuitionistic fuzzy subset of S. Hence, as β > 0, AC is a non-empty βα intuitionistic fuzzy subset of S. Let x,y ∈ S. Then (µ )C (xy) = β·µ (xy)+α ≥β ·µ (y)+α =(µ )C (y) A βα A A A βα and (ν )C (xy) = β·ν (xy)−α ≤ β·ν (y)−α = (ν )C (y). A βα A A A βα Hence AC = ((µ )C ,(ν )C ) is an intuitionistic fuzzy left ideal S. βα A βα A βα Conversely, let AC = ((µ )C ,(ν )C ) be an intuitionistic fuzzy left ideal of S. βα A βα A βα Then AC and hence A is a non-empty intuitionistic fuzzy subset of S. Now for all βα x,y ∈ S, (µ )C (xy)≥ (µ )C (y),i.e., β.µ (xy)+α ≥ β.µ (y)+α,i.e., µ (xy) ≥ µ (y) A βα A βα A A A A and (ν )C (xy)≤ (ν )C (y),i.e., β.ν (xy)−α ≤ β.ν (y)−α,i.e., ν (xy) ≤ν (y). A βα A βα A A A A Hence A = (µ ,ν ) is an intuitionistic fuzzy left ideal of S. Similar is the proof for A A intuitionistic fuzzy right ideal or intuitionistic fuzzy ideal. 8 Theorem 3.10. The intuitionistic fuzzy magnified translation AC = ((µ )C ,(ν )C ) βα A βα A βα of A = (µ ,ν ) is an intuitionistic fuzzy semiprime ideal of S if and only if A = A A (µ ,ν ) is an intuitionistic fuzzy semiprime ideal of S. A A Proof. Let A = (µ ,ν ) be an intuitionistic fuzzy semiprime ideal of S. Then A = A A (µ ,ν ) is a non-empty intuitionistic fuzzy subset of S. Hence AC is a non-empty A A βα intuitionistic fuzzy subset of S(β > 0). Let x ∈S. Then (µ )C (x) = β·µ (x)+α ≥ β ·µ (x2)+α = (µ )C (x2) A βα A A A βα and (ν )C (x) = β ·ν (x)−α ≤β ·ν (x2)−α = (ν )C (x2). A βα A A A βα Hence AC = ((µ )C ,(ν )C ) is an intuitionistic fuzzy semiprime ideal of S. βα A βα A βα Conversely, let AC = ((µ )C ,(ν )C ) be an intuitionistic fuzzy semiprime ideal of βα A βα A βα S. Then AC and hence A is a non-empty intuitionistic fuzzy subset of S. Then for all βα x ∈S, (µ )C (x) ≥ (µ )C (x2),i.e., β.µ (x)+α≥ β.µ (x2)+α,i.e., µ (x) ≥ µ (x2). A βα A βα A A A A and (ν )C (x) ≤ (ν )C (x2),i.e., β.ν (x)−α≤ β.ν (x2)−α,i.e., ν (x) ≤ ν (x2). A βα A βα A A A A Hence A= (µ ,ν ) is an intuitionistic fuzzy semiprime ideal of S. A A Theorem 3.11. Let A = (µ ,ν ) and B = (µ ,ν ) be two intuitionistic fuzzy A A B B semiprime ideals of S. Then A ∩ B is an intuitionistic fuzzy semiprime ideal of S, provided it is non-empty. Proof. Let A = (µ ,ν ) and B = (µ ,ν ) be two intuitionistic fuzzy semiprime ideals A A B B of S and x ∈ S. Then 2 2 (µ ∩µ )(x) = min{µ (x),µ (x)} ≥ min{µ (x ),µ (x )} A B A B A B (since A and B are intuitionistic fuzzy semiprime ideals of S) 2 = (µ ∩µ )(x ) A B and 2 2 (ν ∪ν )(x) = max{ν (x),ν (x)} ≤ max{ν (x ),ν (x )} A B A B A B (since A and B are intuitionistic fuzzy semiprime ideals of S) 2 = (ν ∪ν )(x ). A B Hence A∩B is an intuitionistic fuzzy semiprime ideal of S. Since the intersections of intuitionistic fuzzy semiprime ideals is an intuitionistic fuzzy semiprime ideal, so we have the following corollary. 9 Corollary 3.12. Let A = (µ ,ν ) and B = (µ ,ν ) be two intuitionistic fuzzy A A B B semiprime ideals of S. Then AC ∩BC is an intuitionistic fuzzy semiprime ideal of S, βα βα provided it is non-empty. Notation 3.13. Let A = (µ ,ν ) be an intuitionistic fuzzy subset of a semigroup S A A and AC = ((µ )C ,(ν )C ) be an intuitionistic fuzzy magnified translation of A. If βα A βα A βα for any elements x,y ∈ S, (µ )C (x) = (µ )C (y) and (ν )C (x) = (ν )C (y), then we A βα A βα A βα A βα write AC (x)= AC (y) ∀x,y ∈ S. βα βα Theorem 3.14. If A = (µ ,ν ) is an intuitionistic fuzzy semiprime ideal of a semi- A A group S, then AC (x)= AC (x2) ∀x∈ S. βα βα Proof. Let A = (µ ,ν ) be an intuitionistic fuzzy semiprime ideal of S. Let x ∈ S. A A Then (µ )C (x) = β ·µ (x)+α ≥β ·µ (x2)+α = (µ )C (x2). A βα A A A βα Again (µ )C (x2) = β·µ (x2)+α A βα A ≥ β·µ (x)+α(since A is an intuitionistic fuzzy ideal) A = (µ )C (x). A βα Consequently,(µ )C (x) = (µ )C (x2)∀x ∈S.Similarlywecanshowthat(ν )C (x) = A βα A βα A βα (ν )C (x2) ∀x ∈S. Hence AC (x) = AC (x2) ∀x∈ S. A βα βα βα By routine verification we can have the following theorem. Theorem 3.15. Let χ be the characteristic function of a non-empty subset A of S. Then A is a left(resp. right) ideal of S if and only if (χ,χ) is an intuitionistic fuzzy left(resp. right) ideal of S. Definition 3.16. A semigroup S is called intra-regular[8] if for each element a of S, 2 there exist elements x,y ∈ S such that a = xa y. Theorem 3.17. For a semigroup S the following conditions are equivalent: (1) S is an intra-regular semigroup, (2) for every intuitionistic fuzzy ideal A = (µ ,ν ) of S the intuitionistic fuzzy A A magnified translation AC = ((µ )C ,(ν )C ) of A is an intuitionistic fuzzy semiprime βα A βα A βα ideal of S. Proof. (1) ⇒ (2) : Let A = (µ ,ν ) be an intuitionistic fuzzy ideal of S and m ∈ S. A A 2 Then there exist x,y ∈ S such that m = xm y(since S is intra-regular). Then (µ )C (m)= (µ )C (xm2y)= β ·µ (xm2y)+α A βα A βα A ≥ β ·µ (m2y)+α ≥ β·µ (m2)+α = (µ )C (m2) A A A βα 10