Microsoft Word - 95-106   95  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020       Interval Value Fuzzy k-Ideals of a KU-Semigroup          Article history: Received 23 June 2019, Accepted 6 August 2019, Published in April 2020. Abstract The notion of interval value fuzzy k-ideal of KU-semigroup was studied as a generalization of afuzzy k-ideal of KU-semigroup. Some results of this idea under homomorphism are discussed. Also, we presented some properties about the image (pre- image) for interval~ valued fuzzy~k-ideals of a KU-semigroup. Finally, the~ product of~ interval valued fuzzyk-ideals is established. Keywords: KU-algebra;KU-semigroup; interval valuefuzzy S-ideal; interval value fuzzy k- ideal; interval value fuzzy P-ideal. 1. Introduction PrabpayaknandˑLeerawat [1,2]. Introduced the KU-algebra which is dual of BCK-algebra. Inˑ[3]. Kareem and Hasan introduced the KU-semigroups and defined some types of ideals in this concept. The fuzzy set was initiated by Zadeh, in [4]. Since then this concept has been applied in many distinct branches of mathematics such as groups, vector space, topological space and ring theory. In [5]. The idea of fuzzy KU-algebra was introduced by Mostafa et al. and the fuzzy KU-semigroupwas studied by Elaf and Kareem in [6]. Fuzzy sets extensions such as intuitionistic fuzzy sets, Bipolar-valued fuzzy sets, and interval valued fuzzy sets were studied by many mathematicians see [7-12]. The notion of interval value fuzzy k-ideal of KU- semigroup was studied in this paper and few properties were investigated. Some results of these ideals ina KU-semigroup´under homomorphism are discussed. The image of these ideals in a KU-semigroup was defined. Finally, the product of some ideals was established. 2. Preliminaries In this part, we review some concepts related to KU-semigroup and interval valued fuzzy logic. Definition (1) [1-2]. Algebra ℵ,∗ ,0 is called a KU-algebra if, for all 𝜒, 𝛾, τ ∈ ℵ, Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.2.2430 Sally A. Talib Fatema F. Kareem rim89@gmail.comsallyabdulka fa_sa20072000@yahoo.com Department of Mathematics, College of Education for Pure Science Ibn-Al-Haitham, University of Baghdad, Baghdad, Iraq.    96  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 (ku ) 𝜒 ∗ 𝛾 ∗ 𝛾 ∗ τ ∗ 𝜒 ∗ τ 0, (ku ) 𝜒 ∗0 = 0, (ku ) 0* 𝜒 𝜒, (ku ) 𝜒*𝛾 0 and 𝛾 ∗ 𝜒 implies 𝜒 𝛾 and, (ku ) 𝜒 ∗ 𝜒 0. A binary relation on ℵ is defined by 𝜒 𝛾 ⟺ 𝛾 ∗ 𝜒 0. It follows that (ℵ, is a~ partially ordered set~. Then, ℵ,∗, 0) ~ satisfies the following statements. For all 𝜒, 𝛾, τ ∈ ℵ,  (ᴋu ) 𝛾 ∗ τ ∗ 𝜒 ∗ τ 𝜒 ∗ 𝛾 , (ᴋu )0 𝜒, (ᴋu ) 𝜒 𝛾, 𝛾 𝜒 implies   (ᴋu ) 𝛾 ∗ 𝜒 𝜒 . Example (2)[1]. Let ℵ 0, 𝑎, 𝑏, 𝑐 be a set and ∗ a binary operation defined in the following table It is easy to see that ℵ,∗ ,0 is a KU-algebra. ˑTheorem(ˑ3) ˑ [2]. ˑLet ℵ,∗ ,0 be a KU-algebra. Then, for all 𝜒, 𝛾, τ ∈ ℵ, (1) If 𝜒 𝛾 implies 𝛾 ∗ τ 𝜒 ∗ τ, (2) 𝜒 ∗ 𝛾 ∗ τ 𝛾 ∗ 𝜒 ∗ τ , (3) (( 𝛾 ∗ 𝜒 ∗ 𝜒 𝛾. Definition (4) [1-2]. Let ℵ,∗ ,0 be a KU-algebra and 𝛪 be a non-ˑempty subset ofˑℵ. Then 𝛪 is calledˑˑan ˑideal of ℵ if ˑfor any 𝜒, 𝛾 ∈ ℵ,ˑthen (i) 0 ∈ 𝛪and (ii) if 𝜒 ∗ 𝛾 ∈ 𝛪 and 𝜒 ∈ 𝛪 imply 𝛾 ∈ 𝛪.ˑˑ ˑDefinition (5) [1-2].ˑ Let 𝛪 be a subset of a KU-algebra ℵ,∗ ,0 and 𝜑. Then 𝛪 is named a KU-ideal of ℵ, if (Ι 0 ∈ 𝛪 and (Ι ∀𝜒, 𝛾, 𝜏 ∈ ℵ, 𝜒 ∗ 𝛾 ∗ 𝜏 ∈ 𝐼 an𝑑 𝛾 ∈ 𝛪imply 𝜒 ∗ 𝜏 ∈ 𝛪. Definition (6)[3]. A KU-semigroup is a nonempty set ℵ with two binary operations ~∗,∘and a constant ~0 satisfying~ the following~~ (I) ℵ,∗ ,0 isˑa KU-algebra,   97  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 (II) (ℵ,∘ is aˑsemigroup, (III) The operation ∘ is distributiveˑ (on both sides) over theoperation ∗, i.e. 𝜒 ∘ 𝛾 ∗ τ 𝜒 ∘ 𝛾 ∗ 𝜒 ∘ τ and 𝜒 ∗ 𝛾 ∘ τ 𝜒 ∘ τ ∗ 𝛾 ∘ τ , forall 𝜒, 𝛾, τ ∈ 𝛸. Example (7)[3]. Letˑℵ 0,1,2,3 beˑa set. Define∗ˑ-operation and ∘-operationby the following tables Thenˑ ℵ,∗,∘ ,0 is aˑKU-semigroup. " Definition (8)[3]. AˑsubKU-semigroup is a non-empty subset A of aˑKU-semigroup ℵ and it is satisfied 𝜒 ∗ 𝛾, 𝜒 ∘ 𝛾 ∈ Α, for all 𝜒, 𝛾 ∈ Α. Definition (9)[3]. Let ℵ,∗,∘ ,0 be a KU-semigroup and 𝜑 𝐼 ℵ. Then, 𝐼 is named an S-ideal of ℵ , if i) 𝐼 is an ideal of a KU-algebra ℵ,∗ ,0 , ii) For all 𝜒 ∈ ℵ, 𝑎 ∈ 𝐼, we have 𝜒 ∘ 𝑎 ∈ 𝐼 and𝑎 ∘ 𝜒 ∈ 𝐼. Definition (10)[3]. Let ℵ,∗,∘ ,0 ~be a KU-~semigroup and𝜑 𝐴  ℵ. Then~ Α is said to be a k-ideal of ℵ~, if~ i) Α is an KU~-ideal ~of a~ KU-~algebra~ ℵ,∗ ,0 , ii) For~ all 𝜒 ∈ 𝛸~, 𝑎 ∈ Α,~we have~ 𝜒 ∘ 𝑎 ∈ Α and 𝑎 ∘ 𝜒 ∈ Α~. Definition (11)[3]. Let ℵ,∗,∘ ,0 ,~~be a ~KU-~semigroup and 𝜑 𝐴  ℵ. Then, Α is~ said to~ be a P~-idealof ℵ~, if ~ (p1)~ For any 𝜒, 𝛾, τ ∈ ℵ, τ ∗ 𝜒 ∗ 𝛾 ∈ Α and τ ∗ 𝜒 ∈ Α ⟹ τ ∗ 𝛾 ∈ Α. (p2) ~For all 𝜒 ∈ ℵ, 𝑎 ∈ Α, we have 𝜒 ∘ 𝑎 ∈ Α and 𝑎 ∘ 𝜒 ∈ Α. Definition (12)[3]. Let ℵˑ and ℵ be two KU-semigroups. A mapping ƒ : ℵ ⟶ ℵ ˑis called a KU-semigroup homomorphism if ƒ (𝜒 ∗ 𝛾 ) =ƒ 𝜒 ∗ ƒ 𝛾 and ƒ (𝜒 ∘ 𝛾 )= ƒ 𝜒 ∘ ƒ 𝛾 , for all 𝜒, 𝛾 ∈ ℵ. The kernel of ƒ is denoted by ker ƒ and is defined by { 𝜒 ∈ ℵ: ƒ 𝜒 0 . Moreover, the image of ƒ is denoted by im ƒ and is defined by { ƒ 𝜒 ∈  : 𝜒 ∈ ℵ . We review some concepts of fuzzy logic. A functionˑ𝜇: ℵ ⟶ 0,1 is said to be a fuzzy set of a set ℵ and the set 𝑈 𝜇, 𝑡 𝜒 ∈ ℵ: 𝜇 𝜒 𝑡 ˑis said to be a level set of 𝜇, for 𝑡 in 0,1 .∞   98  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Definition (13)[6]. A fuzzy set 𝜇 of ℵ is called a fuzzy sub KU-semigroup if, for all 𝜒, 𝛾 ∈ ℵ i) ˑ µ ˑ𝜒 ∗ 𝛾ˑ min ˑ µ 𝜒 , ˑµ 𝛾 , ii) ˑ𝜒 ∘ 𝛾ˑ min ˑ µ 𝜒 , ˑµ 𝛾 . Definition (14)[6]. A fuzzy set µ inˑℵ is called a fuzzyˑS-ideal of ℵ if, for all𝜒, 𝛾 ∈ ℵ.ˑ (S1)ˑµ(0) µ 𝜒 , (S2) µ 𝛾 min µ 𝜒 ∗ 𝛾 ,µ 𝜒 (S3) µ 𝜒 ∘ 𝛾 min µ 𝜒 ˑ, ˑµ 𝛾 . Definition (15)[6]. A fuzzy setˑµ in ℵˑis calledˑaˑfuzzyˑk-idealˑof ℵˑif it satisfies the following conditions: for all 𝜒, 𝛾, τ ∈ ℵ. (k1) µ(0) µ  , (k2)µ 𝜒 ∗ τ min µ 𝜒 ∗ 𝛾 ∗ τ , ˑµ 𝛾 (k3)µ 𝜒 ∘ 𝛾 min µ 𝜒 ˑ, ˑµ 𝛾 . Example (16)[6]. Let ℵ 0, 𝑎, 𝑏, 𝑐, 𝑑 ˑˑbe a ~set. Define∗ -~operation and ∘-~operation~ by the ~following ~tablesˑ~ Thenˑ ℵ,∗,∘ ,0 ˑis a~KU-semigroup. Defineˑa fuzzyˑsetˑ𝜇 ∶ ℵ ⟶ 0,1 ˑˑby 𝜇 0 𝜇 𝑎 0.4, 𝜇 (b)= 𝜇 𝑐 0.2ˑ, 𝜇 𝑑 0.1.ˑ . Then by routine calculation we can prove thatˑ𝜇 is afuzzy k-ideal ofℵ. Definition (17) [6]. The Cartesian product of two fuzzy sets 𝜇 and 𝛽of ℵ is denoted by 𝜇 𝛽: ℵ ℵ → 0,1 and defined by 𝜇 𝛽 𝜒, 𝛾 min 𝜇 𝜒 , 𝛽 𝛾 , ∀𝜒, 𝛾 ∈ ℵ. Definition (18)[6]. Let 𝜇 be a fuzzy set in ℵ. If 𝜇 is defined by: ℵ ℵ → 0,1 , then 𝜇 is said to be a fuzzy relation on a set 𝑆´, where S⊆ ℵ. Definition (19)[6]. Let 𝜇 be a fuzzy relation on ℵ and  be a fuzzy subset of ℵ. Then the strongest fuzzy relation on ℵ is denoted by 𝜇 and is defined as follows 𝜇 𝜒, 𝛾 min 𝛽 𝜒 , 𝛽 𝛾 , ∀𝜒, 𝛾 ∈ ℵ. 3. Interval value fuzzy k-ideals in KU-semigroup In this part, we recall the definition of interval valued fuzzy set 𝜇 of ℵ as follows   99  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 𝜇 𝜒, 𝜇 𝜒 , 𝜇 𝜒 : 𝜒 ∈ ℵ , by briefly 𝜇 𝜇 , 𝜇 , where 𝜇 and 𝜇 are two fuzzy sets in ℵ such that 𝜇 𝜒 𝜇 𝜒 , for all 𝜒 ∈ ℵ and the closed~ sub-intervals ̂of ̂[0, 1] is denoted by D [0, 1]̂~. Note that, if 𝜇 𝜒 𝜇 𝜒 𝑐, ~where ~0 𝑐 1, ~then 𝜇 𝜒 𝑐, 𝑐 ~, ~it follows that 𝜇 𝜒 ∈ D [0, 1]~and it is given by 𝜇: ℵ → 𝐷 0,1 , for all χ∈ℵ and D [0, 1]={[𝑎 , 𝑎 : 𝑎 𝑎 𝑓𝑜𝑟 𝑎 , 𝑎 ∈ 0, 1 . Consider, two elements ],[1 UL aaD  and ],[2 UL bbD  in D [0, 1] are defined by )],min(,),[min(),min( 21 UULL babaDDr  and  ),max(,),max(),max( 21 UULL babaDDr  . Definition (20). Let ℵ,∗,∘ ,0 ~be a KU-~semigroup. If 𝜇: ℵ → 𝐷 0,1 , then the level subset of 𝜇~is ~denoted ~by 𝜇 and it is ~defined ~by 𝜇 {𝜒 ∈ ℵ: 𝜇 𝜒 �̃� , for every ~[0,0] �̃� 1,1 . Definition (?21). ~ Let ~ ℵ,∗,∘ ,0 be a KU-~semigroup and 𝜇: ℵ → 𝐷 0,1 . Then ~𝜇 is called an ~interval ~valued~fuzzy sub KU-~semigroup ℵ, if it satisfies the following conditions 𝜇 𝜒 ∗ 𝛾 𝑟𝑚𝑖𝑛 𝜇 𝜒 ˑ, ˑ𝜇 𝛾 ~,𝜇 𝜒 ∘ 𝛾 𝑟𝑚𝑖𝑛 𝜇 𝜒 ˑ, ˑ𝜇 𝛾 , ∀𝜒 , 𝛾 ∈ ℵ. Example (22). Let ℵ 0,1,2,3 ~be a set. We define two operations by the following tables Then ℵ,∗,∘ ,0 ˑis a KU-semigroup. Define 𝜇 𝜒 ˑas follows 𝜇 𝜒 0.2,0.7 𝑖𝑓 𝜒 0,1,2 0.1,0.3 𝑖𝑓 𝜒 3 And by applying definition21, we can prove that 𝜇 𝜒 is an interval valued fuzzysub KU- semi group of ℵˑ. Definition (23). Let ℵ,∗,∘ ,0 be a KU-semigroup and 𝜇: ℵ → 𝐷 0,1 . Then 𝜇 is named an interval valuedfuzzy S-ideal of ℵ if 𝑖 𝜇 (0) 𝜇 𝜒 , ∀𝜒 ∈ ℵ, 𝑖 For all, , 𝛾 ∈ , 𝜇 𝛾 𝑟 min 𝜇 𝜒 ∗ 𝛾 ,𝜇 𝜒 , 𝑖 𝜇 𝜒 ∘ 𝛾 𝑟min 𝜇 𝜒 ˑ, ˑ𝜇 𝛾 . Definition (24). Let ℵ,∗,∘ ,0 be a KU-semigroup and 𝜇: ℵ → 𝐷 0,1 . Then 𝜇~is named ~an interval valued fuzzy k-~´ideal of´ℵ~ if (,ƒ 𝜇 (0) 𝜇 𝜒 , ∀𝜒 ∈ ℵ, (ƒ For all 𝜒, 𝛾, τ ∈ ℵ,𝜇 𝜒 ∗ τ 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 , (ƒ 𝜇 𝜒 ∘ 𝛾 𝑟min 𝜇 𝜒 ˑ, ˑ𝜇 𝛾 .   100  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Example (25). From Example16, we define 𝜇 𝜒 as follows: [0.3,0.9] if {0,a,b,c} ( ) [0.1, 0.6] if d          . By this definition of 𝜇 , we can prove it is an interval valued~ fuzzy k-~ideal. Theorem (26).~Let ℵ,∗,∘ ,0 be a KU-semigroup. Then 𝜇in~ℵ is an interval valuedfuzzy~k- idealif~and only ifit is an interval valued fuzzy S-~ideal ofˑℵ~. Proof. ()~By taking 𝜒 0 in(ƒ , (ƒ and using (ku ), we obtain~~ ∀𝛾, τ ∈ ℵ 𝜇 τ 𝜇 0 ∗ τ 𝑟𝑚𝑖𝑛 𝜇 0 ∗ 𝛾 ∗ τ , 𝜇 𝛾 𝑟𝑚𝑖𝑛 𝜇 𝛾 ∗ τ , 𝜇 𝛾 and 𝜇 𝜒 ∘ 𝛾 𝑟min 𝜇 𝜒 ˑ, ˑ𝜇 𝛾 are satisfied. ( ? ) we have 𝜇 𝜒 ∗ τ 𝑟 𝑚𝑖𝑛 𝜇 𝛾 ∗ 𝜒 ∗ τ , 𝜇 𝛾 and by apply Theorem 3, we get 𝜇 𝜒 ∗ τ 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 and by Definition 24, we have 𝜇 𝜒 ∘ 𝛾 𝑟min 𝜇 𝜒 ˑ, ˑ𝜇 𝛾 . Thus 𝜇 is an interval valuedfuzzy k-ideal of ℵ. Theorem (27). Let ℵ,∗,∘ ,0 be a KU-semigroup, 𝛢 be a nonempty subset of ? ℵ and ? 𝜇 be an interval valued fuzzy set in ? ℵ. We define 𝜇 as follows 𝜇 𝜒 𝑡 , 𝑡 𝜒 ∈ 𝛢 𝛼1, 𝛼1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑤ℎ𝑒𝑟𝑒 𝑡 𝛼1 , 𝑡 𝛼2 𝑎𝑛𝑑 𝛼1, 𝛼1,𝑡 , 𝑡 ∈ D 0,1 . Then 𝛢 is' a k-´´ideal of ℵ´if´ and only ´if 𝜇 is´ an´ interval ´valued' fuzzy k´-´ideal´´of´ ℵ. Moreover,´ℵ Α, such that´ℵ 𝜒 ∈ ℵ: 𝜇 (𝜒) 𝜇 0 .  Proof. ( ) Since 𝜇 0 𝜇 𝜒 , ∀𝜒 ∈ ℵ, we get 𝜇 0 𝑡 , 𝑡 , then 0 ∈ Α. Let 𝜒 ∗ 𝛾 ∗ τ ∈ Α and 𝛾 ∈ Α, for any 𝜒, 𝛾, τ ∈ ℵ. Using )( 2f , we have 𝜇 𝜒 ∗ τ 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 = 𝑟𝑚𝑖𝑛 { [𝑡 ,𝑡 , 𝑡 , 𝑡 𝑡 , 𝑡 and thus 𝜇 𝜒 ∗ τ 𝑡 , 𝑡 , it follows that 𝜒 ∗ τ ∈ Α . Now, let 𝜒 ∈ 𝛢 and 𝛾 ∈ 𝛢 by using )( 3f , we know that 𝜇 𝜒 ∘ 𝛾 𝑟 min 𝜇 𝜒 ˑ, ˑ𝜇 𝛾 𝑟𝑚𝑖𝑛 𝑡 , 𝑡 , 𝑡 , 𝑡 𝑡 , 𝑡 , and thus 𝜇 𝜒 ∘ 𝛾 𝑡 , 𝑡 . Hence 𝜒 ∘ 𝛾 ∈ 𝛢 and it follows that 𝛢 is a k-ideal of ℵ. () Since 0 ∈ 𝛢, it follows that 𝜇 (0) 𝑡 , 𝑡 𝜇 𝜒 for all 𝜒 ∈ ℵ. Let 𝜒, 𝛾, τ ∈ ℵ. If 𝛾 ∉ Α and 𝜒 ∗ τ ∈ Α, then clearly 𝜇 𝜒 ∗ τ 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 . Assume that 𝛾 ∈ 𝛢 and 𝜒 ∗ τ ∉ 𝛢. Then by definition9, we have χ ∗ γ ∗ τ ∉ 𝛢 . Therefore 𝜇 𝜒 ∗ τ 𝛼1, 𝛼1 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 . Also, let 𝜒 ∈ 𝛢, γ ∉ 𝛢 𝑎𝑛𝑑 𝜒 ∘ 𝛾 ∈ 𝛢, then clearly: 𝜇 𝜒 ∘ τ 𝑡 , 𝑡 𝑟 𝑚𝑖𝑛 𝜇 𝜒 , 𝜇 𝛾 . Assume that 𝜒 ∉ 𝛢, 𝛾 ∈ Α and 𝜒 ∘ 𝛾 ∉ 𝛢. Then 𝜇 𝜒 ∘ 𝛾 𝛼 ,𝛼 =𝑟 𝑚𝑖𝑛 𝜇 𝜒 , 𝜇 𝛾 . Hence 𝜇 is a fuzzy k-ideal in ℵ.   101  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Finally, we have ´ℵ 𝜒 ∈ ℵ: 𝜇 (𝜒) 𝜇 0 𝜒 ∈ ℵ:𝜇 (𝜒) 𝑡 , 𝑡 Α. Theorem (28). Let ℵ,∗,∘ ,0 be a KU-semigroup and 𝜇: ℵ → 𝐷 0,1 . Then the level set 𝜇 of 𝜇 is a k-ideal in ℵ iff 𝜇 is an interval valued fuzzy k-ideal. ©?? Proof. ( ) For any �̃� 𝑡 , 𝑡 ∈ D 0,1 , assume 𝜇 is a non empty, then there exists 𝜒 ∈ 𝜇 and 𝜇 𝜒 �̃�. Itˑ followsˑ from Definition 24 that 𝜇 0 𝜇 𝜒 �̃�, so that 0 ∈ 𝜇 . Let 𝜒, 𝛾, τ ∈ ℵ such that 𝜒 ∗ 𝛾 ∗ τ ∈ 𝜇 and t   . We have ( ( )) t       and 𝜇 𝛾 �̃�. From Definition 24, we get the following 𝜇 𝜒 ∗ τ 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 𝑟 𝑚𝑖𝑛 �̃� , �̃� �̃� , thus 𝜒 ∗ τ ∈ 𝜇 . Now, let 𝑎 ∈ 𝜇 and 𝜒 ∈ ℵ, then 𝜇 𝜒 �̃� and 𝜇 𝑎 �̃�. We get 𝜇 𝜒 ∘ 𝑎 𝑟𝑚𝑖𝑛 𝜇 𝜒 , 𝜇 𝑎 �̃� , �̃� �̃�, whichimpliesthat 𝜒 ∘ 𝑎 ∈ 𝜇 . Similarly, 𝑎 ∘ 𝜒 ∈ 𝜇 . Therefore~𝜇 is a k-~ideal of ℵ~??? (⟹ Let μ be a non-empty and a k-ideal of ℵ , we have 𝜇 𝜒 �̃�, for every �̃� ∈ D 0,1 and for any 𝜒 ∈ ℵ. This implies that 𝜒 ∈ μ . And since 0 ∈ μ , then 𝜇 0 �̃� 𝜇 𝜒 . Now, we show that 𝜇 satisifies 𝑘 and 𝑘 . If not, suppose that ∃𝑙 , 𝑚, 𝑛 ∈ ℵ such that 𝜇 𝑙 ∗ 𝑛 𝑟 min 𝜇 𝑙 ∗ 𝑚 ∗ 𝑛 , 𝜇 𝑚 . put �̃� 𝜇 𝑙 ∗ 𝑛 𝑟 min 𝜇 𝑙 ∗ 𝑚 ∗ 𝑛 , 𝜇 𝑚 , for n any integer number, so 𝜇 𝑙 ∗ 𝑛 �̃� 𝑟 min 𝜇 𝑙 ∗ 𝑚 ∗ 𝑛 , 𝜇 𝑚 . Implies that 𝑙 ∗ 𝑚 ∗ 𝑛 ∈ 𝜇 𝑎𝑛𝑑 𝑚 ∈ 𝜇 , but 𝑙 ∗ 𝑛 ∉ 𝜇 , which implies 𝜇 is not a k- ideal of ℵ. Then, it is a contradiction. Let l, m∈ 𝜇 such that 𝜇 𝑙 ∘ 𝑚 𝑟 min 𝜇 𝑙 , 𝜇 𝑚 . Then by taking �̃� 𝜇 𝑙 ∘ 𝑚 𝑟 min 𝜇 𝑙 , 𝜇 𝑚 . We have 𝜇 𝑙 ∘ 𝑚 �̃� 𝑟 min 𝜇 𝑙 , 𝜇 𝑚 . Then, 𝑙, 𝑚 ∈ 𝜇 but 𝑙 ∘ 𝑚 ∉ 𝜇 . It means that 𝜇 is not k-ideal of ℵ and ~this' is~ a ´~contradiction.' The proof is completed. Definition (29). A fuzzy set 𝜇 in ℵ isˑ calledˑ an interval valued fuzzy P-ideal ofˑℵ if, for all 𝜒, 𝛾, τ ∈ ℵ 𝑃 𝜇 (0) 𝜇 𝜒 𝑃 𝜇 τ ∗ 𝛾 𝑟 𝑚𝑖𝑛 𝜇 τ ∗ 𝜒 ∗ 𝛾 , 𝜇 τ ∗ 𝜒 . 𝑃 𝜇 𝜒 ∘ 𝛾 𝑟 min 𝜇 𝜒 ˑ, ˑ𝜇 𝛾 . Example30. Let ℵ 0, 𝑎, 𝑏 be a set. Define ∗-ˑoperation and∘-operation byˑtheˑfollowing tables   102  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Then ℵ,∗,∘ ,0 is a KU-semigroup. Define 𝜇 𝜒 as follows 𝜇 𝜒 0.3,0.8 𝜒 0 0.2,0.4 𝜒 0 . Then 𝜇 𝜒 ~is an interval valued fuzzy P-ideal of ℵˑ~. Theorem31. Let ℵ,∗,∘ ,0 be a KU-semigroup and 𝜇: ℵ → 𝐷 0,1 . If 𝜇´ is ~an interval~valued ´fuzzy ~P-~ideal, then~ it is an interval~ valued´ fuzzy S-´ideal. µ Proof~. By (P2) weget: 𝜇 τ ∗ 𝛾 𝑟 𝑚𝑖𝑛 𝜇 τ ∗ 𝜒 ∗ 𝛾 , 𝜇 τ ∗ 𝜒 , put τ 0, we get: 𝜇 0 ∗ 𝛾 𝑟 𝑚𝑖𝑛 𝜇 0 ∗ 𝜒 ∗ 𝛾 , 𝜇 0 ∗ 𝜒 Thus 𝜇 𝛾 𝑟 𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 , 𝜇 𝜒 . The reverse of Theorem~31is incorrect. The~ example32~ shows ~the reverse. Example 32. Let ℵ 0, 𝑎, 𝑏 beˑaˑset. Define (∗-operation) and (∘-operation) by the following tables Then ℵ,∗,∘ ,0 ˑis a KU-semigroup. Define 𝜇 𝜒 as follows: 𝜇 𝜒 0.4,0.8 𝑖𝑓 𝜒 0, 𝑏 0.1,0.3 𝑖𝑓 𝜒 𝑎 . We can easily prove that~𝜇 is ~an interval~ valued fuzzy ~S-idealof~ℵ, butˑ~it ~is not an interval~ valued fuzzy P-ideal, since 𝜇 0 ∗ 𝑎 0.1,0.3 𝑟 min 𝜇 0 ∗ 𝑏 ∗ 𝑎 , 𝜇 0 ∗ 𝑏 0.4,0.8 . 4. Study of Image (Pre-´image) for interval valued´ fuzzy k-´ideal The image and thepre-image are important topics in modern algebra so we will focus on these two concepts in this part of our paper. We will study these concepts with~ the interval~   103  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 valued fuzzy~k-ideals ina KU-~semigroup~ℵˑunder~ homomorphism. Also, wewill prove that the products of ~interval~ valued~ fuzzy k-~ideals~ are a fuzzy ~k-~ideal of~ a KU-semigroup ℵˑ. Definition33. Let 𝑓: ℵ → 𝑌 be a mapping from KU-semigroup ℵ into KU-semigroup 𝑌 and 𝜇 be an interval valued fuzzy subset of ℵ. We define the image for 𝜇 under𝑓, denoted by 𝑓 (𝜇 as follows 𝑓 𝜇 𝛾 𝑠𝑢𝑝𝜇 𝜒 ∈ , 𝑖𝑓 𝑓 𝛾 𝜒 ∈ ℵ: 𝑓 𝜒 𝛾 ∅ 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 And the pre-image for 𝛽 under f, where 𝛽 is´ an interval´ valued fuzzy´ subset of 𝑌´, denoted´ by 𝑓 𝟏 𝛽 in ℵ by 𝜇 𝜒 𝑓 𝟏 𝛽 𝛽 𝑓 𝜒 ,∀ 𝜒 ∈ ℵ. Lemma34.´Let ƒ be a homomorphism mapping from a KU-semigroup ℵ,∗,∘ ,0 into a KU- semigroup ℵ′,∗′,∘′, 0′ . Then ƒ 𝟏 𝛽 is an interval valued fuzzy k´-ideal of ℵ, if the mapping 𝛽 is an interval valued fuzzy k-ideal of ℵ′.~?? Proof. For all 𝜒 ∈ ℵ, we have 𝜇 0 =𝛽 ƒ 0 𝛽 ƒ 𝜒 𝜇 𝜒 . Let , 𝛾, τ ∈ ℵ, then we have 𝜇 𝜒 ∗ τ 𝛽 ƒ 𝜒 ∗ τ 𝛽 ƒ 𝜒 ∗ 𝑓 τ 𝑟𝑚𝑖𝑛 𝛽 ƒ 𝜒 ∗ ƒ 𝛾 ∗ ƒ τ , 𝛽ƒ 𝛾 = r min 𝛽 ƒ 𝜒 ∗ 𝛾 ∗ τ , 𝛽ƒ 𝛾 = r min 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 Also, we have 𝜇 𝜒 ∘ 𝛾 𝛽 ƒ 𝜒 ∘ 𝛾 𝛽 𝑓 𝜒 ∘′ ƒ τ 𝑟𝑚𝑖𝑛 𝛽 ƒ 𝜒 , 𝛽 ƒ 𝛾 =𝑟𝑚𝑖𝑛 𝜇 𝜒 , 𝜇 𝛾 . Hence´the proof´is completed. Theorem(35). Letƒ: ℵ → ℵ′ be an epimorphism between two KU-semigroups ℵ and ℵ'.ƒ 𝜇 ~is ´an~ interval ´valued ~fuzzy´ k-~ideal of ´ℵ , if 𝜇~is´ an~interval ´valued ~fuzzy´k-~ideal ofℵ´~.µ Proof.' µLet 𝜒′, 𝛾′ ∈ ℵ′ , ~then ∃𝜒, 𝛾 ∈ ℵ such that ƒ 𝜒 𝜒 and ƒ 𝛾 𝛾 . By definition of image, we have ƒ(𝜇 𝜒′ 𝑠𝑢𝑝𝜇 𝜒 ∈ƒ 𝟏 ′ , for some 𝜒 ∈ ℵ, and, ƒ(𝜇 𝛾′ 𝑠𝑢𝑝𝜇 𝛾 ∈ƒ 𝟏 ′ , for some 𝛾 ∈ ℵ. Weˑ have𝜇 (0) 𝜇 𝜒 , ∀𝜒 ∈ ℵ. Then, (i ƒ(𝜇 0′ =𝑠𝑢𝑝𝜇 0 ∈ƒ 𝟏 ′ 𝑠𝑢𝑝𝜇 𝜒 ∈ƒ 𝟏 ′ ƒ(𝜇 𝜒 ′ , for any 𝜒′ ∈ ℵ. (ii) For any𝜒′, 𝛾′, τ′ ∈ ℵ′, let 𝜒 ∈ ƒ 𝟏 𝜒′ , 𝛾 ∈ ƒ 𝟏 𝛾′ , τ ∈ ƒ 𝟏 τ′ , and since f is ahomomorphism, thenƒ(𝜇 𝜒′ ∗ τ′ =𝑠𝑢𝑝𝜇 𝜒 ∗ τ ∗τ ∈ƒ 𝟏 ′∗ τ′ 𝑟𝑚𝑖𝑛 𝑠𝑢𝑝𝜇 𝜒 ∗ 𝛾 ∗ τ ∗ ∗τ ∈ƒ 𝟏 ′∗ ′∗ τ′ , 𝑠𝑢𝑝𝜇 𝛾 ∈ƒ 𝟏 ′ =𝑟𝑚𝑖𝑛 ƒ(𝜇 𝜒′ ∗ 𝛾′ ∗ τ′ , ƒ(𝜇 𝛾′ (iii) ˑForany 𝜒 ′, 𝛾′ ∈ ℵ′ , let 𝜒 ∈ ƒ 𝟏 𝜒′ , 𝛾 ∈ ƒ 𝟏 𝛾′ be such that: ƒ(𝜇 𝜒′ ∘ 𝛾′ =𝑠𝑢𝑝𝜇 𝜒 ∘ 𝛾 ∘ τ ∈ƒ 𝟏 ′∘ τ′ 𝑟𝑚𝑖𝑛 𝑠𝑢𝑝𝜇 𝜒 ∈ƒ 𝟏 ′ , 𝑠𝑢𝑝𝜇 𝛾 ∈ƒ 𝟏 ′   104  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 = 𝑟𝑚𝑖𝑛 {(𝜇 𝜒′ , ƒ(𝜇 𝛾′ Hence´ the ~proof´ is~ completed. Definition (36). If ~𝜇 and 𝛽 are´ two ~interval´ valued ~fuzzy´subsets of a ~set ℵ.´ Then~ the product of 𝜇 and 𝛽 denoted by 𝜇 𝛽 is defined by: 𝜇 𝛽 𝜒, 𝛾 𝑟 min 𝜇 𝜒 , 𝛽 𝛾 ,for all 𝜒, 𝛾 ∈ ℵ ℵ. Theorem (37). The product 𝜇 𝛽 is~ an interval ~valued fuzzy ~k-ideal~ of ℵ ℵ, if 𝜇 𝑎𝑛𝑑 𝛽~are~ interval valued ~fuzzy ~k-´ideals of ~a KU-semigroup ℵ. Proof. Let 𝜒, 𝛾 ∈ ℵ ℵ, we have 𝜇 𝛽 0, 0 𝑟𝑚𝑖𝑛 𝜇 0 , 𝛽 0 𝑟𝑚𝑖𝑛 𝜇 𝜒 , 𝛽 𝛾 𝜇 𝛽 𝜒, 𝛾 . Now, let (𝜒 ,, 𝜒 , 𝛾 , 𝛾 , τ , τ ∈ ℵ ℵ, then 𝜇 𝛽 𝜒 ∗ τ , 𝜒 ∗ τ 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ τ , 𝛽 𝜒 ∗ τ 𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 , 𝑟𝑚𝑖𝑛 𝛽 𝜒 ∗ 𝛾 ∗ τ , 𝛽 𝛾 = 𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝛽 𝜒 ∗ 𝛾 ∗ τ , 𝑟𝑚𝑖𝑛 𝜇 𝛾 , 𝛽 𝛾 =𝑟𝑚𝑖𝑛 𝜇 𝛽 𝜒 ∗ 𝛾 ∗ τ , 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛽 𝛾 , 𝛾 . And, 𝜇 𝛽 𝜒 ∘ 𝜒 𝛾 ∘ 𝛾 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∘ 𝜒 , 𝛽 𝛾 ∘ 𝛾 𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝜇 𝜒 , 𝜇 𝜒 , 𝑟𝑚𝑖𝑛 𝛽 𝛾 , 𝛽 𝛾 = 𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝜇 𝜒 , 𝛽 𝛾 , 𝑟𝑚𝑖𝑛 𝜇 𝜒 , 𝛽 𝛾 =𝑟𝑚𝑖𝑛 𝜇 𝛽 𝜒 , 𝛾 , 𝜇 𝛽 𝜒 , 𝛾 . Definition´(38). Let 𝜇 be an interval~ valued fuzzy set in ℵ. If 𝜇 ´is defined by: 𝜇: 𝑆 𝑆 → 𝐷 0,1 , then 𝜇´ is named an interval~ valued fuzzy relation on a set 𝑆´, where S⊆ ℵ. Definition (39). Let 𝛽 be an interval~ valued fuzzy set in ℵ. Then the strongest interval~ valued fuzzy relation on ℵ by 𝛽 is denoted by 𝜇 and defined as follows 𝜇 𝜒, 𝛾 𝑟𝑚𝑖𝑛 𝛽 𝜒 , 𝛽 𝛾 , for all 𝜒, 𝛾 ∈ ℵ. Lemma (40). If the strongest interval~ valued fuzzy relation on~ℵ is ´an interval'' valued fuzzy 'k-ideal of ℵ ℵ,´~then𝛽 𝜒 𝛽 0 , for all 𝜒 ∈ ℵ and 𝛽 is an interval valued fuzzy set of a KU-semigroupℵ. Proof. ´Let 𝜇 ´´be an interval valued ´fuzzy k-´ideal of ℵ ℵ´, it follows that: 𝛽 𝜒 𝑟𝑚𝑖𝑛 𝛽 𝜒 , 𝛽 𝜒 𝜇 𝜒, 𝜒 𝜇 0,0 𝑟𝑚𝑖𝑛 𝛽 0 , 𝛽 0 𝛽 0 . Hence, 𝛽 𝜒 𝛽 0 .   105  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Theorem(41). The strongest´~ interval valued fuzzy relation~𝜇 on~ℵ is an interval 'valued fuzzy k''-ideal of´~ℵ ℵ iff the mapping 𝛽~is ´an~interval~´valued~fuzzy''k-ideal of´~ℵ.???? Proof. ( )~Since 𝛽~is an interval valued fuzzy~´k-ideal of~ℵ~´, ~then 𝜇 0,0 𝑟𝑚𝑖𝑛 𝛽 0 , 𝛽 0 𝑟𝑚𝑖𝑛 𝛽 𝜒 , 𝛽 𝛾 𝜇 𝜒, 𝛾 . Now, for any(𝜒 ,, 𝜒 𝛾 , 𝛾 τ , τ ∈ ℵ ℵ´, ´we have´: 𝜇 𝜒 , ∗ τ , 𝜒 ∗ τ 𝑟𝑚𝑖𝑛 𝛽 𝜒 , ∗ τ , 𝛽 𝜒 ∗ τ 𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝛽 𝜒 , ∗ 𝛾 ∗ τ , 𝛽 𝛾 , 𝑟𝑚𝑖𝑛 𝛽 𝜒 , ∗ 𝛾 ∗ τ , 𝛽 𝛾 = 𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝛽 𝜒 , ∗ 𝛾 ∗ τ , 𝛽 𝜒 , ∗ 𝛾 ∗ τ , 𝑟𝑚𝑖𝑛 𝛽 𝛾 , 𝛽 𝛾 =𝑟𝑚𝑖𝑛 𝜇 𝜒 , ∗ 𝛾 ∗ τ , 𝜒 , ∗ 𝛾 ∗ τ , 𝜇 𝛾 , 𝛾 . And, 𝜇 𝜒 , ∘ 𝜒 , 𝛾 ∘ 𝛾 𝑟𝑚𝑖𝑛 𝛽 𝜒 ∘ 𝜒 , 𝛽 𝛾 ∘ 𝛾 𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝛽 𝜒 , , 𝛽 𝜒 , 𝑟𝑚𝑖𝑛 𝛽 𝛾 , , 𝛽 𝛾 =𝑟𝑚𝑖𝑛 𝜇 𝜒 , 𝜒 , 𝜇 𝛾 , 𝛾 . () For all, 𝜒 , 𝛾 ∈ ℵ ℵ~, 𝑟𝑚𝑖𝑛 𝛽 0 , 𝛽 0 𝜇 0,0 𝜇 𝜒, 𝛾 𝑟𝑚𝑖𝑛 𝛽 𝜒 , 𝛽 𝛾 .~ Then 𝛽 0 𝛽 𝜒 ´, ∀𝜒 ∈ ℵ. Now, let 𝜒 , 𝜒 𝛾 , 𝛾 τ , τ ∈ ℵ ℵ. Then´ 𝑟𝑚𝑖𝑛 𝛽 𝜒 ∗ τ , 𝛽 𝜒 ∗ τ 𝜇 𝜒 ∗ τ , 𝜒 ∗ τ 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ τ , 𝜇 𝜒 ∗ τ = 𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 , 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝜒 , ∗ 𝛾 ∗ τ , 𝑟𝑚𝑖𝑛 𝜇 𝛾 , 𝜇 𝛾 =𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛𝜇 𝜒 ∗ 𝛾 ∗ τ , 𝜒 ∗ 𝛾 ∗ τ , 𝜇 𝛾 , 𝛾 =𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝛽 𝜒 ∗ 𝛾 ∗ τ , 𝛽 𝜒 ∗ 𝛾 ∗ τ , 𝑟𝑚𝑖𝑛 𝛽 𝛾 , 𝛽 𝛾 = 𝑟𝑚𝑖𝑛 𝑟𝑚𝑖𝑛 𝛽 𝜒 ∗ 𝛾 ∗ τ , 𝛽 𝛾 , 𝑟𝑚𝑖𝑛 𝛽 𝜒 ∗ 𝛾 ∗ τ , 𝛽 𝛾 In particular, if we take 𝜒 𝛾 τ 0, then 𝛽 𝜒 ∗ τ 𝑟 min 𝛽 𝜒 ∗ 𝛾 ∗ τ , 𝛽 𝛾 . Hence, ´ the proof´ is completed´. 5. 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