109 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 Samah H. Asaad Akram S. Mohammed Abstract In this article, we study some properties of anti-fuzzy sub-semigroup, anti fuzzy left (right, two sided) ideal, anti fuzzy ideal, anti fuzzy generalized bi-ideal, anti fuzzy interior ideals and anti fuzzy two sided ideal of regular semigroup. Also, we characterized regular LA-semigroup in terms of their anti fuzzy ideal. Keywords: Fuzzy ideal, regular, anti fuzzy interior ideal, anti fuzzy ideal. 1. Introduction and Basic Concept Fuzzy sub-semigroup and fuzzy interior ideal in semigroup was introduced by Hong, et al., in [1]. And the concept of fuzzy ideal and fuzzy bi-ideals in semigroups was studied by Nobuaki Kuroki in (1981)”, [2]. The concept of the product of two fuzzy subset and anti product of two fuzzy subset was introduced by Shabir and Nawaz [3]. The concept of characterizations of semigroups by their anti fuzzy ideals was studied by Khan and Asif in [4]. The concept of intra-regular (left almost semigroup denoted by LA-semigroups) characterized by their anti fuzzy ideals by Khan and Faisal in [5]. Many other authors interested studied of fuzzy ideal, for example see [6-9]. Through out of this paper we are denoted of a regular semigroup byℵ𝑟 . Definition 1 [1]. A fuzzy subset ζ in a semigroup ℵ is said to be a fuzzy sub-semigroup of ℵ if ζ(wz)≥ min{ζ(w), ζ(z)}, whenever w, z ∈ ℵ. Definition 2 [1]. A fuzzy sub-semigroup ζ of a semigroup ℵ is said to be a fuzzy interior ideal of ℵ if ζ(swr)≥ ζ(w), whenever s, w, r ∈ ℵ. Definition 3 [2]. A fuzzy function ζ of a semigroup ℵ is said to be a fuzzy ideal if ζ(swr)≤max{ζ(s), ζ(r)}={ζ(s)⋁ ζ(r)}, whenever s, w, r ∈ ℵ. Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi:10.30526/32.3.2287 New Properties of Anti Fuzzy Ideals of Regular Semigroups Department of Mathematics, College of Computer Science and Mathematics, University of Tikrit, Iraq. ahmedibrahimsalh89@gmail.com akr-tel@tu.edu.iq Article history: Received 25 March 2019, Accepted 14 April 2019, Publish September 2019. mailto:ahmedibrahimsalh89@gmail.com mailto:tel@tu.edu.iq 110 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 Definition 4 [2]. A fuzzy sub-semigroup ζ of a semigroup ℵ is said to be a fuzzy bi ideal in ℵ if ζ(swr)≥min{ζ(s), ζ(r)}, whenever s, w, r ∈ ℵ. Definition 5 [3]. Let ζ and φ be any fuzzy subsets of a semigroup ℵ then the product ζ ∘ φ is defined by (ζ ∘ φ)(w) = { ⋁ {ζ(s) ∧ φ(r)}, ∃ s, r ∈ ℵ s. t w = srw=sr 0; other wise Definition 6 [3]. Let ζ and φ be any fuzzy subsets of a semigroup ℵ then the anti product ζ ∗ φ is defined by (ζ ∗ φ)(w) = { ⋀ {ζ(s) ∨ φ(r)}, ∃ s, r ∈ ℵ s. t w = s rw=s r 1; other wise Definition 7 [4]. A fuzzy subset ζ of a semigroup ℵ is said to be anti fuzzy sub-semigroup of ℵ if ζ(sr) ≤ ζ(s) ∨ ζ(r), whenever s, r ∈ ℵ. Definition 8 [4]. A fuzzy subset ζ of a semigroup ℵ is said to be anti fuzzy left (right) ideal of ℵ if ζ(sr) ≤ ζ(r), (ζ(sr) ≤ ζ(s)), whenever s, r ∈ ℵ. Definition 9 [4]. A fuzzy subset ζ of a semigroup ℵ is said to be anti fuzzy ideal of ℵ if it is both anti fuzzy left ideal and anti fuzzy right ideal. Definition 10 [4]. A fuzzy subset ζ of a semigroup ℵ is said to be anti fuzzy interior ideal of ℵ if ζ(swr) ≤ ζ(w), whenever s, w, r ∈ ℵ. Definition 11 [4]. A fuzzy subset ζ of a semigroup ℵ is said to be anti fuzzy generalized bi-ideal of ℵ if ζ(swr) ≤ ζ(s) ∨ ζ(r), whenever s, w, r ∈ ℵ. Definition 12 [4]. A fuzzy sub-semigroup ζ is said to be anti fuzzy bi-ideal of ℵ if ζ(swr) ≤ ζ(s) ∨ ζ(r) whenever s, w, r ∈ ℵ. Definition 13 [5]. A fuzzy subset ζ of a LA-semigroup ℵ is said to be a fuzzy LA-sub-semigroup if ζ(sr) ≥ ζ(s) ∧ ζ(r), whenever s, r ∈ ℵ. Definition 14 [5]. A fuzzy subset ζ of a LA-semigroup ℵ is said to be a fuzzy left(right)ideal of ℵ if ζ(sr) ≥ ζ(r), (ζ(sr)≥ ζ(s)), whenever s, r ∈ ℵ. Definition 15 [5]. A fuzzy LA-sub-semigroup ζ of a LA-semigroup ℵ is said to be a fuzzy bi-ideal 111 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 if ζ((sr)t) ≥ ζ(s) ∧ ζ(t), whenever s, r, t ∈ ℵ. Definition 16 [5]. A fuzzy LA-sub-semigroup ζ of a LA-semigroup ℵ is said to be fuzzy interior ideal if ζ((sr)t) ≥ ζ(r), whenever s, r, t ∈ ℵ. 2. New Properties of Anti Fuzzy Ideals of a Regular Semigroup In this section we introduce some properties anti fuzzy ideal Definition 17 ℵ is said to be a regular semigroup if w=wzw, whenever w, z ∈ ℵ or equivalently w ∈ wℵw. Theorem 18 Every fuzzy interior ideal in ℵ𝑟 is idempotent. Proof Suppose that ζ is a fuzzy interior ideal of a semigroup ℵ, then clearly ζ ∘ ζ ⊆ ζ, Let w ∈ ℵ then ∃ z ∈ ℵ s. t w = wzw ⟹ w=wzw =(wz)w(z)w=(wz)w(z)wzw=((wz)w(z))(wzwzw) (ζ ∘ ζ)(w)= ⋁ { ζ(wz)w(z) ∧ ζ(wz)w(zw)}w=((wz)w(z))(wzwzw) ≥ ζ(wz)w(z) ∧ ζ(wz)w(zw) ≥ ζ(w) ∧ ζ(w)=ζ(w) This is implies that ζ ∘ ζ ⊇ ζ, hence ζ ∘ ζ = ζ. Then ζ is idempotent. Theorem 19 Let ζ be a fuzzy subset in ℵ𝑟 then it is an anti fuzzy two sided ideal of ℵ iff it is an anti fuzzy interior ideal of ℵ. Proof ⟹Since ζ be anti fuzzy two sided ideal of ℵ, then obviously, ζ is an anti fuzzy interior ideal of ℵ. ⟸ Suppose that ζ is an anti fuzzy interior ideal of ℵ. Let w, z ∈ ℵ, by by hypotheses so ∃ s, r ∈ ℵ, s. t w=wsw and z=zrz ζ(wz)=ζ((wsw)𝑧)=ζ((ws)wswz ))=ζ((ws)w(swz))≤ ζ(w), and Also ζ(wz)= ζ(w(zrz)) = ζ(wzrzrz) =ζ((wzr)z(rz)) ≤ ζ(z), Hence, ζ is an anti fuzzy two sided ideal of ℵ. Example 20 Let ℵ ={s, r, t, v} be a set with operation as follows: . s r t v s s s s s r s s s s s s s r s v s s r r 112 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 Then we can easily see that (ℵ,.) is not a regular semigroup. Define the fuzzy subset ζ of ℵ as ζ(s) = 0.3, ζ(r) = 0.9, ζ(t) = 0.5, ζ(v) = 0.7. Then clearly, ζ is anti fuzzy interior ideal of ℵ but it is not an anti fuzzy two sided ideal of ℵ, since {s, r} is not a two sided ideal of ℵ. Proposition 21 In regular semigroup ℵ, then i- Every anti fuzzy right ideal is idempotent. ii- Every anti fuzzy interior ideal is idempotent. Proof i- Suppose that ζ is an anti fuzzy right ideal of semigroup ℵ, then clearly ζ ⊆ ζ ∗ ζ. Since ℵ is a regular so whenever w ∈ ℵ, ∃ z ∈ ℵ, s.t w=wzw, so (ζ ∗ ζ)(w)=⋀ {ζ(wz) ∨ ζ(wzw)}w=wzw=wzwzw = ⋀ {ζ(wz) ∨ ζ(wt)}w =(wz)(wzw) where t=zw ≤ ζ(wz) ∨ ζ(wt)≤ ζ(w) ∨ ζ(w)=ζ(w) This implies that ζ ∗ ζ ⊆ ζ. Hence ζ ∗ ζ = ζ. ii- Suppose that ζ is an anti fuzzy interior ideal of semigroup ℵ, then clearly ζ ⊆ ζ ∗ ζ. Since ℵ is a regular so whenever w ∈ ℵ, ∃ z∈ ℵ, s.t w=wzw, so w=wzw=wzwzw=((wz)w(z)) ((wz)w (z w)) (ζ ∗ ζ)(w) =⋀ {ζ(wz)w(z)) ∨ ζ((wz)w(zw))}w=((wz)w(z)) ((wz)w(zw)) ≤ ζ(wz)w(z)) ∨ ζ((wz)w(zw))≤ ζ(w) ∨ ζ(w)=ζ(w). This implies that ζ ∗ ζ ⊆ ζ. Hence ζ ∗ ζ = ζ. Proposition 22 [3]. Let ζ be an anti fuzzy right ideal and μ an anti fuzzy left ideal of a semigroup ℵ. Then ζ ∗ μ ⊇ ζ ∪ μ. It is clear that from Proposition 22. ζ ∗ μ ⊇ ζ ∪ μ , but the converse needs not at all be true. Consider the following example, Example 23 Consider the semigroup ℵ= {s, r, t , v} with the operation as follows: . s r t v s s s s s r s s s s t s s r s v s s r r 113 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 The ideals of ℵ are {s}, {s, r}, {s, r, t} and {s, r, t, v} Let us define two fuzzy subsets ζ and μ of ℵ as follows ζ(s)=0.5, ζ(r)=0.6, ζ(t)=0.7, ζ(v)=0.8. μ(s)=0.6, μ(r)=0.7, μ(t)=0.8, μ(v)=0.9. Then ζ and μ are an anti fuzzy ideal of ℵ, and we note that: (ζ ∗ μ)(r) = ⋀ {ζ(x) ∨ μ(y)}r = xy =⋀{ 0.8, 0.8, 0.9} = 0.8 ≥ (ζ ∪ μ)(r) = 0.7. To consider the converse of proposition 22, we need to strengthen the condition of semigroupℵ. Theorem 24 If ζ, μ are any anti fuzzy two sided ideals of ℵ𝑟 , then ζ ∗ μ = ζ ∪ μ. Proof Let ζ and μ be any anti fuzzy two sided ideals of ℵ, then obviously ζ ∗ μ ⊇ ζ ∪ μ. since ℵ is a regular so whenever element w ∈ ℵ, ∃ z ∈ ℵ, s.t w=wzw, so (ζ ∗ μ)(w) = ⋀ {ζ(wz) ∨ μ(wzw)}w=wzw=wzwzw ≤ ζ(wz) ∨ μ(wzw)≤ ζ(w) ∨ μ(w)=(ζ ∪ μ)(w) Then ( ζ ∗ μ) ⊆ ζ ∪ μ. Hence, ζ ∗ μ = ζ ∪ μ. Example 25 Let ℵ ={s, r, t} be a semigroup with the following table: . s r t s s r t r r r t t t t t Define a fuzzy subset ζ of ℵ by ζ(s)=0.6, ζ(r)=0.5, ζ(t)=0.4. By routine calculation, we can check that ζ is an anti fuzzy ideal, anti fuzzy interior ideal and anti fuzzy bi-ideal of ℵ𝑟 . Now, we give other fuzzy characterizations of a regular semigroup. Proposition 26 A fuzzy subset ζ of ℵ𝑟 , then ζ is anti fuzzy bi-ideal of ℵ iff it is an anti fuzzy generalized bi-ideal of ℵ. Proof ⟹ Suppose that ζ be any anti fuzzy bi-ideal of ℵ, the obviously, ζ is an anti fuzzy generalized bi-ideal of ℵ. ⟸ Suppose that ζ be any anti fuzzy generalized bi-ideal of ℵ, since ℵ is a regular of a semigroup, so whenever w ∈ ℵ, ∃ z ∈ ℵ s.t w=w z w. we have ζ(wr)=ζ(wzwr)=ζ(w t r) ≤ ζ(w) ∨ ζ(r) where 𝑡=𝑧𝑤. Therefore, ζ is an anti fuzzy sub-semigroup of ℵ. Hence, ζ is an anti fuzzy generalized bi-ideal of ℵ. 114 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 Theorem 27 For anti fuzzy generalized bi-ideal ζ and anti fuzzy right ideal μ of ℵ𝑟 , then ζ ∗ μ ⊆ ζ ∪ μ. Proof Let ζ and μ are any anti fuzzy generalized bi-ideal and anti fuzzy right ideal of ℵ, respectively, then whenever w ∈ ℵ, ∃ z ∈ ℵ s.t w=wzw. Then (ζ ∗ μ)(w) = ⋀ {ζ(b) ∨ μ(c)}w = bc ≤ ζ(wzw) ∨ μ(zw) ≤ ζ(w) ∨ μ(w)=(ζ ∨ μ)(w) And so we have ζ ∗ μ ⊆ ζ ∪ μ. Theorem 28 If ζ and μ are any anti fuzzy interior ideals of ℵ𝑟 , then (ζ ∗ μ) ∪ (μ ∗ ζ) ⊆ ζ ∨ μ. Proof Let ζ, μ be any anti fuzzy interior ideals of ℵ, and w ∈ ℵ. Then since ℵ is regular semigroup then, ∃ z ∈ ℵ s.t 𝑤= wzw=((wz)w(z)) (w(zw))=((wz)w(z)) ((wz)w(zw)). Hence (ζ ∗ μ)(w) = ⋀ {ζ(b) ∨ μ(c)}w = bc. ≤ ζ((wz)w(z))∨ μ((w z)w(zw)) ≤ ζ(w) ∨ μ(w)=(ζ ∨ μ)(w) And so we have ζ ∗ μ ⊆ ζ ∪ μ. Similarly, we have (μ ∗ ζ) ⊆ ζ ∪ μ Therefore (ζ ∗ μ) ∪ (μ ∗ ζ) ⊆ ζ ∪ μ. Theorem 29 For every anti fuzzy left ideal α, every anti fuzzy generalized bi-ideal μ, and every anti fuzzy interior ideal ζ of ℵ𝑟 , then μ ∗ α ∗ ζ ⊆ μ ∪ α ∪ ζ. Proof Let α, μ and ζ be any anti fuzzy left ideal, any anti fuzzy generalized bi-ideal and anti fuzzy interior ideal of ℵ𝑟 , respectively, whenever w ∈ ℵ, ∃ z ∈ ℵ. Because ℵ is a regular, s.t w=wzw=wzwzw=(wzw) (zw)zw=((wzw) [(zw) ((z)w(zw)]). Then we have: ( μ ∗ α ∗ ζ)(w)=⋀ {μ((wzw)) ∨ ( α ∗ ζ)((zw)((z)w(zw))}w=((wzw)[(zw))((z)w(zw)) ≤ μ(w) ∨{⋀ {α(zw) ∨ ζ((z)w(zw))}((zw)(z)w(zw)) ≤ μ(w) ∨ α(w) ∨ ζ(w) = (μ ∪ α ∪ ζ)(w) And so we have μ ∗ α ∗ ζ ⊆ μ ∪ α ∪ ζ . Now, we characterized regular (left almost-semigroup for short LA-semigroup) by the properties of their fuzzy left (right, two sided) ideal. Let ℵ be a gropoid. Then 1. ℵ is called LA-semigroup if (wr) j=(jr) w; whenever w, r, j ∈ ℵ. 2. Medial law of a LA-semigroup means (wr) (jv) = (wj) (rv); whenever w, r, j, v ∈ ℵ. 3. In additional if ℵ has a left identity(necessary unique) the paramedical law mean 4. (wr) (jv)=(vr) (jw); whenever w, r, j, v ∈ ℵ. 5. An LA-semigroup with right identity becomes a commutative semigroup with identity. if an LA-semigroup contains left identity, the following law holds w (r j) = r (w j); whenever w, r, j ∈ ℵ. 115 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 Proposition 30 A fuzzy subset ζ of ℵ𝑟 is a fuzzy right ideal iff it is a fuzzy left ideal. Proof ⟹ Suppose that ζ is a fuzzy right ideal of ℵ, since ℵ is a regular so whenever w∈ ℵ, ∃ z ∈ ℵ, s.t w=wzw, so by using (1) ζ(wb) = ζ((wzw) b) =ζ((𝑤𝑧𝑤)(zw)b)) =ζ(b (𝑧𝑤)(wzw)) ≥ ζ(b(zw)) ≥ ζ(b) ⟸ Suppose that ζ is a fuzzy left ideal of ℵ𝑟 , then using (1) ζ(wr) = ζ((wzw)r)=ζ((𝑤𝑧𝑤)(zw)r) =ζ(r(zw)(wzw))≥ ζ(wzw) = ζ((wz)w)≥ ζ((w)w) ≥ ζ(𝑤 2) ≥ ζ(w). Theorem 31 Every fuzzy two sided ideal of a regular LA-semigroup ℵ, with left identity is idempotent. Proof Suppose that ζ is a fuzzy two sided ideal of ℵ, then clearly ζ ∘ ζ ⊆ ζ ∘ ℵ ⊆ ζ. Since ℵ is a regular so whenever w ∈ ℵ, ∃ z ∈ ℵ, s.t w=wzw so by using (1) w=wzw=w(zw)(zw)=(zwzw)w, (ζ ∘ ζ)(w)=⋁ ζ(zwzw) ∧ ζ(w)w=(zwzw)w ≥ ζ(zwzw) ∧ ζ(w) ≥ ζ(w) ∧ ζ(w)=ζ(w). And this implies that ζ ∘ ζ ⊇ ζ, hence ζ ∘ ζ = ζ. Theorem 32 For a fuzzy subset ζ of a regular LA-semigroup ℵ, with left identity then ζ is a fuzzy two sided ideal of ℵ iff it is a fuzzy interior ideal of ℵ. Proof ⟹ Suppose that ζ be a fuzzy two sided ideal of ℵ, then obviously, ζ is a fuzzy interior ideal of ℵ. ⟸ Suppose that ζ be a fuzzy interior ideal of ℵ, and w, r∈ ℵ, then since ℵ is a regular of AL- semigroup, so ∃ z, y ∈ ℵ s.t w=wzw, r=ryr, then ζ(wr) = ζ((w z w)r) using (1) = ζ(r(z w))(w z w) ) using (2) = ζ(rw)((zw)(zw)) = ζ(𝑟𝑤)t) where t= ((z w)(z w)) ≥ ζ(w), Also ζ(wr) =ζ(w(ryr))= ζ(w(ryry)r)) using (4) =ζ((ryry)(wr) )= ζ((ry)r(ywr) )=ζ(jrt) Where j= ry and t= y w r and ≥ ζ(r), Hence, ζ is a fuzzy two sided ideal. 116 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 2. Conclusion From the research / the evidence we conclude that 1. Let ζ be a fuzzy subset in ℵ𝑟 then it is an anti fuzzy two sided ideal of ℵ iff is an anti fuzzy interior ideal of ℵ. 2. In a regular semigroup ℵ, then the following are satisfy the following i) Every anti fuzzy right ideal is idempotent. ii) Every anti fuzzy interior ideal is idempotent. 3. If ζ , μ are an anti fuzzy two sided ideals of ℵ𝑟 , then ζ ∗ μ = ζ ∪ μ. 4. For anti fuzzy generalized bi-ideal ζ and anti fuzzy right ideal μ of ℵ𝑟 , 5. Then ζ ∗ μ ≤ ζ ∨ μ. 6. For every anti fuzzy left ideal α, every anti fuzzy generalized bi-ideal μ, and every anti fuzzy interior ideal ζ of ℵ𝑟 , then μ ∗ α ∗ ζ ⊆ μ ∪ α ∪ ζ. 7. For a fuzzy subset ζ of a regular LA-semigroup ℵ, with left identity then ζ is a fuzzy two sided ideal of ℵ iff it is a fuzzy interior ideal of ℵ. References 1. Hong, S.M.; Jun, Y.B.; Meng, J. IN Fuzzy interior ideals in semigroups. Indain J. Pure appl. Math.1995, 26, 9, 859-863. 2. 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Tawfiq, L.N.M.; Qa'aed, M.M. on Fuzzy Groups and Group Homomorphism, Ibn al- Haitham Journal for Pure and Applied Science.2012, 25, 2, 340-346. 9. Tawfiq, L.N.M. Some Results on Solvable Fuzzy Subgroup of a Group. Journal of Al- Qadisiyah for Pure Science.2011, 16, 4, 17-29. https://www.iasj.net/iasj?func=search&query=au:%22Luma.%20N.%20M.%20Tawfiq%20%20%20%D9%84%D9%85%D9%89%20%D9%86%D8%A7%D8%AC%D9%8A%20%D9%85%D8%AD%D9%85%D8%AF%20%D8%AA%D9%88%D9%81%D9%8A%D9%82%20%20%20%20%22&uiLanguage=en https://www.iasj.net/iasj?func=issues&jId=139&uiLanguage=en https://www.iasj.net/iasj?func=issues&jId=139&uiLanguage=en