International Journal of Analysis and Applications Volume 16, Number 1 (2018), 97-116 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-97 COMPLEX NEUTROSOPHIC SUBSEMIGROUPS AND IDEALS MUHAMMAD GULISTAN1,∗, ASGHAR KHAN2, AMIR ABDULLAH1, NAVEED YAQOOB3 1Department of Mathematics, Hazara University, Mansehra, Pakistan 2Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan 3Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia ∗Corresponding author: azhar4set@yahoo.com Abstract. In this article we study the idea of complex neutrosophic subsemigroups. We define the Cartesian product of complex neutrosophic subsemigroups, give some examples and study some of its related results. We also define complex neutrosophic (left, right, interior) ideal in semigroup. Furthermore, we introduce the concept of characteristic function of complex neutrosophic sets, direct product of complex neutrosophic sets and study some results prove on its. 1. Introduction In 1965, Zadeh, ( [1]) presented the idea of a fuzzy set. Atanassov in 1986, ( [2]) initiated the notion of intuitionistic fuzzy set, which is the generalization of a fuzzy set. Neutrosophic set was first proposed by Smarandache in 1999 ( [5]), which is the generalization of a fuzzy set and intuitionistic fuzzy set. Neutrosophic set is characterized by a truth membership function, an indeterminacy membership function and a falsity membership function. It must be noted that there are lots of researchers that worked at complex number and fuzzy sets, for instance Buckly ( [6]), Nguyen et al. ( [7]) and Zhang et al. ( [10]). On the other hand Ramot Received 19th September, 2017; accepted 5th December, 2017; published 3rd January, 2018. 2010 Mathematics Subject Classification. 03B52. Key words and phrases. complex fuzzy sets; complex neutrosophic sets; fuzzy subsemigroups; complex neutrosophic sub- semigroups; complex neutrosophic ideals. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 97 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-97 Int. J. Anal. Appl. 16 (1) (2018) 98 et al. ( [8]) presented a innovative come close to that is entirely unlike from other researchers, wherever they extensive the variety of membership function to unit circle in the complex plane, unlike the others who limited to. Further to solve enigma they added an extra terms which is called phase term in translating human language to complex valued functions on physical terms and vice versa (for more information, see ( [8]). Abd Uazeez et al. in 2012 ( [12]), added the non-membership term to the idea of complex fuzzy set which is known as complex intuitionistic fuzzy sets, the range of values are extended to the unit circle in complex plan for both membership and non-membership functions instead of [0, 1]. In 2016, Mumtaz Ali et al. ( [14]), more extended the concept of complex fuzzy set, complex intuitionistic fuzzy set, and introduced the concept of complex neutrosophic sets, which is a collection of a complex truth membership function, a complex indeterminacy membership function and a complex falsity membership function. The idea of a fuzzy set in the model of semigroups was first initiated by Kuroki in 1979 ( [3]), and defined fuzzy subsemigroups. Vildan and Halis in 2017 ( [15]), extended the concept of fuzzy subgroups on the base of neutrosophic sets, which is known as neutrosophic subgroups . Due to the motivation and inspiration of the above discussion. In this paper we are initiating the study of complex neutrosophic semigroups. This paper introduce the notion of complex neutrosophic subsemi- groups and Cartesian product of complex neutrosophic subsemigroups with the help of example. We define characteristic function of complex neutrosophic set, direct product of complex neutrosophic sets, complex neutrosophic ideals (left, right, interior) and proved some results. 2. Preliminaries Here in this part we gathered some basic helping materials. Definition 2.1. ( [1]) A function f is defined from a universe X to a closed interval [0, 1] is called a fuzzy set, i.e., a mapping: f : X −→ [0, 1]. Definition 2.2. ( [8]) A complex fuzzy set (CFS) C over the universe X, is defined an object having of the form: C = {(x,µC(x)) : x ∈X} where µC(x) = rC(x)·eiωC(x), here the amplitude term rC(x) and phase term ωC(x), are real valued functions, for every x ∈X , the amplitude term µC(x) : X → [0, 1] and phase term ωC(x) lying in the interval [0, 2π]. Definition 2.3. ( [13]) Let C1 and C2 be any two complex Atanassov’s intuitionistic fuzzy sets (CAIFSs) over the universe X, where C1 = {〈 x,rC1 (x) ·e iνC1 (x) ,kC1 (x) ·e iωC1 (x) 〉 : x ∈X } Int. J. Anal. Appl. 16 (1) (2018) 99 and C2 = {〈 x,rC2 (x) ·e iνC2 (x) ,kC2 (x) ·e iωC2 (x) 〉 : x ∈X } . Then 1. Containment: C1 ⊆C2 ⇔ rC1 (x) ≤ rC2 (x),kC1 (x) ≥ kC2 (x) and νC1 (x) ≤ νC2 (x),ωC1 (x) ≥ ωC2 (x). 2. Equal: C1 = C2 ⇔ rC1 (x) = rC2 (x),kC1 (x) = kC2 (x) and νC1 (x) = νC2 (x),ωC1 (x) = ωC2 (x). Definition 2.4. ( [14]) Let X be a universe of discourse, and x ∈X . A complex neutrosophic set (CNS) C in X is characterized by a complex truth membership function CT (x) = pC(x) ·eiµC(x), a complex indeterminacy membership function CI(x) = qC(x)·eiνC(x) and a complex falsity membership function CF (x) = rC(x)·eiωC(x). The values CT (x),CI(x),CF (x) may lies all within the unit circle in the complex plane, where pC(x), qC(x), rC(x) and µC(x), νC(x) ωC(x) are amplitude terms and phase terms, respectively, and where pC(x), qC(x), rC(x) ∈ [0, 1], such that, 0 ≤ pC(x) + qC(x) + rC(x) ≤ 3 and µC(x), νC(x) ωC(x) ∈ [0, 2π]. The complex neutrosophic set can be represented in the form as: C =   〈 x,CT (x) = pC(x) ·eiµC(x),CI(x) = qC(x) ·eiνC(x), CF (x) = rC(x) ·eiωC(x) 〉 : x ∈X   . Example 2.1. Let X = {x1,x2,x3} be the universe set and C be a complex neutrosophic set which is given by: C =   〈 x1, 0.2e 0.5πi, 0.3e0.6πi, 0.4e0.8πi 〉 , 〈 x2, 0.4e 0.6πi, 0.5e1.3πi, 0.1e0.6πi 〉 ,〈 x3, 0.1e 0.6πi, 0.3e0.9πi, 0.9e0.7πi 〉   . Definition 2.5. ( [3]) A fuzzy subset A of a semigroup S is said to be a fuzzy subsemigroup of S if its satisfy the following condition: A(x ·y) ≥A(x) ∧A(y) ∀ x,y ∈S. Definition 2.6. ( [15]) Let G be any group with multiplication and N be a neutrosophic set on G. Then N is said to be a neutrosophic subgroup (NSG) of G, if its satisfy the following conditions: (NSG1): N(x ·y) ≥N(x) ∧N(y), i.e., TN(x ·y) ≥ TN(x) ∧TN(y), IN(x ·y) ≥ IN(x) ∧ IN(y) and FN(x ·y) ≤ FN(x) ∨FN(y). (NSG2): N(x−1) ≥N(x), i.e., TN(x −1) ≥ TN(x), IN(x−1) ≥ IN(x) and FN(x−1) ≤ FN(x), for all x and y in G. Int. J. Anal. Appl. 16 (1) (2018) 100 Lemma 2.1. ( [16]) For a semigroup S, the following conditions are equivalent. (1) S is regular. (2) R∩L = RL for every right ideal R of S and every left ideal L of S. 3. Complex Neutrosophic Subsemigroup Note: It should be noted that through out in this part we use a capital letter C to denote a complex neutrosophic set; C = {〈 TC = pC ·eiµC,IC = qC ·eiνC,FC = rC ·eiωC 〉} . Definition 3.1. A complex neutrosophic set C = {〈 TC = pC ·eiµC,IC = qC ·eiνC,FC = rC ·eiωC 〉} on a semi- group S is known as a complex neutrosophc subsemigroup (CNSG), if its satisfy the following condition: C(xy) ≥ min{C(x),C(y)} i.e., (i) pC(xy) ·eiµC(xy) ≥ min{pC(x) ·eiµC(x),pC(y) ·eiµC(y)} (ii) qC(xy) ·eiνC(xy) ≥ min{qC(x) ·eiνC(x),qC(y) ·eiνC(y)} (iii) rC(xy) ·eiωC(xy) ≤ max{rC(x) ·eiωC(x),rC(y) ·eiωC(y)}, ∀ x, y ∈S. Example 3.1. Let S = {1, 2, 3} be a semigroup with the following multiplication table: · 1 2 3 1 1 2 3 2 2 1 3 3 3 3 3 Consider a complex neutrosophic set C on S as: C =   〈 1, 0.9e0.7πi, 0.7e0.6πi, 0.5e0.4πi 〉 ,〈 2, 0.8e0.6πi, 0.6e0.5πi, 0.4e0.3πi 〉 ,〈 3, 0.5e0.4πi, 0.4e0.2πi, 0.3e0.2πi 〉   Then clearly C is a complex neutrosophic subsemigroup of S. 3.1. Cartesian Product of Complex Neutrosophic Subsemigroups. Definition 3.2. Let C1 = {〈 C1T = pC1e iµC1 ,C1I = qC1e iνC1 ,C1F = rC1e iωC1 〉} and C2 = {〈 C2T = pC2e iµC2 ,C2I = qC2e iνC2 ,C2F = rC2e iωC2 〉} Int. J. Anal. Appl. 16 (1) (2018) 101 be any two complex neutrosophic subsemigroups of the semigroups S1 and S2 respectively. Then the Cartesian product of C1 and C2 denoted by C1 ×C2 is defined as: C1 ×C2 =   〈 (x,y), (C1 ×C2)T (x,y), (C1 ×C2)I(x,y), (C1 ×C2)F (x,y) 〉 / ∀x ∈S1,y ∈S2   where (C1 ×C2)T (x,y) = min{C1T (x),C2T (y)} , (C1 ×C2)I(x,y) = min{C1I(x),C2I(y)} , (C1 ×C2)F (x,y) = max{C1F (x),C2F (y)} , for all x in S1 and y in S2. Example 3.2. Let S1 = {1, 2, 3} and S2 = {a,b,c} are any two semigroups with the following multiplication tables: · 1 2 3 1 1 2 3 2 2 1 3 3 3 3 3 , · a b c a c b a b b b c c c c c Consider C1 =   〈 1, 0.9e0.7πi, 0.7e0.6πi, 0.5e0.4πi 〉 , 〈 2, 0.8e0.6πi, 0.6e0.5πi, 0.4e0.3πi 〉 ,〈 3, 0.5e0.4πi, 0.4e0.2πi, 0.3e0.2πi 〉   and C2 =   〈 a, 0.8e0.7πi, 0.5e0.3πi, 0.4e0.4πi 〉 , 〈 b, 0.6e0.5πi, 0.5e0.4πi, 0.3e0.2πi 〉 ,〈 c, 0.8e0.7πi, 0.7e0.5πi, 0.3e0.2πi 〉   be any two complex neutrosophic subsemigroups of S1 and S2, respectively. Now let x = 1 and y = a, then C1 ×C2 = {〈(C1 ×C2)T (1,a), (C1 ×C2)I(1,a), (C1 ×C2)F (1,a)〉 , ...} = {〈min{C1T (1),C2T (a)} , min{C1I(1),C2I(a)} , max{C1F (1), C2F (a)}〉 , ...} = { 〈 min{0.9e0.7πi, 0.8e0.7πi}, min{0.7e0.6πi, 0.5e0.3πi} , max{0.5e0.4πi, 0.4e0.4πi} 〉 , ...} = { 〈 0.8e0.7πi, 0.5e0.3πi, 0.5e0.4πi 〉 , ...}. 4. Complex Neutrosophic Ideals In this section, we define some ideals namely complex neutrosophic (left, right, interior) ideal in semigroup, with the help of examples and study some of its related results. Int. J. Anal. Appl. 16 (1) (2018) 102 4.1. Complex Neutrosophic Left Ideal. Definition 4.1. A complex neutrosophic set C = {〈 TC = pC ·eiµC,IC = qC ·eiνC,FC = rC ·eiωC 〉} on a semi- group S is known as a complex neutrosophic left ideal of S, if C(xy) ≥C(y) i.e., (i) pC(xy) ·eiµC(xy) ≥ pC(y) ·eiµC(y) (ii) qC(xy) ·eiνC(xy) ≥ qC(y) ·eiνC(y) (iii) rC(xy) ·eiωC(xy) ≤ rC(y) ·eiωC(y), ∀ x, y ∈S. Example 4.1. Let S = {a,b,c,d} be a semigroup with the following multiplication table: · a b c d a a a a a b a a a a c a a b a d a a b b Consider a complex neutrosophic set C on S as: C =   〈 a, 0.9e0.6πi, 0.8e0.5πi, 0.4e0.3πi 〉 , 〈 b, 0.7e0.5πi, 0.6e0.4πi, 0.5e0.4πi 〉 ,〈 c, 0.6e0.4πi, 0.4e0.3πi, 0.7e0.5πi 〉 , 〈 d, 0.5e0.5πi, 0.4e0.3πi, 0.7e0.5πi 〉   Then C is a complex neutrosophic left ideal of S. 4.2. Complex Neutrosophic Right Ideal. Definition 4.2. A complex neutrosophic set C = {〈 TC = pC ·eiµC,IC = qC ·eiνC,FC = rC ·eiωC 〉} on a semi- group S is known as a complex neutrosophic right ideal of S, if C(xy) ≥C(x) i.e., (i) pC(xy) ·eiµC(xy) ≥ pC(x) ·eiµC(x) (ii) qC(xy) ·eiνC(xy) ≥ qC(x) ·eiνC(x) (iii) rC(xy) ·eiωC(xy) ≤ rC(x) ·eiωC(x), ∀ x, y ∈S. 4.3. Complex Neutrosophic Ideal. Definition 4.3. A complex neutrosophic set C = {〈 TC = pC ·eiµC,IC = qC ·eiνC,FC = rC ·eiωC 〉} on a semi- group S is known as a complex neutrosophic ideal of S, if it is both a complex neutrosophic left ideal and a complex neutrosophic right ideal of S. Int. J. Anal. Appl. 16 (1) (2018) 103 Example 4.2. Let S = {a,b,c} be a semigroup with the following Cayley table: · a b c a a a a b a a a c a a c If we define a complex neutrosophic set C on S as: C =   〈 a, 0.8e0.6πi, 0.6e0.5πi, 0.5e0.4πi 〉 , 〈 b, 0.7e0.6πi, 0.5e0.4πi, 0.6e0.4πi 〉 ,〈 c, 0.7e0.5πi, 0.4e0.3πi, 0.7e0.5πi 〉   Then obviously C is a complex neutrosophic ideal of S. Remark 4.1. Every complex neutrosophic left (resp., right) ideal is a complex neutrosophic subsemigroup. But the converse may not be true as seen in the following example. Example 4.3. Let S = {a,b,c,d} be a semigroup with the following Cayley table: · a b c d a a a a a b a a a a c a a a b d a a b c Take a complex neutrosophic set C on S as: C =   〈 a, 0.8e0.6πi, 0.6e0.5πi, 0.5e0.4πi 〉 , 〈 b, 0.6e0.6πi, 0.5e0.4πi, 0.6e0.4πi 〉 ,〈 c, 0.8e0.5πi, 0.4e0.3πi, 0.7e0.5πi 〉 , 〈 d, 0.4e0.4πi, 0.3e0.3πi, 0.7e0.5πi 〉   Then clearly C is a complex neutrosophic subsemigroup of S. However it is not a complex neutrosophic right ideal of S, because TC (cd) = TC (b) = 0.6e 0.6πi � 0.8e0.5πi = TC (c) . 4.4. Complex Neutrosophic Interior Ideal. Definition 4.4. A complex neutrosophic set C = {〈 TC = pC ·eiµC,IC = qC ·eiνC,FC = rC ·eiωC 〉} on a semi- group S is known as a complex neutrosophic interior ideal of S, if C(xκy) ≥C(κ) i.e., (i) pC(xκy) ·eiµC(xκy) ≥ pC(κ) ·eiµC(κ) (ii) qC(xκy) ·eiνC(xκy) ≥ qC(κ) ·eiνC(κ) (iii) rC(xκy) ·eiωC(xκy) ≤ rC(κ) ·eiωC(κ), ∀ x, κ, y ∈S. Int. J. Anal. Appl. 16 (1) (2018) 104 Example 4.4. Let S = {a,b,c,d} be a semigroup with the following multiplication table: · a b c d a a a a a b a a a a c a a b a d a a b b . Consider a complex neutrosophic set C on S as: C =   〈 a, 0.7e0.6πi, 0.6e0.4πi, 0.3e0.5πi 〉 , 〈 b, 0, 0.5e0.4πi, 0.5e0.6πi 〉 ,〈 c, 0.5e0.4πi, 0.4e0.3πi, 0.7e0.7πi 〉 , 〈 d, 0, 0.3e0.2πi, 0.7e0.7πi 〉   Then C is a complex neutrosophic interior ideal of S. Remark 4.2. Every complex neutrosophic ideal is a complex neutrosophic interior ideal. But the converse may not be true as seen in the Example 4.4. For left TC (dc) = TC (b) = 0 � 0.5e0.4πi = TC (c) right TC (dc) = TC (b) = 0 ≥ 0 = TC (d) . So it is a complex neutrosophic right ideal but not a left ideal. Hence C is not a complex neutrosophic ideal. 5. Characteristic Function of Complex Neutrosophic Set Definition 5.1. Let H be a non-empty subset over the universe X. Then the characteristic complex neu- trosophic function of H in X, defined to be a structure: CH = {〈x,TCH (x),ICH (x),FCH (x)〉 : x ∈ H} where TCH (x) =   1 ·e i2π if x ∈ H 0 otherwise ICH (x) =   1 ·e i2π if x ∈ H 0 otherwise FCH (x) =   0 if x ∈ H1 ·ei2π otherwise . Definition 5.2. The characteristic function of whole complex neutrosophic set S in semigroup S is defined as; CS = {〈 (1̂TCS , 1 ·e i2π), (1̂ICS , 1 ·e i2π), (0̂FCS , 0) 〉 : x ∈S } . Int. J. Anal. Appl. 16 (1) (2018) 105 5.1. Direct Product of Two Complex Neutrosophic Sets. Definition 5.3. Let C1 = 〈 C1T = pC1e iµC1 ,C1I = qC1e iνC1 ,C1F = rC1e iωC1 〉 and C2 = 〈 C2T = pC2e iµC2 ,C2I = qC2e iνC2 ,C2F = rC2e iωC2 〉 be any two complex neutrosophic sets on S, then the product is define as; C1 ⊗C2 =   〈 x, (pC1 ◦pC2 )(x) ·ei(µC1◦µC2 )(x), (qC1 ◦qC2 )(x) ·ei(νC1◦νC2 )(x), (rC1 ◦rC2 )(x) ·ei(ωC1◦ωC2 )(x) 〉 : x ∈S   where (pC1 ◦pC2 )(x) ·e i(µC1◦µC2 )(x) =   sup x=yκ [ min{pC1 (y)eiµC1 (y),pC2 (κ)eiµC2 (κ)} ] if x = yκ for some y,κ ∈S 0 otherwise (qC1 ◦qC2 )(x) ·e i(νC1◦νC2 )(x) =   sup x=yκ [ min{qC1 (y)eiνC1 (y),qC2 (κ)eiνC2 (κ)} ] if x = yκ for some y,κ ∈S 0 otherwise (rC1 ◦rC2 )(x) ·e i(ωC1◦ωC2 )(x) =   inf x=yκ [ max{rC1 (y)eiωC1 (y),rC2 (κ)eiωC2 (κ)} ] if x = yκ for some y,κ ∈S 1 ·ei2π otherwise for all x in S. Proposition 5.1. A complex neutrosophic sets C1,C2 and C3 of a semigroup S, if C1 ⊆ C2, then C1 ⊗C3 ⊆ C2 ⊗C3 and C3 ⊗C1 ⊆C3 ⊗C2. Proof: We are proving C1 ⊗C3 ⊆C2 ⊗C3. Since C1,C2 and C3 are complex neutrosophic sets of S. Let x ∈S. Case 1: If x is not expressed as x = yκ, then (C1 ⊗C3)(x) = 〈 0̂, 0̂, 1̂ 〉 and (C2 ⊗C3)(x) = 〈 0̂, 0̂, 1̂ 〉 . Clearly, C1 ⊗C3 ⊆C2 ⊗C3. Case 2: Assume that there exist y,κ ∈S, such that x = yκ. Then (pC1 ◦pC3 )(x) ·e i(µC1◦µC3 )(x) = sup x=yκ [ min{pC1 (y)e iµC1 (y),pC3 (κ)e iµC3 (κ)} ] ≤ sup x=yκ [ min{pC2 (y)e iµC2 (y),pC3 (κ)e iµC3 (κ)} ] = (pC2 ◦pC3 )(x) ·e i(µC2◦µC3 )(x). Int. J. Anal. Appl. 16 (1) (2018) 106 Similarly, (qC1 ◦qC3 )(x) ·e i(νC1◦νC3 )(x) ≤ (qC2 ◦qC3 )(x) ·e i(νC2◦νC3 )(x). And (rC1 ◦rC3 )(x) ·e i(ωC1◦ωC3 )(x) = inf [ max{rC1 (y)e iωC1 (y),rC3 (κ)e iωC3 (κ)} ] ≥ inf [ max{rC2 (y)e iωC2 (y),rC3 (κ)e iωC3 (κ)} ] = (rC2 ◦rC3 )(x) ·e i(ωC2◦ωC3 )(x). Therefore, C1 ⊗C3 ⊆C2 ⊗C3. Similarly we can proved C3 ⊗C1 ⊆C3 ⊗C2. � Proposition 5.2. Let H and K be any subsets of a semigroup S, we have (1) CH ⊗CK = CHK ⇒〈TCH ◦TCK,ICH ◦ ICK,FCH ◦FCK〉 = 〈TCHK,ICHK,FCHK〉 . (2) CH ∪CK = CH∪K ⇒〈TCH ∪TCK,ICH ∪ ICK,FCH ∩FCK〉 = 〈TCH∪K,ICH∪K,FCH∩K〉 . (3) CH ∩CK = CH∩K ⇒〈TCH ∩TCK,ICH ∩ ICK,FCH ∪FCK〉 = 〈TCH∩K,ICH∩K,FCH∪K〉 . Proof: (1) Let α ∈S. If α ∈ HK, then TCHK (α) = 1.e i2π, ICHK (α) = 1.e i2π and FCHK (α) = 0 and α = mn for some m ∈ H and n ∈ K. Thus, (TCH ◦TCK ) (α) = sup α=xy {min{TCH (x),TCK (y)}} ≥ min{TCH (m),TCK (n)} = 1.e i2π (ICH ◦ ICK ) (α) = sup α=xy {min{ICH (x),ICK (y)}} ≥ min{ICH (m),ICK (n)} = 1.e i2π and (FCH ◦FCK ) (α) = inf α=xy {max{FCH (x),FCK (y)}} ≤ max{FCH (m),FCK (n)} = 0. It follows that, (TCH ◦TCK ) (α) = 1.ei2π, (ICH ◦ ICK ) (α) = 1.ei2π and (FCH ◦FCK ) (α) = 0. Therefore, 〈TCH ◦TCK,ICH ◦ ICK,FCH ◦FCK〉 = 〈TCHK,ICHK,FCHK〉⇒ CH ⊗CK = CHK. Assume that α /∈ HK, then TCHK (α) = 0, ICHK (α) = 0 and FCHK (α) = 1.e i2π. Let y,κ ∈S be such that α = yκ, then we know that y /∈ H or κ /∈ K. Assume that y /∈ H, then Int. J. Anal. Appl. 16 (1) (2018) 107 (TCH ◦TCK ) (α) = sup α=yκ {min{TCH (y),TCK (κ)}} = sup α=yκ {min{0,TCK (κ)}} = 0 = TCHK (α) (ICH ◦ ICK ) (α) = sup α=yκ {min{ICH (y),ICK (κ)}} = sup α=yκ {min{0,ICK (κ)}} = 0 = ICHK (α) and (FCH ◦FCK ) (α) = inf α=yκ {max{FCH (y),FCK (κ)}} = inf α=yκ { max { 1.ei2π,FCK (κ) }} = 1.ei2π = FCHK (α). Similarly, if κ /∈ K, then (TCH ◦TCK ) (α) = 0 = TCHK (α), (ICH ◦ ICK ) (α) = 0 = ICHK (α) and (FCH ◦FCK ) (α) = 1.ei2π = FCHK (α). Therefore CH ⊗CK = CHK. Proof of (2) and (3) are straightforward. � Theorem 5.1. A complex neutrosophic set C on a semigroup S is a complex neutrosophic subsemigroup of S if and only if C⊗C ⊆C. Proof: Let C be a complex neutrosophic subsemigroup of S, and x ∈S. Case 1: If x 6= yκ, for any y,κ ∈S, then obviously C⊗C ⊆C. Case 2: If x = yκ, for any y,κ ∈S, then (pC ◦pC)(x) ·ei(µC◦µC)(x) = sup x=yκ [ min{pC(y)eiµC(y),pC(κ)eiµC(κ)} ] ≤ sup x=yκ [ pC(yκ)e iµC(yκ) ] = pC(x) ·eiµC(x). Similarly, (qC ◦qC)(x) ·ei(νC◦νC)(x) ≤ qC(x) ·eiνC(x). Int. J. Anal. Appl. 16 (1) (2018) 108 And (rC ◦rC)(x) ·ei(ωC◦ωC)(x) = inf x=yκ [ max{rC(y) ·eiωC(y),rC(κ) ·eiωC(κ)} ] ≥ inf x=yκ [rC(yκ) ·eiωC(yκ)] = rC(x) ·eiωC(x). Therefore, C⊗C ⊆C. Conversely, Suppose C⊗C ⊆C, and assume x = yκ, then pC(yκ) ·eiµC(yκ) ≥ (pC ◦pC)(yκ) ·ei(µC◦µC)(yκ) = sup yκ=yκ [ min{pC(y)eiµC(y),pC(κ)eiµC(κ)} ] = min{pC(y)eiµC(y),pC(κ)eiµC(κ)}. Similarly, qC(yκ) ·eiνC(yκ) ≥ min{qC(y)eiνC(y),qC(κ)eiνC(κ)}. And rC(yκ) ·eiωC(yκ) ≤ (rC ◦rC)(yκ) ·ei(ωC◦ωC)(yκ)) = inf yκ=yκ [ max{rC(y)eiωC(y),rC(κ)eiωC(κ)} ] = max{rC(y)eiωC(y),rC(κ)eiωC(κ)}. Hence C is a complex neutrosophic subsemigroup of S. � Proposition 5.3. A complex neutrosophic set C on a semigroup S, the following are equivalent: (1) C is a complex neutrosophic left ideal of S. (2) S⊗C ⊆C. Proof: (1) ⇒ (2) : Assume that C is a complex neutrosophic left ideal of S. Let x ∈ S, such that (S⊗C)(x) = 〈 0̂, 0̂, 1̂ 〉 , then it is clear S⊗C ⊆C. Whenever there exist any two elements y,κ ∈S, such that x = yκ. Then (1̂ST ◦pC ·e iµC )(x) = sup x=yκ [min{1̂ST (y),pC(κ) ·e iµC(κ)}] ≤ sup x=yκ [min{1 ·ei2π,pC(yκ) ·eiµC(yκ)}] = pC(x) ·eiµC(x). Similarly, (1̂SI ◦qC ·e iνC )(x) ≤ qC(x) ·eiνC(x). Int. J. Anal. Appl. 16 (1) (2018) 109 And (0̂SF ◦rC ·e iω)(x) = inf x=yκ [max{0̂SF (y),rC(κ) ·e iω(κ)}] ≥ inf x=yκ [max{0,rC(yκ) ·eiω(yκ)}] = rC(x) ·eiω(x). Therefore, S⊗C ⊆C. Conversely, (2) ⇒ (1) : Suppose that S⊗C ⊆C. For any elements y,κ of S, let x = yκ. Then pC(yκ) ·eiµC(yκ) = pC(x) ·eiµC(x) ≥ (1̂ST ◦pC ·e iµC )(x) = sup x=yκ [min{1̂ST (y),pC(κ) ·e iµC(κ)}] = pC(κ) ·eiµC(κ). Similarly, qC(yκ) ·eiνC(yκ) ≥ qC(κ) ·eiνC(κ). And rC(yκ) ·eiωC(yκ) = rC(x) ·eiωC(x) ≤ (0̂SF ◦rC ·e iωC )(x) = inf x=yκ [max{0̂SF (y),rC(κ) ·e iωC(κ)}] = rC(κ) ·eiωC(κ). Hence C is a complex neutrosophic left ideal of S. � Proposition 5.4. A complex neutrosophic set C on a semigroup S, the following are equivalent: (1) C is a complex neutrosophic right ideal of S. (2) C⊗S ⊆C. Proof: Proof is similar to the Proposition 5.3. � Theorem 5.2. If C is a complex neutrosophic set of a semigroup S, then S⊗C (resp., C⊗S) is a complex neutrosophic left (resp. right) ideal of S. Proof: Since S⊗(S⊗C) = (S⊗S)⊗C ⊆S⊗C, it follows from Proposition 5.3, that S⊗C is a complex neutrosophic left ideal of S. Similarly C⊗S is a complex neutrosophic right ideal of S. � Theorem 5.3. Let S be a left zero subsemigroup of a semigroup S. If C is a complex neutrosophic left ideal of S, then C(x) = C(y) for all x,y ∈ S. Int. J. Anal. Appl. 16 (1) (2018) 110 Proof: Let x,y ∈ S. Then xy = x and yx = y. Thus pC(x) ·eiµC(x) = pC(xy) ·eiµC(xy) ≥ pC(y) ·eiµC(y) = pC(yx) ·eiµC(yx) ≥ pC(x) ·eiµC(x). Similarly, qC(x) ·eiνC(x) = qC(y) ·eiνC(y). And rC(x) ·eiωC(x) = rC(xy) ·eiωC(xy) ≤ rC(y) ·eiωC(y) = rC(yx) ·eiωC(yx) ≤ rC(x) ·eiωC(x). Therefore, C(x) = C(y) for all x,y ∈ S. � Theorem 5.4. Let S be a right zero subsemigroup of a semigroup S. If C is a complex neutrosophic right ideal of S, then C(x) = C(y) for all x,y ∈ S. Proof: Proof is similar to the Theorem 5.3. � Theorem 5.5. Let C is a complex neutrosophic left ideal of a semigroup S. If the set of all idempotent elements of S form a left zero subsemigroup of S, then C(x) = C(y) for all idempotent elements x and y of S. Proof: Let Idm(S) be the set of all idempotent elements of S and assume that Idm(S) is a left zero subsemigroup of S. For any x,y ∈Idm(S), we have xy = x and yx = y. Thus pC(x) ·eiµC(x) = pC(xy) ·eiµC(xy) ≥ pC(y) ·eiµC(y) = pC(yx) ·eiµC(yx) ≥ pC(x) ·eiµC(x) = pC(y) ·eiµC(y). Similarly, qC(x) ·eiνC(x) = qC(y) ·eiνC(y). And rC(x) ·eiωC(x) = rC(xy) ·eiωC(xy) ≤ rC(y) ·eiωC(y) = rC(yx) ·eiωC(yx) ≤ rC(x) ·eiωC(x) = rC(y) ·eiωC(y). Therefore, C(x) = C(y) for all x,y ∈Idm(S). � Int. J. Anal. Appl. 16 (1) (2018) 111 Theorem 5.6. Let C is a complex neutrosophic right ideal of a semigroup S. If the set of all idempotent elements of S form a right zero subsemigroup of S, then C(x) = C(y) for all idempotent elements x and y of S. Proof: Proof is similar to the Theorem 5.5. � Proposition 5.5. If S be a semigroup. Then the following properties are hold. (1) The intersection of two complex neutrosophic subsemigroups of S is a complex neutrosophic subsemi- group of S. (2) The intersection of two complex neutrosophic left (resp., right) ideals of S is a complex neutrosophic left (resp., right) ideal of S. Proof: Let C1 = 〈 C1T = pC1 ·e iµC1 ,C1I = qC1 ·e iνC1 ,C1F = rC1 ·e iωC1 〉 and C2 = 〈 C2T = pC2 ·e iµC2 ,C2I = qC2 ·e iνC2 ,C2F = rC2 ·e iωC2 〉 be any two complex neutrosophic subsemigroups of S. Let x,y ∈S. Then (pC1 ·e iµC1 ∩pC2 ·e iµC2 )(xy) = min{pC1 (xy) ·e iµC1 (xy),pC2 (xy) ·e iµC2 (xy)} ≥ min{min{pC1 (x) ·e iµC1 (x),pC1 (y) ·e iµC1 (y)}, min{pC2 (x) ·e iµC2 (x),pC2 (y) ·e iµC2 (y)}} = min{min{pC1 (x) ·e iµC1 (x),pC2 (x) ·e iµC2 (x)}, min{pC1 (y) ·e iµC1 (y),pC2 (y) ·e iµC2 (y)}} = min{(pC1 ·e iµC1 ∩pC2 ·e iµC2 )(x), (pC1 ·e iµC1 ∩pC2 ·e iµC2 )(y)}. Similarly, (qC1 ·e iνC1 ∩qC2 ·e iνC2 )(xy) ≥ min{(qC1 ·e iνC1 ∩qC2 ·e iνC2 )(x), (qC1 ·e iνC1 ∩qC2 ·e iνC2 )(y)}. Int. J. Anal. Appl. 16 (1) (2018) 112 And (rC1 ·e iωC1 ∪rC2 ·e iωC2 )(xy) = max{rC1 (xy) ·e iωC1 (xy),rC2 (xy) ·e iωC2 (xy)} ≤ max{max{rC1 (x) ·e iωC1 (x),rC1 (y) ·e iωC1 (y)}, max{rC2 (x) ·e iωC2 (x),rC2 (y) ·e iωC2 (y)}} = max{max{rC1 (x) ·e iωC1 (x),rC2 (x) ·e iωC2 (x)}, max{rC1 (y) ·e iωC1 (y),rC2 (y) ·e iωC2 (y)}} = max{(rC1 ·e iωC1 ∪rC2 ·e iωC2 )(x), (rC1 ·e iωC1 ∪rC2 ·e iωC2 )(y)}. Therefore, C1 ∩C2 is a complex neutrosophic subsemigroup of S. (2) Let C1 and C2 be any two complex neutrosophic left ideals of semigroup S, and x,y ∈S. Then (pC1 ·e iµC1 ∩pC2 ·e iµC2 )(xy) = min{pC1 (xy) ·e iµC1 (xy),pC2 (xy) ·e iµC2 (xy)} ≥ min{pC1 (y) ·e iµC1 (y),pC2 (y) ·e iµC2 (y)} = (pC1 ·e iµC1 ∩pC2 ·e iµC2 )(y). Similarly, (qC1 ·e iνC1 ∩qC2 ·e iνC2 )(xy) ≥ (qC1 ·e iνC1 ∩qC2 ·e iνC2 )(y). And (rC1 ·e iωC1 ∪rC2 ·e iωC2 )(xy) = max{rC1 (xy) ·e iωC1 (xy),rC2 (xy) ·e iωC2 (xy)} ≤ max{rC1 (y) ·e iωC1 (y),rC2 (y) ·e iωC2 (y)} = (rC1 ·e iωC1 ∪rC2 ·e iωC2 )(y). Thus C1 ∩C2 is a complex neutrosophic left ideal of semigroup S. The intersection of complex neutrosophic right ideal can be proved in a similar manner. � Proposition 5.6. If S be a semigroup. Then the following properties are hold. (1) The union of two complex neutrosophic subsemigroups of S is a complex neutrosophic subsemigroup of S. (2) The union of two complex neutrosophic left (resp., right) ideals of S is a complex neutrosophic left (resp., right) ideal of S. Proof: Let C1 = 〈 C1T = pC1 ·e iµC1 ,C1I = qC1 ·e iνC1 ,C1F = rC1 ·e iωC1 〉 Int. J. Anal. Appl. 16 (1) (2018) 113 and C2 = 〈 C2T = pC2 ·e iµC2 ,C2I = qC2 ·e iνC2 ,C2F = rC2 ·e iωC2 〉 be any two complex neutrosophic subsemigroups of S. Let x,y ∈S. Then (pC1 ·e iµC1 ∪pC2 ·e iµC2 )(xy) = max{pC1 (xy) ·e iµC1 (xy),pC2 (xy) ·e iµC2 (xy)} ≥ max{min{pC1 (x) ·e iµC1 (x),pC1 (y) ·e iµC1 (y)}, min{pC2 (x) ·e iµC2 (x),pC2 (y) ·e iµC2 (y)}} = pC1 (x) ·e iµC1 (x) ∧pC1 (y) ·e iµC1 (y) ∨ pC2 (x) ·e iµC2 (x) ∧pC2 (y) ·e iµC2 (y) = pC1 (x) ·e iµC1 (x) ∨pC2 (x) ·e iµC2 (x) ∧ pC1 (y) ·e iµC1 (y) ∨pC2 (y) ·e iµC2 (y) = min{(pC1 ·e iµC1 ∪pC2 ·e iµC2 )(x), (pC1 ·e iµC1 ∪pC2 ·e iµC2 )(y)}. Similarly, (qC1 ·e iνC1 ∪qC2 ·e iνC2 )(xy) ≥ min{(qC1 ·e iνC1 ∪qC2 ·e iνC2 )(x), (qC1 ·e iνC1 ∪qC2 ·e iνC2 )(y)}. And (rC1 ·e iωC1 ∩rC2 ·e iωC2 )(xy) = min{rC1 (xy) ·e iωC1 (xy),rC2 (xy) ·e iωC2 (xy)} ≤ min{max{rC1 (x) ·e iωC1 (x),rC1 (y) ·e iωC1 (y)}, max{rC2 (x) ·e iωC2 (x),rC2 (y) ·e iωC2 (y)}} = rC1 (x) ·e iωC1 (x) ∨rC1 (y) ·e iωC1 (y) ∧ rC2 (x) ·e iωC2 (x) ∨rC2 (y) ·e iωC2 (y) = rC1 (x) ·e iωC1 (x) ∧rC2 (x) ·e iωC2 (x) ∨ rC1 (y) ·e iωC1 (y) ∧rC2 (y) ·e iωC2 (y) = max{(rC1 ·e iωC1 ∩rC2 ·e iωC2 )(x), (rC1 ·e iωC1 ∩rC2 ·e iωC2 )(y)}. Therefore, C1 ∪C2 is a complex neutrosophic subsemigroup of S. Int. J. Anal. Appl. 16 (1) (2018) 114 (2) Let C1 and C2 be any two complex neutrosophic left ideals of semigroup S, and x,y ∈S. Then (pC1 ·e iµC1 ∪pC2 ·e iµC2 )(xy) = max{pC1 (xy) ·e iµC1 (xy),pC2 (xy) ·e iµC2 (xy)} ≥ max{pC1 (y) ·e iµC1 (y),pC2 (y) ·e iµC2 (y)} = (pC1 ·e iµC1 ∪pC2 ·e iµC2 )(y). Similarly, (qC1 ·e iνC1 ∪qC2 ·e iνC2 )(xy) ≥ (qC1 ·e iνC1 ∪qC2 ·e iνC2 )(y). And (rC1 ·e iωC1 ∩rC2 ·e iωC2 )(xy) = min{rC1 (xy) ·e iωC1 (xy),rC2 (xy) ·e iωC2 (xy)} ≤ min{rC1 (y) ·e iωC1 (y),rC2 (y) ·e iωC2 (y)} = (rC1 ·e iωC1 ∩rC2 ·e iωC2 )(y). Thus C1 ∪C2 is a complex neutrosophic left ideal of semigroup S. The union of complex neutrosophic right ideal can be proved in a similar manner. � Theorem 5.7. If C1 and C2 be a complex neutrosophic right and left ideals of a semigroup S, respectively. Then C1 ⊗C2 ⊆C1 ∩C2. Proof: Let C1 is complex neutrosophic right ideal and C2 is any complex left neutrosophic ideal of S. Then by Proposition 5.3 and Proposition 5.4 we have C1 ⊗C2 ⊆ C1 ⊗S ⊆ C1 and C1 ⊗C2 ⊆ S ⊗C2 ⊆ C2. Hence C1 ⊗C2 ⊆C1 ∩C2. � Theorem 5.8. If S is regular semigroup, then C1 ⊗C2 = C1 ∩C2 for every complex neutrosophic right ideal C1 = 〈 pC1 ·eiµC1 ,qC1 ·eiνC1 ,rC1 ·eiωC1 〉 and every complex neutrosophic left ideal C2 = 〈 pC2 ·eiµC2 ,qC2 ·eiνC2 ,rC2 ·eiωC2 〉 of S. Proof: Let α be any element of S. Since S is regular, there exist an element x ∈S such that α = αxα. Hence we have (pC1 ·e iµC1 ◦pC2 ·e iµC2 )(α) = sup α=yκ {min{pC1 (y) ·e iµC1 (y),pC2 (κ) ·e iµC2 (κ)}} = sup αxα=yκ {min{pC1 (y) ·e iµC1 (y),pC2 (κ) ·e iµC2 (κ)}} ≥ min{pC1 (αx) ·e iµC1 (αx),pC2 (α) ·e iµC2 (α)} ≥ min{pC1 (α) ·e iµC1 (α),pC2 (α) ·e iµC2 (α)} = (pC1 ·e iµC1 ∩pC2 ·e iµC2 )(α). Int. J. Anal. Appl. 16 (1) (2018) 115 Similarly, (qC1 ·e iνC1 ◦qC2 ·e iνC2 )(α) ≥ (qC1 ·e iνC1 ∩qC2 ·e iνC2 )(α). And (rC1 ·e iωC1 ◦rC2 ·e iωC2 )(α) = inf α=yκ {max{rC1 (y) ·e iωC1 (y),rC2 (κ) ·e iωC2 (κ)}} = inf αxα=yκ {max{rC1 (y) ·e iωC1 (y),rC2 (κ) ·e iωC2 (κ)}} ≤ max{rC1 (αx) ·e iωC1 (αx),rC2 (α) ·e iωC2 (α)} ≤ max{rC1 (α) ·e iωC1 (α),rC2 (α) ·e iωC2 (α)} = (rC1 ·e iωC1 ∪rC2 ·e iωC2 )(α). So C1 ⊗C2 ⊇C1 ∩C2, and C1 ⊗C2 ⊆C1 ∩C2 is true from Theorem 5.7. Hence C1 ⊗C2 = C1 ∩C2. � Theorem 5.9. For a non-empty subset H of a semigroup S. We have (1) H is a subsemigroup of S if and only if the characteristic complex neutrosophic set CH = 〈TCH ,ICH ,FCH〉 of H in S is a complex neutrosophic subsemigroup of S. (2) H is a left (right) ideal of S if and only if the characteristic complex neutrosophic set CH = 〈TCH ,ICH ,FCH〉 of H in S is a complex neutrosophic left (resp., right) ideal of S. Proof: Straightforward. � Theorem 5.10. For every complex neutrosophic right ideal C1 = 〈TC1,IC1,FC1〉 and every complex neutro- sophic left ideal C2 = 〈TC2,IC2,FC2〉 of a semigroup S, if C1 ⊗C2 = C1 ∩C2, then S is regular. Proof: Assume that C1 ⊗C2 = C1 ∩C2 for every complex neutrosophic right ideal C1 = 〈TC1,IC1,FC1〉 and every complex neutrosophic left ideal C2 = 〈TC2,IC2,FC2〉 of a semigroup S. Let R and L be any right and left ideal of S, respectively. In order to see that R∩L⊆RL holds. Let α be any element of R∩L, then the characteristic complex neutrosophic sets CR = 〈TCR,ICR,FCR〉 and CL = 〈TCL,ICL,FCL〉 are a complex neutrosophic right ideal and a complex neutrosophic left ideal of S, respectively, by Theorem 5.9. It follows from the hypothesis and proposition 5.2, that is TCRL (α) = (TCR ◦TCL )(α) = (TCR ∩TCL )(α) = TCR∩L (α) = 1.e i2π ICRL (α) = (ICR ◦ ICL )(α) = (ICR ∩ ICL )(α) = ICR∩L (α) = 1.e i2π Int. J. Anal. Appl. 16 (1) (2018) 116 and FCRL (α) = (FCR ◦FCL )(α) = (FCR ∪FCL )(α) = FCR∪L (α) = 0. So that α ∈RL. Thus R∩L⊆RL. Since the inclusion in the other direction always holds, we obtain that R∩L⊆RL. It follows from Lemma 2.1, that S is regular. � References [1] Zadeh, L. A. Fuzzy Sets. Inf. Control, 8 (1965), 338-353. [2] Atanassov, K. T. Intuitionistic Fuzzy Sets, Fuzzy Sets Syst. 20 (1986), 87-96. [3] Kuroki, N. Fuzzy Bi-ideals in Semigroups. Comment. Math. Univ. St. Pauli, 27 (1979), 17-21. [4] Wang, H. et al. Single Valued Neutrosophic Sets. 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Complex Neutrosophic Ideal 4.4. Complex Neutrosophic Interior Ideal 5. Characteristic Function of Complex Neutrosophic Set 5.1. Direct Product of Two Complex Neutrosophic Sets References