Preethi, D.

Rajareega, S.

Selvachandran, Ganeshsree

Smarandache, F.

Vimala, J.

English

University of New Mexico Digital Repository

Neutrosophic Sets and Systems Volume 31 Article 6 5-2-2020 Some Results on Single Valued Neutrosophic Hypergroup S. Rajareega D. Preethi J. Vimala Ganeshsree Selvachandran F. Smarandache Follow this and additional works at: https://digitalrepository.unm.edu/nss_journal Recommended Citation Rajareega, S.; D. Preethi; J. Vimala; Ganeshsree Selvachandran; and F. Smarandache. "Some Results on Single Valued Neutrosophic Hypergroup." Neutrosophic Sets and Systems 31, 1 (2020). https://digitalrepository.unm.edu/nss_journal/vol31/iss1/6 This Article is brought to you for free and open access by UNM Digital Repository. It has been accepted for inclusion in Neutrosophic Sets and Systems by an authorized editor of UNM Digital Repository. For more information, please contact amywinter@unm.edu, lsloane@salud.unm.edu, sarahrk@unm.edu. Neutrosophic Sets and Systems, Vol. 31, 2020 University of New Mexico  S. Rajareega, D. Preethi, J. Vimala, Ganeshsree Selvachandran and Florentin Smarandache, Some Results on Single Valued Neutrosophic Hypergroup   Some Results on Single Valued Neutrosophic Hypergroup S. Rajareega1, D. Preethi2, J. Vimala 3,*, Ganeshsree Selvachandran4, Florentin Smarandache5 1,2,3    Department of Mathematics, Alagappa University, Karaikudi, Tamilnadu, India. E-mail:  reega948@gmail.com, preethi06061996@gmail.com, vimaljey@alagappauniversity.ac.in 4 Department of Actuarial Science and Applied Statistics, Faculty of Business and Information Science,UCSI University, Jalan Menara Gading, 56000 Cheras, Kuala Lumpur, Malaysia. E-mail: ganeshsree86@yahoo.com or Ganeshsree@ucsiuniversity.edu.my 5   Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, New Mexico, USA.  E-mail: smarand@unm.edu * Correspondence: vimaljey@alagappauniversity.ac.in  Abstract: We introduced the theory of Single valued neutrosophic hypergroup as the initial theory of single valued neutrosophic hyper algebra and also developed some results on single valued neutrosophic hypergroup. Keywords: Hypergroup; Level sets; Single valued neutrosophic sets; Single valued neutrosophic hypergroup.   1. Introduction Florentin Smarandache introduced Neutrosophic sets in 1998 [16], which is the generalization of the intuitionistic fuzzy sets. In some real time situations, decision makers faced some difficulties with uncertainty and inconsistency values. Neutrosophic sets helped the decision makers to deal with uncertainty values. Abdel-Basset et.al. used neutrosophic concept in real life decision-making problems [1-7]. The concept of single valued neutrosophic set was introduced by Wang. et. al [17].  As a generalization of classical algebraic structure, Algebraic hyper structure was introduced by F. Marty [11]. Corsini and Leoreanu-Fotea developed the applications of hyper structure [9]. Algebraic hyperstructures has many applications in fuzzy sets, lattices, artificial intelligence, automation, combinatorics. Corsini introduced hypergroup theory [8]. After while the hyperstructure theory has seen broader applications in many fields. Some of the recent works on hyperstructures related to vague soft groups, vague soft rings and vague soft ideals can be found in [12, 13].  In this paper we develop the theory of single valued neutrosophic hypergroup and also established some results on single valued neutrosophic hypergroup. 2. Preliminaries Definition 2.1 [17] Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function TA, an indeterminancy-membership function IA and a falsity-membership function FA. TA(x), IA(x) and FA(x) are real standard or non-standard subsets of ]0−, 1+[.   𝑇𝐴: 𝑋 →]0−, 1+[  𝐼𝐴: 𝑋 →]0−, 1+[  𝐹𝐴: 𝑋 →]0−, 1+[ Neutrosophic Sets and Systems, Vol. 31, 2020 81    S. Rajareega, D. Preethi, J. Vimala, Ganeshsree Selvachandran and Florentin Smarandache, Some Results on Single Valued Neutrosophic Hypergroup There is no restriction on the sum of 𝑇𝐴(𝑥), 𝐼𝐴(𝑥) and 𝐹𝐴(𝑥), so 0− ≤ 𝑠𝑢𝑝𝑇𝐴(𝑥) + 𝑠𝑢𝑝𝐼𝐴(𝑥) +𝑠𝑢𝑝𝐹𝐴(𝑥) ≤ 3+.  Definition 2.2 [17] Let X be a space of points (objects),with a generic element of X denoted by x. A single valued neutrosophic set (SVNS) A in X is characterized by TA, IA and FA. For each point x in X, TA, IA, FA ∈ [0,1].  Definition 2.3 [17] The complement of a SVNS A is denoted by c(A) and is defined by  Tc(A)(x) = FA(x)  Ic(A)(x) = 1 − IA(x)  Fc(A)(x) = TA(x), for all x in X.   Definition 2.4  [17] A SVNS A is contained in the other SVNS  B, A ⊆ B, if and only if,  TA(x) ≤ TB(x)  IA(x) ≥ IB(x)  FA(x) ≥ FB(x), for all x in X.   Definition 2.5 [17] The union of two SVNS s A and B is a SVNS   C, written as C = A ∪ B, whose truth, indeterminancy and falsity-membership functions are defined by,  TC(x) = max(TA(x), TB(x))  IC(x) = min(IA(x), IB(x))  FC(x) = min(FA(x), FB(x)), for all x in X.   Definition 2.6 [17] The intersection of two SVNS s A and B is a SVNS C, written as C = A ∩ B, whose truth, indeterminancy and falsity-membership functions are defined by,  TC(x) = min(TA(x), TB(x))  IC(x) = max(IA(x), IB(x))  FC(x) = max(FA(x), FB(x)), for all x in X.  Definition 2.7 [17] The falsity-favorite of a SVNS B, written as B∇ A, whose truth and falsity-membership functions are defined by   TB(x) = TA(x)  IB(x) = 0  FB(x) = min{FA(x) + IA(x),1}, for all x in X.  Definition 2.8 [13] A hypergroup 〈H,∘〉 is a set H equipped with an associative hyperoperation (∘): H × H → P(H) which satisfies x ∘ H = H ∘ x = H for all x ∈ H  (Reproduction axiom)   Definition 2.9 [13] A hyperstructure 〈H,∘〉 is called an Hv-group if the following axioms hold:  (i) x ∘ (y ∘ z) ∩ (x ∘ y) ∘ z ≠ ∅ for all x, y, z ∈ H,  (ii) x ∘ H = H ∘ x = H for all x ∈ H. If 〈H,∘〉 only satisfies (i), then 〈H,∘〉  is called a Hv- semigroup.   Definition 2.10 [13] A subset K of H is called a subhypergroup if 〈K,∘〉 is a hypergroup of 〈H,∘〉.  3. Single Valued Neutrosophic Hypergroup. Throughout this section 𝐻 denotes the hypergroup < 𝐻,∘>  Definition 3.1 Let 𝒜 be a single valued neutrosophic set over H. Then 𝒜 is called a single valued neutrosophic hypergroup over H, if the following conditions are satisfied (𝑖)  ∀  𝑝, 𝑞 ∈ 𝐻,𝑚𝑖𝑛{𝑇𝒜(𝑝), 𝑇𝒜(𝑞)} ≤ 𝑖𝑛𝑓{𝑇𝒜(𝑟):  𝑟 ∈ 𝑝 ∘ 𝑞},  𝑚𝑎𝑥{𝐼𝒜(𝑝), 𝐼𝒜(𝑞)} ≥ 𝑠𝑢𝑝{𝐼𝒜(𝑟):  𝑟 ∈ 𝑝 ∘ 𝑞}  𝑎𝑛𝑑 Neutrosophic Sets and Systems, Vol. 31, 2020 82    S. Rajareega, D. Preethi, J. Vimala, Ganeshsree Selvachandran and Florentin Smarandache, Some Results on Single Valued Neutrosophic Hypergroup  𝑚𝑎𝑥{𝐹𝒜(𝑝), 𝐹𝒜(𝑞)} ≥ 𝑠𝑢𝑝{𝐹𝒜(𝑟):  𝑟 ∈ 𝑝 ∘ 𝑞} (𝑖𝑖)  ∀  𝑙, 𝑝 ∈ 𝐻, 𝑡ℎ𝑒𝑟𝑒  𝑒𝑥𝑖𝑠𝑡𝑠  𝑞 ∈ 𝐻  𝑠𝑢𝑐ℎ  𝑡ℎ𝑎𝑡  𝑝 ∈ 𝑙 ∘ 𝑞  𝑎𝑛𝑑  𝑚𝑖𝑛{𝑇𝒜(𝑙), 𝑇𝒜(𝑝)} ≤ 𝑇𝒜(𝑞),  𝑚𝑎𝑥{𝐼𝒜(𝑙), 𝐼𝒜(𝑝)} ≥ 𝐼𝒜(𝑞)  𝑎𝑛𝑑  𝑚𝑎𝑥{𝐹𝒜(𝑙), 𝐹𝒜(𝑝)} ≥ 𝐹𝒜(𝑞) (𝑖𝑖𝑖)  ∀  𝑙, 𝑝 ∈ 𝐻, 𝑡ℎ𝑒𝑟𝑒  𝑒𝑥𝑖𝑠𝑡𝑠  𝑟 ∈ 𝐻  𝑠𝑢𝑐ℎ  𝑡ℎ𝑎𝑡  𝑝 ∈ 𝑟 ∘ 𝑙  𝑎𝑛𝑑  𝑚𝑖𝑛{𝑇𝒜(𝑙), 𝑇𝒜(𝑝)} ≤ 𝑇𝒜(𝑟),  𝑚𝑎𝑥{𝐼𝒜(𝑙), 𝐼𝒜(𝑝)} ≥ 𝐼𝒜(𝑟)  𝑎𝑛𝑑  𝑚𝑎𝑥{𝐹𝒜(𝑙), 𝐹𝒜(𝑝)} ≥ 𝐹𝒜(𝑟)  If 𝒜 satisfies condition (i) then  𝒜 is a single valued neutrosophic semihypergroup over H. Condition (ii) and (iii) represent the left and right reproduction axioms respectively. Then 𝒜 is a single valued neutrosophic subhypergroup of H.  Example 3.2 If the family of t-level sets of SVNS  𝒜 over H   𝒜t = {p ∈ H  |  T𝒜(p) ≥ t, I𝒜(p) ≤ t  and  F𝒜(p) ≤ t}  is a subhypergroup of H then, 𝒜 is a single valued neutrosophic hypergroup over H.    Theorem 3.3 Let 𝒜 be a SVNS  over H. Then 𝒜 is a single valued neutrosophic hypergroup over H iff 𝒜 is a single valued neutrosophic semihypergroup over H and also 𝒜 satisfies the left and right reproduction axioms.   Proof. The proof is obvious from Definition: 3.1    Theorem 3.4 Let 𝒜 be a SVNS  over H. If 𝒜 is a single valued neutrosophic hypergroup over H ,then ∀ t ∈ [0,1] 𝒜t ≠ ∅ is a subhypergroup of H.  Proof. Let 𝒜 be a single valued neutrosophic hypergroup over H and let p, q ∈ 𝒜t, then  T𝒜(p), T𝒜(q) ≥ t, I𝒜(p), I𝒜(q) ≤ t and F𝒜(p), F𝒜(q) ≤ t. Then we have,                 inf{T𝒜(r): r ∈ p ∘ q} ≥ min{T𝒜(p), T𝒜(q)} ≥ min{t, t} = t sup{I𝒜(r): r ∈ p ∘ q} ≤ t and  sup{F𝒜(r): r ∈ p ∘ q} ≤ t This implies r ∈ 𝒜t. Then ∀  r ∈ p ∘ q  , p ∘ q ⊆ 𝒜t. Thus ∀r ∈ 𝒜t, we obtain r ∘ 𝒜t ⊆ 𝒜t Now, Let l, p ∈ 𝒜t, then there exist q ∈ H such that p ∈ l ∘ q and                         {T𝒜(q)} ≥ min{T𝒜(l), T𝒜(p)} ≥ min{t, t} = t {I𝒜(q)} ≤ t and {F𝒜(q)} ≤ t.  This implies q ∈ 𝒜t This proves that 𝒜t ⊆ r ∘ 𝒜t. As such 𝒜t = r ∘ 𝒜t Which proves that 𝒜t is a subhypergroup of H.   Theorem 3.5 Let 𝒜 be a SVNS  over H. Then the following are equivalent, (i) 𝒜 is a single valued neutrosophic hypergroup over H  (ii) ∀ t ∈ [0,1] 𝒜t ≠ ∅ is a subhypergroup of H.  Proof. (i) ⇒ (ii) The proof is obvious from Theorem : 3.4. Neutrosophic Sets and Systems, Vol. 31, 2020 83    S. Rajareega, D. Preethi, J. Vimala, Ganeshsree Selvachandran and Florentin Smarandache, Some Results on Single Valued Neutrosophic Hypergroup (ii) ⇒ (i) Now assume that 𝒜t is a subhypergroup of H. Let p, q ∈ 𝒜t0 and let min{T𝒜(p), T𝒜(q)} = max{I𝒜(p), I𝒜(q)} = max{F𝒜(p), F𝒜(q)} = t0  Since p ∘ q ⊆ 𝒜t0, then for every r ∈ p ∘ q, T𝒜(r) ≥ t0, I𝒜(r) ≤ t0, F𝒜(r) ≤ t0                 min{T𝒜(p), T𝒜(q)} ≤ inf{T𝒜(r):  r ∈ p ∘ q}, max{I𝒜(p), I𝒜(q)} ≥ sup{I𝒜(r):  r ∈ p ∘ q} and max{F𝒜(p), F𝒜(q)} ≥ sup{F𝒜(r):  r ∈ p ∘ q}  Condition (i) is verified. Next, let l, p ∈ 𝒜t1, for every t1 ∈ [0,1] and  let min{T𝒜(l), T𝒜(q)} = max{I𝒜(l), I𝒜(p)} = max{F𝒜(l), F𝒜(q)} = t1  Then there exist q ∈ 𝒜t1  such that p ∈ l ∘ q ⊆ 𝒜t1. Since q ∈ 𝒜t1 , T𝒜(q) ≥ t1 = min{T𝒜(l), T𝒜(q)} I𝒜(q) ≤ t1 = max{I𝒜(l), I𝒜(q)} F𝒜(q) ≤ t1 = max{F𝒜(l), F𝒜(q)} Condition (ii) is verified. Similarly, (iii) .   Theorem 3.6 Let 𝒜 be a SVNS  over H. Then 𝒜 be a single valued neutrosophic hypergroup over H iff  ∀ α, β, γ ∈ [0,1], 𝒜(α,β,γ) is a subhypergroup of H.   Proof. The proof is straight forward.    Theorem 3.7 Let 𝒜 be a single valued neutrosophic hypergroup over H and  ∀ t1, t2 ∈ [0,1] 𝒜t1  and 𝒜t2 be the t-level sets of 𝒜 with t1 ≥ t2, then  𝒜t1 is a subhypergroup of 𝒜t2.   Proof. ∀t1, t2 ∈ [0,1], 𝒜t1  and 𝒜t2 be the t-level sets of 𝒜 with t1 ≥ t2 This implies that 𝒜t1 ⊆ 𝒜t2  By Theorem 3.4. 𝒜t1 is a subhypergroup of 𝒜t2.   Theorem 3.8 Let 𝒜 and ℬ be single valued neutrosophic hypergroups over H. Then 𝒜 ∩ ℬ is a single valued neutrosophic hypergroup over H if it is non-null.   Proof. Suppose 𝒜 and ℬ be single valued neutrosophic hypergroups over H. By Definition: 2.6. 𝒜 ∩ ℬ = {< p, T𝒜∩ℬ(p), I𝒜∩ℬ(p), F𝒜∩ℬ(p) > :  p ∈ H} where T𝒜∩ℬ(p) = T𝒜(p) ∧ Tℬ(p), I𝒜∩ℬ(p) = I𝒜(p) ∨ Iℬ(p) and F𝒜∩ℬ(p) = F𝒜(p) ∨ Fℬ(p) For all p, q ∈ H  (i) min{T𝒜∩ℬ(p), T𝒜∩ℬ(q)} = min{T𝒜(p) ∧ Tℬ(p), T𝒜(q) ∧ Tℬ(q)}  ≤ min{T𝒜(p), T𝒜(q)} ∧ min{Tℬ(p), Tℬ(q)}  ≤ inf{T𝒜(r):  r ∈ p ∘ q} ∧ inf{Tℬ(r):  r ∈ p ∘ q}  ≤ inf{T𝒜(r) ∧ Tℬ(r):  r ∈ p ∘ q}  = inf{T𝒜∩ℬ(r):  r ∈ p ∘ q}  Similarly, we can prove that max{I𝒜∩ℬ(p), I𝒜∩ℬ(q)} ≥ sup{I𝒜∩ℬ(r):  r ∈ p ∘ q}            max{F𝒜∩ℬ(p), FA∩B(q)} ≥ sup{F𝒜∩ℬ(r):  r ∈ p ∘ q} (ii) ∀ l, p ∈ H, there exists q ∈ H such that p ∈ l ∘ q,  min{T𝒜∩ℬ(l), T𝒜∩ℬ(p)} = min{T𝒜(l) ∧ Tℬ(l)}, {T𝒜(p) ∧ Tℬ(p)}             = min{T𝒜(l), T𝒜(p)} ∧ min{Tℬ(l), TB(p)}             ≤ T𝒜(q) ∧ Tℬ(q) = T𝒜∩ℬ(q) Neutrosophic Sets and Systems, Vol. 31, 2020 84    S. Rajareega, D. Preethi, J. Vimala, Ganeshsree Selvachandran and Florentin Smarandache, Some Results on Single Valued Neutrosophic Hypergroup  Therefore, 𝒜 ∩ ℬ is a single valued neutrosophic hypergroup over H.   Theorem 3.9 Let 𝒜 and ℬ be single valued neutrosophic hypergroups over H. Then 𝒜 ∪ ℬ is a single valued neutrosophic hypergroup over H.  Proof. By Definition: 2.5.                                    𝒜 ∪ ℬ = {< p, T𝒜∪ℬ(p), I𝒜∪ℬ(p), F𝒜∪ℬ(p) > :  p ∈ H} where T𝒜∪ℬ(p) = T𝒜(p) ∨ Tℬ(p), I𝒜∪ℬ(p) = I𝒜(p) ∧ Iℬ(p) and F𝒜∪ℬ(p) = F𝒜(p) ∧ Fℬ(p) For all p, q ∈ H,  min{T𝒜∪ℬ(p), T𝒜∪ℬ(q)} = min{T𝒜(p) ∨ Tℬ(p), T𝒜(q) ∨ Tℬ(q)}  ≤ min{T𝒜(p), T𝒜(q)} ∨ min{Tℬ(p), Tℬ(q)}  ≤ inf{T𝒜(r):  r ∈ p ∘ q} ∨ inf{Tℬ(r):  r ∈ p ∘ q}  ≤ inf{T𝒜(r) ∨ Tℬ(r):  r ∈ p ∘ q}  = inf{T𝒜∪ℬ(r):  r ∈ p ∘ q}  Similarly, the other holds.    Theorem 3.10 Let 𝒜 be a single valued neutrosophic hypergroup over H. Then the falsity- favorite of 𝒜 (ie. , ∇𝒜) is also a single valued neutrosophic hypergroup over H.   Proof. By Definition: 2.7. ℬ = ∇𝒜, where the membership values are Tℬ(x) = T𝒜(x), Iℬ(x) = 0 and Fℬ(x) = min{F𝒜(x) + I𝒜(x),1} Then we have to prove for Fℬ, ∀p, q ∈ H  max{Fℬ(p), Fℬ(q)} = max{F𝒜(p) + I𝒜(p) ∧ 1, F𝒜(q) + I𝒜(q) ∧ 1}  = max{F𝒜(p) + I𝒜(p), F𝒜(q) + I𝒜(q)} ∧ 1  ≥ (max{F𝒜(p), F𝒜(q)} + max{I𝒜(p), I𝒜(q)}) ∧ 1  ≥ (sup{F𝒜(r)  ∶   r ∈ p ∘ q} + sup{I𝒜(r)  ∶   r ∈ p ∘ q}) ∧ 1  = sup{F𝒜(r) + I𝒜(r) ∧ 1 ∶   r ∈ p ∘ q}  = sup{Fℬ(r)  ∶   r ∈ p ∘ q})  In similar manner the other conditions holds. 4. Conclusions  In this paper, we have developed the theory of hypergroup for the single-valued neutrosophic set by introducing several hyperalgebraic structures and some results were verified. The future research related to this work involve the development of other hyperalgebraic theory for the single-valued neutrosophic sets and interval-valued neutrosophic sets.  Acknowledgments: The article has been written with the joint financial support of RUSA-Phase 2.0 grant sanctioned vide letter No.F 24-51/2014-U, Policy (TN Multi-Gen), Dept. of Edn. Govt. of India, Dt. 09.10.2018, UGC-SAP (DRS-I) vide letter No.F.510/8/DRS-I/2016(SAP-I) Dt. 23.08.2016, DST-PURSE 2nd Phase programme vide letter No. SR/PURSE Phase 2/38 (G) Dt. 21.02.2017 and DST (FST - level I) 657876570 vide letter No.SR/FIST/MS-I/2018/17 Dt. 20.12.2018.  Conflicts of Interest The authors declare no conflict of interest.  Neutrosophic Sets and Systems, Vol. 31, 2020 85    S. Rajareega, D. Preethi, J. Vimala, Ganeshsree Selvachandran and Florentin Smarandache, Some Results on Single Valued Neutrosophic Hypergroup References  1. Abdel-Basset, M., Mohamed, R., Zaied, A. E. N. H., amp; Smarandache, F. (2019).  A Hybrid Plithogenic Decision- Making Approach with Quality Function Deployment for Selecting Supply Chain Sustainability Metrics, Symmetry, 11(7), 903.  2. 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Marty, Sur une Generalisation de la Notion de Groupe , in Proceedings of the 8th Congress Mathematiciens Scandinaves, pp. 45-49, Stockholm, Sweden, 1934.  12.  G. Selvachandran and A. R. Salleh. Vague soft hypergroups and vague soft hypergroup homomorphism, Advances in Fuzzy Systems, vol. 2014, Article ID 758637, 10 pages, 2014.  13. G.Selvachandran and A. R. Salleh, Algebraic Hyperstructures of Vague Soft Sets Associated with Hyperrings and Hyperideals, Hindawi Publishing Corporation The Scientific World Journal Volume 2015, Article ID 780121, 12 pages.  14.  G. Selvachandran, Introduction to the theory of soft hyperrings and soft hyperring homomorphism , JPJournal ofAlgebra, Number Theory and Applications. In press.  15. G. Selvachandran and A.R. Salleh,  Hypergroup theory applied to fuzzy soft sets. Global Journal of Pure and Applied Mathematics 11(2) (2015) 825-835.  16. Smarandache. F.  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Some Results on Single Valued Neutrosophic Hypergroup

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Some Results on Single Valued Neutrosophic Hypergroup

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