International Journal of Applied Mathematics Volume 31 No. 4 2018, 527-539 IN: 1311-1728 (printed version); IN: 1314-8060 (on-line version) doi: http://dx.doi.org/10.12732/ijam.v31i4.2 HEITANT FUZZY ET APPROACH TO IDEAL THEORY IN TERNARY EMIGROUP A. Fairooze Talee 1, M. Yahya Abbasi 2,. Ali Khan 3 1,2,3 Department of Mathematics Jamia Millia Islamia New Delhi 110 025, INDIA Abstract: In this paper, we introduce hesitant fuzzy ideals on ternary semigroups and define hesitant fuzzy left (resp. lateral, right)ideals on ternary semigroups. Here we define hesitant fuzzy point on ternary semigroups and prove some interesting results on them. We also characterize regular ternary semigroups in terms of hesitant fuzzy left, hesitant fuzzy lateral and hesitant fuzzy right ideals using hesitant fuzzy points. AM ubject Classification: 20M12, 20N99, 08A72 Key Words: ternary semigroups, hesitant fuzzy ideals 1. Introduction and Preliminaries The idea of investigation of n-ary algebras, i.e. the sets with one n-ary operation was given by Kasner [4]. A ternary semigroup is a particular case of an n-ary semigroup for n = 3, [5]. The ternary semigroups are universal algebras with one associative operation. The ideal theory in ternary semigroup was studied by ioson [7]. Zadeh[11] introduced the concept of fuzzy set as an extension of the classical notion of set. The fuzzy set theory can be used in a wide range of domains to manipulate different types of uncertainties. Torra [8, 9] introduced hesitant Received: January 20, 2018 Correspondence author c 2018 Academic Publications
528 A.F. Talee, M.Y. Abbasi,.A. Khan fuzzy sets (briefly HFs) as a new generalization of fuzzy set. Hesitant fuzzy set is used to control hesitant situation which were difficult to handle by previous extensions. Jun et. al. [2] applied the HFs in semigroups. Abbasi et al. [6] applied the notion of HFs to ordered semigroups. Kar and arkar [3] introduced fuzzy left (right, lateral) ideals in ternary semigroups and characterized regular and intra-regular ternary semigroups by using the concept of fuzzy ideals of ternary semigroups. Our main aim in this study is to use the idea of Jun and Kar to introduce the notion of hesitant fuzzy ideals in ternary semigroups and study their related properties. Here we define hesitant fuzzy left (right, lateral) ideals of ternary semigroups and characterize regular and intra-regular ternary semigroups by using the concept of hesitant fuzzy ideals of ternary semigroups. Definition 1. [5] A non-empty set with a ternary operation, written as (x 1,x 2,x 3 ) [x 1,x 2,x 3 ], is called a ternary semigroup if it satisfies the following identity, for any x 1,x 2,x 3,x 4,x 5, [[x 1 x 2 x 3 ]x 4 x 5 ] = [x 1 [x 2 x 3 x 4 ]x 5 ] = [[x 1 x 2 [x 3 x 4 x 5 ]]. = 0 0 a } Example 2. Let 0 b 0 : a,b,c Æ 0}. Then is c 0 0 a ternary semigroup under the usual multiplication of matrices over Æ 0} while is not a semigroup under the same operation. If A = a}, then we write [a}bc] as [abc] and similarly if B = b} or C = c}, we write [AbC] and [ABc], respectively. For the sake of simplicity, we write [x 1 x 2 x 3 ] as x 1 x 2 x 3 and [ABC] as ABC. Definition 3. A non-empty subset T of a ternary semigroup is called a ternary subsemigroup of if TTT T. Definition 4. An element a in a ternary semigroup is called regular if there exists an element x in such that axa = a. Torra defined HFs in terms of a function that returns a set of membership values for each element in the domain. We give some background on the basic notions on HFs. For more details, we refer to references. Let be a reference set. Then we define HF on in terms of a function
HEITANT FUZZY ET APPROACH TO IDEAL... 529 H that when applied to X returns a subset of [0,1]. For a HF H on and x, y, we use the notations H x := H(x) and H x = H y H x H y, H x H y. Let H and G be two HFs on. The hesitant union H G and hesitant intersection H G of H and G are defined to be HFs on by: and H G : P([0,1]), x H x G x H G : P([0,1]), x H x G x For any HFs H and G on, we define H G if H x G x for all x. For a non-empty subset A of and ε, δ P([0,1]) with ε δ, define a map [χ (ε,δ) A ] as follows: [χ (ε,δ) A ] : P([0,1]), x ε, if x A, δ, otherwise. Then [χ (ε,δ) A ] is a HF on, which is called the (ε,δ)-characteristic HF. The HF [χ (ε,δ) ] is called the (ε, δ)-identity HF on. The (ε, δ)-characteristic HF with ε = [0,1] and δ = is called the characteristic HF, and is denoted by [χ A ]. The (ε,δ)-identity HF with ε = [0,1] and δ = is called the identity HF, and is denoted by [χ ]. 2. Hesitant Fuzzy Ideals on Ternary emigroups Definition 5. A non-empty hesitant fuzzy subset H of is called a hesitant fuzzy ternary subsemigroup on if H uvw H u H v H w for all u, v, w. Definition 6. A non-empty hesitant fuzzy subset H of is called a hesitant fuzzy left (resp. lateral, right) ideal on if H uvw H w (H uvw H v, H uvw H u ) for all u, v, w.
530 A.F. Talee, M.Y. Abbasi,.A. Khan Y = Example 7. Let = 0 1 0,Z = O = 1 0 0,W =,X = 0 0 1 0 1 0 1 0 0 0 0 1 },. Then isaternary semigroup under the usual multiplication of matrices. Consider L = O,Z}. Then L is a left ideal of ternary semigroup. Now we define a HF L on by: L : P[0,1], x (0.7,0.9], if x = O or Z,, if x = X or Y or W Hence we can easily show that L is a hesitant fuzzy left ideal on. We say that H is a hesitant fuzzy two sided ideal if H is both a hesitant fuzzy left ideal and a hesitant fuzzy right ideal on. H is called a hesitant fuzzy ideal if H is a hesitant fuzzy left ideal, a hesitant fuzzy lateral ideal and a hesitant fuzzy right ideal on. Lemma 8. Let be a ternary semigroup. If A is a non-empty subset and [χ (ε,δ) A ] is a (ε,δ)-characteristic HF on. Then, (1) A is a ternary subsemigroup on if and only if [χ (ε,δ) A ] is a hesitant fuzzy ternary subsemigroup on. (2) A is a left(resp. lateral, right) ideal on if and only if [χ (ε,δ) A ] is a hesitant fuzzy left(resp. lateral, right)ideal on. Proof. The proof is a straightforward. Definition 9. Let H, I, K HF(), where HF() is the set of all HF on. The product of H, I, K denoted by H ˆ I ˆ K is defined as, for any x, x=uvw (H ˆ I ˆ K) x = H u I v K w } if u, v, w s.t. x = uvw otherwise.
HEITANT FUZZY ET APPROACH TO IDEAL... 531 Lemma 10. Let H HF() such that H u ε for all u and [χ (ε,δ) ] be a (ε,δ)-identity HF on. Then, (1) H is a hesitant fuzzy ternary subsemigroup on if and only if H ˆ H ˆ H H. (2) H is a hesitant fuzzy left ideal on if and only if [χ (ε,δ) ] ˆ H H. (3) H is a hesitant fuzzy lateral ideal on if and only if [χ (ε,δ) H. (4) H is a hesitant fuzzy right ideal on if and only if H ˆ [χ (ε,δ) H. ] ˆ H ˆ [χ (ε,δ) ] ] Proof. (1) Let H be a hesitant fuzzy ternary subsemigroup on and x. Now we have following two cases: Case 1: If h uvw for any u, v, w, then (H ˆ H ˆ H) h = H h. Case 2: If there exists h such that h = uvw for some u, v, w, then (H ˆ H ˆ H) h = = h=uvw h=uvw h=uvw = H h. H u H v H w } H uvw } H h } Hence in all the cases H ˆ H ˆ H H. Conversely, suppose H ˆ H ˆ H H for any H HF() and let u, v, w. As is a ternary semigroup, we have uvw. Let h = uvw. Then, H uvw = H h (H ˆ H ˆ H) h = H r H s H t } h=rst H u H v H w. Therefore H is a hesitant fuzzy ternary subsemigroup on. (2) Let H be a hesitant fuzzy left ideal on and h. Then we have following two cases: Case 1: If h uvw for any u, v, w, then ([χ (ε,δ) ] ˆ H) h = H h.
532 A.F. Talee, M.Y. Abbasi,.A. Khan Case 2: If there exists h such that h = uvw for some u, v, w, then ([χ (ε,δ) ] ˆ H) h = [χ (ε,δ) ] u [χ (ε,δ) ] v H w } h=uvw = ε ε H w } = = h=uvw h=uvw h=uvw h=uvw = H h. H w } as ([χ (ε,δ) H uvw } H h } ] u = [χ (ε,δ) ] v = ε ) Therefore H ˆ H ˆ H H. Conversely, suppose [χ (ε,δ) ] ˆ H H for any H HF() and let u, v, w. As is a ternary semigroup, we have uvw. Let h = uvw. Then, H uvw = H h ([χ (ε,δ) = [χ (ε,δ) h=rst [χ (ε,δ) ] ˆ H) h ] r [χ (ε,δ) ] s H t } ] u [χ (ε,δ) ] v H w = H w (As [χ (ε,δ) Hence H is a hesitant fuzzy left ideal on. (3) and (4) are analogous to (2). ] u = [χ (ε,δ) ] v = ε). Example 11. Consider as given in the Example 7. Let [χ (ε,δ) ] be a (ε,δ)-identity HF on, where ε = [0,1], δ =. Now we define HF R on by: R : P[0,1], x (0.5,0.8], if x = O or X,, if x = Y or Z or W Then it is easy to verify that R ˆ [χ (ε,δ) ] R. Hence R is a hesitant fuzzy right ideal on. Example 12. Consider as given in the Example 7. Let [χ (ε,δ) ] be a (ε,δ)-identity HF on, where ε = [0,1], δ =. Now we define HF M on by:
HEITANT FUZZY ET APPROACH TO IDEAL... 533 M : P[0,1], x For Z, M Z =. Now (0.7,0.8], if x = O, (0.5,0.65], if x = X or Y, if x = Z or W. 0 0 1 0 0 1 0 0 1 Z = = 0 1 0 0 1 0. 1 0 0 1 0 0 1 0 0 It implies Z = WXW. Hence we have ([χ (ε,δ) ] ˆ M ˆ [χ (ε,δ) ]) Z = Then it can be seen that ([χ (ε,δ) Z=UVP [χ (ε,δ) ] U M V [χ (ε,δ) ] P } = [χ (ε,δ) ] W M X [χ (ε,δ) ] W } = ε (0.5,0.65] ε = (0.5,0.65]. ] ˆ M ˆ [χ (ε,δ) ]) Z M Z. Hence [χ (ε,δ) ] ˆ M ˆ [χ (ε,δ) ] not M. Therefore M is not a hesitant fuzzy lateral ideal on. Lemma 13. If H and G are two hesitant fuzzy ternary subsemigroups on, then H G is also a hesitant fuzzy ternary subsemigroup on. Proof. The proof is a straightforward. Lemma 14. If H and G are two hesitant fuzzy left(resp. lateral, right) ideals on and [χ (ε,δ) ] be a (ε,δ)-identity HF on. Then H G is also a hesitant fuzzy left (resp. lateral, right) ideal on. Proof. The proof is a straightforward. Definition 15. Let be a non-empty set and h. Let ε [0,1] such that ε. A hesitant fuzzy point h ε on is a hesitant fuzzy subset of defined by: ( h ε ) x = ε, if x = h,, otherwise, where x.
534 A.F. Talee, M.Y. Abbasi,.A. Khan We will denote the set of all hesitant fuzzy points by HFP(). Proposition 16. Let α, ε, β P([0,1]) and ḡ α, h ε, ī β HFP() such that α ε β, then ḡ α ˆ h ε ˆ ī β = (ghi) α ε β., Proof. The proof is a straightforward. Proposition 17. Let h α HFP() such that α ε. Then for any x (1) ([χ (ε,δ) ] ˆ h α ) x = (2) ([χ (ε,δ) ] ˆ h α ˆ [χ (ε,δ) ]) x = (3) ( h α ˆ [χ (ε,δ) ]) x = α, if x h,, otherwise. α, if x h,, otherwise. α, if x h,, otherwise. Proof. (1) Let x. We have following two cases: Case 1: If x rst for any r, s, t, then ([χ (ε,δ) ] ˆ h α ) x =. Case 2: If x = rst for some r, s, t, then ([χ (ε,δ) ] ˆ h α ) x = x=rst [χ (ε,δ) ] r [χ (ε,δ) ] s ( h α ) t }. ubcase 1: If x uvh for any u, v, then ([χ (ε,δ) ] ˆ h α ) x =. ubcase 2: If there exist x such that x = uvh for some u, v, then x h and hence ([χ (ε,δ) ] ˆ h α ) x = x=rst = [χ (ε,δ) Hence the result is proved. [χ (ε,δ) = ε ε α = α. Proofs of (2) and (3) are analogous to (1). ] r [χ (ε,δ) ] s ( h α ) t }. ] u [χ (ε,δ) ] v ( h α ) h
HEITANT FUZZY ET APPROACH TO IDEAL... 535 Corollary 18. Let h α HFP(), then for any x, ( h α ([χ (ε,δ) ] ˆ h α )) x = Proof. It is easy to show that h α ([χ (ε,δ) α, if x = h or x h,, otherwise. left ideal on containing h α. If x = h, then ( h α ([χ (ε,δ) ( h α ) h ([χ (ε,δ) ] ˆ h α ) h = α = α. If x = uvh for some u, v, then ( h α ([χ (ε,δ) ] ˆ h α ) is a hesitant fuzzy ] ˆ h α )) x = ] ˆ h α )) x = ] ˆ h α ) uvh = α(by the Proposition 17(1)) = α. ( h α ) uvh ([χ (ε,δ) If x h and x uvh for any u, v, then ( h α ([χ (ε,δ) =. Hence proof the result. ] ˆ h α )) x Definition 19. Let H HF() and h α HFP(). Then we say h α H if (H) h α. Proposition 20. For any H HF() and h α, ḡ β HFP(). Then, (1) h α H if and only if h α H. (2) H = h α. h α H (3) h α ḡ β if and only if h = g and α β. (4) h α ˆ H ˆ ḡ β = h α ˆ z γ ˆ ḡ β. z γ H Definition 21. (1) Let be a ternary semigroup and h α be a hesitant fuzzy point on. Then the hesitant intersection of all hesitant fuzzy left ideals on containing h α is a hesitant fuzzy left ideal on containing h α, denoted by < h α > L and defined as < h α > L = H, h α H HFL L () where HFL L () is the set of all hesitant fuzzy left ideals on. < h α > L is said to be hesitant fuzzy left ideal generated by hesitant fuzzy point h α. Analogously, wecandefine< h α > M and< h α > R respectively, thehesitant fuzzy lateral ideal and hesitant fuzzy right ideal generated by the hesitant fuzzy point h α. Proposition 22. Let h α HFP(). For any x,
536 A.F. Talee, M.Y. Abbasi,.A. Khan (< h α > L ) x = Proof. As we know that < h α > L =, (< h α > L ) x = ( H) x = ( h α H HFL L () α, if x < h > L,, otherwise. h α H HFL L () h α H HFL L () If x = h, then (< h α > L ) x = (< h α > L ) h = ( H. Hence for any x )H x. h α H HFL L () )(H) h = α (As h α H, (H) h α). If x = uvh for some u, v, then for any hesitant fuzzy left ideal H on, H uvh H h α and therefore (< h α > L ) x = ( )(H) uvh = α. h α H HFL L () If x h and x uvh for any u, v, then by the Corollary 18, we have (< h α > L ) x =. Hence the result is proved. Proposition 23. Let h α HFP(). For any x, (1) (< h α, if x < h > M, α > M ) x =, otherwise. (2) (< h α, if x < h > R, α > R ) x =, otherwise. Proof. Proofs of (1) and (2) are analogous to Proposition 22. Proposition 24. Let h α HFP(). Then, (1) < h α > L = h α ([χ (ε,δ) ] ˆ h α ). (2) < h α > M = h α ([χ (ε,δ) ˆ [χ (ε,δ) ]). (3) < h α > R = h α ( h α ˆ [χ (ε,δ) ] ˆ h α ˆ [χ (ε,δ) ]). ]) ([χ (ε,δ) ] ˆ h α Proof. It is easy to show that h α ([χ (ε,δ) ] ˆ h α ) is a hesitant fuzzy left ideal on containing h α. Now it remains to show that it is the smallest hesitant fuzzy left ideal on containing h α. For this let G be another hesitant fuzzy left ideal on containing h α such that G h α ([χ (ε,δ) ] ˆ h α ). As h α G, we have [χ (ε,δ) ] ˆ h α [χ (ε,δ) ] ˆ G G (As G is a hesitant fuzzy left ideal on ). It implies h α ([χ (ε,δ) ] ˆ h α ) G. Therefore G = h α ([χ (ε,δ) ] ˆ h α ). This shows that h α
HEITANT FUZZY ET APPROACH TO IDEAL... 537 ([χ (ε,δ) ] ˆ h α ) is a smallest hesitant fuzzy left ideal on containing hesitant fuzzy point h α. Therefore < h α > L = h α ([χ (ε,δ) ] ˆ h α ). Proofs of (2) and (3) are analogous to (1). Definition 25. A ternary semigroup is called regular if for each element a of, there exists an element x in such that axa = a. Theorem 26. Let be a ternary semigroup. Then the following statements are equivalent: (1) is regular; (2) R ˆ M ˆ L = R M L, for every hesitant fuzzy right ideal R, hesitant fuzzy lateral ideal M and hesitant fuzzy left ideal L on and R u, M u, L u ε for all u ; (3) < r α > R ˆ < m α > M ˆ < l α > L = < r α > R < m α > M < l α > L for all r, m, l and α ε such that α ; (4) < h α > R ˆ < h α > M ˆ < h α > L = < h α > R < h α > M < h α > L for all h and α ε such that α. Proof. (1) (2). Let be a regular ternary semigroup and let R, M, L be hesitant fuzzy right ideal, hesitant fuzzy lateral ideal and hesitant fuzzy left ideal on respectively. Now R ˆ M ˆ L R ˆ [χ (ε,δ) ] R, R ˆ M ˆ L [χ (ε,δ) ] ˆ M ˆ [χ (ε,δ) ] M, R ˆ M ˆ L [χ (ε,δ) ] ˆ L L. It implies R ˆ M ˆ L R M L. As is regular, hence for any x, there exist h such that x = xhx and so x = xhx = xhxhx. Now (R ˆ M ˆ L) x = (R) u (M) v (L) w } x=uvw (R) x (M) hxh (L) x (R) x (M) x (L) x = (R M L) x. Therefore R M L R ˆ M ˆ L. Hence R ˆ M ˆ L = R M L. It is easy to verify (2) (3) and (3) (4). (4) (1).
538 A.F. Talee, M.Y. Abbasi,.A. Khan Let h. Then h α < h α > R < h α > M < h α > L. Given that < h α > R ˆ < h α > M ˆ < h α > L = < h α > R < h α > M < h α > L. Now h α < h α > R < h α > M < h α > L = < h α > R ˆ < h α > M ˆ < h α > L h α ˆ [χ (ε,δ) ] ˆ h α = ( h α ˆ z γ ˆ h α ) (By the proposition 20 ). z γ [χ (ε,δ) ] Hence from the above expression there exists z α HFP() such that h α h α ˆ z α ˆ h α = (hzh) α. Thus by Proposition 20, h = hzh for some z. Hence is regular. Definition 27. Let be a ternary semigroup and H HF(). H is said to be idempotent if H = H ˆ H ˆ H. Lemma 28. Let be a regular ternary semigroup. Then every hesitant fuzzy lateral ideal on is idempotent. Proof. The proof is a straightforward. Definition 29. A ternary semigroup is called left (resp. lateral, right) simple if contains no proper left (resp. lateral, right) ideal of. Definition 30. A ternary semigroup is called hesitant fuzzy left (resp. lateral, right) simple if every hesitant fuzzy left (resp. lateral, right) ideal on is constant function. Theorem 31. Let be a ternary semigroup. Then is left (resp. lateral, right) simple if and only if is hesitant fuzzy left (resp. lateral, right) simple ternary semigroup. Proof. uppose is left simple and H is a hesitant fuzzy left ideal on. Let h, k. Then h and k are the left ideal of. As is left simple, it implies h = and k =. o there exists t, u, v, w s.t. k = tuh and h = vwk. It follows that H k = H tuh H h = H vwk H k. Hence H h = H k for all h, k. Therefore H is a constant function and hence is hesitant fuzzy left simple. Conversely, suppose is hesitant fuzzy left simple and A is a left ideal of
HEITANT FUZZY ET APPROACH TO IDEAL... 539. By Lemma 8, [χ (ε,δ) A ] is a hesitant fuzzy left ideal on. Now by assumption, ] is a constant function. As A is a left ideal, it is non-empty. If a A, [χ (ε,δ) A then [χ (ε,δ) A ] a = ε. Therefore for any element h, [χ (ε,δ) A ] h = ε, which implies h A. Hence A. Consequently is simple. References [1] B. Davvaz, A. Zeb and A. Khan, On (α,β)-fuzzy ideals of ternary semigroups, J. Indones. Math. oc., 19, No 2 (2013), 123138. [2] Y.B. Jun, K.J. Lee and.z. ong, Hesitant fuzzy bi-ideals in semigroups, Commun. Korean Math. oc., 30, No 3 (2015), 143154. [3]. Kar and P. arkar, Fuzzy ideals of ternary semigroups, Fuzzy Inf. Eng., 2 (2012), 181-193. [4] E. Kasner, An extension of the group concept, Bull. Amer. Math. oc., 10 (1904), 290-291. [5] D.H. Lehmer, A ternary analogue of abelian groups, AM J. Math., 59 (1932), 329-338. [6] M.Y. Abbasi, A.F. Talee, X.Y. Xie and.a. Khan, Hesitant fuzzy ideal extension in po-semigroups, TWM J. App. Eng. Math., (2017), Accepted. [7] F.M. ioson, Ideal theory in ternary semigroups, Math. Japan., 10 (1965), 63-84. [8] V. Torra and Y. Narukawa, On HFs and decision, in: The 18th IEEE International Conference on Fuzzy ystems, Jeju Island, Korea (2009), 13781382. [9] V. Torra, Hesitant fuzzy sets, Int. J. Intell. yst., 25 (2010), 529539. [10] X.Y. Xie and J. Tang, Fuzzy radicals and prime fuzzy ideals of ordered semigroups, Information ciences, 178 (2008), 43574374. [11] L.A. Zadeh, Fuzzy sets, Inform. Control., 8 (1965), 338-353.
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