Cubic Γ-n normed linear spaces

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1 Malaya Journal of Maemaik, Vol. 6, No. 3, , 18 hps://doi.org/1.6637/mjm63/8 Cubic Γ-n normed linear spaces P. R. Kavyasree1 * and B. Surender Reddy Absrac This paper is aimed o propose he noion of cubic Γ-n-normed linear spaces based on he heory of cubic n-normed linear space, fuzzy n-normed linear space, inerval valued fuzzy n-normed linear space and cubic ses. The concep of convergence and Cauchy sequences in cubic Γ-n-normed linear space are inroduced and we provide some resuls on i. Also, his paper inroduces he noion of compleeness in cubic Γ-n-normed linear Keywords Cubic Γ-n-normed linear space, cubic n-normed linear space, inerval valued fuzzy n-normed linear space, cubic ses. AMS Subjec Classificaion 46-XX, 46S4, 3E7. 1, Deparmen of Mahemaics, Osmania Universiy, Hyderabad-57, India. *Corresponding auhor: 1 kavyasree.anu@gmail.com; bsrmahou@osmania.ac.in Aricle Hisory: Received 8 March 18; Acceped 1 Sepember 18 Conens 1 Inroducion Preliminaries Main Resuls References Inroducion A significan heory on -normed space was iniially inroduced by Gahler []. Consequenly, Misak [9], Kim and Cho, Malceski [8], Hendra Gunwan and Mashadi [3] ook an effor in developing his heory o a grea exen. Zadeh [17] in 1965, firs inroduced he noion of fuzzy ses. This inroducion laid foundaion for he developmen of various srucures in mahemaics. This heory has a wide range of applicaions in several branches of mahemaics such as logic se heory, group heory, real analysis, measure heory, opology ec. Fuzzy groups, fuzzy rings, fuzzy semigroup, fuzzy opology, fuzzy norm and so on are few ineresing opics emerged afer he developmen of fuzzy ses. Fuzzy conceps also play a vial role in image processing, Paern recogniion, medical diagnosis, neural nework heory on so on. Laer on, he noion of inerval-valued fuzzy ses was inroduced by Zadeh [18] in 1975, as an exension of fuzzy ses, ha is, fuzzy ses wih inerval valued membership funcions. Kasaras and Liu [7] inroduced he conceps of fuzzy vecor and fuzzy opological vecor spaces. In sudying fuzzy opological vecor spaces, c 18 MJM. Kasaras in 1984 [6], firs inroduced he noion of fuzzy norm on a linear In [1] Vijayabalaji inroduced he noion of fuzzy n-normed linear space as a generalisaion of n-normed space by Gunwan and Mashadi. The concep of inuionisic n-normed linear space, inerval valued fuzzy linear space and inerval valued fuzzy n-normed linear space are discussed in [14], [13]. Jun e al.[4] have inroduced a noiceable heory of cubic ses which comprises of inerval-valued fuzzy se and a fuzzy se. A deailed heory of cubic linear space can be found in [16], [15]. The concep of Γ-ring was inroduced by Nobusawa [11] more general han a ring. Barnes [1] gave he definiion of Γ-ring as a generalisaion of a ring and he has developed some oher conceps of Γ-rings such as Γ-homomorphism, prime and primary ideals, m-sysems ec. The noion of Γ-vecor spaces was inroduced by Sabur Uddin and Payer Ahamed [1]. Inspired by he above heories Vijayabalaji [5] consruced -normed and n-normed lef Γ-linear space as a generalisaion of n-normed linear He also inroduced he noion of n-funcional in n-normed lef Γ-linear Inspiried by he above heory we inroduce he noion of cubic Γ-n-normed linear space and also define convergen and cauchy sequences in cubic Γ-n-normed linear. Preliminaries Definiion.1. Le M and Γ be wo addiive abelian groups. Suppose ha here is a mapping from M Γ M M (send-

2 Cubic Γ-n normed linear spaces 644/647 Example.7. Le V = R, be a lef Γ-linear space over a division Γ ring = R. Le Γ = Z be an addiive abelian group and define k.,...,.k on V by ing (x, α, y) ino xαy) such ha (1) (x + y)αz = xαz + yαz () x(α + β )z = xαz + xβ z (3) xα(y + z) = xαy + xαz (4) (xαy)β z = xα(yβ z) where x, y, z M and α, β Γ. Then M is called a Γ-ring. kδ1 γv1, δ γv,..., δn γvn k =... de(δi γvki ) k1 Definiion.. A subse A of he Γ-ring M is a lef (righ) ideal of M if A is an addiive abelian subgroup of M and MΓA = {cαa c M, α Γ, a A} (AΓM = {aαc a A, α Γ, c M}) is conained in A. If A is boh a lef and a righ ideal of M, hen we say ha A is an ideal or wo sided ideal of M. kn Then (V, k.,...,.k) is called an n-normed lef Γ-linear space over. Definiion.8. A sequence {δn γvn } in an n-normed lef Γlinear space (V, k.,...,.k) is said o converge o δ γv V if lim kδ1 γv1, δ γv,..., δn 1 γvn 1, δn γvn δ γvk =. n Definiion.9. A sequence {δn γvn } in an n-normed lef Γlinear space (V, k.,...,.k) is called a cauchy sequence if Definiion.3. Le M be a Γ-ring. Then M is called a division Γ-ring if i has an ideniy elemen and is only non zero ideal is iself. lim kδ1 γv1, δ γv,..., δn 1 γvn 1, δn γvn δk γvk k =. Definiion.4. Le (V, +) be an abelian group. Le be a division Γ-ring wih ideniy 1 and le ϕ : Γ V V, where we denoe ϕ(δ, γ, v) by (δ γv). Then V is called a lef Γ-vecor space over, if for all δ1, δ, v1, v V and β, γ Γ, he following hold (1) δ1 γ(v1 + v ) = δ1 γv1 + δ γv () (δ1 + δ )γv1 = δ1 γv1 + δ γv (3) (δ1 β δ )γv1 = δ1 β (δ γv1 ) (4) 1γv1 = v1 for some γ Γ We call he elemens v of V are vecors and he elemens δ of are scalars. We also call δ γv he scalar muliple of v by δ. Similarly, we can also define righ Γ-vecor space over. Definiion.5. Le V be a lef Γ- linear space over. A real valued funcion k.,.k : V V [, ) saisfying he following properies. (1) kδ1 γv1, δ γv k = if and only if v1 and v are linearly Γ dependen over () kδ1 γv1, δ γv = kδ γv, δ1 γv1 (3) kδ1 γv1, αδ γv k = α kδ1 γv1, δ γv k for any α Γ (4) kδ1 γv1, δ γv +δ3 γv3 k kδ1 γv1, δ γv k+kδ1 γv1, δ3 γv3 k for all δ1, δ, δ3, v1, v, v3 V, γ Γ. is called -norm on lef Γ-linear space V and he pair ( V,.,. ) is called an -normed lef Γ-linear space over. Definiion.6. Le V be a lef Γ- linear space over. A real valued funcion on V n saisfying he following four properies: (1) kδ1 γv1, δ γv,..., δn γvn k = if any only if v1,v,..., vn are linearly Γ-dependen over () kδ1 γv1, δ γv,..., δn γvn k is invarian under any permuaion of v1, v,...,vn (3) kδ1 γv1, δ γv,..., αδn γvn k = α kδ1 γv1, δ γv,..., δn γvn k, for any α Γ (4) kδ1 γv1, δ γv,..., δn 1 γvn 1, δ γy + δ γzk kδ1 γv1,..., δn 1 γvn 1, δ γyk + kδ1 γv1,..., δn 1 γvn 1, δ γzk for all δ1, δ,..., δn, δn, δ, δ, v1, v,..., vn, y, z V, γ Γ is called an n-norm on lef Γ-linear space V and he pair (V, k.,...,.k) is called an n-normed lef Γ-linear space over. Definiion.1. An n-normed lef Γ-linear space is said o be complee if every cauchy sequence in i is convergen. Definiion.11. An inerval number on [, 1], say a, is a closed sub inerval of [, 1], ha is a = [a, a+ ], where a a+ 1. Le D[, 1] denoe he family of all closed subinervals of [, 1], ha is, D[, 1] = {a = [a, a+ ] : a a+ and a, a+ [, 1]}. Definiion.1. Le X be a se. A mapping A : X D[, 1] is called an inerval valued fuzzy se (briefly, an i-v fuzzy se) of X, where A(x) = [A (x), A+ (x)], for all x X, and A and A+ are fuzzy ses in X such ha A (x) A+ (x) for all x X. Definiion.13. Le X be a nonempy se. A cubic se A in a se X is a srucure A ={hx, µ A (x), λ (x)i : x X} which is briefly denoed by A =hµ A, λ i where µ A = [µa, µa+ ] is an inerval valued fuzzy se (briefly, IVF) in X and λ : X [, 1] is a fuzzy se in X. Definiion.14. Le V be a linear space over a field F, (V, µ) be an inerval-valued fuzzy linear space and (V, η) be a fuzzy linear space of V. A cubic se A = hµ, ηi in V is called a cubic linear space of V if i saisfies for all x, y V and α, β F: (a) µ(αx β y) min{µ(x), µ(y)}, (b) η(αx β y) max{η(x), η(y)}. Definiion.15. A binary operaion : [, 1] [, 1] [, 1] is a coninuous -norm if saisfies he following condiions: (1) is commuaive and associaive. () is coninuous. (3) a 1 = a for all a [, 1]. (4) a b c d whenever a c and b d and a, b, c, d [, 1]. Definiion.16. A binary operaion : [, 1] [, 1] [, 1] is a coninuous -co-norm if saisfies he following condiions: (1) is commuaive and associaive. () is coninuous. (3) a = a, for all a [, 1]. (4) a b c d whenever a c and b d and a, b, c, d [, 1]. 644

3 Cubic Γ-n normed linear spaces 645/647 r < 1, here exiss an ineger n N such ha N(δ1 γv1, δ γv,..., δn γvn δ γv,) < r and N(δ1 γv1, δ γv,..., δn γvn δ γv,) > 1 r for all n n 3. Main Resuls : V n [, ) Le V be a lef Γ- linear space over. Le N [, 1] and N : X n [, ) [, 1] be a fuzzy se and an inervalvalued fuzzy se respecively. A srucure C = (V, N, N) is a cubic Γ-n-normed linear space (or) briefly cubic Γ-n- NLS if i saisfies he following properies: (1) N(δ1 γv1, δ γv,..., δn γvn,) >. () N(δ1 γv1, δ γv,..., δn γvn,) = if and only if v1, v,..., vn are linearly dependen. (3) N(δ1 γv1, δ γv,..., δn γvn,) is invarian under any permuaion of v1, v,..., vn. (4) N(δ1 γv1, δ γv,..., cδn γvn,) = N(δ1 γv1, δ γv,..., δn γvn, c ), if c 6=, c Γ. Theorem 3.. In a cubic Γ n-nls C = (V, N, N) a sequence {δn γvn } converges o δ γv if and only if N(δ1 γv1, δ γv,..., δn γvn δ γv,) and N(δ1 γv1, δ γv,..., δn γvn δ γv,) 1, as n Proof. Fix >. Suppose {δn γvn } converges o δ γv. Then for a given r, < r < 1, here exiss an ineger n N such ha N(δ1 γv1, δ γv,..., δn γvn δ γv,) < r and N(δ1 γv1, δ γv,..., δn γvn δ γv,) > 1 r. Thus N(δ1 γv1, δ γv,..., δn γvn δ γv,) < r and 1 N(δ1 γv1, (5) N(δ1 γv1, δ γv,..., δn γvn +δn γvn, s+) N(δ1 γv1, δ γv, δ γv,..., δn γvn δ γv,) < r and hence N(δ1 γv1, δ γv,...,..., δn γvn, s) N(δ1 γv1, δ γv,..., δn γvn,). δn γvn δ γv,) and (6) N(δ1 γv1, δ γv,..., δn γvn,) is lef coninuous and nonn(δ1 γv1, δ γv,..., δn γvn δ γv,) 1, as n. increasing funcion of R such ha conversely, if for each >, N(δ1 γv1, δ γv,..., δn γvn δ γv,) and N(δ1 γv1, δ γv,..., δn γvn δ γv,) 1, as lim N(δ1 γv1, δ γv,..., δn γvn,) =. n, hen for every r, < r < 1, here exiss an ineger n such ha N(δ1 γv1, δ γv,..., δn γvn δ γv,) < r and (7) N(δ1 γv1, δ γv,..., δn γvn,) >. 1 N(δ1 γv1, δ γv,..., δn γvn δ γv,) < r for all n n. (8) N(δ1 γv1, δ γv,..., δn γvn,) = 1 if and only if v1, v,..., vn Thus N(δ1 γv1, δ γv,..., δn γvn δ γv,) < r and N(δ1 γv1, are linearly dependen. δ γv,..., δn γvn δ γv,) > 1 r for all n n. Hence {δn γvn } (9) N(δ1 γv1, δ γv,..., δn γvn,) is invarian under any perconverges o δ γv in C = (V, N, N). muaion of v1, v,..., vn. (1) N(δ1 γv1, δ γv,..., cδn γvn,) = N(δ1 γv1, δ γv,..., Definiion 3.3. A sequence {δn γvn } in a cubic Γ-n-NLS C = δn γvn, c ), if c 6=, c Γ. (V, N, N) is said o be cauchy sequence if given ε >, wih (11) N(δ1 γv1, δ γv,..., δn γvn + δn γvn, s + ) N(δ1 γv1, < ε < 1, > here exiss an ineger n N such ha δ γv,..., δn γvn, s) N(δ1 γv1, δ γv,..., δn γvn,). N(δ1 γv1, δ γv,..., δn γvn δk γvk,) < ε and N(δ1 γv1, (1) N(δ1 γv1, δ γv,..., δn γvn,) is lef coninuous and non- δ γv,..., δn γvn δk γvk,) > 1 ε for all n, k n. decreasing funcion of R such ha Theorem 3.4. In a cubic Γ-n-NLS C = (V, N, N) every conlim N(δ1 γv1, δ γv,..., δn γvn,) = 1. vergen sequence is a cauchy sequence. Example Le (V, k.,...,.k) be an n-normed lef Γ-linear space over. Define a b = min{a, b} and a b = max{a, b} for a, b [, 1]. Also define N(δ1 γv1, δ γv,..., δn γvn,) = kδ1 γv1, δ γv,..., δn γvn k + kδ1 γv1, δ γv,..., δn γvn k and N(δ1 γv1, δ γv,..., δn γvn,) =. + kδ1 γv1, δ γv,..., δn γvn k Then C = (V, N, N) is a cubic Γ-n-normed linear Noion of Convergen sequence and Cauchy sequence in a cubic Γ-n-normed linear space Definiion 3.1. A sequence {δn γvn } in C = (V, N, N) a cubic Γ-n-NLS is said o converge o δ γv if given r >, >, < Proof. Le {δn γvn } be a convergen sequence in C = (V, N, N). Suppose {δn γvn } converges o δ γv. Le > and ε (, 1). Choose r (, 1) such ha r r < ε and (1 r) (1 r) > 1 ε. Since {δn γvn } converges o δ γv, we have an ineger n N 3 N(δ1 γv1, δ γv,..., δn γvn δ γv, ) < r and N(δ1 γv1, δ γv,..., δn γvn δ γv, ) > 1 r for all n n. Now, N(δ1 γv1, δ γv,..., δn γvn δk γvk,) = N(δ1 γv1, δ γv,..., δn γvn δ γv + δ γv δk γvk, + ) N(δ1 γv1, δ γv,..., δn γvn δ γv, ) N(δ1 γv1, δ γv,..., δn γvn δ γv, ) < r r f orall n, k n < ε f orall n, k n. Also, N(δ1 γv1, δ γv,..., δn γvn δk γvk,) = N(δ1 γv1, δ γv,..., δn γvn δ γv + δ γv δk γvk, + ) N(δ1 γv1, δ γv,..., δn γvn δ γv, ) N(δ1 γv1, δ γv,..., δn γvn δ γv, ) > (1 r) (1 r) f orall n, k n 645

4 Cubic Γ-n normed linear spaces 646/647 δ γv. We need o prove ha {δn γvn } converges o δ γv. Le > and ε (, 1). Choose r (, 1) suchha r r < ε and (1 r) (1 r) > 1 ε. Given ha {δn γvn } is a cauchy sequence, here exiss an ineger n N 3 N(δ1 γv1, δ γv,..., δn γvn δk γvk, ) < r and N(δ1 γv1, δ γv,..., δn γvn δk γvk, ) > 1 r for all n, k n. Also since {δn γvnk } converges o δ γv, here is a posiive ik > n 3 N(δ1 γv1, δ γv,..., δn γvik δ γv, ) < r and N(δ1 γv1, δ γv,..., δn γvik -δ γv, ) > 1 r Now, N(δ1 γv1, δ γv,..., δn γvn δ γv,) = N(δ1 γv1, δ γv,..., δn γvn δn γvik + δn γvik δ γv, + ) N(δ1 γv1,..., δn γvn δn γvik, ) N(δ1 γv1,..., δn γvik δ γv, ) < r r < ε. Also N(δ1 γv1, δ γv,..., δn γvn δ γv,) = N(δ1 γv1, δ γv,..., δn γvn δn γvik + δn γvik δ γv + ) N(δ1 γv1,..., δn γvn δn γvik, ) N(δ1 γv1,..., δn γvik δ γv, ) > (1 r) (1 r) > 1 ε. Therefore {δn γvn } converges o δ γv in C = (V, N, N) and hence i is complee. > 1 ε f orall n, k n Therefore {δn γvn } is a cauchy sequence in C = (V, N, N). Definiion 3.5. A cubic Γ-n-NLS C = (V, N, N) is said o be complee if every cauchy sequence in i is convergen. Remark 3.6. The following example shows ha here may exis cauchy sequence in cubic Γ-n-NLS C = (V, N, N) which is no convergen. Example 3.7. Consider a cubic Γ-n-NLS C = (V, N, N) as in he previous example Le {δn γvn } be a sequence in C = (V, N, N) hen (a) {δn γvn } is a cauchy sequence in (V, k.,...,.k) if and only if {δn γvn } is a cauchy sequence in C = (V, N, N). (b) {δn γvn } is a convergen sequence in (V, k.,...,.k) if and only if {δn γvn } is convergen in C = (V, N, N). Proof. (a) {δn γvn } is a cauchy sequence in (V, k.,...,.k) lim kδ1 γv1, δ γv,..., δn γvn δk γvk k = lim N(δ1 γv1, δ γv,..., δn γvn δk γvk ) kδ1 γv1,δ γv,...,δn γvn δk γvk k +kδ1 γv1,δ γv,...,δn γvn δk γvk k = and = lim N(δ1 γv1, δ γv,..., δn γvn δk γvk ) = +kδ γv,δ γv,...,δ =1 n γvn δk γvk k 1 1 N(δ1 γv1, δ γv,..., δn γvn δk γvk,) and N(δ1 γv1, δ γv,..., δn 1 γvn 1, δn γvn δk γvk,) 1 as n, k N(δ1 γv1, δ γv,..., δn γvn δk γvk,) < r and N(δ1 γv1, δ γv,..., δn γvn δk γvk,) > 1 r, r (, 1), n, k n {δn γvn } is a cauchy sequence in C (b) {δn γvn } is a convergen sequence in (V, k.,...,.k) lim kδ1 γv1, δ γv,..., δn γvn δ γvk = Acknowledgmen This work is financially suppored by Council of Scienific and Indusrial Research(CSIR). References [1] n lim N(δ1 γv1, δ γv,..., δn γvn δ γv) [] = = and lim N(δ1 γv1, δ γv,..., δn γvn δ γv) [3] kδ1 γv1,δ γv,...,δn γvn δ γvk +kδ1 γv1,δ γv,...,δn γvn δ γvk [4] = +kδ γv,δ γv,...,δn γvn δ γvk = N(δ1 γv1, δ γv,..., δn γvn δ γv,) and N(δ1 γv1, δ γv,..., δn γvn δ γv,) 1 as n N(δ1 γv1, δ γv,..., δn γvn δ γv,) < r and N(δ1 γv1, δ γv,..., δn γvn δ γv,) > 1 r, r (, 1), n n {δn γvn } is a convergen sequence in C Thus if here exiss an n-normed lef Γ-linear space (V, k.,...,.k) which is no complee, hen he cubic Γ-n norm induced by such a crisp n-norm k.,...,.k on an incomplee n-normed lef Γ linear space V is an incomplee cubic Γ-n normed linear [5] [6] [7] [8] [9] Theorem 3.8. A cubic Γ-n-NLS C = (V, N, N) in which every cauchy sequence has a convergen subsequence is complee. [1] Proof. Le {δn γvn } be a cauchy sequence in C = (V, N, N) and {δn γvnk } be a subsequence of {δn γvn } ha converges o 646 W.Barnes, On he Γ-rings of Nobusawa, Pacific Journal of Mahemaics, 18 (1966), S Gahler, Lineare -normiere raume,mah. Nachr., 8(1965), H. Gunawan and Mashadi, On n-normed spaces, In. J. Mah. Mah. Sci., 7(1)(1), YB Jun, CS Kim, and MS Kang, Cubic subalgebras and ideals of bck/bci-algebras, Far Eas Journal of Mahemaical Sciences 44(1), S Kalaiselvan and S Sivaramakrishnan. S. vijayabalaji, n-normed lef Γ-linear space, Inernaional Journal of Applied Engineering Research, 1(7)(15), A. K. Kasaras, Fuzzy opological vecor spaces, Fuzzy ses and sysems, 1 (1984), A. K. Kasaras and Dar B Liu, Fuzzy vecor spaces and fuzzy opological vecor spaces, Journal of Mahemaical Analysis and Applicaions, 58(1977), A. Malcheski, Srong convex n-normed spaces, Ma. Bilen, 1(1997), A. Misiak, n-inner produc spaces, Mah. Nachr, 14(1989), AL Narayanan and S Vijayabalaji, Fuzzy n-normed linear space, Inernaional Journal of Mahemaics and Mahemaical Sciences, 4(5),

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