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1 RPN Journal of Systems Software JSS Journal. ll rghts reserved Characterzng -Semgroups by Intutonstc Fuzzy Ponts Muhammad kram Department of Informaton Technology, Hgher College of Technology, Muscat, Oman E-mal: STRCT In ths paper, we consder the ntutonstc fuzzy sets n a -semgroup the correspondng sets of ntutonstc fuzzy ponts. We analyze some relatons between the ntutonstc fuzzy -deals the sets of ntutonstc fuzzy ponts of these -deals of a -semgroup S. Keywords: -semgroup, Intutonstc Fuzzy set, Intutonstc Fuzzy Ponts, Intutonstc Fuzzy Interor, Prme Semprme - deals. non-empty subset of a -semgroup S s called a 1. INTRODUCTION In 1965, Zadeh [1] ntroduced the concept of fuzzy sets. In 1986, tanassov [2] ntroduced the noton of ntutonstc fuzzy sets as a generalzaton of fuzzy sets also see [3,4]. Many concepts n fuzzy set theory were also extended to ntutonstc fuzzy set theory, such as ntutonstc fuzzy relatons, ntutonstc L- fuzzy sets, ntutonstc fuzzy mplcatons, ntutonstc fuzzy logcs, ntutonstc fuzzy rough sets etc. In 1981, Sen [5] ntroduced the noton of -semgroup as a generalzaton of semgroup ternary semgroup. In 1986, Sen Saha [6,7] modfed the defnton of Sen's -semgroup, whch s known as one sded - semgroup. -semgroups have been analyzed by a lot of mathematcans. The noton of fuzzy ponts was ntroduced by Pu Lu [8] n Wang et al. [9] Km [10] characterzed fuzzy deals n semgroups by fuzzy ponts. Jun Song [11] ntroduced the noton of ntutonstc fuzzy ponts. In [12] Sardar et al., defned some relatons between the ntutonstc fuzzy deals of a semgroup S the set of all ntutonstc fuzzy ponts of S. In [13] kram characterzed ntutonstc fuzzy deals n ternary semgroups by ntutonstc fuzzy ponts. lso see [14-29]. In ths paper, we analyze some relatons between the ntutonstc fuzzy deals the correspondng subsets of ntutonstc fuzzy ponts of a -semgroup. 2. PRELIMINRIES Let S { x, y, z,...} {,,,...} be two non-empty sets. Then S s called a -semgroup f t satsfes x y S xy z x yz, for all x, y, z S,. -subsemgroup of S f. left rgh - deal of a -semgroup S s a non-empty subset of S such that S S a two sded - deal or smply a -deal s that whch s both a left a rght -deal of S. -subsemgroup of a - semgroup S s called a b- -deal of S f S. -subsemgroup of a -semgroup S s called an nteror -deal of S f S S. n deal I of a -semgroup S s called a prme - deal f for any deals of S, I mples that I or I s called semprme -deal f I mples that I. n element x of a -semgroup S s called regular f there exst an element s S, such that x xsx S s called a regular -semgroup f every element of S s regular. fuzzy set n a non-empty set X s a functon, : X 0,1 the complment of, denoted by s the fuzzy set n X gven by 1 for all x X. n ntutonstc fuzzy set brefly, IFS n a non-empty set X s an object havng the form, x, x X, where the functons, : X 0,1 : X 0,1 denotes the degree of membershp the degree of nonmembershp respectvely, for all x X, s 0. n ntutonstc fuzzy set x, x X n X can be dentfed by X X an ordered par, n I I. For smplcty, we shall use IFS for ntutonstc fuzzy set, for IFS x, x X. Defnton 2.1: [ 11] Let a, b[0,1] wth a b 1 an ntutonstc fuzzy pont wrtten as x s defned to be an ntutonstc fuzzy subset of S, gven by 359

2 RPN Journal of Systems Software JSS Journal. ll rghts reserved x a, b y 0,1 f x y otherwse. [mn{ t x y 0 x, y }] f t xy otherwse, 3. INTUITIONISTIC FUZZY POINTS ND IDELS IN -SEMIGROUPS In what follows, let S denotes a -semgroup unless otherwse specfed. Defnton 3.1: [ 27] n IFS, n a - semgroup S s called an ntutonstc fuzzy - subsemgroup of S f xy mn{ y} xy max{ y}, for all x, y S,. Defnton 3.2: [ 27] n IFS, n a - semgroup S s called an ntutonstc fuzzy left rgh -deal of S f xy y xy xy y for all x, ys,. xy, Defnton 3.3: [ 27] n IFS, n a - semgroup S s called an ntutonstc fuzzy -deal of S f t s both an ntutonstc fuzzy left, an ntutonstc fuzzy rght -deal of S. It s clear that any ntutonstc fuzzy left rgh -deal of S s an ntutonstc fuzzy subsemgroup of S but the converse s not true. Defnton 3.4: [ 27] n ntutonstc fuzzy - subsemgroup, of S s called an ntutonstc fuzzy nteror -deal of S f xzy z xzy z, for all x, y, z S,. Defnton 3.5: [ 27] n ntutonstc fuzzy - subsemgroup, of S s called an ntutonstc fuzzy b- -deal of S f xzy mn{ y} xzy max{ y}, for all, x, y, z S,. Let S be the set of all ntutonstc fuzzy subsets of a -semgroup S. For any two ntutonstc fuzzy sets,, of S, we defne,, where, [max{ x, y }] f t xy t x y 1 otherwse, for all x, y S,. It s easy to verfy that S s a -semgroup. For any,, C S f C C C. If, I be a collecton of ntutonstc fuzzy subsets of S ther ntersecton, s an ntutonstc fuzzy subset I I I of S where, nf I, t S I sup I, t S. I For -semgroup S, we denotes the set of all of ts ntutonstc fuzzy ponts by S. Then clearly S s a - subsemgroup of S, as for any s, t, u d e, f S,,, we have s t d s a bd S, s t d u e, f s, t, u,. If, s an ntutonstc fuzzy subset of S denotes the set of all ntutonstc fuzzy ponts contaned n that s { s S : a b}. If s S, a 0 b 1. Proposton 3.6: Let S be a -semgroup, for ntutonstc fuzzy subsets,, of S, followng holds. Proof: Let s { s S : a b} s S : max a mn b 360

3 RPN Journal of Systems Software JSS Journal. ll rghts reserved s S : a or a or b s S : a b or x S : a b s or s s b. Hence. Smlarly one can prove. s t d : s, s, t d s a bd : a b, c d s a bd : a c, b d s a bd : mn{ a max{ b d s a bd : sup[mn{ ] a nf[max{ ] b d v a bd : v a v b d, for v xy S. Hence. Proof: We suppose that, s an ntutonstc fuzzy -subsemgroup of S s, t d, a 0, c 0 b 1, d 1. Snce, for s, t S. We have s mn a c s max b d, so that s t d s ac, bd. Whch mples that. Hence s a -subsemgroup of S. Conversely, let be a -subsemgroup of S s, t S,. If 0 y 1 s 0 mn s 1 max. If 0 1 s, y,. Snce s a - subsemgroup of S, s s s t,,, Corollary 3.7: If, 1,2,3,4,..., n, be the ntutonstc fuzzy subsets of a -semgroup S, Where 1,2,3,4,..., n. The followng lemma s easy to prove. Lemma 3.8: Le be the characterstc functon of a non-empty subset of a -semgroup S. Then s a left rght, two sded -deal of S f only f, s an ntutonstc fuzzy left rght, two sded -deal of S. Lemma 3.9: non-empty ntutonstc fuzzy subset, of a -semgroup S s an ntutonstc fuzzy -subsemgroup of S f only f s a - subsemgroup of S. whch mples that s mn{ s max{. Hence s an ntutonstc fuzzy -subsemgroup of S. Lemma 3.10: non-empty ntutonstc fuzzy subset, of a -semgroup S s an ntutonstc fuzzy b- -deal of S f only f s a b- -deal of S. Proof: We suppose that, be an ntutonstc fuzzy b- -deal of S, t s an ntutonstc fuzzy -subsemgroup of S by Lemma 3.9, s a - subsemgroup of S. Let s a, b, t c, d, u e, f S, a 0, c 0, b 1, d 1. s s an ntutonstc fuzzy b- -deal, we have, 361

4 RPN Journal of Systems Software JSS Journal. ll rghts reserved su mn{ a e c snce e 0 su max{ b f d snce d 1 Hence s u e, f t d su ae b f d Ths mples that S. Hence s a b- - deal of S. Conversely, we suppose that s a b- -deal of S s a -subsemgroup of S by lemma 3.9, s an ntutonstc fuzzy -subsemgroup of S. Let s, t, u S. If 0 1 su 0 mn{ su 1 max{. If s, 0 s, 1 s, t,,. s s a b- -deal of S so we have, su, su, s u t,. Whch mples that su mn{ su max{, for all s, t, u S,. Hence, s an ntutonstc fuzzy b- - deal of S. Lemma 3.11: non-empty ntutonstc fuzzy subset, of a -semgroup S s an ntutonstc fuzzy nteror -deal of S f only f s an nteror -deal of S. Proof: Straghtforward. Lemma 3.12: non-empty ntutonstc fuzzy subset, of a -semgroup S s an ntutonstc fuzzy left rgh -deal of S f only f s a left rgh -deal of S. Proof: Straghtforward. Remark 3.13: It s clear that any -deal of a - semgroup S s an nteror -deal of S any ntutonstc fuzzy -deal of S s an ntutonstc fuzzy nteror -deal of S. Theorem 3.14: non-empty ntutonstc fuzzy subset, of a regular -semgroup S s an ntutonstc fuzzy rght lef -deal of S f only f s an nteror -deal of S. Proof: We suppose that, s an ntutonstc fuzzy rght lef -deal of S by Re mark 3.13, s an nteror -deal of S by Lemma 3.11, s an nteror -deal of S. Conversely, we suppose that s an nteror -deal of S. Let s, t S. Snce S s a regular -semgroup s sws, for some w S,. If 0 1 s 0 s 1. If 0 s1 s s s, t S. Snce s a rght -deal of S we have, s sws sws sw,,, s t SS. Snce s an nteror - deal Whch mples that, s s. Hence s an ntutonstc fuzzy rght -deal of S. Lemma 3.15: -semgroup S s regular f only f the -semgroup S s regular. Proof: Let S be a regular -semgroup s S for s S. Snce S s regular for s S there exst w S,, such that s sws w S s w s sw aa bbb sw b s b Whch mples that S s a regular -semgroup. 362

5 RPN Journal of Systems Software JSS Journal. ll rghts reserved Conversely, we suppose that S s a regular - semgroup. Let s S for any a, b[0,1] there exst an element w d S, such that s s w s b d sw ac b sws. a bd d b Hence s sws for w S,, whch mples that S s a regular -semgroup. Theorem 3.16: 25] semgroup S are equvalent. [ The followng condtons on a - S s regular. R L RL, for every rght -deal R every left -deal L of S. Theorem 3.17: [ 26] For a -semgroup S the followng condtons are equvalent. S s regular,, for every ntutonstc fuzzy left -deal, ntutonstc fuzzy rght - deal, of S. Theorem 3.18: For a -semgroup S the followng condtons are equvalent. S s regular,, for every ntutonstc fuzzy left -deal, ntutonstc fuzzy rght - deal, of S. Proof: : Let S be a regular -semgroup, be the ntutonstc fuzzy left ntutonstc fuzzy rght -deals of S. Then by Lemma 3.15, S s regular by Lemma 3.12, be the left rght -deals of S. Hence by Theorem 3.16,. : We suppose that, for every ntutonstc fuzzy left -deal ntutonstc fuzzy rght -deal of S. Let s S. If 0 or 0 1 or 1 0. e. 1. e. Ths mples that. Now f s, 0 s, 1 ut Imples,,, so we have, s, Then Whch mples that. lso obvously. Hence by Theorem 3.17, S s regular. Defnton 3.19: n ntutonstc fuzzy -subsemgroup, of S s called an ntutonstc fuzzy prme -deal of S f xy max{ y} xy mn{ y}, for all x, ys,. Defnton 3.20: n ntutonstc fuzzy -subsemgroup, of S s called an ntutonstc fuzzy semprme -deal of S f x x for all x S,. Theorem 3.21: non-empty ntutonstc fuzzy subset, of a -semgroup S s an ntutonstc fuzzy semprme -deal of S f only f s a semprme -deal of S. Proof: We suppose that, s an ntutonstc fuzzy semprme -deal of S. Then, for all s S, we have s s Now, let s s s. Then s a s b. Whch gves s a s b so a b. Whch mples that s a, b. Hence s a semprme -deal of S. 363

6 RPN Journal of Systems Software JSS Journal. ll rghts reserved Conversely, we suppose that be a semprme - deal of S. Let s a s b s a, b s a, b s a, b, for mples that s because s a semprme -deal of S. Then a b, whch mples that s s. Hence s an ntutonstc fuzzy semprme -deal of S. -deal of S. Then s max{ s mn{, for all s, t S. Hence, we have s max{ s mn{, for all s, t S. Whch mples that, s an ntutonstc fuzzy prme -deal of S. REFERENCES Theorem 3.22: non-empty ntutonstc fuzzy subset, of a -semgroup S s an ntutonstc fuzzy prme -deal of S f only f s a prme -deal of S. Proof: We suppose that, s an ntutonstc fuzzy prme -deal of S, for all s, t S, we have s max{ sy mn{. Let s a, b t a, b but s t s a b s mples that s. Then s a s b, whch mples that max{ a mn{ b. Then a or a b or b mples that a or a mples that s or t. Hence s a prme - deal of S Conversely, we suppose that s a prme -deal of S. Let s a s b. Then s t s mples that s or t, as s a prme -deal of S. Whch mples that a or a.e. a or a b or b.e. max{ a mn{ b, mples that max{ a s mn{ b s. Snce s a prme -deal of S so s a -deal of S by Lemma 3.12,, s an ntutonstc fuzzy [1] Zadeh, L Fuzzy sets, Informaton Control, 8, [2] tanassov, K.T Intutonstc fuzzy sets, Fuzzy sets Systems, 20, [3] tanassov, K.T New operatons defned over the ntutonstc fuzzy sets, Fuzzy sets Systems, 61, [4] tanassov, K.T Intutonstc fuzzy sets, Theory pplcatons Studes n Fuzzness Soft Computng, 35, Physca-Verlag, Hedelberg. [5] Sen, M.K On -semgroups, Proceedngs of the Internatonal Conference on lgebra ts pplcatons. Decker Publcaton, New York, 301. [6] Sen, M.K., Sah N.K On - semgroups I, ulletn of Calcutta Mathematcal Socety, 78, [7] Sah M.K On -semgroups II, ulletn of Calcutta Mathematcal Socety, 79, [8] Pu, P.M., Lu, Y.M., Fuzzy topology I, neghborhood structure of a fuzzy pont Moore-Smth convergence, Journal of Mathematcs pplcatons,762, [9] Wang, X.P., Mo, Z.W., Lu, W.J Fuzzy deals generated by fuzzy ponts n semgroups, Schuan Shfan Daxue Xuebao Zran Kexue an,154, [10] Km, K.H On fuzzy ponts n semgroups, Internatonal Journal of Mathematcs Mathematcal Scences, 2611, [11] Jun, Y.., Song, S.Z Intutonstc fuzzy sempreopen sets ntutonstc fuzzy semprecontnuous mappngs, Journal of ppled Mathematcs Computng, 191-2, [12] Sardar, S.K., Mal, M., Majumder, S.K Intutonstc fuzzy ponts n semgroups, World cademy of Scences, Engneerng Technology, 75, [13] kram, M Intutonstc fuzzy ponts deals of ternary semgroups, Internatonal Journal of lgebra Statstcs, 11,

RPN Journal of Systems Software 2009-2012 JSS Journal. ll rghts reserved [14] slam, M., roob, T., Yaqoob, N. 2013. On cubc Γ-hyperdeals n left almost Γ- semhypergroups.

7 RPN Journal of Systems Software JSS Journal. ll rghts reserved [14] slam, M., roob, T., Yaqoob, N On cubc Γ-hyperdeals n left almost Γ- semhypergroups. nnals of Fuzzy Mathematcs Informatcs, 51, [15] slam, M., bdullah, S., Davvaz,., Yaqoob, N Rough M-hypersystems fuzzy M-hypersystems n Γ-semhypergroups. Neural Computng pplcatons, 211, [16] Fasal., Yaqoob, N., Chnram, R note on, qk-fuzzy Γ-deals of Γ- semgroups. ppled Mathematcal Scences, 6101, [17] Fasal., Yaqoob, N., Hl K On fuzzy 2,2 regular ordered Γ-G**-groupods. UP Scentfc ulletn, Seres, 742, [18] Fasal., Ullah, K., Haseeb,., Yaqoob, N Some fuzzfcaton n ordered G**- groupods, RPN Journal of Systems Software, 22, [19] Yaqoob, N., Chnram, R., Ghareeb,., slam, M Left almost semgroups characterzed by ther nterval valued fuzzy deals. frka Matematk [20] Yaqoob, N., bdullah, S., Rehman, N., Naeem, M Roughness fuzzness n ordered ternary semgroups. World ppled Scences Journal, 1712, [21] Yaqoob, N., slam, M., nsar, M Structures of N-Γ-hyperdeals n left almost Γ-semhypergroups. World ppled Scences Journal, 1712, [22] Yaqoob, N., slam, M., Hl K Rough fuzzy hyperdeals n ternary semhypergroups. dvances n Fuzzy Systems, 2012, 9 pages. [23] Yaqoob, N., slam, N., Fasal On soft Γ-hyperdeals over left almost Γ- semhypergroups, Journal of dvanced Research n Dynamcal Control Systems, 41, [24] Davvaz,., Majumder, S.K tanassov's Intutonstc fuzzy nteror deals of -semgroups, U.P.. Sc. ull., Seres, 733, [25] Hl K On regular semprme quas-reflexve -semgroup mnmal quas-deals, Lobachevsk Journal of Mathematcs, 293, [26] Sardar, S.K., Mal, M., Majumder, S.K tanassov's ntutonstc fuzzy deals of -semgroups, Internatonal Journal of lgebr 57, [27] Uckun, M., Ozturk, M.., Jun, Y Intutonstc fuzzy sets n -semgroups, ulletn of the Korean Mathematcal Socety, 442, [28] kram, M. Sanjay K. T. Intutonstc Fuzzy Prme -Ideals of Ternary Semgroups, Inter. Jour. of Fuzzy Mathematcs Systems, Vol. 22,2012, [29] kram, M. Intutonstc Fuzzy Prme - - Ideals of -semgroups, In press for publcaton n "nnals of Fuzzy Mathematcs Informatcs". UTHOR PROFILE Muhammad kram dd hs M.Sc wth dstncton from Unversty of The Punjab, Lahore M.Phl from Quad-zam Unversty, Islamabad, Pakstan. He s a permanent faculty member at the Unversty of Gujrat, Pakstan. Presently he s workng n the Informaton Technology Department at Hgher College of Technology, Muscat, Oman. Hs research nterests are algebrac structures, fuzzy ntutonstc fuzzy algebrac structures etc. He publshed several research papers n hs nterest area of research. 365

PROPERTIES OF BIPOLAR FUZZY GRAPHS

PROPERTIES OF BIPOLAR FUZZY GRAPHS Internatonal ournal of Mechancal Engneerng and Technology (IMET) Volume 9, Issue 1, December 018, pp. 57 56, rtcle ID: IMET_09_1_056 valable onlne at http://www.a aeme.com/jmet/ssues.asp?typeimet&vtype