RPN Journal of Systems Software 2009-2012 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: makram_69@yahoo.com 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 1980. 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
RPN Journal of Systems Software 2009-2012 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
RPN Journal of Systems Software 2009-2012 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
RPN Journal of Systems Software 2009-2012 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
RPN Journal of Systems Software 2009-2012 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
RPN Journal of Systems Software 2009-2012 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.. 1965. Fuzzy sets, Informaton Control, 8, 338-353. [2] tanassov, K.T. 1986. Intutonstc fuzzy sets, Fuzzy sets Systems, 20, 87-96. [3] tanassov, K.T. 1994. New operatons defned over the ntutonstc fuzzy sets, Fuzzy sets Systems, 61,137-172. [4] tanassov, K.T. 1999. Intutonstc fuzzy sets, Theory pplcatons Studes n Fuzzness Soft Computng, 35, Physca-Verlag, Hedelberg. [5] Sen, M.K. 1981. On -semgroups, Proceedngs of the Internatonal Conference on lgebra ts pplcatons. Decker Publcaton, New York, 301. [6] Sen, M.K., Sah N.K. 1986. On - semgroups I, ulletn of Calcutta Mathematcal Socety, 78, 180-186. [7] Sah M.K. 1987. On -semgroups II, ulletn of Calcutta Mathematcal Socety, 79, 331-335. [8] Pu, P.M., Lu, Y.M., 1980. Fuzzy topology I, neghborhood structure of a fuzzy pont Moore-Smth convergence, Journal of Mathematcs pplcatons,762, 571-599. [9] Wang, X.P., Mo, Z.W., Lu, W.J. 1992. Fuzzy deals generated by fuzzy ponts n semgroups, Schuan Shfan Daxue Xuebao Zran Kexue an,154, 17-24. [10] Km, K.H. 2001. On fuzzy ponts n semgroups, Internatonal Journal of Mathematcs Mathematcal Scences, 2611, 707-712. [11] Jun, Y.., Song, S.Z. 2005. Intutonstc fuzzy sempreopen sets ntutonstc fuzzy semprecontnuous mappngs, Journal of ppled Mathematcs Computng, 191-2, 467-474. [12] Sardar, S.K., Mal, M., Majumder, S.K. 2011. Intutonstc fuzzy ponts n semgroups, World cademy of Scences, Engneerng Technology, 75, 1107-1112. [13] kram, M. 2012. Intutonstc fuzzy ponts deals of ternary semgroups, Internatonal Journal of lgebra Statstcs, 11, 74-82. 364
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