Continuous Indexed Variable Systems

Transcription

1 Ieraoal Joural o Compuaoal cece ad Mahemacs. IN Volume 3, Number 4 (20), pp Ieraoal Research Publcao House hp:// Couous Idexed Varable ysems. Pouhassa ad F. Mohammad ghjeh Mashhad Islamc zad Uversy, Parad Brach, Tehra, Ira E-mal: apouhassa@gmal.com, mohammad_h@yahoo.com bsrac I hs paper, we roduce he heory o Couous Idexed Varable ysems as a ew srucure o Idexed Varable ysems. lso, we sudy wh he vew o Topology ad lgebra ad we prove some esseal properes abou he caegory o Idexable ses. Keywords: Couous Idexed Varable ysem, Caegory, Covara, Coravara, Fucor, Idexable se, Local Topology. Iroduco dea or roducg Idexed Varable ysems ( IV or shor) ca be oud [3], [6]. ccordg hs, a IV coas a oempy se ad a amly o Ξ, where R s a subse o erval [0,] ad or membershp-cogrue relaos { } each x, y r r R he ollowg codos hold: () here exss r R such ha x = r y, () x = r y, he y = r x, () x y uary. = ad oly x ad y are o wo dere objecs ; vz, {, } x y s Moslemy ad Pouhassa [3] proved ha each IV s a merc space ad vce versa, urhermore, hey proved ha Idexed Idey relao s o a equvalece relao. lso hey roduced he coceps o local opology ad opologcal Eropy or a IV ad provded some properes abou hem [4]. ccordg hs properes, Pouhassa [5] gave a ew srucure or IG ha ca be useul or urher vesgaos he uure. I he ollowg secos, we wll dee Couous Idexed Varable ysems (CIV or shor) wh respec o prevous sudes ad meawhle reed prevous

2 402. Pouhassa ad F. Mohammad ghjeh Mashhad resuls, some properes ad cosequeces hs opc specally algebrac properes - wll be preseed. Idexable es We sar hs seco by he ollowg deo ha expresses he deo o IV wh ucos. Deo II.. Le be a arbrary oempy se. uco : [0,] s called Couous Idexed Varable ysem (CIV or shor) over he ollowg codos hold or each x, y.. There exss r [0,] such ha ( x, y) = r.. ( x, y) = ( y, x).. ( x, y ) = ad oly x = y. I here exss such uco, he s called a Idexable se. Here we prese some examples o such sysems. Example II.2. () Le = (0, ), he he uco 2xy ( x, y) = s a CIV over. 2 2 x + y π π (2) Le = (, ), he he uco ( x, y) = Cos x y s a CIV over. 4 4 (3) ( x, y) = s a CIV over. x y + I he res o hs seco, we assume ha s a arbrary oempy se. Lemma II.3. I ad g are CIVs over he so s. g, where (. g )( x, y ) = ( x, y ). g ( x, y ) or each x, y. lso every e produc o CIVs over s CIV over. Proo. The rs assero easly ollows rom Deo II. ad he secod oe, ca be deduce by several mes usg he rs par. Lemma II.4. Le be a CIV over ad x, y arbrary ad xed elemes o. Cosder he sequece ( ( x, y)) =, where ( x, y) = ( x, y). ( x, y) or each x = y. The lm ( x, y) = δ ( x, y), where δ ( x, y) =. 0 x y

3 Couous Idexed Varable ysems 403 Proo. I s rval by he Deo II.. Lemma II.5. Le :[0,] [0,] be a jecve uco ad g be a CIV over. The og s a CIV over ad oly () =. Proo. I og s a CIV over, he x = y mples ha og ( x, y ) = ad so g( x, y) = (). I he oher had, x = y mples ha g( x, y ) =, hus () =. Vce versa, he rs wo codos o CIV s deo hold mmedaely or og over. lso or each x, y, we have og ( x, y ) = ad oly g( x, y ) =. I ollows ha og sases he hrd codo o CIV s deo. Example II.6. I he secod par o Example II.2, oe has g ( x, y) = cos x y s a π π x + CIV over = (, ). Le :[0,] [0,] such ha ( x ): =. Hece cos x y + s a CIV over by Lemma II.5. 2 Local Topologes o he Idexable es I hs seco, be a arbrary oempy se ad s a CIV over. The s a Idexable se. For each x ad [0,], we deoe N ( x) = { y ; ( x, y) }. By usg hs oao, we cosruc a opology or he ex resul. Theorem III.. Le x ad τ ( x ) cosss o all ses N ( x ), where [0,]. The s a opologcal space wh τ ( x ). Proo. I s rval ha τ ( x ) cosss ad empy se. uppose ha { N ( x)} I be a arbrary subse o τ ( x ). e : =. Oe ca see ha I N ( x) = N ( x) τ ( x). I lso, { N ( x )} = be a e subse o τ ( x ), he N ( x) = N ( ) ( ) s x τ x, = where s = sup. Thereore τ ( x ) s a opology o.

4 404. Pouhassa ad F. Mohammad ghjeh Mashhad The above opology proposed as local opology geeraed by x (or depeds o ( x )). Theorem III.2. Le Y be a arbrary oempy se ad φ : Y s a bjecve uco. The Y s a Idexable se ad φ s a homomorphsm bewee local opologes geeraed by x ad φ ( x ) or each x (.e. φ ad s verse are boh couous). Proo. Dee he uco g : Y Y [0,] by g ( y, y 2) = ( x, x), where φ ( x ) = y ad φ ( x 2) = y 2. The g s a CIV over Y, because g ( y, y ) = ( x, x ) ad 2 2 = ( x 2, x) = g ( y, y ) 2 g( y, y ) = ( x, x ) = 2 2 x = x 2 y = y. 2 For he secod assero, le x 0, φ( x 0) = y 0ad [0,]. The φ( N ( x )) = φ({ x ; ( x, x) }) 0 0 = {( φ x ); ( x, x) } = { y Y ; g( y, y) } = g N ( y ) mples ha φ s couous. mlarly, proo. φ s also couous ad complees he Remark III.3. I he Theorem III.2, τ ( φ ( x )) are homomorphc. g = Y, he local opologes τ ( x ) ad Theorem III.4. Le be a e dmeso vecor space o he eld o real umbers. ssume ha = { a,..., a } be a oempy subse o ad L deoes he subspace o ha s geeraed by. The L s a Idexable se. Proo. ce every geeraor se o a vecor space coas a bass, he whou loss

5 Couous Idexed Varable ysems 405 o geeraly, we may ad do assume ha s a base or L. We dee θ : L L [0,], such ha or each r, r. θ ( ra, ra ) : = { ( ra, ra ); =,..., } = = We show ha s a CIV over L. I s clear ha θ s well deed ad sases he rs ad secod codos rom he deo o CIV. The, or compleg he proo, we eed o show ha or each eleme uvo, L, θ ( uv, ) = ad oly u = v. s uv, L, here exss r, r such ha u = ra ad v ce s a bass or L ad s a CIV over, we have θ ( uv, ) = ( ra, ra ) = =,..., ra = ra =,..., r = r =,..., u = v. = = ra. = The L s a Idexable se, as clamed. The Caegory o Idexable es I hs seco, we sudy Idexable ses rom a lgebrac po o vew. o, we cosruc he caegory o Idexable ses whch s a useul laguage ad provdes a eld or more sudes abou Idexable ses ad CIVs. For more ormao abou caegores, we reer he reader o [], [2]. Deo IV.. caegory s a class C o objecs such as, B, C,... ad a dsjo amly o morphsms or each eleme, B o C whch s deoed by HomC (, B ) (elemes o HomC (, B ) are cosdered as : B ) such ha here s a uco or each eleme, B, D o C such as θ : Hom ( B, ) Hom ( BD, ) Hom ( D, ) BD C C C ha θ ( BD, g ) = go, where he composo go s deed ad sases he ollowg codos:. or each : B D, g : B ad h: E, we have ( og ) oh = o( goh) (ssocavy);

6 406. Pouhassa ad F. Mohammad ghjeh Mashhad. or each eleme o C such as B, here exss Hom ( B, B ) such ha or : B ad g : B, oe has B o = ad go B B C = g (Idey). Deo Iv.2. Le B ad C be wo caegores. B s called a subcaegory o C sases he ollowg codo:. each objec o B s a objec o C ;. or each objec ad Y B, Hom (, Y ) Hom (, Y ) ;. v. he combao o morphsms B ad C are exacly he same; or each objec o B, he dey morphsms B ad C are exacly he same. lso, he subcaegory B o C s called ull subcaegory or all ad Y B, oe has Hom (, Y ) = Hom (, Y ). B C Example ad Noao IV.3. Le be a class o all ses. For each, B, assume ha Hom (, B ) s he se o all ucos : B ad he combao bewee ucos s he same usual combao. The s a caegory. Oe ca see ha he class o all Idexable ses s a ull subcaegory o ha we deoe by. The ollowg Lemma shows ha he se o all morphsms bewee wo Idexable ses s Idexable se, oo. B C Lemma IV.4. Le ad Y. The Hom (, Y ). Proo. Le be a CIV over Y. We dee h : Hom (, Y ) Hom (, Y ) [0,] such ha h( α, β) : = { ( α ( x), β ( x)); x } or all α, β Hom (, Y ). The by he smlar proo o Theorem III.3, he assero s ollowed. Oe o he eresg properes o caegores s exsece o produc hem. We show ha he produc exss or each amly o objecs. For hs mea, we eed he ollowg deo. Deo IV.5. Le C be a caegory ad { ; I} a amly o objecs C. produc or { ; I} s a objec P o C wh a amly o morphsms such as { ϕ : P ; I} such ha or each objec B o C ad amly { ψ : B ; I} o morphsms, here exss a uque morphsm φ : B P such ha ϕoφ = ψ or each I.

7 Couous Idexed Varable ysems 407 Theorem IV.6. Each amly o objecs has a produc. Proo. ssume ha { ; I} s a amly o objecs ad produc o hem ( we recall ha = {( a) I; a, I} ). I s a Caresa I The s a produc caegory by []. ce s a ull subcaegory o, s suce o prove ha s a Idexable se. For hs mea, we dee I h: [0,] I I uch ha h(( a) I,( b) I) : = { h( a, b); I}, where h s a CIV over ad a, b or each I. s h( a, b) = h( b, a) or all a, b, he h(( a) I,( b) I) = h(( b) I,( a) I). lso we have h(( a) I,( b) I) = ad oly h( a, b ) = ad oly a = b or each I. Thus h s a CIV o. Remark IV.7. I produc exss or a amly o objecs a arbrary caegory, he s uque up o somorphsm, see []. Fally, we coclude hs paper by deg maps ha preserve he srucure o Idexable ses. For hs mea, we eed he deo o covara ad coravare ucors. Deo IV.8. ssume ha C ad D are wo arbrary caegores. covara (resp. coravara) ucor s a par o ucos ha boh o hem are deoed by T such ha oe o hem gves o each objec o C, oe objec T ( ) o D ad he oher gves o each morphsm : B o C a morphsm such as T ( ): T ( ) T ( B) (resp. T ( ): T ( B) T ( ) ) o D ha:. or each objec o C, T ( ) = T ( ) ;. or each wo morphsms ad g o C ha her combao ca be deed, oe has T ( go ) = T ( g ) ot ( )(resp. T ( go ) = T ( ) ot ( g )). Theorem IV.9. (a) Le. The h ( Y ): = Hom (, Y ), I h (.) : s deed by ad h ( )( g): = og or all Y, Z, Hom ( Y, Z ) ad g Hom (, Y ) s a covara ucor.

8 408. Pouhassa ad F. Mohammad ghjeh Mashhad (b) Le Y Y. The h (.) : s deed by Y h ( ): = Hom (, Y ) ad or each Y h ( )( g ): = go, Hom (, Z ) ad, Z g Hom ( Z, Y ) s a coravare ucor. Proo. (a) By Lemma IV.4, h (.) s well deed. lso, by he deo o h (.), oe ca see ha h ( Y ) = h ( Y ) or each Y. Hece, we jus show ha h ( go ) = h ( g ) oh ( ), Where g : Y T, : Z Y ad Y, Z, T. We have h ( go ): Hom (, Z ) Hom (, T ) ha h ( go )( h) = ( go ) oh = go( h ( )( h)) = ( h ( g) oh ( ))( h) or each h Hom (, Z ). Thus h ( go ) = h ( g ) oh ( ) ad hs complees he proo. (b) By usg coravara ucor sead o covara ucor, he proo proceed exacly (a), so we leave o he reader. ckowledgme Ths research was par suppored by a gra rom Islamc zad Uversy, Parad brach, Ira.

9 Couous Idexed Varable ysems 409 Reereces [] T. W. Hugerord, lgebra, prger Verlag, 974, ch.0 ad 0. [2]. Maclae, Caegory or he workg mahemacas, New York: prger- Verlag, 97. [3] M. H. Moslemy Koupae ad. Pouhassa, Idexed dey ad uzzy se heory, Proceedg o he 6-h Iraa Coerece o Fuzzy ysems ad -h Islamc World Coerece o Fuzzy ysems (Persa Verso), [4] M. H. Moslemy Koupae ad. Pouhassa, Topologcal Eropy o a IV, Iraa Mahemacal Coerece, [5]. Pouhassa, ome dscree proposos IVs, Proceedg o WET, vol. 37, 2009, pp [6].. a a, Idexed dey ad uzzy se heory, arv:99082.