Fuzzy derivations KU-ideals on KU-algebras BY

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1 Fu ervaons KU-eals on KU-algebras BY Sam M.Mosaa, Ahme Ab-elaem 2 sammosaa@ahoo.com ahmeabelaem88@ahoo.com,2deparmen o mahemacs -Facul o Eucaon -An Shams Unvers Ro, Caro, Egp Absrac. In hs manuscrp, we nrouce a new concep, whch s calle u le rgh ervaons KU- eals n KU-algebras. We sae an prove some heorems abou unamenal properes o. Moreover, we gve he conceps o he mage an he pre-mage o u le rgh ervaons KU-eals uner homomorphsm o KU- algebras an nvesgae some s properes. Furher, we have prove ha ever he mage an he pre-mage o u le rgh ervaons KU-eals uner homomorphsm o KU- algebras are u le rgh ervaons KU-eals. Furhermore, we gve he concep o he Caresan prouc o u le rgh ervaons KU - eals n Caresan prouc o KU algebras. AMS Subjec Classcaon: 3G25, 6F35 Kewors. KU-algebras,u le rgh ervaons o KU-eals, he mage an he per- mage o u le rgh ervaons KU eals, he Caresan prouc o u le rgh ervaons KU eals. Corresponng Auhor : Sam M. Mosaa sammosaa@ahoo.com. Inroucon As s well known, BCK an BCI-algebras are wo classes o algebras o logc. The were nrouce b Ima an Isek [,,2] an have been eensvel nvesgae b man researchers. I s known ha he class o BCK-algebras s a proper sub class o he BCI-algebras.The class o all BCK-algebras s a quasvare. Is ek pose an neresng problem solve b Wro nsk [24] wheher he class o BCK-algebras s a vare. In connecon wh hs problem, Komor [5] nrouce a noon o BCC-algebras, an Duek [7] reene he noon o BCC-algebras b usng a ual orm o he ornar enon n he sense o Komor. Duek an Zhang [8] nrouce a new noon o eals n BCCalgebras an escrbe connecons beween such eals an congruences.

2 C.Prabpaak an U.Leerawa [22 ], [23 ] nrouce a new algebrac srucure whch s calle KU - algebra. The gave he concep o homomorphsms o KUalgebras an nvesgae some relae properes. Several auhors [2,3,5,6,9,4] have sue ervaons n rngs an near rngs. Jun an Xn [3] apple he noon o ervaons n rng an near-rng heor o BCI-algebras, an he also nrouce a new concep calle a regular ervaon n BCI -algebras. The nvesgae some o s properes, ene a -ervaon eal an gave conons or an eal o be -ervaon. Laer, Hama an Al-Shehr [], ene a le ervaon n BCI-algebras an nvesgae a regular le ervaon. Zhan an Lu [27 ] sue -ervaons n BCI-algebras an prove some resuls. G. Muhun el [2,2] nrouce he noon o, -ervaon n a BCIalgebra an nvesgae relae properes. The prove a conon or a, - ervaon o be regular. The also nrouce he conceps o a, - nvaran, -ervaon an α-eal, an hen he nvesgae her relaons. Furhermore, he obane some resuls on regular, - ervaons. Moreover, he sue he noon o -ervaons on BCI-algebras an obane some o s relae properes. Furher, he characere he noon o p-semsmple BCI-algebra X b usng he noon o -ervaon. Laer, Mosaa e al [8,9], nrouce he noons o, r - r, -ervaon o a KU-algebra an some relae properes are eplore. The concep o u ses was nrouce b Zaeh [26]. In 99, X [25] apple he concep o u ses o BCI, BCK, MValgebras.Snce s ncepon, he heor o u ses, eal heor an s ucaon has evelope n man recons an s nng applcaons n a we vare o els. Mosaa e al, n 2[7] nrouce he noon o u KU-eals o KU-algebras an hen he nvesgae several basc properes whch are relae o u KU-eals. In hs paper, we nrouce he noon o u le rgh ervaons KU- eals n KU - algebras. The homomorphc mage premage o u le rgh - ervaons KU- eals n KU - algebras uner homomorprhsm o a KU-algebras are scusse. Man relae resuls have been erve. 2

3 2. Prelmnares In hs secon, we recall some basc enons an resuls ha are neee or our work. Denon 2. [22,23 ] Le X be a se wh a bnar operaon an a consan. X,, s calle KU-algebra he ollowng aoms hol :,, X : KU [ ] KU 2 KU 3 KU 4 mples Dene a bnar relaon b :, we can prove ha X, s a parall orere se. Throughou hs arcle, X wll enoe a KU-algebra unless oherwse menone Corollar 2.2 [7,22] In KU-algebra he ollowng enes are rue or all,, X : I mples ha v v [ ] Denon 2.3 [22,23] A subse S o KU-algebra X s calle sub algebra o X S, whenever, S Denon 2.4 [22,23 ] Anon emp subse A o KU-algebra X s calle eal o X s sase he ollowng conons: A A, A mples A, X. 3

4 Denon 2.5 [7] A non - emp subse A o a KU-algebra X s calle an KU eal o X sases he ollowng conons : A, 2 * * A, A mples * A, or all,, X Denon 2.6[7] Le X be a KU - algebra, a u se µ n X s calle u sub-algebra sases: S µ µ, S 2 µ {µ *, µ } or all, X. Denon 2.7 [7] Le X be a KU-algebra, a u se µ n X s calle a u KU-eal o X sases he ollowng conons: F µ µ, F 2 µ * mn {µ * *, µ }. Denon 2.8 For elemens an o KU-algebra X,,, we enoe. Denon 2.9[8] Le X be a KU-algebra. A sel map : X X s a le rgh ervaon brel,, r -ervaon o X sases he en, X I sases he en, X s calle rgh-le ervaon brel, r, -ervaon o X. Moreover, s boh, r an r, ervaon hen s calle a ervaon o X. Denon 2.[8] A ervaon o KU-algebra s sa o be regular. 4

5 Lemma 2.[8] A ervaon o KU-algebra X s regular. Eample 2.2 [8] Le X = {,,2,3. 4 } be a se n whch he operaon s ene as ollows: Usng he algorhms n Appen A, we can prove ha X, *, s a KU-algebra. Dene a map : X X b 4,,2,3 4 Then s eas o show ha s boh a, r an r, -ervaon o X. Eample 2.2. Le on bnar relaon on be he se o all posve negers an. The operaon * s ene as ollows: *=,where " " he mnus operaon.dene a b :. Then,*, X s a KU-algebra. We ene a map : X X b = or all.then, X,we have *= =.I, * = = =+ an * = = =, bu * * =+ * * = [ + ]= = + II From I an II, s no, r ervaon o X. On oher han 5

6 *= =, *= = =+, bu * * = [ ** *]* *= [ ] =. III From I an III, s r, ervaon o X. Hence r, -ervaon an, r ervaon are no conce. Proposon 2.3[8] Le X be a KU-algebra wh paral orer, an le be a ervaon o X. Then he ollowng hol, X :... v. v { X } s a sub algebra o X. Denon 2.4 [8] Le X be a KU-algebra an be a ervaon o X. Denoe F X { X : }. Proposon 2.5[8] Le X be a KU-algebra an be a ervaon o X.Then F X s a sub algebra o X. 6

7 3. Fu ervaons KU- eals o KU-algebras In hs secon, we wll scuss an nvesgae a new noon calle u- le ervaons KU - eals o KU - algebras an su several basc properes whch are relae o u le ervaons KU - eals. Denon 3. Le X be a KU-algebra an : X X be sel map.a non - emp subse A o a KU-algebra X s calle le ervaons KU eal o X sases he ollowng conons: A, 2 * * A, A mples * A, or all,, X Denon 3.2 Le X be a KU-algebra an : X X be sel map.a non - emp subse A o a KU-algebra X s calle rgh ervaons KU eal o X sases he ollowng conons: A, 2 * * A, A mples * A, or all,, X. Denon 3.3 Le X be a KU-algebra an : X X be sel map.a non - emp subse A o a KU-algebra X s calle ervaons KU -eal o X sases he ollowng conons: A, 2 * * A, A mples * A, or all,, X Denon 3.4 Le X be a KU-algebra an : X X be sel map. A u se : X [,] n X s calle a u le ervaons KU-ealbrel, F, -ervaon o X sases he ollowng conons: F µ µ, FL 2 µ * mn{ µ**, µ }. 7

8 Denon 3.5 Le X be a KU-algebra an : X X be sel map. A u se : X [,] n X s calle a u rgh ervaons KU-ealbrel, F, r o X sases he ollowng conons: F µ µ. -ervaon FR 2 µ * mn { µ**, µ }. Denon 3.6 Le X be a KU-algebra an : X X be sel map. A u se : X [,] n X s calle a u ervaons KU-eal, sases he ollowng conons F µ µ. F 2 µ * mn{ µ* *, µ }. Remark3.7 I I s e, he enons 3., 3.2,3.3 gves he enon KU-eal. II I s e, he enons 3.4,3.5, 3.6 gves he enon u KU-eal. Eample 3.8 Le X = {,,2,3. 4 } be a se n whch he operaon s ene as ollows: Usng he algorhms n Appen A, we can prove ha X, *, s a KU-algebra. * Dene a sel map : X X b 8

9 4,,2,3 4. Dene a u se µ : X [,],b µ =, µ =µ 2 =, µ 3 = µ 4 = 2, where,, 2 [,] wh > > 2.Roune calculaons gve ha µ s a u le rgh- ervaons KU- eal o KU- algebra X. Lemma 3.9 Le µ be a u le ervaons KU - eal o KU - algebra X, he nequal, * hols n X, hen µ mn {µ, µ }. Proo. Assume ha he nequal * hols n X, hen * * =, * * =, snce rom Proposon 2.3 an bfl 2, we have µ * mn{ µ**,µ }= mn{ µ, µ }= µ Pu =, we have µ * = µ mn{ µ*,µ }, bu µ * mn {µ * *, µ } = mn {µ * *, µ } =mn {µ, µ } = µ From,, we ge µ mn {µ, µ }, hs complees he proo. Lemma 3. I µ s a u le ervaons KU - eal o KU - algebra X an, hen μ μ. Proo. I,hen * =, *= snce rom Proposon 2.3 hs ogeher wh * = an μ μ,we ge µ * = µ mn {µ * *, µ } = mn {µ *, µ } = = mn {µ,µ } = µ. Proposon 3. The nersecon o an se o u le ervaons KU - eals o KU algebra X s also u le ervaons KU - eal. Proo. le be a aml o u le ervaons KU - eals o KU- algebra X, hen or an,, X, n n an 9

10 mn * n * nmn * *, n * *,n mn * *,. Ths complees he proo. Lemma 3.2 The nersecon o an se o u rgh ervaons KU - eals o KU algebra X s also u rgh ervaons KU - eal. proo. Clear Theorem3.3 Le µ be a u se n X hen µ s a u le ervaons KU- eal o X an onl sases : For all α [,],U μ, α φ mples Uμ,α s KU- eal o X A, where U μ, α = { X / μ α}. Proo. Assume ha µ s a u le ervaons KU- eal o X, le α [, ] be such ha U μ, α φ, an, X such ha U μ, α, hen µ α an so b FL 2, µ * = µ mn { µ * *, µ }= mn{µ *, µ } = mn {µ, µ } = α, hence U μ, α. Le * * U μ, α, U μ, α, I ollows romfl 2 ha µ * mn {µ * *, µ } = α, so ha * U µ, α. Hence U μ, α s KU - eal o X. Conversel, suppose ha µ sases A, le,, X be such ha µ * < mn {µ * *, µ },akng β = /2 {µ * + mn {µ * *, µ }, we have β [,] an µ * < β < mn {μ * *, µ }, ollows ha * * U μ, β an * U μ, β, hs s a conracon an hereore µ s a u le ervaons KU - eal o X. Theorem3.4 Le µ be a u se n X hen µ s a u rgh ervaons KU- eal o X an onl sases : For all α [,],U μ, α φ mples Uμ,α s KU- eal o X. Proposon 3.5 I µ s a u le ervaons KU - eal o X, hen µ * * µ

11 proo. Takng = * n FL2 an usng ku2 an F, we ge µ * * mn { µ * * *, µ } = mn {µ * * *, µ } = mn {µ * *, µ }= mn {µ, µ } = µ. Denon3.6 Le µ be a u le ervaons KU - eal o KU - algebra X,.he KU - eals, [,] are calle level KU - eal o µ. Corollar3.7 Le I be an KU - eal o KU - algebra X, hen or an e number n an open nerval,, here es a u le ervaons KU eal µ o X such ha = I. proo. The proo s smlar he corollar 4.4 [7]. 4 Image Pre-mage o u ervaons KU-eals uner homomorphsm In hs secon, we nrouce he conceps o he mage an he pre-mage o u le ervaons KU-eals n KU-algebras uner homomorphsm. Denon 4. Le be a mappng rom he se X o a se Y. I s a u subse o X, hen he u subse β o Y ene b sup, { X, } oherwse s sa o be he mage o uner. Smlarl β s a u subse o Y, hen he u subse µ = β n X.e he u subse ene b µ = β or all X s calle he prmage o β uner.

12 Theorem 4.2 An ono homomorphc premage o a u le ervaons KU - eal s also a u le ervaons KU - eal. Proo.Le : X X` be an ono homomorphsm o KU - algebras, β a u le ervaons KU - eal o X` an µ he premage o β uner, hen β = µ, or all X. Le X, hen µ = β β = µ. Now le,, X, hen µ * = β * mn {β *` *`, β } = mn { β * *,β }= mn {µ * *, µ }. The proo s complee. Denon 4.3 [4 ] A u subse µ o X has sup proper or an subse T o X, here es T such ha, SUP. T Theorem 4.4 Le : X Y be a homomorphsm beween KU - algebras X an Y. For ever u le ervaons KU - eal µ n X, µ s a u le ervaons KU - eal o Y. Proo. B enon sup or all Y an sup.we have o prove ha mn{, }, `, `, `Y. Le : X Y be an ono a homomorphsm o KU - algebras, µ a u le ervaons KU - eal o X wh sup proper an β he mage o μ uner, snce µ s a u le ervaons KU - eal o X, we have µ µ or all X. Noe ha `, where, ` are he ero o X an Y respecvel 2

13 3 Thus,, sup or all X, whch mples ha, sup or an Y. For an Y,,,le,, be Such ha sup, sup \ an sup } sup } {. Then sup }, mn{ =, sup mn sup \ = }, mn{. Hence β s a u le ervaons KU-eal o Y. Theorem 4.5 An ono homomorphc premage o a u rgh ervaons KU - eal s also a u rgh ervaons KU - eal Theorem 4.6 Le : X Y be a homomorphsm beween KU - algebras X an Y. For ever u rgh ervaons KU - eal µ n X, µ s a u rgh ervaons KU - eal o Y. proo. Clear

14 5. Caresan prouc o u le ervaons KU-eals Denon 5.[4] A u µ s calle a u relaon on an se S, µ s a u subse µ : S S [,]. Denon 5.2 [4] I µ s a u relaon on a se S an β s a u subse o S, hen μ s a u relaon on β μ, mn {β, β },, S. Denon 5.3 [4] Le µ an β be u subse o a se S, he Caresan prouc o μ an β s ene b μ β, = mn {μ, β },, S. Lemma 5.4[4] Le μ an β be u subse o a se S,hen s a u relaon on S. = or all [,]. Denon 5.5 I µ s a u le ervaons relaon on a se S an β s a u le ervaons subse o S, hen µ s a u le ervaons relaon on β µ, mn {β, β },, S. Denon 5.6 [4] Le µ an β be u le ervaons subse o a se S, he Caresan prouc o µ an β s ene b µ β, = mn {µ, β },, S Lemma 5.7[4] Le µ an β be u subse o a se S,hen s a u relaon on S, = or all [,]. 4

15 Denon 5.8 I β s a u le ervaons subse o a se S, he sronges u relaon on S, ha s a u ervaons relaon on β s µ β gven b µ β, = mn {β, β },, S. Lemma 5.9 For a gven u le ervaons subse S, le µ β be he sronges u le ervaons relaon on S,hen or [,], we have µ β = β β. Proposon 5. For a gven u subse β o KU - algebra X, le µ β be he sronges le u ervaons relaon on X. I µ β s a u le ervaons KU - eal o X X, hen β β = β or all X. Proo. Snce µ β s a u le ervaons KU- eal o X X, ollows rom F ha µ β, = mn {β, β } β, = mn {β, β }, where, X X hen β β = β. Remark5. Le X an Y be KU- algebras, we ene * on X Y b : For ever,, u, vx Y,, * u, v = * u, * v, hen clearl X Y, *,, s a KU- algebra. Theorem 5.2 Le µ an β be a u le ervaons KU- eals o KU - algebra X,hen µ β s a u le ervaons KU-eal o X X. Proo : or an, X X,we have, µ β, = mn {µ, β }= mn {µ, β} mn {µ, β } = µ β,. Now le, 2,, 2,, 2 X X, hen, µ β *, 2 * 2 = mn {µ,, β 2, 2 } mn{mn {µ * *, µ }}, mn {β 2 * 2 * 2, β 2 }} 5

16 = mn{mn{µ * *, µ 2 * 2 * 2 }, mn{µ, β 2 }} = mn{µ β * *, 2 * 2 * 2, µ β, 2 }. Hence µ β s a u le ervaons KU- eal o X X. Analogous o heorem 3.2 [ 6], we have a smlar resuls or u le ervaons KUeal, whch can be prove n smlar manner, we sae he resuls whou proo. Theorem 5.3 Le µ an β be a u le ervaons subse o KU-algebra X,such ha µ β s a u le ervaons KU-eal o X X, hen Eher µ µ or β β or all X, I µ µ or all X, hen eher µ β or β β, I β β or all X, hen eher µ µ or β µ, v Eher µ or β s a u le ervaons KU- eal o X. Theorem 5.4 Le β be a u subse o KU- algebra X an le µ β be he sronges u le ervaons relaon on X, hen β s a u le ervaons KU - eal o X an onl µ β s a u le ervaons KU- eal o X X. proo : Assume ha β s a u le ervaons KU- eal X, we noe rom F ha : µ β, = mn {β, β } =mn {β, β } mn {β, β } = µ β,. Now, or an, 2,, 2,, 2 X X, we have rom F 2 : µ β *, 2 * 2 = mn {β *, β 2 * 2 } mn {mn{β * *, β }, mn {β 2 * 2 * 2, β 2 }} = mn{mn{β * *, β 2 * 2 * 2 }, mn {β, β 2 }} = mn {µ β * *, 2 * 2 * 2, µ β, 2 }. 6

17 Hence µ β s a u le ervaons KU - eal o X X. Conversel. For all, X X, we have Mn {β, β } = µ β, = mn {β, β }. I ollows ha β β or all X, whch prove F. Now, le, 2,, 2,, 2 X X, hen mn {β *, β 2 * 2 } = µ β *, 2 * 2 mn {µ β, 2 *, 2 *, 2, µ β, 2 } = mn {µ β * *, 2 * 2 * 2, µ β, 2 } = mn {mn {β * *, β 2 * 2 * 2 }, mn {β, β 2 }} = mn {mn {β * *, β }, mn {β 2 * 2 * 2, β 2 }} In parcular, we ake 2 = 2 = 2 =, hen, β * mn { β * *, β } Ths prove FL 2 an complees he proo. Concluson Dervaon s a ver neresng an mporan area o research n he heor o algebrac srucures n mahemacs. In he presen paper, he noon o u le ervaons KU - eal n KU-algebra are nrouce an nvesgae he useul properes o u le ervaons KU - eals n KU-algebras. In our opnon, hese enons an man resuls can be smlarl eene o some oher algebrac ssems such as BCI-algebra, BCH-algebra,Hlber algebra,bf-algebra -Jalgebra,WS-algebra,CI-algebra, SU-algebra,BCL-algebra,BP-algebra,Coeer algebra,bo-algebra,pu- algebras an so orh. The man purpose o our uure work s o nvesgae: The nerval value, bpolar an nuonsc u le ervaons KU - eal n KUalgebra. 2 To conser he cubc srucure o le ervaons KU - eal n KU-algebra. We hope he u le ervaons KU - eals n KU-algebras, have applcaons n eren branches o heorecal phscs an compuer scence. 7

18 Algorhm or KU-algebras Inpu X : se, : bnar operaon Oupu X s a KU-algebra or no Begn I X hen go o.; En I I X hen go o.; En I Sop: =alse; : ; Whle X an no Sop o I hen Sop: = rue; En I j : Whle j X an no Sop o I hen j Sop: = rue; En I En I k : Whle k X an no Sop o I hen j j k k 8

19 Sop: = rue; En I En Whle En Whle En Whle I Sop hen. Oupu X s no a KU-algebra Else Oupu X s a KU-algebra En I En. Reerences [] H. A. S. Abujabal an N. O. Al-Shehr, On le ervaons o BCI-algebras, Soochow Journal o Mahemacs, vol. 33, no. 3, pp , 27. [2] H. E. Bell an L.-C. Kappe, Rngs n whch ervaons sas ceran algebrac conons, Aca Mahemaca Hungarca, vol. 53, no. 3-4, pp , 989. [3] H. E. Bell an G. Mason, On ervaons n near-rngs, near-rngs an Near-els, Norh-Hollan Mahemacs Sues, vol. 37, pp. 3 35, 987. [4] P.Bhaachare an N.P.Mukheree, Fu relaons an u group norm, sc,36985, [5] M. Breˇsar an J. Vukman, On le ervaons an relae mappngs, Proceengs o he Amercan Mahemacal Soce, vol., no., pp. 7 6, 99. [6] M. Breˇsar, On he sance o he composon o wo ervaons o he generale ervaons, Glasgow Mahemacal Journal, vol. 33, no., pp , 99. [7] W. A. Duek, The number o subalgebras o ne BCC-algebras, Bull. Ins. Mah. Aca. Snca, 2 992, [8] W. A. Duek an X. H. Zhang, On eals an congruences n BCCalgebras, Cechoslovak Mah. J., , [9] B. Hvala, Generale ervaons n rngs, Communcaons n Algebra, vol. 26, no. 4, pp , 998 9

20 [] Y.Ima an Isek K: On aom ssems o Proposonal calcul, XIV, Proc. Japan Aca. Ser A, Mah Sc., 42966,9-22. [] k.isek: An algebra relae wh a proposonal calcul, Proc. Japan Aca. Ser A Mah. Sc., , [2] K Isek an Tanaka S: An nroucon o heor o BCK-algebras, Mah. Japo., [3] Y. B. Jun an X. L. Xn, On ervaons o BCI-algebras, Inormaon Scences, vol. 59, no. 3-4, pp , 24. [4]A. A. M. Kamal, σ-ervaons on prme near-rngs, Tamkang Journal o Mahemacs, vol. 32, no. 2, pp , 2 [5]Y. Komor, The class o BCC-algebras s no a vare, Mah. Japonca, , [6 ] D.S.Malk an J.N.Moreson, Fu relaon on rngs an groups, u ses an ssems, 4399,7-23. [ 7] S. M. Mosaa M. A. Ab-Elnab- M.M. Youse, Fu eals o KU algebra Inernaonal Mahemacal Forum, Vol. 6, 2.no. 63, [8] S. M.Mosaa, R. A. K. Omar, A. Ab-elaem, Properes o ervaons on KUalgebras,gournal o avances n mahemacs Vol.9, No. [9]S. M. Mosaa, F. F.Kareem, Le e maps an α-ervaons o KU-algebra, gournal o avances n mahemacs Vol.9, No 7 [2] G. Muhun an Abullah M. Al-Roq On, -Dervaons n BCI-Algebras.Dscree Dnamcs n Naure an Soce Volume 22. [2] G. Muhun an Abullah M. Al-roq,On -Dervaons o BCI-Algebras Absrac an Apple Analss Volume 22, Arcle ID , 2 pages [22] C.Prabpaak an U.Leerawa, On eals an congruence n KU-algebras, scena Magna n- ernaonal book seres, Vol. 529, No., [23] C.Prabpaak an U.Leerawa, On somorphsms o KU-algebras, scena Magna nernaonal book seres, 529, no.3, [24] A. Wronsk, BCK-algebras o no orm a vare, Mah. Japonca, , [25] O.G.X, u BCK-algebras, Mah. Japan,

21 [26] L.A.Zaeh, Fu ses, norm. an conrol,8965, [27] J. Zhan an Y. L. Lu, On -ervaons o BCI-algebras, Inernaonal Journal o Mahemacs an Mahemacal Scences, no., pp , 25. Sam M. Mosaa sammosaa@ahoo.com Deparmen o Mahemacs, Facul o Eucaon, An Shams Unvers, Ro, Caro, Egp. Ahme Ab-elaem ahmeabelaem88@ahoo.com Deparmen o Mahemacs, Facul o Eucaon, An Shams Unvers, Ro, Caro, Egp. 2