Types Ideals on IS-Algebras

Transcription

1 Ieraioal Joural of Maheaical Aalyi Vol. 07 o IARI Ld hp://doi.org/0.988/ija Type Ideal o IS-Algebra Sudu Najah Jabir Faculy of Educaio ufa Uiveriy Iraq Copyrigh 07 Sudu Najah Jabir. Thi aricle i diribued uder he Creaive Coo Aribuio Licee which peri urericed ue diribuio ad reproducio i ay ediu provided he origial work i properly cied. Abrac I hi paper we udy I-ideal ad P-ideal o IS-algebra ad prove oe reul abou hi. eyword: BCI- algebra eigroup IS-algebra I-ideal P-ideal Iroducio The cla of BCI- algebra which wa iroduced by.ieki [] i 966 i a ipora cla of logical algebra which ha wo origi.oe of he oivaio for heir ue i baed o e heory he oher o claical ad oclaical propoiioal calculu.by defiiio [6] iroduced a ew cla of algebra relaed o BCI- algebra ad eigroup called a BCI- eigroup. Fro ow o we reae i a a IS- algebra for he coveiece of udy. Preliiary We review oe defiiio ad properie ha will be ueful i our reul. Defiiio. A Seigroup i a ordered pair G where G i a o epy e ad "." i a aociaive biary operaio o G. [3] Defiiio. A BCI- algebra i riple G 0 where G i a o epy e "" i biary operaio o G 0 G i a elee uch ha he followig axio are aified for all r G : r r = 0 = 0

2 636 Sudu Najah Jabir 3 = 0 4 = 0 ad = 0 iplie = If 0 = 0 for all G he G i called BC-algebra. [] Defiiio.3 A IS-algebra i a o epy e wih wo biary operaio "" ad "." ad coa 0 aifyig he axio:. G 0 i a BCI-algebra.. G. i a Seigroup 3.. r =..r ad.r =.r.r for all r G. [9] Exaple.4 le G={0v} defie "" operaio ad uliplicaio "." by he followig able: 0 v v 0 c 0 v v v v 0. 0 v v 0 0 v The by rouie calculaio we ca ee ha G i a IS-algebra.[9] Exaple.5 le G={0vu} defie "" operaio ad uliplicaio "." by he followig able: 0 v u v v v v 0 0 u u u u u 0. 0 v u v v u 0 v u The G i a IS-algebra. [9]

3 Type ideal o IS-algebra 637 Defiiio.6 A o- epy ube of a eigroup G i aid o be lef rep. righ able if rep. wheever G ad boh lef ad righ able i wo ided able or iply able. [9] Defiiio.7 A o- epy ube of IS-algebra G i called a lef rep. righ I-ideal of G if: i a lef rep. righ able of G. For ay G ad iply ha. Boh lef ad righ I-ideal i called a wo ided I-ideal or iply a I-ideal. [9] Defiiio.8 A biary relaio o G by leig 0 i a parial ordered e. [0] 3 Mai Reul if ad oly if I hi ecio we fid oe reul abou I-ideal ad P-ideal o IS-algebra. Propoiio 3. Le ad are lef rep. righ I-ideal of G The lef rep. righ I-ideal of G. Le ad be lef I-ideal of G ad le G ad The ad o ad he Now le ad he ad o ad herefore ece i a lef I-ideal. [ice are lef I-ideal ] [ice are lef I-ideal] Propoiio 3. Le ad are lef rep. righ I-ideal of G The lef rep. righ I-ideal If or. Suppoe ha ad are lef I-ideal of G Wihou lo of geeraliy we ay aue ha he ice i a lef I-ideal o i a lef I-ideal. i a i a

4 638 Sudu Najah Jabir Defiiio 3.3 le ad be a ideal i IS-algebra The } : { i a ideal where he biary operaio " " ad " " are defie by he followig: for all. Propoiio 3.4 Le ad be a lef rep. righ I-ideal of IS-eigroup G. The i a lef rep. righ I-ideal of G G. Le ad are lef I-ideal of IS-eigroup G Le G G he ] [i Now o The ideal I lef are ce ad Sice he o ideal I lef are ce ad he ad he ad he ad if G G where ad le ] [i ece i a lef I-ideal. Defiiio 3.5 Le G ad R be IS-algebra a appig R G : i called a ISalgebra hooorphi briefly hooorphi if ad for all G. Le R G : IS-algebra hooorphi. The he e } 0 : { G i called he kerel of ad deoe by ker. Moreover he e } : { G R i called he iage of ad deoe by I.

5 Type ideal o IS-algebra 639 Defiiio 3.6 Le G ad R be a IS-algebra uch ha hooorphi he : i a ooorphi iff oe o oe hooorphi. i a epiorphi iff oo hooorphi. 3 i a ioorphi iff bijecive hooorphi. : G R IS-algebra Propoiio 3.7 Le : G R be a IS-eigroup hooorphi The ker i a I-ideal of G. Le : G R o prove ker i a able le G ad ker be a IS-eigroup hooorphi o 0 he.0 0 [ice i a hooorphi] hu ker he ker i a able. Now le G uch ha ker ad ker o 0 ad 0 o 0 ad 0 [ice i a hooorphi] he 0 0 [ice ker he ker hu 0 ece ker i a I- ideal. ] Propoiio 3.8 Le : G R be a IS-eigroup epiorphi ad le i a lef rep. righ I-ideal i G.The i a lef rep. righ I-ideal i R. Le be a lef I-ideal of G le ad R where ice oo he here exi G uchha

6 640 Sudu Najah Jabir To prove ice [i ce i a lef I ideal ] o G ad bu [ i epiorphi ] herefore i able. Now uppoe ha for oe uch ha ad To prove ice i a hooorphi he ad ice hu [ice i I-ideal] herefore ece i a lef I-ideal i R. Theore 3.9 Le : G R be a IS-eigroup hooorphi if i a I- { G/ i a I-ideal of G coaiig ker ideal of R he }. Le : G R be a IS-eigroup hooorphi Suppoe ha i a I-ideal of R he r Le G ad r ad r r ad r r Sice i able ad r r Thu i able. Now

7 Type ideal o IS-algebra 64 G uch ha ad ad The he [Sice i a I-ideal ] Tha i Moreover herefore i a I-ideal of G. { 0} iplie ha ker {0}. Defiiio 3.0 Le ad be ideal of IS-algebra G defie { : }. Lea 3. Le ad be a I-ideal of IS-eigroup G wih uiy uch ha The i a I-ideal of G. Aue ha ad be a I-ideal Le G ad he ad ] Thu i able ube of G. Now Suppoe ha [ice G uch ha ad ad he ice ad be a I-ideal ad hu ad o ece i a I-ideal of G. Defiiio 3. A o-epy ube of IS-algebra G i called a lef rep. righ P-ideal of G if: i a lef rep. righ able of G. If r ad r iply ha r for all r G. Boh lef ad righ P-ideal i called a wo ided P-ideal or iply a P-ideal.

8 64 Sudu Najah Jabir Propoiio 3.3 Le ad are lef rep. righ P-ideal of G The lef rep. righ P-ideal of G. Le ad be a lef P-ideal of G ad le G ad The ad o ad he Now le r ad r he r r ad o r ad r herefore r ece i a lef P-ideal. [ice are lef P-ideal ] [ice are lef P-ideal] Propoiio 3.4 Le ad are lef rep. righ P-ideal of G The lef rep. righ P-ideal If or. Suppoe ha ad are lef P-ideal of G Wihou lo of geeraliy we ay aue ha he ice i a lef P-ideal o i a lef P-ideal. i a i a Propoiio 3.5 Le ad be lef rep. righ P-ideal of IS-eigroup G. The G. i a lef rep. righ P-ideal of G Le ad are lef P-ideal of IS-eigroup G Le GG he o Sice The Now ad [ice are lef P ideal]

9 Type ideal o IS-algebra 643 le r if [ he he he he o he ece [ r r ad r r r r ad ] r r r r r r where r r r r r i a lef P-ideal. GG ] ad ad ad ad r r r r r r r r [ice are lef Pideal] Defiiio 3.6 A BCI-algebra i aid o be Poiive iplicaive if r r r for all r G. Propoiio 3.7 Le : G R be a IS-eigroup hooorphi uch ha G i poiive iplicaive The ker i a P-ideal of G. Le : G R o prove ker i a able le G ad ker be a IS-eigroup hooorphi o 0 he.0 0 [ice i a hooorphi] hu ker he ker i a able. Now le r G uch ha r ker ad r ker By a propery of a BCI algebra r r ker ad r ker ece 0 [ r] ad 0 r r

10 644 Sudu Najah Jabir By defiiio [.8] hi iplie ha r ad r By raiiviy r r ad o [ r] r 0 Sice i a hooorphi ad G i i poiive iplicaive 0 [ r r] [ r r r] [ r0] r Thu Therefore r ker ker i a P-ideal of G. Propoiio 3.8 Every P-ideal of a IS-eigroup i a I-ideal bu he covere i o rue. Le i a P-ideal of a IS-eigroup ad Le r G uch ha r ad r Pu r 0 we obai i a I-ideal of G. Propoiio 3.9 Le : G R be a IS-eigroup epiorphi ad le i a lef rep. righ P-ideal i G The i a lef rep. righ P-ideal i R. Le be a P-ideal of G he i a I-ideal of G herefore i a I-ideal [ by Propoiio 3.5 ] Thu i a able ube of R Now Suppoe ha r for oe r uch ha [ ] r ad r ice i a hooorphi he [ r] ad ice r hu r r r [ice i P-ideal] herefore r r ece i a lef P-ideal i R.

11 Type ideal o IS-algebra 645 ~ Defiiio 3.0 Le be a ideal of IS-algebra G he relaio " " o G defied by: relaio. ~ if ad oly if ad i a equivalece Defiiio 3. Le G be a IS-algebra ad be a ideal of G deoe a he equivalece cla coaiig G ad G / a he e of all equivalece ~ clae of G wih repec o " " ha i { : ~ G } ad G / { : G}. Propoiio 3. If i a ideal of IS-eigroup G The G / 0 i a IS-eigroup uder he biary operaio ad for all G /. We are eay o prove ha G / 0 i a BCI-algebra fir we how ha i well-defied. Le u ad v he we have v v ad v v hu ~ v had ad o. O he oher v uv u v ad uv v u v hece v ~ uv uv herefore G/ i eigroup.oreover for ay r G / we obai r r r r r Siilarly we ge r r r ece G / i a IS-eigroup. r Referece [] Y. Iai ad. Ieki O Axio Sye of Propoiioal Calculi XIV Proc. Japa Acad hp://doi.org/0.379/pja/95569 []. Ieki A Algebra Relaed wih a Propoiioal Calculu Proc. Japa Acad hp://doi.org/0.379/pja/9557

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