International Journal of Mathematical Analysis

Vol. 11, 2017, no. 13, 635 - 646

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ijma.2017.7466

Types Ideals on IS-Algebras

Sundus Najah Jabir

Faculty of Education

Kufa University, Iraq

Copyright © 2017 Sundus Najah Jabir. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

In this paper we study I-ideals and P-ideals on IS-algebra and prove some

results about this.

Keywords: BCI- algebras, semigroup, IS-algebra, I-ideals, P-ideals

1 Introduction

The class of BCI- algebras which was introduced by K.Iseki [1] in

1966 is an important class of logical algebras which has two origins.one of the

motivations for their use is based on set theory, the other on classical and non-

classical propositional calculus.by definition, [6] introduced a new class of

algebras related to BCI- algebras and semigroups called a BCI- semigroup. From

now on, we rename it as an IS- algebra for the convenience of study.

2 Preliminary

We review some definitions and properties that will be useful in our results.

Definition 2.1 A Semigroup is an ordered pair ),( G , where G is a non empty set

and " ." is an associative binary operation on G. [3]

Definition 2.2 A BCI- algebra is triple (G ,* ,0) where G is a non empty set "*" is

binary operation on G , G0 is an element such that the following axioms are

satisfied for all Grts ,, :

1) ((s ∗ t) ∗ (s ∗ r)) ∗ (r ∗ t) = 0,

2) (s ∗ (s ∗ t) ∗ t = 0,

636 Sundus Najah Jabir

3) s ∗ s = 0,

4) s ∗ t = 0 and t ∗ s = 0 implies s = t

If 0 ∗ s = 0 for all Gs then G is called BCK-algebra. [12]

Definition 2.3 An IS-algebra is a non empty set with two binary operation "*"

and "." and constant 0 satisfying the axioms:

1. (G ∗, 0) is a BCI-algebra.

2. (G, .) is a Semigroup,

3. s.(t ∗ r) = (s.t ∗ (s.r) and (s ∗ t).r = (s.r) ∗ (t.r), for all s, t, r G . [9]

Example 2.4 let G={0,n,m,v} define "*" operation and multiplication "." by the

following tables:

. 0 n m v

0 0 0 0 0

n 0 n 0 n

m 0 0 m m

v 0 0 m v

Then by routine calculations we can see that G is an IS-algebra.[9]

Example 2.5 let G={0,n,m,v,u} define "*" operation and multiplication "." by the

following tables:

. 0 n m v u

0 0 0 0 0 0

n 0 0 0 0 0

m 0 0 0 0 m

v 0 0 0 m v

u 0 n m v u

Then G is an IS-algebra. [9]

* 0 n m v

0 0 0 v m

n n 0 c m

m m m 0 v

v v v m 0

* 0 n m v u

0 0 0 0 0 0

n n 0 n n 0

m m m 0 0 0

v v v v 0 0

u u u u u 0

Types ideals on IS-algebras 637

Definition 2.6 A non- empty subset K of a semigroup G is said to be left (resp.

right ) stable if ).( KnsrespKsn whenever KnandGs both

left and right stable is two sided stable or simply stable. [9]

Definition 2.7 A non- empty subset K of IS-algebra G is called a left (resp. right)

I-ideal of G if:

1) K is a left (resp. right) stable of G.

2) For any KsthatimplyKtandKtsGts *,, .

Both left and right I-ideal is called a two sided I-ideal or simply an I-ideal. [9]

Definition 2.8 A binary relation on G by letting ts

if and only if

0* ts is a partial ordered set . [10]

3 Main Results

In this section, we find some results about I-ideals and P-ideals on IS-algebra.

Proposition 3.1 Let K and H are left (resp. right) I-ideals of G Then HK is a

left (resp. right) I-ideals of G.

Proof: Let K and H be left I-ideals of G ,

and let Gs and HKn Then

HnandKn

HsnandKsnso [since K , H are left I-ideals ]

HKsnthen

HKtandHKtsletNow *,

HtKtandHtsKtsthen ,*,*

HsandKsso [since K , H are left I-ideals]

HKstherefore

HKHence is a left I-ideal.

Proposition 3.2 Let K and H are left (resp. right) I-ideals of G Then HK is a

left (resp. right) I-ideals KHorHKIf .

Proof: Suppose that K and H are left I-ideals of G

Without loss of generality we may assume that

HK then HHK

since H is a left I-ideal

so HK is a left I-ideal.

638 Sundus Najah Jabir

Definition 3.3 let K and H be an ideals in IS-algebra Then

},:),{( HmKnmnHK is an ideal where the binary operations ""

and " * " are define by the following:

),(),(),( 21212211 mmnnmnmn ,

)*,*(),(*),( 21212211 mmnnmnmn for all HKmnmn ),(),,( 2211 .

Proposition 3.4

Let K and H be a left (resp. right ) I-ideals of IS-semigroups G . Then HK

is a left (resp. right) I-ideal of GG .

Proof:

Let K and H are left I-ideal of IS-semigroups G

Let HKnnGGss ),(,),( 2121 then

,

),(),(

),(

],[sin

),(),(),(

2121

21

2211

22112121

11

Now

HKnnssso

HKnsnsThen

idealsIleftareHKceHnsandKnsSince

nsnsnnss

HKsthen

HKssso

idealsIleftareHKceHsandKsthen

HtKtandHtsKtsthen

HKttandHKtststhen

HKttandHKttssif

GGtttsss

whereHKtandHKtslet

),(

],[sin

,*,*

),()*,*(

),(),(*),(

),(,),(

,)*(

21

21

212211

212211

212121

2121

Hence HK is a left I-ideal.

Definition 3.5 Let G and R be IS-algebra a mapping RG : is called a IS-

algebra homomorphism (briefly homomorphism) if )(*)()*( tsts and

)()()( tsts for all Gts , .

Let RG : IS-algebra homomorphism. Then the set }0)(:{ sGs is

called the kernel of , and denote by ker . Moreover, the set

}:)({ GsRs is called the image of and denote by Im .

Types ideals on IS-algebras 639

Definition 3.6 Let G and R be an IS-algebra such that RG : IS-algebra

homomorphism then :

1) is a monomorphism iff one to one homomorphism .

2) is an epimorphism iff onto homomorphism .

3) is an isomorphism iff bijective homomorphism .

Proposition 3.7 Let RG : be an IS-semigroup homomorphism Then ker (

) is a I-ideal of G.

Proof:

Let RG : be a IS-semigroup homomorphism,

to prove ker ( ) is a stable

let )(ker nandGs so 0)( n then

00).()()()( snsns [since is a homomorphism]

thus kersn

then ker ( ) is a stable .

Now,

let Gts , such that

kerker* tandts ,

so 0)(0)*( tandts

so 0)(0)(*)( tandts [since is a homomorphism]

then 00*)( s [since kert ]

thus 0)( s then kers

Hence ker ( ) is a I- ideal.

Proposition 3.8 Let RG : be an IS-semigroup epimorphism and let K is a

left (resp. right) I-ideal in G .Then )(K is a left (resp. right) I-ideal in R.

Proof:

Let K be a left I-ideal of G

let KnwhereRtandKnn )()(

since onto then there exists tsthatsuchGs )(

640 Sundus Najah Jabir

To prove )(Ktn

since KnandGsidealIleftaisKceKsn ][sin

so )()( Ksn

but tnnsns )()()(

[ is epimorphism ]

therefore )(K is stable .

Now, suppose that KtssomeforKts ,)()(),( ,

such that )()()()(*)( KtandKts

To prove )()( Ks

since is a homomorphism then

)(*)( ts ,)()(sin)()*( KtceandKts

thus KsKtKts ,* [since K is I-ideal]

therefore )()( Ks

Hence )(K is a left I-ideal in R .

Theorem 3.9 Let RG : be an IS-semigroup homomorphism if K is an I-

ideal of R then })(/{)(1 KnGnK is an I-ideal of G containing ker

( ) .

Proof:

Let RG : be an IS-semigroup homomorphism

Suppose that K is an I-ideal of R

Let )(1 KrandGs then Kr )( and

KsrrsandKrssr )()()()()()(

Since K is stable and )(1, Krssr

Thus )(1 K is stable.

Now,

Types ideals on IS-algebras 641

)(1)(1*, KtandKtsthatsuchGts then

)(*)( ts KtandKts )()*(

Then Ks )( [Since K is an I-ideal ]

That is )(1 Ks therefore )(1 K is an I-ideal of G.

Moreover K}0{ implies that ker ( )(1})0({1) K .

Definition 3.10 Let K and H be ideals of IS-algebra G define

},:{ HmKnKHnmKH .

Lemma 3.11 Let K and H be an I-ideals of IS-semigroup G with unity such that

HK 1 Then KH is an I-ideal of G.

Proof:

Assume that K and H be an I-ideals

Let KHnmandGs then

KHmsnsnm )()( and KHmsnnms )()( [since

KsnHsm , ]

Thus KH is stable subset of G .

Now,

Suppose that Gts , such that KHtandKHts * then

nmstsandnmt ** since K and H be an I-ideals and

HnmKnm , thus HsandKs so KHs

Hence KH is an I-ideal of G .

Definition 3.12 A non-empty subset K of IS-algebra G is called a left (resp. right)

P-ideal of G if:

1) K is a left (resp. right) stable of G.

2) If KrsthatimplyKrtandKrts ***)*( , Grtsallfor ,, .

Both left and right P-ideal is called a two sided P-ideal or simply a P-ideal.

642 Sundus Najah Jabir

Proposition 3.13 Let K and H are left (resp. right) P-ideals of G Then HK is a

left (resp. right) P-ideals of G.

Proof: Let K and H be a left P-ideals of G ,

and let Gs and HKn Then

HnandKn

HsnandKsnso [since K , H are left P-ideals ]

HKsnthen

HKrtandHKrtsletNow **)*(,

HtKtandHrtsKrtsthen ,*)*(,*)*(

HrsandKrsso ** [since K , H are left P-ideals]

HKrstherefore *

HKHence is a left P-ideal.

Proposition 3.14 Let K and H are left (resp. right) P-ideals of G Then HK is a

left (resp. right) P-ideals KHorHKIf .

Proof: Suppose that K and H are left P-ideals of G

Without loss of generality we may assume that

HK then HHK

since H is a left P-ideal

so HK is a left P-ideal .

Proposition 3.15 Let K and H be left (resp. right) P-ideals of IS-semigroups G.

Then HK is a left (resp. right) P-ideal of GG .

Proof:

Let K and H are left P-ideal of IS-semigroups G Let

HKnnGGss ),(,),( 2121 then

,

),(),(

),(

],[sin

),(),(),(

2121

21

2211

22112121

Now

HKnnssso

HKsnsnThen

idealsPleftareHKceHnsandKnsSince

ssnnss nn

Types ideals on IS-algebras 643

HKrsthen

HKrsrsso

idealsPleftareHKceHrsandKrsthen

HrtKrtandHrtsKrtsthen

HKrtrtandHKrtsrtsthen

HKrtrtandHKrrtststhen

HKrrttandHKrrttssif

GGrrrtttsss

whereHKrtandHKrtslet

*

)*,*(

],[sin**

*,**)*(,*)*(

)*,*(]*)*(,*)*[(

)*,*(),(*)*,*(

),(*),(),(*]),(*),([

),(),(,),(

,**)*(

21

21

2121

2121

2121

2121

21

21

21

212211

212211

212211

212121

2121

Hence HK is a left P-ideal .

Definition 3.16 A BCI-algebra is said to be Positive implicative if

Grtsallforrtrsrts ,,)*(*)*(*)*( .

Proposition 3.17 Let RG : be an IS-semigroup homomorphism such that G

is positive implicative Then ker ( ) is a P-ideal of G.

Proof:

Let RG : be an IS-semigroup homomorphism,

to prove ker ( ) is a stable

let )(ker nandGs so 0)( n then

00).()()()( snssn [since is a homomorphism]

thus kersn

then ker ( ) is a stable .

Now,

let Grts ,, such that

ker*ker*)*( rtandrts ,

By a property of a BCI algebra

ker*ker*)*(*)*( rtandrtstrs

Hence )(*)()*(0)(*)](*)([0 rtrtandtrs

644 Sundus Najah Jabir

By definition [2.8] this implies that )()()()(*)( rtandtrs

By transitivity 0)(*)](*)([)()(*)( rrssoandrrs

Since is a homomorphism and G is is positive implicative

)*(]0*)*[()]*(*)*[(]*)*[(0 rsrsrrrsrrs

Thus ker* rs

Therefore ker is a P-ideal of G.

Proposition 3.18 Every P-ideal of an IS-semigroup is an I-ideal but the converse

is not true.

Proof:

Let K is a P-ideal of a IS-semigroup and

Let Grts ,, such that KrtandKrts **)*(

Put 0r we obtain K is a I-ideal of G.

Proposition 3.19 Let RG : be an IS-semigroup epimorphism and let K is a

left (resp. right) P-ideal in G Then )(K is a left (resp. right) P-ideal in R.

Proof:

Let K be a P-ideal of G then K is an I-ideal of G therefore

)(K is an I-ideal [ by Proposition 3.5 ]

Thus )(K is a stable subset of R

Now ,

Suppose that KrtssomeforKrts ,,)()(,)(),( ,

such that )()(*)()()(*)](*)([ KrtandKrts

since is a homomorphism then

)()*(sin)(]*)*[( KrtceandKrts

thus KrsKrtKrts **,*)*( [since K is P-ideal]

therefore )()*()(*)( Krsrs

Hence )(K is a left P-ideal in R.

Types ideals on IS-algebras 645

Definition 3.20 Let K be an ideal of IS-algebra G the relation "~

" K on G

defined by: tKs~

if and only if KstandKts ** is an equivalence

relation.

Definition 3.21 Let G be an IS-algebra and K be an ideal of G denote sK as

the equivalence class containing Gs and KG / as the set of all equivalence

classes of G with respect to "~

" K that is }~

:{ tKsGtKs and

}:{/ GsKKG s .

Proposition 3.22 If H is an ideal of IS-semigroup G Then ),,,/( 0HHG is

an IS-semigroup under the binary operations tsts HHH * and

tsts HHH for all HGHH ts /, .

Proof:

We are easy to prove that ),,,/( 0HHG is a BCI-algebra first we show

that is well-defined. Let vtus HHandHH then we have

KtvsstsvandKvtssvst )*(*)*(* thus svHst~

. On the other

hand HvsusvuvandHvusuvsv )*(*)*(* hence uvHsv~

and so uvst HH therefore ),/( HG is semigroup .moreover for any

HGHHH rts /,, we obtain

)()(

)( *)*(*

rsts

srstsrstrtsrtsrts

HHHH

HHHHHHHHH

Similarly we get )()()( rtrsrts HHHHHHH

Hence HG / is an IS-semigroup.

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646 Sundus Najah Jabir

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Received: May 11, 2017; Published: July 5, 2017