ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2012, Vol. 33, No. 2, pp. 170–174. c© Pleiades Publishing, Ltd., 2012.

On (m, (p, q), n)(m, (p, q), n)(m, (p, q), n)-Bi-Ideals in Ternary Semirings

Manish Kant Dubey1* and Anuradha2**

(Submitted by M.M. Arslanov)1SAG, DRDO, Metcalf House, Delhi, 110054 India

2Department of Mathematics, University of Delhi, Delhi, 110007 IndiaReceived September 15, 2011; in final form, November 14, 2011

Abstract—In this paper, we introduce the notion of (m, (p, q), n)-bi-ideals and minimal(m, (p, q), n)-bi-ideals in ternary semirings and give their characterizations.

DOI: 10.1134/S1995080212020072

Keywords and phrases: Ternary semiring, Bi-ideals, (m, (p, q), n)-Quasi-ideals.

1. INTRODUCTION AND PRELIMINARIESThe notion of bi-ideal in semigroup was introduced by Good and Hughes [3]. S. Kar [4] has studied

widely the notion of quasi-ideals and bi-ideals in ternary semirings and characterized them in manyways. In [2] M.K. Dubey and Anuradha introduced the notion of (m, (p, q), n)-quasi-ideal in ternarysemiring and give their characterizations. In this paper, we introduce the concept of (m, (p, q), n)-bi-ideal in ternary semiring and generalized the results in terms of the same. We also give the example suchthat every (m, (p, q), n)-quasi-ideal is (m, (p, q), n)-bi-ideal of ternary semiring but converse need nottrue.

Recall [4], the following:A non empty set S together with a binary operation called addition and ternary multiplication, denoted

by juxtaposition is said to be a ternary semiring if S is an additive commutative semigroup satisfying thefollowing conditions:

(i) (abc)de = a(bcd)e = ab(cde),(ii) (a + b)cd = acd + bcd,(iii) a(b + c)d = abd + acd,(iv) ab(c + d) = abc + abd, for all a, b, c, d, e ∈ S.Let S be a ternary semiring. If there exists an element 0 ∈ S such that 0 + x = x and 0xy = x0y =

xy0 = 0 for all x, y ∈ S then ‘0’ is called the zero element or simply the zero of the ternary semiring S.In this case we say that S is a ternary semiring with zero.

Throughout this paper, S will always denote a ternary semiring with zero. An additive subsemigroupT of S is called a ternary subsemiring if t1t2t3 ∈ T for all t1, t2, t3 ∈ T . An element a of a ternary semiringS is called regular if there exists an element x in S such that axa = a. A ternary semiring S is calledregular if all of its elements are regular. Let A,B,C be three subsets of S. Then by ABC, we mean theset of all finite sums of the form

∑aibici with ai ∈ A, bi ∈ B, ci ∈ C. A ternary subsemiring B of a

ternary semiring S is called a bi-ideal of S if BSBSB ⊆ B.Definition 1.1 ([2]). Let S be a ternary semiring. Then a subsemiring(i) R of S is called an m-right ideal of S if R(SS)m ⊆ R,(ii) M of S is called an (p, q)-lateral ideal of S if SpMSq + SpSMSSq ⊆ M ,(iii) L of S is called an n-left ideal of S if (SS)nL ⊆ L, where m,n are positive integers and p + q is

an even positive integer.

*E-mail: kantmanish@yahoo.com**E-mail: anuvikaspuri@yahoo.co.in

170

ON (m, (p, q), n)-BI-IDEALS IN TERNARY SEMIRINGS 171

2. (m, (p, q), n)-BI-IDEALS IN TERNARY SEMIRING

In this section, we define (m, (p, q), n)-bi-ideals in ternary semirings and give their characterizations.Definition 2.1 ([2]). A sub-semiring Q of a ternary semiring S is called an (m, (p, q), n)-quasi-ideal

of S if Q(SS)m ∩ (SpQSq + SpSQSSq) ∩ (SS)nQ ⊆ Q, where m,n, p, q are positive integers greaterthan 0 and p + q = even.

Example 2.2. Let Z− \ {−1} be the set of all negative integers excluding {0}. Then Z− \ {−1}is a ternary semiring with following binary addition and ternary multiplication, a + b = max{a, b} anda · b · c = abc (usual ternary multiplication). Consider Q = {−3,−27} ∪ {k ∈ Z− : k ≤ −72}. Clearly Qis non empty subsemiring of S and also Q is (2, (1, 1), 3)-quasi-ideal of S but it is not a quasi-ideal of S.

Definition 2.3. A ternary subsemiring B of a ternary semiring S is called an (m, (p, q), n)-bi-idealof S if B(SS)m−1SpBSq(SS)n−1B ⊆ B, where m,n, p, q are positive integers greater than 0 and p andq are odd.

Remark. Every bi-ideal of a ternary semiring S is (1, (1, 1), 1)-bi-ideal of S. But every (m, (p, q), n)-bi-ideal of a ternary semiring S need not be a bi-ideal of S which is illustrated by the following example.

Example 2.4. Let Z− \ {−1} be the set of all negative integers excluding {0}. Then Z− \ {−1} is aternary semiring under following binary addition and ternary multiplication:

a + b = max{a, b} and a · b · c = abc (usual ternary multiplication).

Consider B = {−3,−27} ∪ {k ∈ Z− : k ≤ −110}. Clearly B is a non empty ternary subsemiring of Sand also B is (3, (1, 1), 4)-bi-ideal of S. Now −108 ∈ BSBSB. But −108 �∈ B. Therefore BSBSB �B. Hence B is not a bi-ideal of Z− \ {−1}.

Theorem 2.5. Let S be a ternary semiring and Bi be an (m, (p, q), n)-bi-ideals of S such that⋂

i∈I

Bi �= ∅. Then⋂

i∈I

Bi is an (m, (p, q), n) bi-ideal of S.

Proof. It is straight forward. �

Remark. Let Z− be the set of all negative integers. Then Z− is a ternary semiring under usual binaryaddition and ternary multiplication and Bi = {k ∈ Z− : k ≤ −i} for all i ∈ I. Then Bi is an (3, (1, 1), 4)-bi-ideal of Z− for all i ∈ I. But

⋂

i∈I

Bi = ∅. So condition⋂

i∈I

Bi �= ∅ is necessary.

Theorem 2.6. Every (m, (p, q), n)-quasi-ideal of a ternary semiring S is an (m, (p, q), n)-bi-ideal of S.

Proof. Let Q be an (m, (p, q), n)-quasi-ideal of S. Then

Q(SS)m−1SpQSq(SS)n−1Q ⊆ Q(SS)m−1SpSSq(SS)n−1S ⊆ Q(SS)m.

Similarly,

Q(SS)m−1SpQSq(SS)n−1Q ⊆ S(SS)m−1(SpQSq)(SS)n−1S ⊆ Sp+1QSq+1.

Again {0} ⊆ SpQSq. So

Q(SS)m−1SpQSq(SS)n−1Q ⊆ SpQSq + Sp+1QSq+1.

Also,

Q(SS)m−1SpQSq(SS)n−1Q ⊆ S(SS)m−1SpSSq(SS)n−1Q ⊆ (SS)nQ.

Consequently,

Q(SS)m−1SpQSq(SS)n−1Q ⊆ Q(SS)m ∩ (SpQSq + Sp+1QSq+1) ∩ (SS)nQ ⊆ Q.

Hence Q is an (m, (p, q), n)-bi-ideal of S. �

Remark. Every (m, (p, q), n)-bi-ideal need not be an (m, (p, q), n)-quasi-ideal of S which is illus-trated by the following example.

Example 2.7. Consider the ternary semiring Z− \ {−1} defined in Example 2.4 and let B ={−3,−27} ∪ {k ∈ Z− : k ≤ −194}. Clearly, B is non empty ternary subsemiring of S and also B

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172 DUBEY, ANURADHA

is (2, (1, 1), 3)-bi-ideal of S. Now, −192 ∈ B(SS)2 ∩ (SBS + SSBSS) ∩ (SS)3B. But −192 /∈ B.Therefore B(SS)2 ∩ (SBS + SSBSS) ∩ (SS)3B � B. Hence B is not (2, (1, 1), 3)-quasi-ideal of S.

Theorem 2.8. A ternary subsemiring B of a regular ternary semiring S is an (m, (p, q), n)-bi-ideal of S if and only if B = BSB.

Proof. Suppose B is an (m, (p, q), n)-bi-ideal of a regular ternary semiring S. Let b ∈ B. Then thereexists x ∈ S such that b = bxb. This implies that b ∈ BSB. Hence B ⊆ BSB. Now,

BSB ⊆ BSBSBSBSB ⊆ B(SS)(SBS)(SS)B ⊆ B.

Therefore B = BSB.Conversely, if B = BSB, then

B(SS)m−1SpBSq(SS)n−1B ⊆ B(SS)m−1SpSSq(SS)n−1B ⊆ BSB = B.

Hence B is an (m, (p, q), n)-bi-ideal of S. �

Theorem 2.9. Let S be a regular ternary semiring. Then every (m, (p, q), n)-bi-ideal of S is an(m, (p, q), n)-quasi-ideal of S.

Proof. Let B be an (m, (p, q), n)-bi-ideal of S. Let a ∈ B(SS)m ∩ (SpBSq + SpSBSSq) ∩(SS)nB. Then a ∈ B(SS)m, a ∈ SpBSq + SpSBSSq and a ∈ (SS)nB. Thus a = bi(SS)m = Spb′iS

q +SpSb′′i SSq = (SS)nb′′′i for some bi, b

′i, b

′′i , b

′′′i ∈ B. Since S is regular, therefore for a ∈ S there exists an

element x in S such that a = axa. Then

a = axa = axaxa = bi(SS)mx(Spb′iSq + SpSb′′i SSq)x(SS)nb′′i

∈ B(SS)mS(SpBSq + SpSBSSq)S(SS)nB

= B(SS)mSSpBSqS(SS)nB + B(SS)mSSpSBSSqS(SS)nB

⊆ B[(SS)mSSpSSqS(SS)n]B + B[(SS)mSSpSSSSqS(SS)n]B⊆ BSB + BSB = B + B ⊆ B.

Thus a ∈ B. Therefore B(SS)m ∩ (SpBSq + SpSBSSq) ∩ (SS)nB ⊆ B. Hence B is an (m, (p, q), n)-quasi-ideal of S. �

It is easy to prove the following propositions:Proposition 2.10. The intersection of an (m, (p, q), n)-bi-ideal B of a ternary semiring S with

a ternary subsemiring T of S is either empty or an (m, (p, q), n)-bi-ideal of T .Proposition 2.11. Let B be an (m, (p, q), n)-bi-ideal of a ternary semiring S and T1, T2 are two

ternary subsemirings of S. Then BT1T2, T1BT2 and T1T2B are (m, (p, q), n)-bi-ideals of S.Proposition 2.12 Let B1, B2 and B3 are three (m, (p, q), n)-bi-ideals of a ternary semiring S.

Then B1B2B3 is an (m, (p, q), n)-bi-ideal of S.Proposition 2.13. Let Q1, Q2 and Q3 are three (m, (p, q), n)-quasi-ideals of a ternary semiring

S. Then Q1Q2Q3 is an (m, (p, q), n)-bi-ideal of S.Proposition 2.14. Let R be an m-right, M be an (p, q)-lateral and L be an n-left ideal of a

ternary semiring S. Then the ternary subsemiring B = RML of S is an (m, (p, q), n)-bi-ideal of S.Theorem 2.15. Let S be a regular ternary semiring. If B is an (m, (p, q), n)-bi-ideal of S, then

B(SS)m−1SpBSq(SS)n−1B = B.Proof. Let B be an (m, (p, q), n)-bi-ideal of S. Let a ∈ B. Then a ∈ S. Since S is regular, therefore

there exists x ∈ S such that a = axa. Now a = axa = a(xa)(xax)(ax)a ∈ B(SS)(SBS)(SS)B.Similarly, by property of regularity it is easy to show that a ∈ B(SS)m−1SpBSq(SS)n−1B. Thus, B ⊆B(SS)m−1SpBSq(SS)n−1B. Since B is an (m, (p, q), n)-bi-ideal of S, thereforeB(SS)m−1SpBSq(SS)n−1B ⊆ B. Hence B(SS)m−1SpBSq(SS)n−1B = B �

Corollary 2.16. Let S be a regular ternary semiring. If Q is an (m, (p, q), n)-quasi-ideal of S, thenQ(SS)m−1SpQSq(SS)n−1Q = Q.

Proof. Since every (m, (p, q), n)-quasi-ideal of S is an (m, (p, q), n)-bi-ideal of S, therefore resultfollows directly. �

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ON (m, (p, q), n)-BI-IDEALS IN TERNARY SEMIRINGS 173

3. MINIMAL (m, (p, q), n)-Bi-IDEALS

In this section, we introduce the concept of minimal (m, (p, q), n)-bi-ideals in ternary semirings.Definition 3.1. An (m, (p, q), n)-bi-ideal B of a ternary semiring S is called minimal (m, (p, q), n)-

bi-ideal of S if B does not properly contain any (m, (p, q), n)-bi-ideal of S.Lemma 3.2. Let S be a ternary semiring and a ∈ S. Then the following holds:(i) a(SS)m−1 is an m-right ideal of S.(ii) SpaSq is an (p, q)-lateral ideal of S.(iii) (SS)n−1a is an n-left ideal of S.(iv) a(SS)m−1SpaSq (SS)n−1a is an (m, (p, q), n)-bi-ideal of S.Proof. (i), (ii) and (iii) are obvious and (iv) follows from (i), (ii), (iii) and Proposition 2.14. �

Theorem 3.3. Let S be a ternary semiring and B be an (m, (p, q), n)-bi-ideal of S. Then B isminimal if and only if B is the product of some minimal m-right ideal R, minimal (p, q)-lateralideal M and minimal n-left ideal L of S.

Proof. Suppose B is minimal (m, (p, q), n)-bi-ideal of S. Let a ∈ B. Then by above Lemma,a(SS)m−1 is an m-right ideal, SpaSq is an (p, q)-lateral ideal, (SS)n−1a is an n-left ideal anda(SS)m−1SpaSq(SS)n−1a is an (m, (p, q), n)-bi-ideal of S. Now a(SS)m−1SpaSq(SS)n−1a ⊆B(SS)m−1SpB Sq(SS)n−1B ⊆ B. Since B is minimal, therefore a(SS)m−1SpaSq(SS)n−1a = B. Nowto show that a(SS)m−1 is minimal m-right ideal of S. Let R be an m-right ideal of S contained ina(SS)m−1. Then R(SpaSq)(SS)n−1a ⊆ a(SS)m−1(SpaSq)(SS)n−1a = B. Since R SpaSq(SS)n−1ais an (m, (p, q), n)-bi-ideal of S and B is minimal, therefore R(SpaSq)(SS)n−1a = B. This impliesB ⊆ R. Therefore a(SS)m−1 ⊆ B(SS)m−1 ⊆ R(SS)m−1 ⊆ R. Thus a(SS)m−1 is minimal. Similarlywe can prove that SpaSq is minimal (p, q)-lateral ideal of S and (SS)n−1a is minimal n-left ideal of S.

Conversely, assume that B = RML for some minimal m-right ideal R, minimal (p, q)-lateral idealM and minimal n-left ideal L. So B ⊆ R, B ⊆ M and B ⊆ L. Let B′ be an (m, (p, q), n)-bi-ideal ofS contained in B. Then B′(SS)m−1 ⊆ B(SS)m−1 ⊆ R(SS)m−1 ⊆ R. Similarly, SpB′Sq ⊆ SpBSq ⊆SpMSq ⊆ M and (SS)n−1B′ ⊆ (SS)n−1B ⊆ (SS)n−1L ⊆ L. Now, B′(SS)m−1(SS)m ⊆ B′(SS)m−1.So B′(SS)m−1 is an m-right ideal of S. Similarly SpB′Sq is an (p, q)-lateral ideal and (SS)n−1B′ isan n-left ideal of S. Since R, M and L are minimal m-right ideal, minimal (p, q)-lateral ideal andminimal n-left ideal of S respectively, therefore B′(SS)m−1 = R, SpB′Sq = M and (SS)n−1B′ = L.Thus B = RML = B′(SS)m−1SpB′Sq(SS)n−1B′ ⊆ B′. Hence B = B′. Consequently, B is minimal(m, (p, q), n)-bi-ideal of S. �

Definition 3.4. Let S be a ternary semiring. Then S is called a bi-simple ternary semiring if S is theunique (m, (p, q), n)-bi-ideal of S.

Theorem 3.5. Let S be a ternary semiring and B be an (m, (p, q), n)-bi-ideal of S. Then B is aminimal (m, (p, q), n)-bi-ideal of S if and only if B is a bi-simple ternary semiring.

Proof. Suppose B is a minimal (m, (p, q), n)-bi-ideal of S. Let C be an (m, (p, q), n)-bi-ideal of B.Then C(BB)m−1BpCBq(BB)n−1C ⊆ C ⊆ B. By Proposition 2.11, BCC is an (m, (p, q), n)-bi-idealof S. Therefore (BCC)(SS)m−1Sp(BCC)Sq(SS)n−1BCC ⊆ BCC ⊆ BBB ⊆ B. Since B is minimal,therefore BCC = B. It is easy to show that C(BB)m−1BpCBq(BB)n−1C is an (m, (p, q), n)-bi-idealof S.

Since B is minimal, therefore C(BB)m−1BpCBq(BB)n−1C = B. This implies B =C(BB)m−1BpCBq(BB)n−1C ⊆ C. Hence C = B. Consequently, B is a bi-simple ternary semiring.

Conversely, suppose B is a bi-simple ternary semiring. Let C be an (m, (p, q), n)-bi-ideal of S suchthat C ⊆ B. Then C(BB)m−1BpCBq(BB)n−1C ⊆ C(SS)m−1SpCSq(SS)n−1C ⊆ C which impliesthat C is an (m, (p, q), n)-bi-ideal of B. Since B is bi-simple ternary semiring, therefore C = B. HenceB is minimal. �

ACKNOWLEDGMENT

The authors are thankful to Dr. S.K. Bhambri for his stimulant discussion and valuable suggestionsduring the preparation of the paper.

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REFERENCES1. F. Szasz, Generalized bi-ideals of Rings I, Diese Nachr. 47, 355 (1970).2. M. K. Dubey and Anuradha, On (m, (p, q), n)-Quasi-ideals in Ternary Semirings, International Journal of

Pure and Applied Mathematics, 73 (4), 443, 2011.3. R. A. Good and D. R. Hughes, Associated Groups for a Semigroup, Bull. Amer. Math. Soc. 58, 624 (1952).4. S. Kar, On Quasi-ideals and bi-ideals in Ternary Semirings, Int. J. Math. Math. Sci. 18, 3015 (2005).5. T. K. Dutta and S. Kar, On regular ternary semirings, Advances in Algebra, Proceedings of the ICM

Satellite conference in Algebra and Related Topics (World Scientific, New Jersey, 2003), p. 343.

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