Research Article Characterizations of Regular Ordered Semirings by Ordered Quasi-Ideals

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1 International Mathematics and Mathematical Sciences Volume 2016, Article ID , 8 pages Research Article Characterizations of Regular Ordered Semirings by Ordered Quasi-Ideals Pakorn Palakawong na Ayutthaya 1,2 and Bundit Pibaljommee 1,2 1 Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand 2 Centre of Excellence in Mathematics CHE, Si Ayutthaya Road, Bangkok 10400, Thailand Correspondence should be addressed to Bundit Pibaljommee; banpib@kku.ac.th Received 27 October 2015; Revised 22 December 2015; Accepted 27 December 2015 AcademicEditor:HowardE.Bell Copyright 2016 P. Palakawong na Ayutthaya and B. Pibaljommee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce the notion of an ordered quasi-ideal of an ordered semiring and show that ordered quasi-ideals and ordered bi-ideals coincide in regular ordered semirings. Then we give characterizations of regular ordered semirings, regular ordered duo-semirings, and left (right) regular ordered semirings by their ordered quasi-ideals. 1. Introduction The concept of a quasi-ideal was defined first by Steinfeld for semigroups and for rings [1 3] as a generalization of a right ideal and a left ideal. Then Iséki [4] introduced the notion of a quasi-ideal in a semiring without zero and investigated some of its properties. In 1994, Dönges [5] studied quasi-ideals of a semiring with zero, investigated connections between left (right) ideals, bi-ideals, and quasi-ideals and characterized regular semirings using their quasi-ideals. Later, Shabir et al. [6] have studied some properties of quasi-ideals, using quasiideals to characterize regular and intraregular semirings and regular duo-semirings. As a generalization of quasi-ideals of semirings the quasi-ideals of Γ-semirings were investigated by many authors; see, for example, [7 9]. In 2011, the notion of an ordered semiring was introduced by Gan and Jiang [10] as a semiring with a partially ordered relation on the semiring such that the relation is compatible to the operations of the semiring. In the paper, the concept of a left (right) ordered ideal, a minimal ordered ideal, and a maximal ordered ideal was defined. Then Mandal [11] studied fuzzy ideals in an ordered semiring with the least element zero and gave a characterization of regular ordered semirings by their fuzzy ideals. In this paper, we introduce the notion of an ordered quasi-ideal of an ordered semiring and show that ordered quasi-ideals and ordered bi-ideals coincide in regular ordered semirings. Then characterizations of regular ordered semirings, regular ordered duo-semirings, and left (right) regular ordered semirings by their ordered quasi-ideals have been investigated. 2. Preliminaries An ordered semiring is a system (S,+,, ) consisting of a nonempty set S such that (S, +, ) is a semiring, (S, ) is a partially ordered set, and for any a, b, x S the following conditions are satisfied: (i) if a bthen a+x b+xand x+a x+b; (ii) if a bthen ax bx and xa xb. An ordered semiring S is said to be additively commutative if a+b = b+afor all a, b S. Anelement0 Sis said to be an absorbing zero if 0a=0=a0and a+0 = a = 0+a for all a S. In this paper we assume that S is an additively commutative ordered semiring with an absorbing zero 0. For any subsets A, B of S and a S, we denote (A] = {x S x afor some a A}, AB = {ab S a A,b B},

2 2 International Mathematics and Mathematical Sciences ΣA = { a i S a i i I Aand I is a finite subset of N}, ΣAB = { a i b i S a i A,b i i I Band I is a finite subset of N}, Proof. Assume that A (ΣA 2 + ΣASA].Then ΣA 2 Σ(ΣA 2 + ΣASA] A Σ((ΣA 2 )A+(ΣASA) A] Σ (ΣA 3 + ΣASA] =(Σ(ΣA 3 )+Σ(ΣASA)] =(ΣA 3 + ΣASA] = (ΣAAA + ΣASA] (ΣASA + ΣASA] = (ΣASA]. (2) Na =Σ{a}. Now, we mention some properties of finite sums on an ordered semiring. Remark 1. For any subsets A, B of S,thefollowingstatements hold: (i) Σ(A] (ΣA]; (ii) Σ(ΣA) = ΣA; (iii) A(ΣB) ΣAB and (ΣA)B ΣAB; (iv) Σ(AΣB) ΣAB and Σ(ΣA)B ΣAB; (v) Σ(A + B) = ΣA + ΣB. We note that, for any A S, ΣA = A if and only if A+A A ((A, +) is a subsemigroup of (S, +)). Now, we give the basic properties of the operator (] which are not difficult to verify. Lemma 2. Let A, B, C besubsetsofanorderedsemirings. Then the following statements hold: (i) A (A] and ((A]] = (A]; (ii) If A Bthen (A] (B]; (iii) A(B] (A](B] (AB] and (A]B (A](B] (AB]; (iv) A+(B] (A]+(B] (A+B] and (A]+B (A]+(B] (A + B]; (v) A(B + C] (AB + AC] and (A + B]C (AC + BC]; (vi) (A B] = (A] (B]; (vii) (A B] (A] (B]. In (vii) of the above lemma, we have (A B] = (A] (B] when (A] = A and (B] = B. Lemma 3. Let S be an ordered semiring and 0 A (ΣA 2 + ΣASA] then ΣA 2 (ΣASA]. (1) =A S.If Definition 4 (see [10]). Let S be an ordered semiring and 0 = A S.ThenA is said to be a left ordered ideal (right ordered ideal) if the following conditions are satisfied. (1) A is a left ideal (right ideal) of S. (2) If x afor some a Athen x A(i.e., A = (A]). We call A an ordered ideal if it is both left ordered ideal and right ordered ideal of S. Example 5 (see [10]). Let [0, 1] be the unit interval of real numbers. Define binary operations and on [0, 1] by letting a, b [0, 1], a b=max {a, b}, a b=max {a+b 1,0}, and an ordered relation is the natural order on real numbers. It is easy to show that L = ([0,1],,, ) is an ordered semiring. Let I=[0,1/2]. Then we can prove that I is an ordered ideal of L. Lemma 6. Let A beanonemptysubsetofanorderedsemiring S.Then (i) (ΣSA] is a left ordered ideal of S; (ii) (ΣAS] is a right ordered ideal of S; (iii) (ΣSAS] is an ordered ideal of S. Proof. (i) Let x, y (ΣSA]. Thenx x and y y for some x,y ΣSA.Itisclearthatx+y x +y ΣSA, and so x+y (ΣSA].ByRemark1andLemma2,weobtain S(ΣSA] (SΣSA] (ΣSSA] (ΣSA]. Wehave((ΣSA]] = (ΣSA].Hence,(ΣSA] is a left ordered ideal of S. (ii) and (iii) can be proved similar to (i). Corollary 7. Let S be an ordered semiring. Then, for any a S, (i) (Sa] is a left ordered ideal of S; (ii) (as] is a right ordered ideal of S; (iii) (ΣSaS] is an ordered ideal of S. Let A be a nonempty subset of an ordered semiring S. We denote L(A), R(A) and I(A) as the smallest left ordered ideal, right ordered ideal, and ordered ideal of S containing (3)

3 International Mathematics and Mathematical Sciences 3 A, respectively. In particular, we can show that if A is a left ideal (right ideal, ideal) of S then (A] is the smallest left ordered ideal (resp., right ordered ideal and ordered ideal) of S containing A. Lemma 8. Let A beanonemptysubsetofanorderedsemiring S.Then (i) L(A) = (ΣA + ΣSA]; (ii) R(A) = (ΣA + ΣAS]; (iii) I(A) = (ΣA + ΣSA + ΣAS + ΣSAS]. Proof. (i) Since S hasanabsorbingzero,wehave,forevery a A, a = a + 0 ΣA + ΣSA (ΣA + ΣSA]. Hence,A (ΣA + ΣSA].Letx, y (ΣA + ΣSA].Thenx x and y y for some x,y ΣA+ΣSA.Thusx =a 1 +b 1 and y =a 2 +b 2 for some a 1,a 2 ΣAand b 1,b 2 ΣSA.Itiseasytoshowthat a 1 +a 2 ΣA and b 1 +b 2 ΣSA.Itfollowsthatx+y x +y ΣA + ΣSA,andsox+y (ΣA+ΣSA].ByRemark1 and Lemma 2, we obtain S (ΣA + ΣSA] (S (ΣA + ΣSA)] (SΣA + SΣSA] (ΣSA + ΣSSA] (ΣSA + ΣSA] = (ΣSA] (ΣA + ΣSA]. Since ((ΣA + ΣSA]] = (ΣA + ΣSA], L is a left ordered ideal of S. LetK be any left ordered ideal of S containing A. Itturns out ΣA K and ΣSA K,soΣA + ΣSA K.Itfollowsthat (ΣA + ΣSA] (K] = K. Therefore, (ΣA + ΣSA] is the smallest left ordered ideal of S containing A. (ii) and (iii) can be proved similar to (i). As a special case of Lemma 8, if A = {a} then we have the following corollary. Corollary 9. Let S be an ordered semiring. Then, for any a S, (i) L(a) = (Na+Sa]; (ii) R(a) = (Na+aS]; (iii) I(a) = (Na + Sa + as + ΣSaS]. An element e of an ordered semiring S is said to be an identity if ea=a=ae for all a S.IfS has an identity, then we denote 1 as the identity of S. It is not difficult to show that if S has an identity, then L(A) = (ΣSA], R(A) = (ΣAS] and I(A) = (ΣSAS] for any A S.Inparticularcase,wehaveL(a) = (Sa], R(a) = (as] and I(a) = (ΣSaS] for any a S. 3. Ordered Quasi-Ideals in Ordered Semirings Here, we present a notion of an ordered quasi-ideal of an ordered semiring. Then, in ordered semiring with an identity, we show that every ordered quasi-ideal can be expressed as an intersection of an ordered left ideal and an ordered right ideal. Definition 10. Let (S,+,, ) be an ordered semiring and let (Q, +) be a subsemigroup of (S, +). ThenQ is said to be (4) an ordered quasi-ideal of S if the following conditions are satisfied: (1) (ΣSQ] (ΣQS] Q; (2) if x qfor some q Qthen x Q(i.e., Q = (Q]). It is clear that every left ordered ideal (right ordered ideal and ordered ideal) of an ordered semiring S is an ordered quasi-ideal of S. Moreover, each ordered quasi-ideal of S is asubsemiringofs; indeed, QQ (QQ] (SQ] (QS] (ΣSQ] (ΣQS] Q. Example 11. Let S = {a, b, c, d}. Define binary operations + and on S by the following equations: + a b c d a a b c d b b b b b, c c b c d d d b d d a b c d a a a a a b a b b b. c a c c c d a b b b Then (S,+, )is an additively commutative semiring with an absorbing zero a. Define a binary relation on S by (5) fl {(a, a), (b, b), (c, c), (d, d), (b, d)}. (6) We give the covering relation and the figure of S: a fl {(b, d)}. (7) d b Now, (S,+,, ) is an ordered semiring. Let Q = {a, b}. We have (ΣSQ] (ΣQS] = {a, b, c} {a, b} = Q and (Q] = Q. Hence, Q is an ordered quasi-ideal of S but is not a left ordered ideal of S,sinceSQ = {a, b, c} Q. Lemma 12. Let S be an ordered semiring and let {Q i i I}be a family of ordered quasi-ideals of S.Then i I Q i is an ordered quasi-ideal of S. Let A be a nonempty subset of an ordered semiring S.We denote Q(A) the smallest ordered quasi-ideal of S containing A. Theorem 13. Let S be an ordered semiring and let A be a nonempty subset of S.ThenQ(A) = (ΣA + ((ΣSA] (ΣAS])]. c

4 4 International Mathematics and Mathematical Sciences Proof. Let Q = (ΣA + ((ΣSA] (ΣAS])]. SinceS has an absorbing zero, we have a = a+0 ΣA+((ΣSA] (ΣAS]) Q for every a A.Hence,A Q.Letx, y Q. Thenx x and y y for some x,y ΣA + ((ΣSA] (ΣAS]). Thus x = a 1 +b 1 and y = a 2 +b 2 for some a 1,a 2 ΣA and b 1,b 2 (ΣSA] (ΣAS]. Clearly,a 1 +a 2 ΣA and b 1 +b 2 (ΣSA] (ΣAS]. Itfollowsthatx+y x +y ΣA + ((ΣSA] (ΣAS]), andsox+y Q. By Remark 1 and Lemma 2, we obtain (ΣSQ] (ΣQS] (ΣSQ] = (ΣS (ΣA + ((ΣSA] (ΣAS])]] (ΣS (ΣA + (ΣSA]]] (Σ (SΣA + S (ΣSA]]] (Σ (ΣSA + (ΣSSA]]] (Σ (ΣSA + ΣSSA]] (Σ (ΣSA + ΣSA]] (Σ (ΣSA]] ((ΣSA]] = (ΣSA]. Similarly, we can show that (ΣSQ] (ΣQS] (ΣAS]. Thus (ΣSQ] (ΣQS] (ΣSA] (ΣAS] ΣA+((ΣSA] (ΣAS]) Q. Since (Q] = Q, we obtain that Q is an ordered quasi-ideal of S containing A. LetK be any ordered quasi-ideal of S containing A.Itfollowsthat(ΣSA] (ΣAS] (ΣSK] (ΣKS] K.SoΣA + ((ΣSA] (ΣAS]) K.Hence,Q = (ΣA + ((ΣSA] (ΣAS])] (K] = K. Therefore, Q is the smallest ordered quasi-ideal of S containinga. As a special case of Theorem 13, if A = {a} then we have the following corollary. Corollary 14. Let S be an ordered semiring. Then Q(a) = (Na + ((Sa] (as])] for any a S. If S hasanidentity,thenitiseasytocheckthatq(a) = (ΣSA] (ΣAS] for any A S.Inparticularcase,wehave Q(a) = (Sa] (as] for any a S. Let Q(S) be the set of all ordered quasi-ideals of an ordered semiring S. Using Lemma 12, we define the operations and on Q(S) by letting P 1,P 2 Q(S), P 1 P 2 =P 1 P 2, P 1 P 2 =Q(P 1 P 2 ). Then we obtain the following theorem. Theorem 15. Let S be an ordered semiring. Then (Q(S),, ) is a complete lattice. Theorem 16. The intersection of a left ordered ideal L and a right ordered ideal R of an ordered semiring S is an ordered quasi-ideal of S. (8) (9) Proof. It is easy to show that L Ris a subsemigroup of (S, +). By Remark 1 and Lemma 2, we obtain (ΣS (L R)] (Σ (L R) S] (ΣS (L R)] = (Σ (SL SR)] (ΣSL] L, (ΣS (L R)] (Σ (L R) S] (Σ (L R) S] = (Σ (LS RS)] (ΣRS] R. (10) Hence, (ΣS(L R)] (Σ(L R)S] L R.Lets Ssuch that s xfor some x L R.Thens (L R] (L] (R] = L R. The converse of Theorem 16 is not true as Example 2.1 page 8 in [2] given by A. H. Clifford. Corollary 17. Let S be an ordered semiring. Then the following statements hold. (i) (ΣSA] (ΣAS] is an ordered quasi-ideal of S, forany A S. (ii) (Sa] (as] is an ordered quasi-ideal of S,foranya S. Proof. (i) By Lemma 6, we have (ΣSA] and (ΣAS] aleftanda right ordered ideal of S,respectively.ThenbyTheorem16,we have that (ΣSA] (ΣAS] is an ordered quasi-ideal of S. (ii) It is a particular case of (i). Now, we will show that the converse of Theorem 16 is true if S contains an identity as the following theorem. Theorem 18. Let S be an ordered semiring with identity. Then every ordered quasi-ideal Q of S can be written in the form Q= R Lfor some right ordered ideal R andleftorderedideall of S. Proof. Assume that S has an identity. Let Q be an ordered quasi-ideal of S.ThenR(Q) = (ΣQS] and L(Q) = (ΣSQ].We obtain Q R(Q) L(Q) and R(Q) L(Q) = (ΣQS] (ΣSQ] Q.Hence,Q = R(Q) L(Q). 4. Regular Ordered Semirings In this section, we show that in regular ordered semirings the converse of Theorem 16 is true and ordered quasi-ideals coincide with ordered bi-ideals. Then we give characterizations of regular ordered semirings, regular ordered duo-semirings, and left regular and right regular ordered semirings by their ordered quasi-ideals. Definition 19 (see [11]). An element a of an ordered semiring S is said to be regular if a axafor some x S. An ordered semiring S is said to be regular if every element a S is regular. The following lemma is characterizations of regular ordered semiring which directly follows Definition 19.

5 International Mathematics and Mathematical Sciences 5 Lemma 20. Let S be an ordered semiring. Then the following statements are equivalent: (i) S is regular; (ii) A (ΣASA] for each A S; (iii) a (asa] for any a S. Now, we will show that the converse of Theorem 16 is true in regular ordered semirings. Theorem 21. Every ordered quasi-ideal of a regular ordered semiring S can be written in the form Q=R Lfor some right ordered ideal R and left ordered ideal L of S. Proof. Let Q be an ordered quasi-ideal of S.ByLemma8,we have R(Q) = (ΣQ + ΣQS] and L(Q) = (ΣQ + ΣSQ].Now,Q R(Q) L(Q).Letq Q.SinceS is regular, there exists x S such that q qxq ΣQS.SoQ (ΣQS]. SinceQ+Q Q, ΣQ = Q.Itfollowsthat (ΣQS] (ΣQ + ΣQS] = (Q + ΣQS] ((ΣQS] +ΣQS] (ΣQS]. (11) This implies that R(Q) = (ΣQS]. Similarly, we can show that L(Q) = (ΣSQ]. Hence,R(Q) L(Q) = (ΣQS] (ΣSQ] Q. Therefore, Q = R(Q) L(Q). Definition 22. Let (S,+,, ) be an ordered semiring. A subsemigroup (B, +) of (S, +) is said to be an ordered bi-ideal of S if the following conditions hold: (1) BSB B; (2) if x bfor some b B,thenx B(i.e., B = (B]). We note that condition (1) of Definition 22 is equivalent to ΣBSB B. Theorem 23. Every ordered quasi-ideal of an ordered semiring S is an ordered bi-ideal of S. Proof. Let Q be an ordered quasi-ideal of S. ThenΣQSQ ΣQS (ΣQS] and QSQ ΣSQ (ΣSQ]. So,ΣQSQ (ΣSQ] (ΣQS] Q.Hence,Q is an ordered bi-ideal of S. The converse of Theorem 23 is not generally true as the following example. Example 24. Let S = {a, b, c, d, e}. Define binary operations + and by the following equations: + a b c d e a a b c d e b b b d d d, c c d d d d d d d d d d e e d d d e a b c d e a a a a a a b a a a a a. c a a b b b d a a b b b e a a b b b (12) Then (S,+, )is an additively commutative semiring with an absorbing zero a. Define a binary relation on S by fl {(a, a), (b, b), (c, c), (d, d), (e,e), (a, b), (a, c), (a, e), (a, d), (b, d), (c, d), (e,d)}. We give the covering relation and the figure of S: (13) fl {(a, b), (a, c), (a, e), (b, d), (c, d), (e,d)}. (14) b c d a Then (S,+,, )is an ordered semiring but not regular, since d dxdfor any x S.LetB = {a, e}. Itiseasytoshowthat B is an ordered bi-ideal but not an ordered quasi-ideal of S, since (ΣSB] (ΣBS] = {a, b} B. Now, we show that in regular ordered semirings, ordered bi-ideals and ordered quasi-ideals coincide as the following theorem. Theorem 25. Let S be a regular ordered semiring. Then ordered bi-ideals and ordered quasi-ideals coincide in S. Proof. By Theorem 23, we have that every ordered quasiideal of S is an ordered bi-ideal of S. Now,weshowthat every ordered bi-ideal of S is an ordered quasi-ideal of S. Let B be an ordered bi-ideal of S. Leta (ΣSB] (ΣBS]. By Lemma 20, Remark 1, and Lemma 2, we obtain a (asa] ((ΣBS]S(ΣSB]] ((ΣBSS](ΣSB]] ((ΣBS)(ΣSB)] (Σ(BS(ΣSB))] (Σ(ΣBSSB)] (ΣBSB] B. Hence,B is an ordered quasi-ideal of S. Theorem 26. Let S be an ordered semiring. Then the following statements are equivalent: (i) S is regular; (ii) (ΣRL] = R L for every right ordered ideal R and left ordered ideal L of S; (iii) B = (ΣBSB] for each ordered bi-ideal B of S; (iv) Q = (ΣQSQ] for each ordered quasi-ideal Q of S. e

6 6 International Mathematics and Mathematical Sciences Proof. (i) (ii): assume that S is regular and let R and L be a right ordered ideal and a left ordered ideal of S, respectively. So, (ΣRL] (ΣR] = R and (ΣRL] (ΣL] = L.Hence, (ΣRL] R L. Leta R L.SinceS is regular, a axa for some x S.Sincea R, xa L. Itfollowsthat a a(xa) RL.Thismeansa (RL] (ΣRL]. Therefore, (ΣRL] = R L. (ii) (iii): assume that (ii) holds. Let B be an ordered bi-ideal of S. Itisclearthat(ΣBSB] B. By assumption, B R(B) L(B) = (ΣR(B)L(B)]. By Lemma 8, Remark 1, and Lemmas 2 and 3, we have B (ΣB + ΣBS] (ΣB + ΣSB] = (Σ ((ΣB + ΣBS] (ΣB + ΣSB])] (Σ ((ΣB + ΣBS)(ΣB + ΣSB))] (Σ (ΣB (ΣB + ΣSB) +ΣBS(ΣB + ΣSB))] (Σ(ΣB 2 + ΣBSB + ΣBSB + ΣBSSB)] (Σ(ΣB 2 + ΣBSB)] = (Σ (ΣB 2 )+Σ(ΣBSB)] =(ΣB 2 + ΣBSB] ((ΣBSB] +ΣBSB] (ΣBSB]. (15) (iii) (iv): it follows from Theorem 23. (iv) (i): leta S.ThenQ(a) = (Q(a)SQ(a)]. By Corollary 14, Remark 1, and Lemma 2, we have a (Na+((Sa] (as])] = (Σ ((Na+((Sa] (as])] S (Na+((Sa] (as])])] (Σ ((Na+(aS]] S (Na+(Sa]])] (Σ ((Na+aS] S (Na+Sa])] (Σ (((Na+aS) S] (Na+Sa])] (Σ ((as] (Na+Sa])] (Σ (as (Na+Sa))] (ΣaSa] = (asa]. By Lemma 20, S is regular. (16) Theorem 27. Let S be a regular ordered semiring. Then the following statements hold: (i) every ordered quasi-ideal Q of S can be written in the form Q = R L = (RL] for some right ordered ideal R and left ordered ideal L of S; (ii) (Q 2 ]=(Q 3 ] for each ordered quasi-ideal Q of S. Proof. (i) It is obvious by Theorems 21 and 26. (ii) Let Q be an ordered quasi-ideal of S. Clearly, ((QQ)Q] (QQ]. Letx (QQ].Thenx q 1 q 2 for some q 1,q 2 Q.SinceS is regular, there exists s Ssuch that x q 1 q 2 (q 1 q 2 )s(q 1 q 2 ) QQSQQ.Hence,x (Q(QSQ)Q] (QQQ]. Therefore, (Q 2 ]=(Q 3 ]. Theorem 28. Let S be an ordered semiring. Then S is regular if and only if B I L (BIL] for every ordered bi-ideal B, every ordered ideal I, and every left ordered ideal L of S. Proof. Let B, I,andL be an ordered bi-ideal, an ordered ideal, and a left ordered ideal of S,respectively.Leta B I L.Since S is regular, a axa axaxaxa BIL. Hence,B I L (BIL]. Conversely, assume that B I L (BIL] for every ordered bi-ideal B, every ordered ideal I, and every left ordered ideal L of S.ThenweobtainR L = R S L (RSL] (RL] (ΣRL] foreveryrightorderedidealr and left ordered ideal L of S.On the other hand, we have (ΣRL] R L.Hence,(ΣRL] = R L. By Theorem 26, S is regular. Definition 29. An ordered semiring S is said to be an ordered duo-semiring if every one-sided (right or left) ordered ideal of S is an ordered ideal of S. We note that every multiplicatively commutative ordered semiring is an ordered duo-semiring, but the converse is not generally true. Now, we give an example of a multiplicatively noncommutative ordered semiring which is an ordered duosemiring. Example 30. Let S = {a, b, c, d, e}. Define binary operations + and by the following equations: + a b c d e a a b c d e b b c c c c, c c c c c c d d c c c c e e c c c c a b c d e a a a a a a b a e c e c. c a c c c c d a c c e c e a c c c c (17) Then (S,+, )is an additively commutative semiring with an absorbing zero a. Define a binary relation on S by fl {(a, a), (b, b), (c, c), (d, d), (e,e),(e,c)}. (18) We give the covering relation andthefigureofs: fl {(e,c)}. (19) c e a b d Then (S,+,, ) is an ordered semiring which is not multiplicatively commutative, since bd =db.wehaveallone-sided ordered ideals of S which are as follows: {a}, {a, c}, {a, c, e}, {a, b, c, e}, {a, c, d, e},s. (20)

7 International Mathematics and Mathematical Sciences 7 It is not difficult to check that all of them are ordered ideals of S.ThisshowsthatS is an ordered duo-semiring. Lemma 31. Let S be an ordered semiring. Then the following conditions are equivalent: (i) S is an ordered duo-semiring; (ii) R(A) = L(A) for each A S; (iii) R(a) = L(a) for each a S. Proof. (i) (ii) and (ii) (iii) are obvious. (iii) (i): letl be a left ordered ideal of S and let x L,s S. By assumption, we have xs R(x)S R(x) = L(x) L(L) = L.ItfollowsthatL is a right ordered ideal of S. Similarly, we have that every right ordered ideal of S is a left ordered ideal of S.Hence,S is an ordered duo-semiring. Theorem 32. Let S be an ordered duo-semiring. Then S is regular if and only if (ΣQ 1 Q 2 ]=Q 1 Q 2 for each two ordered quasi-ideals Q 1 and Q 2 of S. Proof. Assume that S is a regular ordered semiring. Let Q 1 and Q 2 be ordered quasi-ideals of S. ByTheorem21,Q 1 and Q 2 canbewrittenintheforms Q 1 =R 1 L 1, (21) Q 2 =R 2 L 2 for some R 1,R 2 and L 1,L 2 which are right ordered ideals and left ordered ideals of S,respectively.SinceS is an ordered duosemiring, R 1,R 2,L 1,andL 2 are ordered ideals of S.Itfollows that Q 1 and Q 2 are ordered ideals of S. ByTheorem26,we have (ΣQ 1 Q 2 ]=Q 1 Q 2. Conversely, assume that (ΣQ 1 Q 2 ] = Q 1 Q 2 for each two ordered quasi-ideals Q 1 and Q 2 of S. LetA S. By assumption, A Q(A) Q(A) = (ΣQ(A)Q(A)]. By Theorem 13, Remark 1, and Lemmas 2 and 3, we have A (Σ ((ΣA + ((ΣSA] (ΣAS])] (ΣA + ((ΣSA] (ΣAS])])] (Σ ((ΣA + (ΣAS]] (ΣA + (ΣSA]])] (Σ ((ΣA + ΣAS] (ΣA + ΣSA])] (Σ ((ΣA + ΣAS)(ΣA + ΣSA))] (Σ (ΣA (ΣA + ΣSA) +ΣAS(ΣA + ΣSA))] (Σ(ΣA 2 + ΣASA + ΣASA + ΣASSA)] (Σ(ΣA 2 + ΣASA)] = (Σ (ΣA 2 )+Σ(ΣASA)] =(ΣA 2 + ΣASA] ((ΣASA] +ΣASA] (ΣASA]. By Lemma 20, S is a regular ordered semiring. (22) Theorem 33. Let S be an ordered duo-semiring. Then the following conditions are equivalent: (i) S is regular; (ii) (ΣL 1 L 2 ]=L 1 L 2 and (ΣR 1 R 2 ]=R 1 R 2 for each two left ordered ideals L 1,L 2 andrightorderedideals R 1,R 2 of S; (iii) (ΣRL] = R L = (ΣLR], for each right ordered ideal R and left ordered ideal L of S. Proof. It is obvious by Theorem 26. Definition 34. Let S be an ordered semiring. Then an element a Sis said to be left regular (right regular) if a xa 2 (a a 2 x) for some x S. An ordered semiring S is said to be left regular (right regular) if every element a S is left regular (right regular). Example 35. Let S = {a, b, c, d, e, f}. Define binary operations +and on S by the following equations: + a b c d e f a a b c d e f b b b b b b b c c c c c c c, d d d d d d d e e e e e e e f f f f f f f a b c d e f a a a a a a a b a b b b b b c a b b b b b. d a b b d b d e a e e e e e f a e e f e f (23) Then (S,+, )is a semiring with an absorbing zero a.definea binary relation on S by fl {(a, a), (b, b), (c, c), (d, d), (e,e),(f,f),(b, c), (24) (b, d), (b, e),(b,f),(c, d), (c, e),(c,f),(d,f),(e,f)}. We give the covering relation andthefigureofs: fl {(b, c), (c, d), (c, e), (d, f), (e,f)}. (25) d f Now, (S,+,, )is an ordered semiring. Clearly, a, b, d, e,and f are left regular. We consider c ec 2 =eb=e. This implies c b e a

8 8 International Mathematics and Mathematical Sciences that S is left regular. Since there does not exist x Ssuch that c cxc, S is not regular. Example 36. Consider the ordered semiring S = (N {0}, max, min, ) where is the natural order relation on numbers. Since n min{n, n, n} for any n N, wegets a regular, left regular, and right regular ordered semiring. In case A (ΣA 2 +ΣSA 2 ],weget ΣA 2 ΣA(ΣA 2 +ΣSA 2 ] Σ(AΣA 2 + AΣSA 2 ] Σ(ΣA 3 +ΣASA 2 ] Σ(ΣA 3 +ΣSA 2 ] (Σ(ΣA 3 ) + Σ (ΣSA 2 )] (ΣA 3 +ΣSA 2 ] (27) The following lemmas can be easily proved using Definition 34. Lemma 37. Let S be an ordered semiring. Then the following statements are equivalent: (i) S is left regular; (ii) A (ΣSA 2 ] for each A S; (iii) a (Sa 2 ] for each a S. Lemma 38. Let S be an ordered semiring. Then the following statements are equivalent: (i) S is right regular; (ii) A (ΣA 2 S] for each A S; (iii) a (a 2 S] for each a S. Definition 39. Let T be a nonempty subset of an ordered semiring S. ThenT is said to be semiprime if for any a S, if a 2 T,thena T. We note that a nonempty subset T of S is semiprime if and only if, for any 0 =A S, A 2 Timplies A T.Because, if T is semiprime, 0 =A T,anda Athen a 2 Tand so a T;thatis,A T. Theorem 40. Let S be an ordered semiring. Then S is left regular and right regular if and only if every ordered quasi-ideal of S is semiprime. Proof. Let Q be an ordered quasi-ideal of S. LetA be a nonempty subset of S such that A 2 Q.SinceS is left regular and by Lemma 37, we have A (ΣSA 2 ].SinceS is right regular and by Lemma 38, we have A (ΣA 2 S]. Hence, A (ΣSA 2 ] (ΣA 2 S] (ΣSQ] (ΣQS] Q. Therefore, Q is semiprime. Conversely, assume that every ordered quasi-ideal of S is semiprime. Let A S.ByTheorem13,wehaveQ(A 2 )= (ΣA 2 + ((ΣSA 2 ] (ΣA 2 S])].SinceA 2 Q(A 2 ) is semiprime, A Q(A 2 ) = (ΣA 2 + ((ΣSA 2 ] (ΣA 2 S])].Thenweobtain A (ΣA 2 +((ΣSA 2 ] (ΣA 2 S])] (ΣA 2 + (ΣSA 2 ]] (ΣA 2 +ΣSA 2 ], A (ΣA 2 +((ΣSA 2 ] (ΣA 2 S])] (ΣA 2 +(ΣA 2 S]] (ΣA 2 +ΣA 2 S]. (26) (ΣSA 2 +ΣSA 2 ]=(ΣSA 2 ] and so A (ΣA 2 +ΣSA 2 ] ((ΣSA 2 ] + ΣSA 2 ] (ΣSA 2 ]. By Lemma 37, S is left regular. Similarly, in case A (ΣA 2 + ΣA 2 S],wegetA (A 2 S].ByLemma38,S is right regular. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgment This work has been supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. References [1] O. Steinfeld, On ideal-quotients and prime ideals, Acta Mathematica Academiae Scientiarum Hungaricae,vol.4,pp , [2] O. Steinfeld, Quasi-Ideals in Rings and Semigroups, vol. 10 of Disquisitiones Mathematicae Hungaricae, Akadémiai Kiadó, Budapest,Hungary,1978. [3] O. Steinfeld, Über die Quasiideale von Halbgruppen, Publicationes Mathematicae Debrecen, vol. 4, pp , [4] K. Iséki, Quasi-ideals in semirings without zero, Proceedings of the Japan Academy, vol. 34, pp , [5] C. Dönges, On Quasi-ideals of semirings, International Journal of Mathematics and Mathematical Sciences,vol.17,no.1,pp , [6] M. Shabir, A. Ali, and S. Batool, A note on quasi-ideals in semirings, Southeast Asian Bulletin of Mathematics,vol.27, no. 5,pp ,2004. [7] R. Chinram, A note on quasi-ideals in Γ-semirings, International Mathematical Forum, vol.3,no.25 28,pp , [8] R. Chinram, A note on quasi-ideals in regular Γ-semirings, International Contemporary Mathematical Sciences, vol. 3, no. 35, pp , [9] R. D. Jagatap and Y. S. Pawar, Quasi-ideals and minimal quasiideals in Γ-semirings, Novi Sad Mathematics, vol. 39, no. 2, pp , [10] A. P. Gan and Y. L. Jiang, On ordered ideals in ordered semirings, Mathematical Research & Exposition, vol. 31, no. 6, pp , [11] D. Mandal, Fuzzy ideals and fuzzy interior ideals in ordered semirings, Fuzzy Information and Engineering, vol. 6, no. 1, pp , 2014.

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