We introduce the notion of ordered quasi-Γ-ideals of regular ordered Γ-semigroups and study the basic properties of ordered quasi-Γ-ideals of ordered Γ-semigroups. We also characterize regular ordered Γ-semigroups in terms of their ordered quasi-Γ-ideals, ordered right Γ-ideals, and left Γ-ideals. Finally, we have shown that (i) a partially ordered Γ-semigroup S is regular if and only if for every ordered bi-Γ-ideal B, every ordered Γ-ideal I, and every ordered quasi-Γ-ideal Q, we have B∩I∩Q⊆BΓIΓQ and (ii) a partially ordered Γ-semigroup S is regular if and only if for every ordered quasi-Γ-ideal Q, every ordered left Γ-ideal L, and every ordered right-Γ-ideal R, we have that R∩Q∩L⊆RΓQΓL.

1. Introduction

Steinfeld [1–3] introduced the notion of a quasi-ideal for semigroups and rings. Since then, this notion has been the subject of great attention of many researchers and consequently a series of interesting results have been published by extending the notion of quasi-ideals to Γ-semigroups, ordered semigroups, ternary semigroups, semirings, Γ-semirings, regular rings, near-rings, and many other different algebraic structures [4–15].

It is a widely known fact that the notion of a one-sided ideal of rings and semigroups is a generalization of the notion of an ideal of rings and semigroups and the notion of a quasi-ideal of semigroups and rings is a generalization of a one-sided ideal of semigroups and rings. In fact the concept of ordered semigroups and Γ-semigroups is a generalization of semigroups. Also the ordered Γ-semigroup is a generalization of Γ-semigroups. So the concept of ordered quasi-ideals of ordered semigroups is a generalization of the concept of quasi-ideals of semigroups. In the same way, the notion of an ordered quasi-ideal of ordered semigroups is a generalization of a one-sided ordered ideal of ordered semigroups. Due to these motivating facts, it is naturally significant to generalize the results of semigroups to Γ-semigroups and of Γ-semigroups to ordered Γ-semigroups.

In 1998, the concept of an ordered quasi-ideal in ordered semigroups was introduced by Kehayopulu [16]. He studied theory of ordered semigroups based on ordered ideals analogous to the theory of semigroups based on ideals. The concept of po-Γ-semigroup was introduced by Kwon and Lee in 1996 [17] and since then it has been studied by several authors [18–22]. Our purpose in this paper is to examine many important classical results of ordered quasi-Γ-ideals in ordered Γ-semigroups and then to characterize the regular ordered Γ-semigroups through ordered quasi-Γ-ideals, ordered bi-Γ-ideals and ordered one-sided Γ-ideals.

2. Preliminaries

We note here some basic definitions and results that are relevant for our subsequent results.

Let S and Γ be two nonempty sets. Then S is called a Γ-semigroup if S satisfies (aγb)μc=aγ(bμc) for all a, b, c∈S and γ, μ∈Γ. A nonempty subset N of a Γ-semigroup S is called a sub-Γ-semigroup of S if aαb∈N for all a, b∈N and α∈Γ. For any nonempty subsets A, B of S, AΓB={aαb:a∈A, b∈B and α∈Γ}. We also denote {a}ΓB, AΓ{b}, and {a}Γ{b}, respectively, by aΓB, AΓb, and aΓb. Many classical results of semigroups have been generalized and extended to Γ-semigroups [23–25]. By an ordered Γ-semigroup S (also called po-Γ-semigroups), we mean an ordered set (S,≤), at the same time a Γ-semigroup satisfying the following conditions:
(1)a≤b⟹aγc≤bγc,cγa≤cγb∀a,b,c∈S,γ∈Γ.

Throughout this paper, S will stand for an ordered Γ-semigroup unless otherwise stated. An ordered Γ-semigroup S is called regular if for each s∈S and for each α, β∈Γ there exists a∈S such that s≤sαaβs. Equivalent definitions of regular ordered Γ-semigroup are as follows: (i) A⊆(AΓSΓA] for each A⊆S and (ii) s∈(sΓSΓs] for each s∈S. Let (S;≤) be an ordered Γ-semigroup and N a sub-Γ-semigroup of S; then (N;≤) is an ordered Γ-semigroup. Let A be a nonempty subset of N. Then similarly to [26], we write (A]N={n∈N:n≤a for some a∈A} and A∪a=A∪{a}. We also write (A]N by simply (A] if N=S (see [27]). A nonempty subset I of an ordered Γ-semigroup S is called an ordered right-Γ-ideal (left-Γ-ideal) of S if IΓS⊆I(SΓI⊆I), and for any x∈I, (x]⊆I. I is called an ordered Γ-ideal of S if it is both a left and a right Γ-ideals of S. Also for any s∈S, we have that (SΓs] is an ordered left Γ-ideal of S and (sΓS] is an ordered right Γ-ideal of S [18]. A nonempty subset Q of S is called an ordered quasi-Γ-ideal of S if (i) (QΓS]∩(SΓQ]⊆Q and (ii) (Q]⊆Q. A sub-Γ-semigroup B of an ordered Γ-semigroup S is called an ordered bi-Γ-ideal of S if BΓSΓB⊆B and for any x∈B, (x]⊆B.

Let X be a nonempty subset of S. Then the least right (left) ordered Γ-ideal of S containing X is given by R(X)=(X∪XΓS](L(X)=(SΓX∪X]). If X={s}, s∈S, we write R{s} and L{s}, respectively, by R(s) and L(s), and R(s)=(s∪sΓS], L(s)=(SΓs∪s] and the ideal generated by s∈S is given by I(s)=(s∪SΓs∪sΓS∪SΓsΓS]. Also, the least quasi-Γ-ideal of S containing X is denoted by Q(X). Moreover, we willl need some notations as follows: (i) NQ={Q:Q≠∅, where Q⊆S and (Q]⊆Q}, (ii) RI is a set of ordered right Γ-ideals of S, (iii) LI is a set of ordered left Γ-ideals of S, and (iv) IT is a two-sided Γ-ideal of S.

Now for any two elements Q1, Q2∈NQ, we define an operation * in NQ as follows:
(2)Q1*Γ*Q2=(Q1ΓQ2].

Further, let N be a sub-Γ-semigroup of S. Then we can easily observe here the following (see [16, 18, 21, 28–30]):

A⊆(A]N⊆(A]=((A]] for A⊆N,

for A⊆N and B⊆N, we have (A∪B]=(A]∪(B],

for A⊆N and B⊆N, we have (A∩B]⊆(A]∩(B],

for a and b∈N with a≤b, we have (aΓN]⊆(bΓN] and (NΓa]⊆(NΓb],

(A]Γ(B]⊆(AΓB],

for every left (right, two-sided) ideal L of S, (L]=L,

if A and B are ordered Γ-ideals of S, then (AΓB] and A∪B are also ideals of S,

In this section, we study some classical properties of the ordered Γ-semigroup S. We start with the following lemma.

Lemma 1.

Let S be an ordered Γ-semigroup. Then,

(NQ,*,⊆) is an ordered Γ-semigroup;

(LI,*,⊆), (RI,*,⊆), and (IT,*,⊆) are sub-Γ-semigroups of (NQ,*,⊆).

Proof.

(i) Suppose P, Q, R∈NQ. Since PΓQ∈(PΓQ], we obtain ((PΓQ)ΓR]⊆((PΓQ]ΓR]. Next, we have (P*Γ*Q)*Γ*R=(PΓQΓR] by using (P*Γ*Q)*Γ*R=(PΓQ]*Γ*R=((PΓQ]ΓR]⊆((PΓQ)ΓR]=(PΓQΓR]. In a similar way, we can show that P*Γ*(Q*Γ*R)=(PΓQΓR] and therefore (P*Γ*Q)*Γ*R=P*Γ*(Q*Γ*R). Hence (NQ,*) is a Γ-semigroup. Suppose P⊆Q. Then P*Γ*R=(PΓQ]⊆(QΓR]=Q*Γ*R and R*Γ*P=(RΓP]⊆(RΓQ]=R*Γ*Q. Hence (NQ,*,⊆) is an ordered Γ-semigroup.

(ii) We have that LI, RI, and IT are nonempty subsets of NQ. Suppose L1, L2∈LI. Then, obviously, we have (L1*Γ*L2]=((L1ΓL2]]=(L1ΓL2]. Moreover, using
(3)SΓ(L1*Γ*L2)=SΓ(L1ΓL2]⊆(SΓ(L1ΓL2]]⊆((SΓL1)ΓL2]⊆(L1ΓL2]=L1*Γ*L2,
we infer that L1*Γ*L2 is a left Γ-ideal of S; that is, L1*Γ*L2∈LI. Thus (LI,*,⊆) is a sub-Γ-semigroup of (NQ,*,⊆).

Dually, we can prove that (RI,*,⊆) is a sub-Γ-semigroup of (NQ,*,⊆). Since IT=LI∩RI, it follows that (IT,*,⊆) is a sub-Γ-semigroup of (NQ,*,⊆).

Let QI={Q:Q is an ordered quasi-Γ-ideal of S}. Then, obviously we have LI∪RI⊆QI⊆NQ. This implies that every one-sided Γ-ideal of an ordered Γ-semigroup is a quasi-Γ-ideal of S. Thus the class of ordered quasi-Γ-ideals of S is a generalization of the class of one-sided ordered Γ-ideals of S.

Lemma 2.

Each ordered quasi-Γ-ideal Q of an ordered Γ-semigroup S is a sub-Γ-semigroup of S.

Proof.

Proof is straightforward. In fact, we have QΓQ⊆QΓS∩SΓQ⊆(QΓS]∩(SΓQ]⊆Q.

Lemma 3.

For every ordered right Γ-ideal R and an ordered left Γ-ideal L of an ordered Γ-semigroup S, R∩L is an ordered quasi-Γ-ideal of S.

Proof.

As RΓL⊆SΓL⊆L and RΓL⊆RΓS⊆R, we obtain RΓL⊆R∩L, so R∩L≠∅. Now the fact that R∩L is an ordered quasi-Γ-ideal of S follows from the following:

(R∩L]⊆(R]∩(L]⊆R∩L,

((R∩L)ΓS]∩(SΓ(R∩L)]⊆(RΓS]∩(SΓL]⊆(R]∩(L]⊆R∩L.

Lemma 4.

Let Q be an ordered quasi-Γ-ideal of S, then one obtains Q=L(Q)∩R(Q)=(SΓQ∪Q]∩(Q∪QΓS].

Proof.

The following relation
(4)Q⊆(SΓQ∪Q]∩(Q∪QΓS]isobvious.

Conversely, suppose a∈(SΓQ∪Q]∩(Q∪QΓS]. Then a≤b or a≤xαu and a≤vβy for some b, u, v∈Q, x, y∈S, and α, β∈Γ. As Q is an ordered quasi-Γ-ideal of S, the former case implies that a∈(Q]⊆Q and the latter case implies that a∈(SΓQ]∩(QΓS]⊆Q. Therefore (SΓQ∪Q]∩(Q∪QΓS]=Q.

We recall here that if X is a nonempty subset of an ordered Γ-semigroup S, then we write the least quasi-ideal of S containing X by Q(X). If X={a}, we write Q({a}) by Q(a).

Theorem 5.

Suppose S is an ordered Γ-semigroup. Then one has the following:

for every s∈S, Q(s)=L(s)∩R(s)=(SΓs∪s]∩(s∪sΓS],

let ∅≠X⊆S, Q(X)=L(X)∩R(X)=(SΓX∪X]∩(X∪XΓS].

Proof of (i).

Suppose s∈S. Using Lemma 3, L(s)∩R(s) is a quasi-Γ-ideal of S containing s; therefore Q(s)⊆L(s)∩R(s), and by Lemma 4, we obtain
(5)L(s)∩R(s)=(SΓs∪s]∩(s∪sΓS]⊆(SΓQ(s)∪Q(s)]∩(Q(s)∪Q(s)ΓS]=Q(s).

Hence Q(s)=L(s)∩R(s).

Proof of (ii).

Its proof can be given as (i).

The notion of a bi-Γ-ideal of Γ-semigroups is a generalization of the notion of a quasi-Γ-ideal of Γ-semigroups. Similarly, the class of ordered quasi-Γ-ideals of ordered Γ-semigroups is a particular case of the class of ordered bi-Γ-ideals of ordered Γ-semigroups. This is what we have shown in the following result.

Theorem 6.

Suppose I is a two-sided ordered Γ-ideal of an ordered Γ-semigroup S and Q is a quasi-Γ-ideal of I; then Q is an ordered bi-Γ-ideal of S.

Proof.

Since Q is an ordered quasi-Γ-ideal of I and Q⊆I, we obtain
(6)QΓQ⊆QΓSΓI=QΓ(SΓI)⊆QΓI⊆(QΓI]⊆(SΓI]⊆(I]⊆I,QΓSΓQ⊆IΓQΓS=(IΓS)ΓQ⊆IΓQ⊆(IΓQ]⊆(IΓS]⊆(I]⊆I,
and q∈(Q]⇒ There exists q1∈Q⊆I such that q≤q1⇒q∈(I]=I and q∈(Q]⇒q∈I∩(Q]=(Q]I⊆Q.

Therefore,
(7)QΓSΓQ⊆(I∩(IΓQ])∩(I∩(QΓI])=(IΓQ]I∩(QΓI]I⊆Q,(Q]⊆Q.
Hence applying these facts together with Lemma 2, we have shown that Q is an ordered bi-Γ-ideal of S.

In this section, we use the concept of ordered quasi-Γ-ideals to characterize regular ordered Γ-semigroups.

Lemma 7.

Let S be an ordered Γ-semigroup. Then the ordered sub-Γ-semigroup of (NQ,*) generated by (LI,*) and (RI,*) is in the following form:
(8)〈LI∪RI〉=LI∪RI∪(RI*Γ*LI).

Proof.

One can easily see that
(9)〈LI∪RI〉={Y1*Γ1*Y2*⋯*Yn-1*Γn-1*Yn∣Yj∈LIorYj∈RI,j=1⋯n,n∈Z+,Γj∈Γ}.

Suppose Yj,Yj+1∈LI∪RI. Then the conditions that arise are as follows: (i) Yj,Yj+1∈LI: in this condition by Lemma 1, we obtain Yj*Γ*Yj+1∈LI; (ii) Yj, Yj+1∈RI: in this condition, Yj*Γ*Yj+1∈RI by also Lemma 1; (iii) Yj∈LI, Yj+1∈RI: in this condition, Yj*Γ*Yj+1=(YjΓYj+1] is an ordered Γ-ideal of S, so Yj*Γ*Yj+1∈IT=LI∩RI; (iv) Yj∈RI, Yj+1∈LI: in this condition, Yj*Γ*Yj+1∈RI*Γ*LI in (NQ,*). Therefore for any Y1,…,Yn∈LI∪RI, where n∈Z+, using (i)–(iv), there arise three conditions as follows.

If Y1∈LI, then Y1*Γ1*Y2*⋯*Yn-1*Γn-1*Yn∈LI.

If Yn∈RI, then Y1*Γ1*Y2*⋯*Yn-1*Γn-1*Yn∈RI.

If Y1∈RI and Yn∈LI, where n≥2, then Y1*Γ1*Y2*⋯*Yn-1*Γn-1*Yn∈RI*Γ*LI. Hence the lemma holds.

Theorem 8.

Let S be an ordered Γ-semigroup. Then the following assertions on S are equivalent.

S is a regular ordered Γ-semigroup.

For every ordered left Γ-ideal L and every ordered right Γ-ideal R, one has
(10)(RΓL]=R∩L.

For every ordered right Γ-ideal R and ordered left Γ-ideal L of S,

(RΓR]=R,

(LΓL]=L,

(RΓL] is an ordered quasi-Γ-ideal of S.

(LI,*) and (RI,*) are ordered idempotent Γ-semigroups and (QI,*) is the sub-Γ-semigroup of (NQ,*) generated by (LI,*) and (RI,*).

(QI,*) is a regular ordered sub-Γ-semigroup of the Γ-semigroup (NQ,*).

Every ordered quasi-Γ-ideal Q of S is given by Q=(QΓSΓQ].

(QI,*,⊆) is a regular sub-Γ-semigroup of the ordered Γ-semigroup of (NQ,*,⊆).

Proof.

(i)⇒(ii) Suppose R and L are ordered right and left Γ-ideals of S, respectively; then we have
(11)(RΓL]⊆R∩L.
Let S be regular; we need to prove only that R∩L⊆(RΓL]. Suppose a∈R∩L. Since S is regular, we obtain a≤aαxβa for some x∈S and α, β∈Γ, and so a∈R and xαa∈L; therefore aαxβa∈RΓL. Therefore a∈(RΓL], and thus R∩L⊆(RΓL].

(ii)⇒(iii)(RΓL] is an ordered quasi-Γ-ideal of S that follows directly from Lemma 3 and the condition (ii). As the ordered two-sided Γ-ideal of S is generated by R=(R∪SΓR], the condition (ii) implies that
(12)R=R∩(R∪SΓR]=(RΓ(R∪SΓR]];therefore(RΓR]⊆(RΓ(R∪SΓR]]=R.
Conversely, suppose a∈(RΓ(R∪SΓR]]. Then a≤rαb for r∈R and b∈(R∪SΓR]. From b∈(R∪SΓR], we have b≤c, where c=r′∈R or c=sαr′′ for some s∈S and r′′∈R. Therefore a≤rαc=rαr′∈RΓR or a≤rαc=rα(sβr′′)=(rαs)γr′′∈RΓR for α,β,γ∈Γ; thus a∈(RΓR]. Thus R⊆(RΓR], so that (RΓR]=R. Similarly we can prove that (LΓL]=L dually.

(iii)⇒(iv) The conditions (1), (2) in (iii) and Lemma 7 show that (LI,*) and (RI,*) are idempotent Γ-semigroups, respectively. Applying (iii) (3), we obtain RI*Γ*LI⊆QI; therefore 〈LI∪RI〉⊆QI in (NQ,*).

Conversely, suppose Q∈QI. Then (Q∪SΓQ] is the ordered left Γ-ideal of S generated by Q. The condition (iii) (2) implies that
(13)Q⊆(Q∪SΓQ]=((Q∪SΓQ]Γ(Q∪SΓQ]]⊆(QΓQ∪SΓQΓQ∪QΓSΓQ⊆⊆∪(SΓQ)Γ(SΓQ)]⊆(SΓQ].

We can dually prove that Q⊆(QΓS]. Therefore using these facts and Lemma 4, it follows that
(14)(a)Q⊆(SΓQ]∩(QΓS]⊆(SΓQ∪Q]∩(Q∪QΓS]=Q.
Therefore for Q∈QI, we have Q=(SΓQ]∩(QΓS], and the condition (iii) (3) together with (a) implies that
(15)(b)(RΓL]=(SΓ(RΓL]]∩((RΓL]ΓS].hhhhhhhhhhhhhhh

Moreover, by the assertion (iii) (2), we have S=(SΓS] and
(16)(SΓQ]=((SΓQ]2]=((SΓQ]Γ(SΓQ]]=((SΓQ]Γ((SΓS]ΓQ]]⊆(SΓQΓSΓSΓQ]⊆(SΓ(QΓS]Γ(SΓQ]]⊆(SΓ((QΓS]Γ(SΓQ]]]⊆(SΓ(QΓSΓSΓQ)]⊆(SΓQ].

Therefore (SΓQ]=(SΓ((QΓS]Γ(SΓQ]]]. Dually, we can prove that
(17)(QΓS]=(((QΓS]Γ(SΓQ]]ΓS].

From these facts, (a) and (b), we obtain
(18)(c)Q=(QΓS]∩(SΓQ]=(((QΓS]Γ(SΓQ]]ΓS]∩(SΓ((QΓS]Γ(SΓQ]]]=((QΓS]Γ(SΓQ]]=(QΓS]*Γ*(SΓQ]∈RI*Γ*LI⊆〈LI∪RI〉
by Lemma 7. Therefore QI⊆〈LI∪RI〉. Hence QI=〈LI∪RI〉 in (NQ,*).

(iv)⇒(iii) It is a consequence of Lemma 7.

(iii)⇒(v) By (iii)⇒(iv), we have (b) and (c). Suppose Q1, Q2 are two ordered quasi-Γ-ideals of S. Then (SΓ(Q1ΓQ2]∪(Q1ΓQ2]] is the least ordered left Γ-ideal of S containing (Q1ΓQ2]. Then the condition (iii) (2) implies that
(19)(Q1ΓQ2]⊆(SΓ(Q1ΓQ2]∪(Q1ΓQ2]]=((SΓ(Q1ΓQ2]∪(Q1ΓQ2]]2]⊆(SΓ(Q1ΓQ2]]=((SΓS]Γ(Q1ΓQ2]]⊆(SΓ(SΓ(Q1ΓQ2]]].

Dually one can prove that (Q1ΓQ2]⊆((Q1ΓQ2]∪(Q1ΓQ2]ΓS]⊆(((Q1ΓQ2]ΓS]ΓS]. These facts together with (b) show that
(20)(Q1ΓQ2]⊆(SΓ(Q1ΓQ2]∪(Q1ΓQ2]]∩((Q1ΓQ2]∪(Q1ΓQ2]ΓS]⊆(SΓ(SΓ(Q1ΓQ2]]]∩(((Q1ΓQ2]ΓS]ΓS]=(((Q1ΓQ2]ΓS]Γ(SΓ(Q1ΓQ2]]]⊆((Q1Γ(Q2ΓSΓS)ΓQ1)ΓQ2]⊆(Q1ΓQ2].

By Theorem 5 (ii), (Q1ΓQ2]=(SΓ(Q1ΓQ2]∪(Q1ΓQ2]]∩((Q1ΓQ2]∪(Q1ΓQ2]ΓS] is an ordered quasi-Γ-ideal of S; therefore Q1*Γ*Q2∈QI. Hence (QI,*) is a sub-Γ-semigroup of (NQ,*). For every Q∈QI, by (c), we obtain Q=((QΓS]Γ(SΓQ]]⊆(QΓSΓSΓQ]⊆(QΓSΓQ]⊆Q, and so Q=(QΓSΓQ]=Q*Γ*S*Γ*Q, where S∈QI. Thus (QI,*) is a regular sub Γ-semigroup of (NQ,*).

(v)⇒(vi) Suppose Q is an ordered quasi-Γ-ideal of S. Applying the condition (iv), there is an ordered quasi-Γ-ideal Q1 of S so that, by Lemma 4,
(21)Q=Q*Γ*Q1*Γ*Q=(QΓQ1ΓQ]⊆(QΓSΓQ]⊆(SΓQ]∩(QΓS]⊆(SΓQ∪Q]∩(Q∪QΓS]=Q,

and therefore Q=(QΓSΓQ].

(vi)⇒(vii) It is straightforward.

(vii)⇒(i) For every s∈S, using Theorem 5, R(s)∩L(s) is an ordered quasi-Γ-ideal of S containing s. By (vii), there exists Q∈QS so that
(22)s∈R(s)∩L(s)⊆(R(s)∩L(s))*Γ*Q*Γ*(R(s)∩L(s))=((R(s)∩L(s))ΓQΓ(R(s)∩L(s))]⊆(R(s)ΓSΓL(s)]=((s∪sΓS]ΓSΓ(SΓs∪s]]⊆(sΓSΓs].

Hence S is a regular ordered Γ-semigroup.

Lemma 9.

Every two-sided ordered Γ-ideal I of a regular ordered Γ-semigroup S is a regular sub-Γ-semigroup of S.

Proof.

Suppose i∈I. As S is regular, there exists s∈S so that, for α,β,γ,δ∈Γ, we have
(23)i≤iαsβi≤iαsβiγsδi=iα(sβiγs)δi.
As sαiβs∈SΓIΓS⊆I, we observe that i∈(iΓIΓi]I.

Theorem 10.

Suppose S is a regular ordered Γ-semigroup. Then the following statements are true.

Every ordered quasi-Γ-ideal of S can be expressed as follows:
(24)Q=R∩L=(RΓL],
where R and L are, respectively, the ordered right and left Γ-ideals of S generated by Q.

Let Q be an ordered quasi-Γ-ideal of S; then (QΓQ]=(QΓQΓQ].

Every ordered bi-Γ-ideal of S is an ordered quasi-Γ-ideal of S.

Every ordered bi-Γ-ideal of any ordered two sided-Γ-ideal of S is a quasi-Γ-ideal of S.

For every L1, L2∈LI and R1, R2∈RI, one obtains
(25)L1∩L2⊆(L1ΓL2],R1∩R2⊆(R1ΓR2].

Proof.

Because S is a regular ordered Γ-semigroup, then by Lemma 4 and Theorem 8, the statement (i) is done. Since (QΓQΓQ]⊆(QΓQ] is always true, we need to show that (QΓQ]⊆(QΓQΓQ]. We have that (QΓQ] is also an ordered quasi-Γ-ideal of S by Theorem 8. Moreover we have the following equation:(26)(QΓQ]=(QΓQΓSΓQΓQ]=(QΓ(QΓSΓQ)ΓQ]⊆(QΓQΓQ].

Suppose Q1 is an ordered bi-Γ-ideal of S. Then (SΓQ1] is an ordered left Γ-ideal and (Q1ΓS] is an ordered right Γ-ideal of S. Applying Theorem 8, we obtain
(27)(SΓQ1]∩(Q1ΓS]=((Q1ΓS]Γ(SΓQ1]]⊆(Q1ΓSΓQ1]⊆(Q1]⊆Q1.
Therefore Q1 is an ordered quasi-Γ-ideal of S.

Suppose I is a two-sided ordered Γ-ideal of S and B is an ordered bi-Γ-ideal of I. By the relation (iii) and Lemma 9, B is an ordered quasi-Γ-ideal of I; therefore using Theorem 6, B is an ordered bi-Γ-ideal of S. Also from the relation (iii) again, we obtain B as an ordered quasi-Γ-ideal of S.

Lastly, suppose L1, L2∈LI. Because S is regular and L1∩L2 is an ordered quasi-Γ-ideal of S, using Theorem 8, we obtain
(28)L1∩L2=((L1∩L2)ΓSΓ(L1∩L2)]⊆(L1Γ(SΓL2)]⊆(L1ΓL2].

Dually, we can prove that R1∩R2⊆(R1ΓR2] for all R1, R2∈RI.

Theorem 11.

A partially ordered Γ-semigroup S is regular if and only if for every ordered bi-Γ-ideal B, every ordered Γ-ideal I, and every ordered quasi-Γ-ideal Q, one has
(29)B∩I∩Q⊆(BΓIΓQ].

Proof.

Let S be regular. Then for any a∈B∩I∩Q there exists s∈S such that
(30)a≤aαsβa≤(aαsβa)γsδ(aαsβa)=(aαsβa)γ(sαaβs)δa∈(BΓB)Γ(SΓIΓS)ΓQ⊆BΓIΓQ.
Hence a∈(BΓIΓQ], where α,β,γ,δ∈Γ.

Conversely, let B∩I∩Q⊆(BΓIΓQ] for every ordered bi-Γ-ideal B, every ordered Γ-ideal I, and every ordered quasi-Γ-ideal Q of S. Suppose s∈S. Let B(s) and Q(s) be the ordered bi-Γ-ideal and ordered quasi-Γ-ideal of S generated by s, respectively. So we have the following:
(31)s∈B(s)∩I(s)∩Q(s)⊆(B(s)ΓI(s)ΓQ(s)]⊆((s∪sΓSΓs]ΓSΓ(s∪(SΓs∩sΓS)]]⊆((s∪sΓSΓs]ΓSΓ(s∪SΓs]]⊆(sΓSΓs].
Hence S is regular.

Next consider R in place of Q in Theorem 11 to obtain the following.

Corollary 12.

An ordered Γ-semigroup S is regular if and only if for every ordered bi-Γ-ideal B, every ordered Γ-ideal I, and every right Γ-ideal R of S,
(32)B∩I∩R⊆(BΓIΓR].

Theorem 13.

A partially ordered Γ-semigroup S is regular if and only if for every ordered quasi-Γ-ideal Q, every ordered left Γ-ideal L, and every ordered right-Γ-ideal R, one has
(33)R∩Q∩L⊆(RΓQΓL].

Proof.

Let S be regular; then for any a∈R∩Q∩L, there exists s∈S such that a≤aαsβa≤(aαsβa)γsδ(aθsλa)=(aαs)βaγ(sδaθsλa)∈(RΓS)ΓQΓ(SΓLΓSΓL)⊆RΓQΓL, for α,β,γ,θ,δ,λ∈Γ. Hence a∈(RΓQΓL].

Conversely, let
(34)R∩Q∩L⊆(RΓQΓL],
for every ordered right Γ-ideal R, every ordered quasi-Γ-ideal Q, and every ordered left Γ-ideal L of S. Suppose s∈S. So we have
(35)s∈R(s)∩Q(s)∩L(s)⊆(R(s)ΓQ(s)ΓL(s)]⊆(R(s)ΓSΓL(s)]⊆(R(s)ΓL(s)]⊆((s∪sΓS]Γ(s∪SΓs]]⊆((sΓs∪sΓSΓs]].
So for α,β,γ∈Γ, s≤sαs or s≤sαxβs for some x∈S. If s≤sαs, then s≤sαs≤(sαs)β(sγs)=sα(sβs)2γs∈sΓSΓs. If s≤sαxβs for some x∈S, then s∈sΓSΓs. So, finally we obtain s∈(sΓSΓs]. Hence S is regular.

Corollary 14.

If one considers an ordered left Γ-ideal L (or an ordered right Γ-ideal R) in place of the ordered quasi-Γ-ideal Q in Theorem 13, one obtains
(36)L∩R⊆(RΓL].

Acknowledgment

The authors are grateful to the referee for the useful comments and valuable suggestions.

SteinfeldO.On ideal-quotients and prime idealsSteinfeldO.Über die Quasiideale von HalbgruppenSteinfeldO.ChinramR.A note on Quasi-ideals in Γ-semiringsChinramR.On quasi gamma-ideals in Γ-semigroupsDöngesC.On Quasi-ideals of semiringsCliffordA. H.Remarks on o-minimal Quasi-ideals in semigroupsChoosuwanP.ChinramR.A study on Quasi-ideals in ternary semigroupsDixitV. N.DewanS.Minimal Quasi-ideals in ternary semigroupDixitV. N.DewanS.A note on quasi and bi-ideals in ternary semigroupsIsekiK.Quasi-ideals in semirings without zeroJagatapR. D.PawarY. S.Quasi-ideals and minimal quasi-ideals in Γ-semiringsKehayopuluN.LajosS.LepourasG.A note on bi- and Quasi-ideals of semigroups, ordered semigroupsLajosS.On quasiideals of regular ringYakabeI.Quasi-ideals in near-ringsKehayopuluN.On completely regular ordered semigroupsKwonY. I.LeeS. K.Some special elements in ordered Γ-semigroupsIampanA.SiripitukdetM.On minimal and maximal ordered left ideals in PO-Γ-semigroupsIampanA.Characterizing ordered bi-ideals in ordered Γ-semigroupsIampanA.Characterizing ordered Quasi-ideals of ordered Γ-semigroupsKwonY. I.LeeS. K.The weakly semi-prime ideals of po-Γ-semigroupsSiripitukdetM.IampanA.On the least (ordered) semilattice congruence in ordered Γ-semigroupsSahaN. K.On Γ-semigroup IISenM. K.SahaN. K.On Γ-semigroup ISenM. K.On Γ-semigroupsKehayopuluN.On prime, weakly prime ideals in ordered semigroupsCaoY.XinzhaiX.Nil-extensions of simple po-semigroupsKehayopuluN.Note on Green's relations in ordered semigroupsKehayopuluN.On regular ordered semigroupsKehayopuluN.On weakly prime ideals of ordered semigroups