EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 4, 2017, 620-630 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On intra-regular ordered Γ-semigroups Niovi Kehayopulu University of Athens, Department of Mathematics, Panepistimiopolis, Greece Abstract. We use the definition of intra-regularity (left regularity) of po-Γ-semigroups introduced in 2016 in Armenian Journal of Mathematics. Being able to describe the form of the elements of the principal filter by using this definition, we study the decomposition of an intra-regular po- Γ-semigroup into simple components. Then we prove that a po-Γ-semigroup M is intra-regular and the ideals of M form a chain if and only if M is a chain of simple semigroups. Moreover, a po-Γ-semigroup M is intra-regular and the ideals of M form a chain if and only if the ideals of M are prime. Finally, for an intra-regular po-Γ-semigroup M , the set {(x)N | x ∈ M} coincides with the set of all maximal simple subsemigroups of M . A decomposition of some left regular po-Γ-semigroups into their left simple components is also given. 2010 Mathematics Subject Classifications: AMS 20M99 Key Words and Phrases: po-Γ-semigroup, intra-regular, ideal, prime ideal, semilattice (chains) of simple semigroups 1. Introduction and prerequisites The notion of a Γ-ring, a generalization of the concept of associative rings, has been introduced and studied by Nobusawa in [11]. Γ-rings have been also studied by Barnes in [1]. Luh studied many properties of simple Γ-rings and primitive Γ-rings in [10]. The concept of a Γ-semigroup has been introduced by Sen in 1981 as follows: Given two nonempty sets S and Γ, S is called a Γ-semigroup if the following assertions are satisfied: (1) aαb ∈ S and αaβ ∈ Γ and (2) (aαb)βc = a(αbβ)c = aα(bβc) for all a, b, c ∈ S and all α, β ∈ Γ [13]. In 1986 Sen and Saha gave a second definition of Γ-semigroups as follows: Let S = {a, b, c, ......} and Γ = {α, β, γ, ......} be two nonempty sets. Then S is called a Γ-semigroup if (1) aαb ∈ S and (2) (aαb)βc = aα(bβc) for all a, b, c ∈ S and all α, β ∈ Γ [14] (the sets S and Γ should not be denumerable). One can find this definition of Γ-semigroups in [17] where the notion of a radical in Γ- semigroups and the notion of ΓS-act over a Γ-semigroup have been introduced and in [15] Email address: nkehayop@math.uoa.gr (N. Kehayopulu) http://www.ejpam.com 620 c© 2017 EJPAM All rights reserved. N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (4) (2017), 620-630 621 and [16] where the notions of regular and orthodox Γ-semigroups have been introduced and studied. But still we cannot say that Γ is a set of binary operations on the set M . Probably this is why Saha defines in [12] the Γ-semigroup as follows: Given two nonempty sets S and Γ, S is called a Γ-semigroup if there exists a mapping S × Γ× S → S | (a, γ, b)→ aγb such that (aαb)βc = aα(bβc) for all a, b, c ∈ S and all α, β ∈ Γ and remarks that most usual semigroup concepts, in particular regular and inverse semigroups, have their analogous for Γ-semigroups. Defining the Γ-semigroup via mappings, in an expression of the form a1γ1a2γ2a3 .... anγn, we can put parentheses in any place beginning with some ai and ending in some aj . The ordered Γ-semigroups have been first considered by Sen and Seth in [18]. Let M and Γ be two nonempty sets. Denote by MΓM the set of (all) elements of the form aγb, where a, b ∈M and γ ∈ Γ. That is, MΓM := {aγb | a, b ∈M,γ ∈ Γ}. Then M is called a Γ-semigroup [5, 6] if the following assertions are satisfied: (1) MΓM ⊆M ; (2) if a, b, c, d ∈M , γ, µ ∈ Γ, a = b, γ = µ and c = d, then aγc = bµd; (3) aγ(bµc) = (aγb)µc ∀ a, b, c ∈M ∀ γ, µ ∈ Γ. In other words, Γ is a set of binary operations on M and the following condition is satisfied: aγ(bµc) = (aγb)µc ∀ a, b, c ∈M ∀ γ, µ ∈ Γ. A Γ-semigroup endowed with an order relation “≤” such that a ≤ b implies aγc ≤ bγc and cγa ≤ cγb for every c ∈M and every γ ∈ Γ is called an ordered Γ-semigroup (shortly, po-Γ-semigroup). For a po-Γ-semigroup M and a subset H of M we denote by (H] the subset of M defined by (H] = {t ∈ M | t ≤ a for some a ∈ H}. We have M = (M ], and for any two subsets A, B of M , we have A ⊆ (A]; if A is a right (or left) ideal of M , then A = (A]; (A]Γ(B] ⊆ (AΓB]; if A ⊆ B, then (A] ⊆ (B]; ((A]] = (A]. Let M be a po-Γ-semigroup. A nonempty subset A of M is called a subsemigroup of M if for every a, b ∈ A and every γ ∈ Γ we have aγb ∈ A, that is, if AΓA ⊆ A. A nonempty subset A of M is called a left (resp. right) ideal of M if (1) MΓA ⊆ A and (2) if a ∈ A and M 3 b ≤ a, then b ∈ A. It is called an ideal of M if it is both a left ideal and a right ideal of M . Clearly, every left (resp. right) ideal of M is a subsemigroup of M . For an element a of M , we denote by L(a), R(a) and I(a) the left ideal, the right ideal and the ideal of M , respectively, generated by a, and we have L(a) = (a ∪MΓa], R(a) = (a ∪ aΓM ], and I(a) = (a ∪MΓa ∪ aΓM ∪MΓaΓM ]. We denote by L the equivalence relation on M defined by aLb if and only if L(a) = L(b), by R the equivalence relation on M defined by aRb if and only if R(a) = R(b) and by I the equivalence relation on M defined by aIb if and only if I(a) = I(b). A subsemigroup F of M is called a filter of M if (1) a, b ∈ F and γ ∈ Γ such that aγb ∈ F implies a ∈ F and b ∈ F and (2) if a ∈ F and M 3 c ≥ a, then c ∈ F . An equivalence relation σ on M is called congruence if (a, b) ∈ σ implies (aγc, bγc) ∈ σ and (cγa, cγb) ∈ σ for any c ∈ M and any γ ∈ Γ. A congruence σ on M N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (4) (2017), 620-630 622 is called semilattice congruence if (aγb, bγa) ∈ σ and (aγa, a) ∈ σ for every a, b ∈ M and every γ ∈ Γ. If σ is a semilattice congruence on M , then the σ-class (a)σ of M containing a is a subsemigroup of M for every a ∈ M . A semilattice congruence σ on M is called complete if a ≤ b implies (a, aγb) ∈ σ for every γ ∈ Γ. We denote by N the relation on M defined by aN b if and only if the filters of M generated by the elements a and b of M coincide. As in semigroups, the relation N is a semilattice congruence on M . So, if z ∈ M and γ ∈ Γ, then we have (zγz, z) ∈ N , (zγzγz, zγz) ∈ N , (zγzγzγz, zγzγz) ∈ N and so on. In particular, exactly as in ordered semigroups, the relation N is a complete semilattice congruence on M . A subsemigroup T of a po-Γ-semigroup M is called left (resp. right) simple if for every left (resp. right) ideal A of T we have A = T , that is if T is the only left (resp. right) ideal of T . It is called simple if it is both left and right simple. A subsemigroup T of a po-Γ-semigroup M is called maximal simple if for any simple subsemigroup A of M such that A ⊇ T , we have A = T . A po-Γ-semigroup M is said to be a semilattice of simple (resp. left simple) semigroups if there exists a semilattice congruence σ on M such that the σ-class (x)σ of M containing x is a simple (resp. left simple) subsemigroup of M for every x ∈ M . A po-Γ-semigroup M is called a chain of simple semigroups if there exists a semilattice congruence σ on M such that (x)σ is a simple subsemigroup of M for every x ∈M and the set M/σ of all σ-classes of M endowed with the order relation (x)σ � (y)σ ⇔ (x)σ = (xγy)σ ∀ γ ∈ Γ is a chain. Many results on Γ-semigroups (or po-Γ-semigroups) can be obtained from semigroups (ordered semigroups) just putting a “Gamma” in the appropriate place. But there are also results for which the transfer is not so easy. Both for a Γ-semigroup or po-Γ-semigroup M the filter of M generated by an element a of M plays an essential role in the structure, in particular, in the decomposition of M . So it is important to get the form of its elements. The definition of intra-regularity of a po-Γ-semigroup in the bibliography was as follows: A po-Γ-semigroup M is intra-regular if a ∈ (MΓaΓaΓM ] for every a ∈ M . With this definition is not possible to describe the form of the elements of the N(x) (x ∈ S). To overcome this difficulty, the following new concept of intra-regularity has been introduced in [9]: We say that a po-Γ-semigroup M is intra-regular if a ∈ (MΓaγaΓM ] for every a ∈ M and every γ ∈ Γ. Using this definition, we first give a structure theorem referring to the decomposition of a po-Γ-semigroup into simple components. Then, for a po-Γ- semigroup M , we prove the following: The ideals of M are weakly prime if and only if they are idempotent and they form a chain. The ideals of M are prime if and only if they form a chain and M is intra-regular. M is intra-regular and the ideals of M form a chain if and only if M is a chain of simple semigroups. For an intra-regular po-Γ-semigroup M the set {(x)N | x ∈ M} coincides with the set of all maximal simple subsemigroups of M . Keeping the new definition of left (right) regularity of po-Γ-semigroups introduced in [9], we also give a structure theorem related to the decomposition of a po-Γ-semigroup M which is left regular and satisfies the relation (xΓM ] ⊆ (MΓx] for every x ∈ M into left simple components. The results of this paper are based on the corresponding results on ordered semigroups considered in [3] and [4], and the aim of writing this paper is to show the importance of these new concepts of intra-regularity and left (right) regularity in the investigation. N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (4) (2017), 620-630 623 2. Main results Let M be a po-Γ-semigroup. A subset A of M is called idempotent, if A = (AΓA]. A subset T of M is called prime if a, b ∈ M and γ ∈ Γ such that aγb ∈ T implies a ∈ T or b ∈ T . The set T is called semiprime if a ∈M and γ ∈ Γ such that aγa ∈ T implies a ∈ T [9]. A subset T of M is called weakly prime if the following assertion is satisfied: If A, B are ideals of M such that AΓB ⊆ T , then A ⊆ T or B ⊆ T . For a subset T of M , we consider the statements: (1) a, b ∈M , γ ∈ Γ, aγb ∈ T =⇒ a ∈ T or b ∈ T . (2) A,B ⊆M , AΓB ⊆ T =⇒ A ⊆ T or B ⊆ T . Then (1) ⇒ (2). In fact: Let A,B ⊆ M , AΓB ⊆ T , A * T and b ∈ B. Take an element a ∈ A such that a 6∈ T and an element γ ∈ Γ (Γ 6= ∅). Since aγb ∈ AΓB ⊆ T , by (1), we have a ∈ T or b ∈ T . Since a 6∈ T , we get b ∈ T . We have the following: (a) If T is a prime subset of M , then T is a semiprime subset of M . (b) If T is a prime subset of M , then T is a weakly prime subset of M . Definition 1. [9] A po-Γ-semigroup M is called intra-regular if x ∈ (MΓxγxΓM ] for every x ∈M and every γ ∈ Γ. Proposition 2. If M is an intra-regular po-Γ-semigroup then, for every x, y ∈ M and every γ ∈ Γ, we have (MΓxγyΓM ] = (MΓyγxΓM ]. Proof. Let x, y ∈M and γ ∈ Γ. Since xγy ∈MΓM ⊆M and M is intra-regular, we have xγy ∈ ( MΓ(xγy)γ(xγy)ΓM ] ⊆ ( (MΓM)Γ(yγx)Γ(MΓM) ] ⊆ (MΓyγxΓM ]. Then we have MΓ(xγy)ΓM ⊆ (M ]Γ(MΓyγxΓM ]Γ(M ] ⊆ (MΓMΓyγxΓMΓM ] ⊆ (MΓyγxΓM ], from which (MΓ(xγy)ΓM ] ⊆ ( (MΓyγxΓM ] ] = (MΓyγxΓM ]. Since M is intra-regular and yγx ∈M , by symmetry, we get (MΓyγxΓM ] ⊆ (MΓxγyΓM ], thus we have (MΓxγyΓM ] = (MΓyγxΓM ]. � Lemma 3. [9] A po-Γ-semigroup M is intra-regular if and only if, for every x ∈ M , we have N(x) = {y ∈M | x ∈ (MΓyΓM ]}. In a similar way as in [2] and [7], we can prove the following lemma. Lemma 4. If M is a po-Γ-semigroup, then I ⊆ N and I ⊆ L. N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (4) (2017), 620-630 624 Lemma 5. [9] A po-Γ-semigroup M is intra-regular if and only if the ideals of M are semiprime. The proof of the following lemma is easy. Lemma 6. If M is a po-Γ-semigroup, then the set (MΓaΓM ] (resp. (MΓa]) is an ideal (resp. left ideal) of M , and the set (aΓM ] is a right ideal of M for every a ∈M . Definition 7. A po-Γ-semigroup S is said to be a semilattice of simple (resp. left simple) semigroups if there exists a semilattice congruence σ on M such that the σ-class (x)σ of M containing x is a simple (resp. left simple) subsemigroup of M for every x ∈M . Theorem 8. Let M be a po-Γ-semigroup. The following are equivalent: (1) M is intra-regular. (2) N(x) = {y ∈M | x ∈ (MΓyΓM ]} for every x ∈M . (3) N = I. (4) For every ideal I of M, we have I = ⋃ x∈I (x)N . (5) (x)N is a simple subsemigroup of M for every x ∈M . (6) M is a semilattice of simple semigroups. (7) Every ideal of M is semiprime. Proof. The implication (1)⇒ (2) follows from Lemma 3, the proof of (3)⇒ (4) is similar with the corresponding result for semigroups without order in [8], (5)⇒ (6) since N is a semilattice congruence on M and (7)⇒ (1) by Lemma 5. (2) =⇒ (3). Let (a, b) ∈ N . Since a ∈ N(a) = N(b), by (2), we have b ∈ (MΓaΓM ] ⊆ (a∪MΓa∪aΓM ∪MΓaΓM ] = I(a), so I(b) ⊆ I(a). Since b ∈ N(a), by symmetry, we get I(a) ⊆ I(b), so I(a) = I(b), and (a, b) ∈ I. Then N ⊆ I, on the other hand by Lemma 4, we have I ⊆ N , thus I = N . (4) =⇒ (5). Let x ∈M . SinceN is a semilattice congruence on M , (x)N is a subsemigroup of M . Let I be an ideal of (x)N . Then I = (x)N . Indeed: Let y ∈ (x)N . Take an element z ∈ I and an element γ ∈ Γ (I,Γ 6= ∅). Since zγzγz ∈ (MΓM)ΓM ⊆ MΓM ⊆ M , by Lemma 6, (MΓzγzγzΓM ] is an ideal of M . By hypothesis, we have (MΓzγzγzΓM ] = ⋃ t∈(MΓzγzγzΓM ] (t)N . Since z ∈ I ⊆ (x)N , we have (x)N = (z)N . Since y ∈ (x)N = (z)N = (zγzγzγzγz)N ⊆ (MΓzγzγzΓM ], we have y ≤ aδzγzγzξb = (aδz)γzγ(zξb) for some a, b ∈M, δ, ξ ∈ Γ. Using the fact that N is a complete semilattice congruence on M , in a similar way as in [8], we prove that aδz ∈ (x)N and zξb ∈ (x)N . Then, since I is an ideal of (x)N and z ∈ I, we have we have (aδz)γzγ(zξb) ∈ (x)NΓIΓ(x)N ⊆ I, and y ∈ I. Hence (x)N ⊆ I, and so I = (x)N . (6) =⇒ (7). Let σ be a semilattice congruence on M such that (x)σ is a simple subsemi- group of M for every x ∈ M . Let I be an ideal of M , x ∈ M and γ ∈ Γ such that xγx ∈ I. The set I ∩ (x)σ is an ideal of (x)σ. Indeed: Taking into account the proof of the implication (6)⇒ (7) in [8], it is enough to prove the following: Let a ∈ I ∩ (x)σ and N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (4) (2017), 620-630 625 (x)σ 3 b ≤ a, then b ∈ I∩(x)σ. Since M ∈ b ≤ a ∈ I and I is an ideal of M , we have b ∈ I, then b ∈ I ∩ (x)σ. Since (x)σ is a simple subsemigroup of M , we have I ∩ (x)σ = (x)σ, then x ∈ I. Thus M is semiprime. � Lemma 9. Let M be a po-Γ-semigroup. The ideals of M are idempotent if and only if for any ideals A,B of M, we have A ∩B = (AΓB]. Proof. =⇒. Let A,B be ideals of M . Then (AΓB] ⊆ (AΓM ] ⊆ (A] = A and (AΓB] ⊆ (MΓB] ⊆ (B] = B, thus (AΓB] ⊆ A ∩ B. On the other hand, A ∩ B is an ideal of M . Indeed: Take an element a ∈ A, an element b ∈ B and an element γ ∈ Γ (A,B,Γ 6= ∅). Then aγb ∈ AΓB ⊆ AΓM ⊆ A and aγb ∈ AΓB ⊆MΓB ⊆ B, so aγb ∈ A∩B, so A∩B is a nonempty subset of M . We also have (A∩B)ΓM ⊆ AΓM ⊆ A, MΓ(A∩B) ⊆MΓB ⊆ B, and if x ∈ A ∩ B and M 3 y ≤ x then, since x ∈ A we have y ∈ A and since x ∈ B we have y ∈ B, so y ∈ A ∩ B. Since A ∩ B is an ideal of M , by hypothesis, we have A ∩B = ( (A ∩B)Γ(A ∩B) ] ⊆ (AΓB]. Hence we have A ∩B = (AΓB]. ⇐=. Let A be an ideal of M . By hypothesis, we have A = (AΓA], so A is idempotent. � Theorem 10. Let M be a po-Γ-semigroup. The ideals of M are weakly prime if and only if they are idempotent and they form a chain. Proof. =⇒. Let A, B be ideals of M . One can easily prove that (AΓB] is an ideal of M . Since A,B, (AΓB] are ideals of M , AΓB ⊆ (AΓB] and (AΓB] is weakly prime, we have A ⊆ (AΓB] ⊆ (MΓB] ⊆ (B] = B or B ⊆ (AΓB] ⊆ (AΓM ] ⊆ (A] = A, thus the ideals of M form a chain. Furthermore, since A and (AΓA] are ideals of M , AΓA ⊆ (AΓA] and (AΓA] is weakly prime, we have A ⊆ (AΓA] ⊆ (MΓA] ⊆ (A] = A, thus we get A = (AΓA], and A is idempotent. ⇐=. Let A,B, T be ideals of M such that AΓB ⊆ T . By hypothesis, we have A ⊆ B or B ⊆ A. If A ⊆ B then, by Lemma 9, A = A ∩ B = (AΓB] ⊆ (T ] = T . If B ⊆ A, then B = A ∩B = (AΓB] ⊆ (T ] = T , thus M is weakly prime. � Lemma 11. Let M be a po-Γ-semigroup. If M is intra-regular, then I(x) = (MΓxΓM ] for every x ∈M. Proof. Let x ∈M . Since (MΓxΓM ] is an ideal of M , by Lemma 5, it is semiprime. Take an element γ ∈ Γ (Γ 6= ∅). Since xγx ∈ M , (xγx)γ(xγx) ∈ (MΓxΓx] and (MΓxΓM ] is a semiprime subset of M , we have xγx ∈ (MΓxΓx], then x ∈ (MΓxΓx], and I(x) ⊆ (MΓxΓx]. On the other hand, (MΓxΓx] ⊆ I(x), thus we get I(x) = (MΓxΓM ]. � Lemma 12. If M is a po-Γ-semigroup, x, y ∈M and γ ∈ Γ, then I(xγy) ⊆ I(x) ∩ I(y). In particular, if M is intra-regular, then I(xγy) = I(x) ∩ I(y). Proof. Let x, y ∈M and γ ∈ Γ. Since I(x) is an ideal of M , we have xγy ∈ I(x)ΓM ⊆ I(x) and xγy ∈MΓI(y) ⊆ I(y). Thus we have I(xγy) ⊆ I(x) ∩ I(y). N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (4) (2017), 620-630 626 Let now M be intra-regular and t ∈ I(x) ∩ I(y). Then, by Lemma 11, we have t ∈ (MΓxΓM ] and t ∈ (MΓyΓM ]. Thus we have t ≤ aµxρb and t ≤ cξyζd for some a, b, c, d ∈M, µ, ρ, ξ, ζ ∈ Γ. Then tγt ≤ (cξyζd)γ(aµxρb) = cξ(yζdγaµx)ρb. In addition, we have yζdγaµx ∈ I(xγy). Indeed, by Lemma 11, (yζdγaµx)γ(yζdγaµx) ∈MΓ(xγy)ΓM ⊆ ( MΓ(xγy)ΓM ] = I(xγy). Since M is intra-regular and I(xγy) is an ideal of M , by Lemma 5, I(xγy) is semiprime. So we get yζdγaµx ∈ I(xγy). Since I(xγy) is an ideal of M , we have cξ(yζdγaµx)ρb ∈ MΓI(xγy)ΓM ⊆ I(xγy)ΓM ⊆ I(xγy), then tγt ∈ I(xγy). Since I(xγy) is semiprime, we have t ∈ I(xγy). Thus we get I(x) ∩ I(y) ⊆ I(xγy) and so I(xγy) = I(x) ∩ I(y). � Theorem 13. Let M be a po-Γ-semigroup. The ideals of M are prime if and only if they form a chain and M is intra-regular. Proof. =⇒. The ideals of M are prime, so they are weakly prime and semiprime. Since they are weakly prime, by Theorem 10, they form a chain. Let now a ∈ M and γ ∈ Γ. Since (MΓaγaΓM ] is an ideal of M , (aγa)γ(aγa) ∈ (MΓaγaΓM ] and (MΓaγaΓM ] is semiprime, we have aγa ∈ (MΓaγaΓM ], then a ∈ (MΓaγaΓM ], thus M is intra-regular. ⇐=. Suppose M is intra-regular and the ideals of M form a chain. Let now T be an ideal of M , a, b ∈ M and γ ∈ Γ such that aγb ∈ T . We have I(a) ⊆ I(b) or I(b) ⊆ I(a). If I(a) ⊆ I(b) then, by Lemma 12, we have a ∈ I(a) = I(a) ∩ I(b) = I(aγb) ⊆ I(T ) = T . If I(b) ⊆ I(a), then b ∈ I(b) = I(a)∩ I(b) = I(aγb) ⊆ T . Thus the ideals of M are prime. � Proposition 14. Let M be an intra-regular po-Γ-semigroup such that the ideals of M form a chain. Then, for every x, y ∈M and every γ ∈ Γ, we have x ∈ (MΓxγyΓM ] or y ∈ (MΓxγyΓM ]. Proof. Let x, y ∈ M and γ ∈ Γ. Since M is intra-regular and the ideals of M form a chain, by Theorem 13, the ideals of M are prime. Since (MΓxγyΓM ] is an ideal of M , (MΓxγyΓM ] is prime. Since (xγx)γ(yγy) ∈ (MΓxγyΓM ], we have xγx ∈ (MΓxγyΓM ] or yγy ∈ (MΓxγyΓM ]. If xγx ∈ (MΓxγyΓM ] then, since (MΓxγyΓM ] is prime, we have x ∈ (MΓxγyΓM ]. If yγy ∈ (MΓxγyΓM ], then y ∈ (MΓxγyΓM ]. � Definition 15. A po-Γ-semigroup M is called a chain of simple semigroups if there exists a semilattice congruence σ on M such that (x)σ is a simple subsemigroup of M for every x ∈M and the set M/σ of all σ-classes of M endowed with the order relation (x)σ � (y)σ ⇔ (x)σ = (xγy)σ ∀ γ ∈ Γ is a chain. In other words, for any x, y ∈M and any γ ∈ Γ we have (x)σ = (xγy)σ or (y)σ = (xγy)σ. N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (4) (2017), 620-630 627 Theorem 16. A po-Γ-semigroup M is intra-regular and the ideals of M form a chain if and only if M is chain of simple semigroups. Proof. =⇒. Since M is intra-regular and N is a semilattice congruence on M , by Theorem 8(1)⇒ (5), (x)N is a simple subsemigroup of M for every x ∈M , so M is a semilattice of simple semigroups. Let now x, y ∈M and γ ∈ Γ. Then (x)N = (xγy)N or (y)N = (xγy)N . In fact: Since M is intra-regular and the ideals of M form a chain, by Proposition 14, we have x ∈ (MΓxγyΓM ] or y ∈ (MΓxγyΓM ]. If x ∈ (MΓxγyΓM ], then N(x) 3 x ≤ aµ(xγy)ρb for some a, b ∈M, µ, ρ ∈ Γ. Since N(x) is a filter of M , we have aµ(xγy)ρb ∈ N(x), then (xγy)ρb ∈ N(x), xγy ∈ N(x) and N(xγy) ⊆ N(x). If y ∈ (MΓxγyΓM ], then N(y) 3 y ≤ cξ(xγy)ζd for some c, d ∈M, ξ, ζ ∈ Γ, then cξ(xγy)ζd ∈ N(y), xγy ∈ N(y), and N(xγy) ⊆ N(y). On the other hand, since xγy ∈ N(xγy), we have x ∈ N(xγy) and y ∈ N(xγy), so N(x) ⊆ N(xγy) and N(y) ⊆ N(xγy). Thus we get N(x) = N(xγy) or N(y) = N(xγy), then (x)N = (xγy)N or (y)N = (xγy)N . Therefore M is a chain of simple semigroups. ⇐=. Let σ be a semilattice congruence on M such that (x)σ is a simple subsemigroup of M for every x ∈ M and (M/σ,�) is a chain. By Theorem 13 it is enough to prove that the ideals of M are prime. So, let I be an ideal of M , a, b ∈ M and γ ∈ Γ such that aγb ∈ I. The set (aγb)σ ∩ I is an ideal of (aγb)σ. Indeed: ∅ 6= (aγb)σ ∩ I ⊆ (aγb)σ (aγb ∈ (aγb)σ, aγb ∈ I),( (aγb)σ ∩ I ) Γ(aγb)σ ⊆ (aγb)σΓ(aγb)σ ∩ IΓ(aγb)σ ⊆ (aγb)σ ∩ IΓ(aγb)σ ⊆ (aγb)σ ∩ IΓM ⊆ (aγb)σ ∩ I, (aγb)σΓ ( (aγb)σ ∩ I ) ⊆ (aγb)σΓ(aγb)σ ∩ (aγb)σΓI ⊆ (aγb)σ ∩MΓI ⊆ (aγb)σ ∩ I. Let x ∈ (aγb)σ ∩ I and (aγb)σ 3 y ≤ x. Since M 3 y ≤ x ∈ I and I is an ideal of M , we have y ∈ I, then y ∈ (aγb)σ ∩ I. Since (aγb)σ is simple, we have (aγb)σ ∩ I = (aγb)σ. By hypothesis, (a)σ = (aγb)σ or (b)σ = (aγb)σ. Then we have a ∈ I or b ∈ I, thus I is a prime. � Lemma 17. Let M be a po-Γ-semigroup, T a subsemigroup of M and x ∈ T . Then the set (MΓxΓM ] ∩ T is an ideal of T. Proof. First of all, the set (MΓxΓM ] ∩ T is a nonempty subset of T . Indeed: Take an element γ ∈ Γ (Γ 6= ∅). Then we have xγxγx ∈ MΓxΓM and xγxγx ∈ (TΓT )ΓT ⊆ TΓT ⊆ T . Moreover,( (MΓxΓM ] ∩ T ) ΓT ⊆ (MΓxΓM ]ΓT ∩ TΓT ⊆ (MΓxΓM ]Γ(M ] ∩ T N. Kehayopulu / Eur. J. Pure Appl. Math, 10 (4) (2017), 620-630 628 ⊆ ( MΓxΓ(MΓM) ] ∩ T ⊆ (MΓxΓM ] ∩ T. In a similar way, we have TΓ ( (MΓxΓM ]∩T ) ⊆ (MΓxΓM ]∩T. Let now a ∈ (MΓxΓM ]∩T and T 3 b ≤ a. Since a ∈ (MΓxΓM ], there exist u, v ∈ M and ξ, ζ ∈ Γ such that a ≤ uξxζv. Then we have b ≤ uξxζv ∈MΓxΓM , and b ∈ (MΓxΓM ], thus b ∈ (MΓxΓM ]∩T . � Theorem 18. Let M be an intra-regular po-Γ-semigroup. Then the set (x)N is a max- imal simple subsemigroup of M for every x ∈ M . Moreover, if T is a maximal simple subsemigroup of M, then there exists x ∈M such that T = (x)N . Proof. Let x ∈ M . By the Theorem 8(1)⇒ (5), the set (x)N is a simple subsemigroup of M . Let now T be a simple subsemigroup of M such that T ⊇ (x)N . Then T = (x)N . Indeed: Let y ∈ T . Since x ∈ T and T is a subsemigroup of M , by Lemma 17, the set (MΓxΓM ] ∩ T is an ideal of T . Since T is a simple subsemigroup of M , we have (MΓxΓM ]∩ T = T , then y ∈ (MΓxΓM ]. Since M is intra-regular, by Lemma 3, we have x ∈ N(y), then N(x) ⊆ N(y). On the other hand, since y ∈ T and T is a subsemigroup of M , by Lemma 17, the set (MΓyΓM ] ∩ T is an ideal of T . So (MΓyΓM ] ∩ T = T , x ∈ (MΓyΓM ], y ∈ N(x), and N(y) ⊆ N(x). Therefore we have N(x) = N(y), then y ∈ (x)N , and T ⊆ (x)N . Then we have T = (x)N , thus the class (x)N is a maximal simple subsemigroup of M . Let now T be a maximal simple subsemigroup of M . Take an element x ∈ T (T 6= ∅). Exactly as in the proof of the “⇒”-part of the theorem given above, we prove that T ⊆ (x)N . Since M is intra-regular, by Theorem 8(1)⇒ (5), (x)N is a simple subsemigroup of M . Since T ⊆ (x)N and T is a maximal simple subsemigroup of M , we have T = (x)N . � Corollary 19. For an intra-regular po-Γ-semigroup M, the set {(x)N | x ∈M} coincides with the set of all maximal simple subsemigroup of M. Definition 20. [9] A Γ-semigroup M is called left (resp. right) regular if x ∈ (MΓxγx] (resp. x ∈ (xγxΓM ]) for every x ∈M and every γ ∈ Γ. Taking into account the Theorem 6 in [8], in a similar way as in the Theorem 8 above, we can prove the following theorem Theorem 21. Let M be a po-Γ-semigroup. The following are equivalent: (1) M is left regular and (xΓM ] ⊆ (MΓx] for every x ∈M . (2) N(x) = {y ∈M | x ∈ (MΓy]} for every x ∈M . (3) N = L. (4) For every left ideal L of M, we have L = ⋃ x∈L (x)N . (5) (x)N is a left simple subsemigroup of M for every x ∈M . (6) M is a semilattice of left simple semigroups. REFERENCES 629 (7) Every left ideal of M is semiprime and two-sided. The right analogue of the above theorem also holds. With my best thanks to the referee for reading the paper carefully, and his prompt reply. References [1] W.E. Barnes. 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