Semilattices of Nil-extensions of Simple Regular ... - Springer Link

Algebra Colloquium 10:1 (2003) 81–90

Algebra Colloquium c AMSS CAS 2003

Semilattices of Nil-extensions of Simple Regular Semigroups Stojan Bogdanovi´ c University of Niˇs, Faculty of Economics Trg Kralja Aleksandra 11, 18000 Niˇs, Yugoslavia E-mail: [email protected]

´ c Miroslav Ciri´ University of Niˇs, Faculty of Sciences and Mathematics Viˇsegradska 33, P.O. Box 224, 18000 Niˇs, Yugoslavia E-mail: [email protected]

Melanija Mitrovi´ c University of Niˇs, Faculty of Mechanical Engineering Beogradska 14, 18000 Niˇs, Yugoslavia E-mail: [email protected] Received 20 May 2001 Revised 24 April 2002 Communicated by K.P. Shum Abstract. Semigroups decomposable into a semilattice of Archimedean semigroups, having certain additional properties such as the intra-, left, right and complete π-regularity, have been investigated in many papers. In the present paper, we study the π-regular ones. Among other things, we characterize them as semilattices of nil-extensions of simple regular semigroups. The obtained results ´ c, and generalize some results given by Munn, Putcha, Shevrin, Bogdanovi´c, Ciri´ others. 2000 Mathematics Subject Classification: 20M10, 20M17 Keywords: simple semigroups, regular semigroups, nil-extensions, semilattice decompositions

1 Introduction Tamura began in 1956 the study of semigroups through their greatest semilattice decompositions, and his work has been since continued by Petrich, ´ c, and others. Much attention in this area has Putcha, Bogdanovi´c, Ciri´ been aimed to semigroups decomposable into a semilattice of Archimedean semigroups. In the general case, these semigroups have been studied by

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´ c and Bogdanovi´c [11], and othPutcha [19], Tamura [21], Kmeˇt [18], Ciri´ ers. Intra-π-regular semigroups from this class have been investigated by ´ c [8], and completely Putcha [19], left π-regular ones by Bogdanovi´c and Ciri´ π-regular ones by Shevrin [20], Veronesi [22], Bogdanovi´c [2, 3], Bogdanovi´c ´ c [4, 8], etc. More information about all the mentioned types of and Ciri´ semigroups can be found in survey papers [5, 12]. The main purpose of this paper is to study semigroups which are πregular and are decomposable into a semilattice of Archimedean semigroups. In Section 2, we give some preliminary results. The most interesting ones are Theorems 2.2 and 2.4, which can be viewed as generalizations of the well-known Munn’s Theorem (see Theorem 2.55 in [13]). The main theorems are proved in Section 3, where we describe semilattices and chains of nil-extensions of simple regular semigroups. We also give some new characterizations of semilattices and chains of completely Archimedean semigroups by their idempotent-generated subsemigroups. Throughout this paper, N will denote the set of positive integers. By Reg(S) (resp., Gr(S), Intra(S), E(S)), we denote the set of all regular (resp., completely regular, intra-regular, idempotent) elements of a semigroup S. For a semigroup S and a ∈ Reg(S), V (a) denotes the set of all inverses of a. Let us denote by hE(S)i the subsemigroup of S generated by E(S). The division relation | on a semigroup S is defined by a|b ⇐⇒ (∃ x, y ∈ S 1 ) b = xay. The concept of π-regularity, in its various forms, appeared first in the ring theory as a natural generalization of the regularity. In the semigroup theory, this concept attracts a great attention both as a generalization of the regularity and a generalization of the finiteness and the periodicity. On the other hand, there are specific relations between the π-regularity (intraπ-regularity) and Archimedeaness as was shown by Putcha in [19] and in many papers of the first two authors. A semigroup S is intra-π-regular (resp., left π-regular, π-regular ) if for every a ∈ S, there exists n ∈ N such that an ∈ Sa2n S (resp., an ∈ San+1 , an ∈ an San ). A semigroup S is completely π-regular if for any a ∈ S, there exist n ∈ N and x ∈ S such that an = an xan and an x = xan , or equivalently, if for each element of S, some its power lies in a subgroup of S. A semigroup S is Archimedean if for all a, b ∈ S, there exists n ∈ N such that an ∈ SbS. An Archimedean semigroup with a primitive idempotent is completely Archimedean. The expression S = S 0 means that S is a semigroup with the zero 0. If S = S 0 , then an element a ∈ S is nilpotent if there exists n ∈ N such that an = 0. By N il(S), we denote the set of all nilpotent elements of S. A semigroup S = S 0 is a nil-semigroup if S = N il(S). An ideal extension S of a semigroup T is a nil-extension of T if S/T is a nil-semigroup.

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If ρ is a relation on a semigroup S, then τ (ρ) denotes a relation on S defined by (a, b) ∈ τ (ρ) ⇐⇒ (∃ n ∈ N) an ρbn for a, b ∈ S (see [9]). For undefined notions and relations, we refer to [2, 4, 13, 17]. 2 Preliminary Results Theorem 2.1. Let E(S) 6= ∅. Then the following conditions on a semigroup S are equivalent: (i) (∀a ∈ S)(∀e ∈ E(S)) a|e ⇒ a2 |e. (ii) (∀a, b ∈ S)(∀e ∈ S) a|e & b|e ⇒ ab|e. (iii) (∀e, f, g ∈ E(S)) e|g & f |g ⇒ ef |g. (iv) S is a semilattice Y of semigroups Sα (α ∈ Y ), where each Sα has a kernel Kα such that E(Sα ) ⊆ Kα or E(Sα ) = ∅. Proof. (i)⇒(ii). Let a, b ∈ S and e ∈ E(S) such that a|e and b|e, i.e., e = xay = ubv for some x, y, u, v ∈ S 1 . By hypothesis, we have e = ee = ubvxay ∈ S(bvxa)2 S ⊆ SabS. Hence, ab|e. (ii)⇒(i) and (ii)⇒(iii). This is obvious. (iii)⇒(ii). Let a, b ∈ S and e ∈ E(S) such that a|e and b|e. Then e = xay = ubv for some x, y, u, v ∈ S 1 . It is easy to verify that (yxa)2 , (bvu)2 ∈ E(S) and e = xa(yxa)2 y = u(bvu)2 bv. Now by (iii), (yxa)2 (bvu)2 |e, whence ab|e. (i)⇔(iv). This was proved in Theorem 2.4 of [19]. 2 The next theorem describes π-regular simple semigroups. In particular, we will see that the π-regularity on simple semigroups fails down to the regularity. Theorem 2.2. The following conditions on a semigroup S are equivalent: (i) S is π-regular and simple. (ii) S is regular and simple. (iii) (∀a, b ∈ S) a ∈ aSbSa. Proof. (i)⇒(ii). Suppose S is π-regular and simple. Let a ∈ S. Then there exist x, y ∈ S such that a = xay = xn ay n for every n ∈ N. For some n ∈ N and v ∈ S, we have y n = y n vy n , and then a = xn ay n vy n = avy n . So by the simplicity of S, we obtain a ∈ aSa2 S. From this, it follows that a = apa2 q for some p, q ∈ S, whence a = (apa)n aq n for every n ∈ N. Since S is πregular, we then have a = (apa)n aq n = (apa)n u(apa)n aq n = (apa)n ua for some n ∈ N and u ∈ S. Therefore, a ∈ aSa and we have proved that S is a regular semigroup. (ii)⇒(i). This is obvious. (ii)⇒(iii). Let a, b ∈ S. Then a ∈ SbS, and also, there exists x ∈ S such that a = axa. But then we have a = axaxa ∈ axSbSxa ⊆ aSbSa. (iii)⇒(ii). This is obvious. 2 Corollary 2.3. The following conditions on a semigroup S are equivalent:

´ c, M. Mitrovi´c S. Bogdanovi´c, M. Ciri´

84 (i) S is π-regular and intra-regular. (ii) S is regular and intra-regular. (iii) (∀a ∈ S) a ∈ aSa2 Sa.

Using Theorem 2.2, we characterize nil-extensions of simple regular semigroups as follows. Theorem 2.4. The following conditions on a semigroup S are equivalent: (i) S is π-regular and Archimedean. (ii) S is a nil-extension of a simple regular semigroup. (iii) (∀a, b ∈ S)(∃ n ∈ N) an ∈ an SbSan . Proof. (i)⇒(ii). If S is a π-regular Archimedean semigroup, then E(S) 6= ∅, and by Theorem 1.9 of [5], S is a nil-extension of a simple semigroup K such that E(S) 6= ∅. Assume a ∈ K. Then there exist x ∈ S and n ∈ N such that an = an xan , so we have an = an xan = an xan xan ∈ an SKSan ⊆ an Kan . By Theorem 2.2, K is regular and simple. Thus, S is a nil-extension of a simple regular semigroup. (ii)⇒(i). Let S be a nil-extension of a simple regular semigroup K. By Theorem 3.13 of [4], S is an Archimedean semigroup. For a ∈ S, there exists n ∈ N such that an ∈ K. But K is a regular semigroup, so we have an ∈ an Kan ⊆ an San , and S is a π-regular semigroup. (ii)⇒(iii). Let S be a nil-extension of a simple regular semigroup K and a, b ∈ S. Then there exists n ∈ N such that an , an b ∈ K, so an ∈ Kan bK, and there exists x ∈ K such that an = an xan = an xan xan ∈ an xKan bKxan ⊆ an KbKan ⊆ an SbSan , which was to be proved. (iii)⇒(i). It is obvious that S is a π-regular semigroup. Assume a, b ∈ S. Then there exists n ∈ N such that an ∈ an SbSan ⊆ SbS, so S is an Archimedean semigroup. 2 Before we give a new characterization of completely Archimedean semigroups in terms of their idempotents, we quote a lemma which will be used throughout the rest of this paper. Lemma 2.5. [14] If a semigroup S is (completely) π-regular, then hE(S)i is (completely) π-regular. Theorem 2.6. A semigroup S is completely Archimedean if and only if S is completely π-regular and hE(S)i is a (completely) simple semigroup. Proof. Let S be a completely Archimedean semigroup. By Theorem 2.2.1 of [2], S is completely π-regular and Reg(S) = Gr(S) is a completely simple kernel of S. By Theorem 3.5 of [15], hE(S)i is a completely simple semigroup.

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Conversely, if S is a completely π-regular semigroup and hE(S)i is a simple semigroup, then by Lemma 2.5, hE(S)i is completely π-regular. By Munn’s Theorem (see [13]), hE(S)i is completely simple, and by Theorem 3.14 of [4], S is a completely Archimedean semigroup. 2 3 The Main Results Now we are ready to prove the main result of this paper, which is a generalization of a result of Shevrin [20]. Theorem 3.1. The following conditions on a semigroup S are equivalent: (i) S is a semilattice of nil-extensions of simple regular semigroups. (ii) S is a band of nil-extensions of simple regular semigroups. (iii) S is π-regular and is a semilattice of Archimedean semigroups. (iv) (∀a, b ∈ S)(∃ n ∈ N) (ab)n ∈ (ab)n Sa2 S(ab)n . (v) S is π-regular, and (∀a ∈ S)(∀e ∈ E(S)) a|e ⇒ a2 |e. (vi) S is π-regular, and (∀a, b ∈ S)(∀e ∈ E(S)) a|e & b|e ⇒ ab|e. (vii) S is π-regular, and (∀e, f, g ∈ E(S)) e|g & f |g ⇒ ef |g. (viii) S is π-regular, and in every homomorphic image with zero of S, the set of all nilpotent elements is an ideal. (ix) S is π-regular and every J -class of S containing an idempotent is a subsemigroup of S. (x) S is intra-π-regular and every J -class of S containing an intra-regular element is a regular subsemigroup of S. (xi) S is π-regular and τ (J ) is a semilattice (or a band ) congruence on S. (xii) S is a semilattice of nil-extensions of simple semigroups and Intra(S) = Reg(S). Proof. (i)⇔(ii). This is evident. (i)⇒(iii). Clearly, S is π-regular, and by Theorem 2.12 of [19], S is a semilattice of Archimedean semigroups. (iii)⇒(i). Let S be a π-regular semigroup which is a semilattice Y of Archimedean semigroups Sα (α ∈ Y ). Then Sα is also π-regular, and by Theorem 2.4, Sα is a nil-extension of a simple regular semigroup for every α∈Y. (i)⇒(iv). Let S be a semilattice Y of nil-extensions of simple regular semigroups Sα (α ∈ Y ). Let a, b ∈ S. Then ab, a2 b ∈ Sα for some α ∈ Y . Now by Theorem 2.4, there exists n ∈ N such that (ab)n ∈ (ab)n Sα a2 bSα (ab)n ⊆ (ab)n Sa2 S(ab)n . (iv)⇒(iii). Let a, b ∈ S. Then there exists n ∈ N such that (ab)n ∈ (ab)n Sa2 S(ab)n ⊆ Sa2 S,

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and by Theorem 5.12 of [4], S is a semilattice of Archimedean semigroups. It is clear that S is π-regular. (v)⇔(vi)⇔(vii). This follows by Theorem 2.1. (v)⇒(iii). Let a, b ∈ S. Then (ab)n = (ab)n x(ab)n for some x ∈ S and n ∈ N. Since a|(ab)n x, we have a2 |(ab)n x, whence (ab)n = (ab)n x(ab)n ∈ Sa2 S, and by Theorem 1 of [11], S is a semilattice of Archimedean semigroups. (iii)⇔(viii). This follows by Theorem 1.1 of [5]. (i)⇒(x). Let S be a semilattice Y of semigroups Sα (α ∈ Y ), and for each α ∈ Y , let Sα be a nil-extension of a simple regular semigroup Kα . By Theorem 4 of [8], S is an intra-π-regular semigroup and every J -class containing an intra-regular element is a subsemigroup of S. Let a ∈ Intra(S). Then a = xa2 y for some x, y ∈ S 1 , and a ∈ Sα for some α ∈ Y , whence xa, ay ∈ Sα and a = (xa)n ay n for each n ∈ N. But xa ∈ Sα yields (xa)n ∈ Kα for some n ∈ N, whence a = (xa)n ay n ∈ Kα Sα ⊆ Kα . This means that Kα is the J -class of a, which completes the proof. (x)⇒(iii). By Theorem 4 of [8], S is an intra-π-regular semigroup and a semilattice of Archimedean semigroups. Let a ∈ S. Then there exists n ∈ N such that an ∈ Intra(S). If we denote by J the J -class of an , then J is a regular semigroup and an ∈ an Jan ⊆ an San . Thus, S is a π-regular semigroup. (iii)⇒(ix). Since S is a π-regular semigroup, by the proof of (iii)⇔(x), we have that each J -class of S containing an idempotent is a regular subsemigroup. (ix)⇒(iii). Let a, b ∈ S. Then there exist x ∈ S and n ∈ N such that (ab)n = (ab)n x(ab)n and (ab)n x, x(ab)n ∈ E(S). It is also true that (ab)n = (ab)n x(ab)n = (ab)n x(ab)n x(ab)n ∈ Sx(ab)n S and x(ab)n = x(ab)n x(ab)n ∈ S(ab)n S. Thus, (ab)n J x(ab)n , and in a similar way, we can show that (ab)n J (ab)n xJ (ab)2n . Therefore, (ab)n ∈ S(ab)2n S ⊆ S(ba)n+1 S and (ba)n+1 ∈ S(ab)n S, which implies that (ab)n , (ba)n+1 ∈ J(ab)n . Since the J -class J(ab)n contains an idempotent, it is a subsemigroup of S. Now (ba)n+1 (ab)n ∈ J(ab)n , whence (ab)n ∈ S(ba)n+1 (ab)n S ⊆ Sa2 S. By Theorem 1 of [11], S is a semilattice of Archimedean semigroups. (i)⇒(xi)⇒(iii). This follows by Theorem 3 of [9]. (i)⇒(xii). Let S be a semilattice Y of semigroups Sα which are nilextensions of simple regular semigroups Kα (α ∈ Y ). Consider an arbitrary a ∈ Reg(S). Then a ∈ Sα for some α ∈ Y , and there exists x ∈ S such that a = axa. Let x ∈ Sβ for some β ∈ Y . Then α = αβ = βα. Thus, xa ∈ Sα and xa ∈ E(Sα ) = E(Kα ), whence (xa)x ∈ Kα Sβ ⊆ Sα Sβ ⊆ Sαβ = Sα . Now a = a(xax)a ∈ aSα a, so a ∈ Reg(Sα ) ⊆ Kα ⊆ Intra(S). Therefore, Reg(S) ⊆ Intra(S).

(1)

Conversely, let a ∈ Intra(S). Then there exists α ∈ Y such that a ∈ Sα , and by Lemma 3.1 of [19], a ∈ Intra(Sα ), i.e., there exist u, v ∈ Sα such

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that a = ua2 v = uk a(av)k for every k ∈ N. Since Sα is a nil-extension of a simple regular semigroup Kα , there exists n ∈ N such that un , (av)n ∈ Kα . Hence, a = un+1 a2 y(ay)n ∈ Kα a2 Kα ⊆ Kα ⊆ Reg(S). Thus, Intra(S) ⊆ Reg(S).

(2)

By (1) and (2), we have Intra(S) = Reg(S). (xii)⇒(i). Let S be a semilattice Y of semigroups Sα (α ∈ Y ), and for each α ∈ Y , let Sα be a nil-extension of a simple semigroup Kα . For any a ∈ S, there exists n ∈ N such that an ∈ Kα ⊆ Intra(S) = Reg(S). Thus, S is a π-regular semigroup, and using (i)⇔(iii), we have that S is a semilattice of nil-extensions of simple regular semigroups. 2 Lemma 3.2. [20] If K is a subsemigroup of a completely π-regular semigroup S and it is also completely π-regular, then Gr(K) = K ∩ Gr(S). Theorem 3.3. A semigroup S is a semilattice of completely Archimedean semigroups if and only if S is completely π-regular, Reg(hE(S)i) = Gr(hE(S)i), and for all e, f ∈ E(S), f |e in S implies f |e in hE(S)i. Proof. Let S be a semilattice Y of completely Archimedean semigroups Sα (α ∈ Y ), where every Sα is a nil-extension of a completely simple semigroup Kα . Consider e, f ∈ E(S) such that e ∈ Sf S, and let e ∈ Sα and f ∈ Sβ for some α, β ∈ Y . Then α = αβ = βα and ef ∈ Sα , whence ef = eef ∈ Kα Sα ⊆ Kα . Now there exists x ∈ Kα such that ef = ef xef . Thus, xef ∈ E(Sα ). By Theorem 2.6, hE(Kα )i is (completely) simple, whence e ∈ hE(Kα )ixef hE(Kα )i = hE(Kα )ixef f hE(Kα ) ⊆ hE(S)if hE(S)i, which was to be proved. Using Lemma 2.5, hE(S)i is completely π-regular, and by Lemma 3.2, we have ReghE(S)i = S ∩ Reg(S) = S ∩ Gr(S) = GrhE(S)i. Conversely, let S be completely π-regular. By Lemma 2.5, hE(S)i is completely π-regular, and by Theorem 4 of [8], hE(S)i is a semilattice of completely Archimedean semigroups. Consider e, f, g ∈ E(S) such that e|g and f |g in S. By hypothesis, we have e|g and f |g in hE(S)i. By Theorem 3.1, ef |g in hE(S)i (and also in S). Again by Theorem 3.1, S is a semilattice of Archimedean semigroups. Since S is completely π-regular, by Theorem 4 of [8], S is a semilattice of completely Archimedean semigroups. 2 Further, we will consider chains of nil-extensions of simple regular semigroups. Note that chains of (completely) Archimedean semigroups were characterized in [3, 6, 7].

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Theorem 3.4. The following conditions on a semigroup S are equivalent: (i) S is a chain of nil-extensions of simple regular semigroups. (ii) (∀a, b ∈ S)(∃ n ∈ N) an ∈ an SabSan or bn ∈ bn SabSbn . (iii) S is π-regular and (∀e, f ∈ E(S)) ef |e or ef |f . (iv) S is π-regular and Reg(S) is a chain of simple regular semigroups. Proof. (i)⇒(ii). Let S be a chain Y of nil-extensions of simple regular semigroups Sα (α ∈ Y ). Let a, b ∈ S. Then a ∈ Sα and b ∈ Sβ for some α, β ∈ Y . If αβ = α, then a, ab ∈ Sα , and by Theorem 2.4, there exists n ∈ N such that an ∈ an SabSan . In a similar way, from αβ = β, we obtain bn ∈ bn SabSbn for some n ∈ N. (ii)⇒(i). It is clear that S is π-regular. Let a, b ∈ S. Then by hypothesis, there exists n ∈ N such that an ∈ SabS or bn ∈ SabS. By Theorem 1 of [1] and Theorem 1 of [6], S is a chain of Archimedean semigroups. Since S is π-regular, by Theorem 3.1, S is a chain of nil-extensions of simple regular semigroups. (ii)⇒(iii). This is clear. (iii)⇒(i). Let S be a π-regular semigroup and e, f, g ∈ E(S) such that f |e and g|e. Then there exist x, y, u, v ∈ S 1 such that e = xf y = ugv, whence (yxf )2 , (gvu)2 ∈ E(S). Now we have (yxf )2 ∈ S(yxf )2 (gvu)2 S or (gvu)2 ∈ S(yxf )2 (gvu)2 S. If (yxf )2 ∈ S(yxf )2 (gvu)2 S, then (yxf )2 ∈ Sf gS. Thus, e = eee = xf yxf yxf y = xf (yxf )2 y ∈ xf Sf gSy ⊆ Sf gS, so f g|e in S. If (gvu)2 ∈ S(yxf )2 (gvu)2 S, then f g|e in S. Now by Theorem 3.1, S is a semilattice Y of nil-extensions of simple regular semigroups Sα (α ∈ Y ). Let α, β ∈ Y and e, f ∈ E(S) such that e ∈ Sα and f ∈ Sβ . If ef |e in S, then αβ = α, and if ef |f , then αβ = β. Therefore, Y is a chain and S is a chain of simple regular semigroups. (i)⇒(iv). Let S be a chain Y of semigroups Sα (α ∈ Y ), and for α ∈ Y , let Sα be a nil-extension of a simple regular semigroup Kα . Let a, b ∈ Reg(S). Then a ∈ Sα and b ∈ Sβ for some α, β ∈ Y . It is clear that a ∈ Kα and b ∈ Kβ . Since Y is a chain, αβ = α or αβ = α. Suppose αβ = α. Then ab ∈ Sα , whence ab ∈ Kα Sα ⊆ Kα , i.e., ab ∈ Reg(S). Similarly, we can prove that αβ = β implies ab ∈ Reg(S). Hence, Reg(S) is a subsemigroup of S, and clearly, [ [ Reg(S) = Reg(Sα ) = Kα . α∈Y

α∈Y

Therefore, Reg(S) is a chain Y of simple regular semigroups Kα (α ∈ Y ). (iv)⇒(iii). Let S be π-regular and let Reg(S) be a chain Y of simple regular semigroups Kα (α ∈ Y ). Consider arbitrary e, f ∈ E(S). Then e ∈ Kα and f ∈ Kβ for some α, β ∈ Y . Since Y is a chain, e, ef ∈ Kα or f, ef ∈ Kβ , whence ef |e or ef |f . 2

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Chains of completely Archimedean semigroups were studied in [7]. Here, we give some new characterizations of these semigroups in terms of their idempotents. Theorem 3.5. The following conditions are equivalent on a semigroup S: (i) S is a chain of completely Archimedean semigroups. (ii) S is completely π-regular, and (∀e, f ∈ E(S)) e ∈ ef hE(S)if e or f ∈ f ehE(S)ief . (iii) S is completely π-regular, and (∀e, f ∈ E(S)) e ∈ ef hE(S)i or f ∈ hE(S)ief . (iv) S is completely π-regular and hE(S)i is a chain of completely simple semigroups. Proof. (i)⇒(ii). Let S be a chain Y of completely Archimedean semigroups Sα (α ∈ Y ). Clearly, S is a completely π-regular semigroup. Let e, f ∈ E(S), e ∈ Sα , and f ∈ Sβ for some α, β ∈ Y . Since Y is a chain, αβ = α or αβ = β. If αβ = α, then e, ef ∈ Sα , so by Theorem 1 and Lemma 1 of [10], ef e = e(ef )e ∈ eSα e = Ge . Thus, e, ef e ∈ Ge , whence e ∈ ef eGe ef e, i.e., e = ef exef e = ef xf e for some x ∈ Ge . Therefore, e = ef (f xf e)(ef xf )f e ∈ ef hE(S)if e. Similarly, we can prove that αβ = β implies f ∈ f ehE(S)ief . (ii)⇒(iii). This follows immediately. (iii)⇒(i). Let a, b ∈ S. Then (ab)m , (ba)n ∈ Reg(S) for some m, n ∈ N. Let x ∈ V ((ab)m ) and y ∈ V ((ba)n ). Then y(ba)n , (ab)m x ∈ E(S). By (iii), we obtain y(ba)n ∈ y(ba)n (ab)m xhE(S)i or (ab)m x ∈ hE(S)iy(ba)n (ab)m x, whence (ba)n ∈ (ba)n (ab)m xS or (ab)m ∈ Sy(ba)n (ab)m . Therefore, (ab)n+1 ∈ Sa2 S or (ab)m ∈ Sa2 S. By Theorem 1 of [11], it follows that S is a semilattice Y of completely Archimedean semigroups Sα (α ∈ Y ). If α, β ∈ Y and e ∈ E(Sα ), f ∈ E(Sβ ), then e ∈ ef hE(S)i implies αβ = α, and f ∈ hE(S)ief implies αβ = β. Thus, Y is a chain by (iii). (iv)⇒(ii). Since hE(S)i is a chain of completely simple semigroups, by (i)⇔(ii), we have the assertion. (i)⇒(iv). By Theorem 3 of [7], Reg(S) is a chain of completely simple semigroups. By the result of Fitzgerald [16], hE(S)i is a union of groups, whence by (i)⇔(ii), we obtain that hE(S)i is a chain of completely simple semigroups. 2 References [1] S. Bogdanovi´c, A note on strongly reversible semiprimary semigroups, Publ. Inst. Math. (Belgrade) 28 (42) (1980) 19–23. [2] S. Bogdanovi´c, Semigroups with a System of Subsemigroups, Institute of Mathematics, Novi Sad, 1985.

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