Advances in Pure Mathematics, 2014, 4, 62-70 Published Online February 2014 (http://www.scirp.org/journal/apm) http://dx.doi.org/10.4236/apm.2014.42010
Applications of Homomorphism on the Structure of Semigroups*
Huishu Yuan, Xiangzhi Kong School of Science, Jiangnan University, Wuxi, China Email: [email protected]
Received December 4, 2013; revised January 4, 2014; accepted January 10, 2014
Copyright © 2014 Huishu Yuan, Xiangzhi Kong. 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. In accordance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of the intellectual property Hui shu Yuan, Xiang zhi Kong. All Copyright © 2014 are guarded by law and by SCIRP as a guardian.
ABSTRACT By utilizing homomorphisms and G -strong semilattice of semigroups, we show that the Green (∗∼, ) -relation ∗∼, is a regular band congruence on a r -ample semigroup if and only if it is a G -strong semilattice of com- pletely ∗∼, -simple semigroups. The result generalizes Petrich’s result on completely regular semigroups with Green’s relation a normal band congruence or a regular band congruence from the round of regular semi- groups to the round of r -ample semigroups.
KEYWORDS Homomorphism; Natural Partial Order; Green’s (∗∼, ) -Relation; Semilattice Decomposition
1. Introduction and Preliminaries It is well known that the usual Green’s relations on a semigroup S play an important role in the study of the structure of regular semigroups [1-7]. Especially, the well-known theorem of A. H. Clifford states that a semi- group is a completely regular semigroup if and only if it can be expressed as a semilattice of completely simple semigroups (see [1]), where a completely regular semigroup is a semigroup whose -class contains an idem- potent. By using this result, A. H. Clifford, M. Petrich both showed that a completely regular semigroup S with its Green’s relation is a normal band congruence if and only if S is a strong semilattice of complete simple semigroups [3]. On the other hand, J. B. Fountain generalized the Clifford theorem by showing that an abundant semigroup is a superabundant semigroup, that is, an abundant semigroup S with every * -class of S contains an idempotent of S if and only if S is a semilattice of completely * -simple semigroups. We first recall some of the generalized Green’s relations which are frequently used to study the structure of abundant semigroups. The following Green ∗ -relations on a semigroup S were originally due to F. Pastijn [8] and were extensively used by J. B. Fountain to study the so called abundant semigroups in [9]. Let S be an arbitrary semigroup. Then, we define *1={(a, b) ∈× S S :,( ∀ x y ∈ S) ax = ay ⇔ bx = by} . Dually, we define ∗ ={(ab,) ∈× SS :,( ∀ xy ∈ Sxa1 ) = ya ⇔ xb = yb} and define **= ∗ , **= ∨ *, while *={(ab,:) ∈× S S J**( a) = J( b)} , where Ja* ( ) is the smallest ideal containing element a saturated by * and * , that is, Ja* ( ) is a union of some * -classes and also a union of some * -classes of S . It was given by M. V. Lawson in [10] the definition of on a semigroup S as
*The research is supported by the national natural science foundation of China (11371174, 11301227) and natural science foundation of Jiansu (BK20130119).
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H. S. YUAN, X. Z. KONG 63
ab ⇔∀∈( eESeaa( )) = ↔ ebb = , where ES( ) is the idempotents set of S . It can be easily seen that * ⊆ and for any regular elements ab, of a semigroup S , ab if and only if ab . In order to further investigate the structure of non-regular semigroups, we have to generalize the usual Green’s relations. For this purpose, J. B. Fountain and F. Pastijn both generalized the Green’s relations to the so called Green ∗ -relations in [8] and [9], respectively and by using these Green ∗ -relations, many new results of rpp -semigroups and abundant semigroups have been obtained by many authors in [10-18]. For the results of all other generalized Green’s relations and their mutual relationships, the reader is referred to a recent paper of Shum, Du and Guo [17]. In this paper, we introduce the concept of the Green (∗∼, ) -relations which is a common generalization of the Green ∗ -relations and the relation. We also introduce the concept of the G -strong semilattice of se- migroups and give the semilattice decomposition of a r -ample semigroup whose ∗∼, is a congruence. By using this decomposition, we will show that a semigroup S is a r -ample semigroup whose ∗∼, is a regular band congruence if and only if S is a G -strong semilattice of completely ∗∼, -simple semigroups. Our result extends and enriches the results of A. H. Clifford, M. Petrich and J. B. Fountain in the literature. We first generalize the usual Green’s relations and the Green ∗ -relations to the Green (∗∼, ) -relations on a semigroup S . ∗∼,,= ∗, ∗∼ = R ∗∼,,= ∗∼ ∗∼ ,,,,, ∗∼= ∗∼ ∨ ∗∼ ∗∼,={(ab,:) ∈× S S J∗∼,,( a) = J∗∼( b)} . where Ja∗∼, ( ) is the smallest ideal containing a saturated by ∗∼, and ∗∼, . We can easily see that ∗∼, is a right congruence on S while ∗∼, is only an equivalence relation on S . One can immediately see that ∗∼, ∗∼, there is at most one idempotent contained in each -class. If e∈ Ha ES( ) , for some aS∈ , then we 0 ∗∼, ∗∼, 00 write e as x , for any xH∈ a . Clearly, for any xH∈ a with aS∈ , we have x= xx = x x . If a semigroup S is a regular semigroup, then every -class of S contains at least one idempotent, and so does every -class of S . If S is a completely regular semigroup, then every -class of S contains an idempotent, in such a case, every -class is a group. A semigroup S is called an abundant semigroup by J. B. Fountain in [9] if every * - and * -class of S contains an idempotent. One can easily see that * = on the regular elements of a semigroup. Therefore, all regular semigroups are obviously abundant semigroups. As an analogy of the orthodox subsemigroup in a regular semigroup, a subsemigroup in an abundant semigroup is called a superabundant semigroup [9] if each of the * -classes of the abundant semigroup S contains an idempotent, in such a case, every * -class of S is a cancellative monoid, which is the generalization of com- pletely regular semigroups within the classes of abundant semigroups. The concept of ample semigroups was first mentioned in the paper of G. Gomes and V. Gould [19]. We now call a semigroup r -ample if each ∗∼, - class and each ∗∼, -class contain an idempotent, the concept was first mentioned by Y. Q. Guo, K. P. Shum and C. M. Gong [3]. Certainly, an abundant semigroup is a r -ample semigroup, but the converse is not true, and an example can be found in [3]. We now call a semigroup S a super r -ample semigroup if each ∗∼, - class of S contains an idempotent and ∗∼, is a congruence. It is clear that every ∗∼, -class of such super r -ample semigroup forms a left cancellative monoid which is a generalization of the completely regular se- migroups and the superabundant semigroups in the classes of r -ample semigroups. It is recalled that a regular band is a band that satisfies the identity axya= axaya . For further notations and terminology, such as strong semilattice decomposition of semigroups, the readers refer to [2,3]. For some other concepts that have already appeared in the literature, we occasionally use its alternatives, though equivalent definitions.
2. Properties of r-Ample Semigroups A completely simple semigroup is a -simple completely regular semigroup whose Green’s relation is a congruence on S , as a natural generalization of this concept, we call a r -ample semigroup S a completely ∗∼, -simple semigroup if it is a ∗∼, -simple semigroup and the Green (∗∼, ) -relation ∗∼, is a congruence on S . We first state the following crucial lemma.
OPEN ACCESS APM 64 H. S. YUAN, X. Z. KONG
Lemma 1 Let S be a r -ample semigroup with each ∗∼, -class contains an idempotent. Then the Green 0 0 ∗∼, relation on S is a congruence on S if and only if for any ab, ∈ S, (ab) = ( a00 b ) . Proof. Necessity. Let ab, ∈ S. Then, aa∗∼,0 and bb∗∼,0. Since ∗∼, is a congruence on S , 0 0 0 ab∗∼, a 00 b . But ab∗∼, ( ab) and so, (ab) = ( a00 b ) since every ∗∼, -class contains a unique idempotent. Sufficiency. Since ∗∼, is an equivalence on S , we only need to show that ∗∼, is compatible with the 0000 multiplication of S . Let (ab, )∈ ∗∼, and cS∈ . Then (ca) =( c00 a) =( c 00 b) = ( cb) and so that ∗∼, is left compatible with the multiplication on S . Similarly, ∗∼, is right compatible wit the multipication on S and thus ∗∼, is a congruence on S . Lemma 2 If ef, are ∗∼, -related idempotents of a r -ample semigroup S with each ∗∼, -class contains an idempotent, then ef . ∗∼, Proof. Since ef , there are elements aa1,, k of S such that ∗∼,, ∗∼ ∗∼ , ea12 aa k f. ∗∼,0,0 ∗∼ 0,∗∼ * Since S is r -ample, ea12 aa k f. Thus, ef since for regular elements = = and = * . Corollary 3 If S is a r -ample semigroup with each ∗∼, -class contains an idempotent, then ∗∼,= ∗∼ , ∗∼ , = ∗∼ ,, ∗∼. Proof. Let ab, ∈ S and ab∗∼, . Then, by Lemma 2, ab00 . Thus, there exist elements cd, in S with a00 cb and a00 db. Then ac∗∼,, ∗∼ b and a∗∼,, db ∗∼ and the result follows. Lemma 4 Let ef, be idempotents in a r -ample semigroup S with each ∗∼, -class contains an idempotent. if eJf , then eDf . 0 Proof. Since SeS= SfS , there are elements xyst, ,, in S such that f= set, e = xfy . Let h= ( fy) and 0 k= ( se) . Then hfy= fy = ffy so that h= h2 = fh , and sek= se = see so that k= k2 = ke . It follows that hf, ek are idempotents with hf h and ek k . Hence ehf eh and ekf kf . Now eh= xfyh = xfy = e and kf= kset = set = f so that e ef f , that is, ef . Proposition 5 If a is an element of a r -ample semigroup S , then J∗∼,0( a) = Sa S . Proof. Certainly, aa0,∈ ∗∼( ) so that Sa0, S⊆ ∗∼( a) . We now show that the ideal Sa0 S is actually an ideal which is saturated by ∗∼, and ∗∼, , since a= aa00 ∈ Sa S , the result follows. Let 0 0 bxaySaSxyS=∈∈00( , ) and k= ( ay0 ) . Then aay00= kay 0 so that a00( ay) = k 2 = k. Also since
0 0 ∗∼, is a congruence on S , xa0, y∗∼ xk . Now let h=( xk) = ( xa0 y) . Then xkh= xkk so that 2 00 ∗∼,, ∗∼ 0 0 h= h = hk = ha k ∈ Sa S . Hence if cL∈∈bb, dR, then c= ch, d = hd ∈ Sa S so that Sa S is indeed an ideal saturated by ∗∼, and ∗∼, , as required. Proposition 6 On a completely ∗∼, -simple semigroup S , ∗∼,,= ∗∼. Proof. Suppose that ab, ∈ S with ab ∗∼, . Then, by Proposition 5, Sa00 S= Sb S . By Lemma 4, ab00 and so a∗∼,0 ab 0∗∼ , b, which implies that ab∗∼, and hence ∗∼,,⊆ ∗∼. Conversely, let ab, ∈ S with ab∗∼, . Now, by Corollary 3, there exists cS∈ such that ac∗∼,, ∗∼ b. Thus ac00 b 0 and so Sa000 S= Sc S = Sb S . By Proposition 5, (ab, )∈ ∗∼, and hence ∗∼,,⊆ ∗∼. Now we have ∗∼,,= ∗∼. Proposition 7 A completely ∗∼, -simple S is primitive for idempotents. Proof. Let ef, be idempotents in S with ef≤ . Since S is a completely ∗∼, -simple semigroup, it follows from Proposition 5 that f∈ SeS . Now by the first part of Exercise 3 of [1,§ 8.4] there is an idempotent g of S such that fg and ge≤ . Let aS∈ be such that fag . Then fa0 g and since gf≤ we have a00= ga( gf) a 0= g( fa 0) = gf = g.
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Now we have gf≤ and gf and so f= fg = g. But ge≤ so that ef= and all idempotent of S are primitive. Lemma 8 In a completely ∗∼, -simple semigroupm S , the regular elements of S generate a completely simple subsemigroup. Proof. Let ab, be regular elements of S . Since S consists of a single ∗∼, -class(by Proposition 6), it follows from Corollary 3 that there is an element cS∈ with ac∗∼,, ∗∼ b. Hence, we have ac∗∼,0 ∗∼ , b. Thus, cb0 = b and ac 0 since a is regular. Now we see that ab b and the regularity of ab follows from that of b . The property of completely simple of the subsemigroup generated by regular elements follows Proposition 6, lemma 2 and Corollary 3 easily. Theorem 9 Let S be a r -ample semigroup.Then S is a semilattice Y of completely ∗∼, -simple ∗∼,,∗∼ ∗∼,,∗∼ semigroups SYα (α ∈ ) such that for α ∈Y and aS∈ α , aa(SS) = ( α ) , aa(SS) = ( α ) . Proof. If aS∈ , then aa∗∼,2 so that by Proposition 5, JaJa∗∼,( ) = ∗∼ ,2( ) . Now for ab, ∈ S, 2 (ab) ∈ SbaS , and so
2 J∗∼,,( ab) = J∗∼(( ab) ) ⊆ J∗∼ ,( ba) Now, by symmetry, we get J∗∼,,( ab) = J∗∼( ba) . By Proposition 5, J∗∼,0( a) = Sa S , J∗∼,0( b) = Sb S so that if cJ∈ ∗∼,,( a) J ∗∼( b) , we have c= xa00 y = zb t for some xyzt,,,∈ S. Now c2=∈⊆ zbtxay 00 SbtxaS 00 J∗∼ ,( btxa 00) and J∗∼,( b 0 txa 0) = J∗∼ ,( a 00 b tx) and by the preceding paragraph. we have c2∈ J∗∼ ,( ab 00) and since cc∗∼,2, c∈ J∗∼,( ab 00). Since aa∗∼,0, bb∗∼,0 and ∗∼, is a congruence on S , and so ab∗∼, a 00 b . Now, c∈ ∗∼, ( ab) , we have JaJbJab∗∼,,( ) ∗∼( ) ⊆ ∗∼ ,( ) and since the opposite inclusion is clear, we conclude that JaJbJab∗∼,,,( ) ∗∼( ) = ∗∼( ) . Because the set Y of all ideals J∗∼, ( aaS)( ∈ ) forms a semilattice under the usual set intersection and that the map aJa ∗∼, ( ) is a homomorphism from S onto Y . The inverse image of Ja∗∼, ( ) is just the ∗∼, ∗∼, ∗∼, -class Ja which is thus a subsemigroup of S . Hence S is a semilattice Y of the semigroups Ja . Now let ab, be elements of ∗∼, -class J ∗∼, and suppose that (ab, )∈ ∗∼,,( J ∗∼) . Certainly ab00, ∈ J∗∼ , so that we have (ab00, )∈ ∗∼ ,( J ∗∼ ,) , that is, ab00= a 0, ba 00 = b 0 and (ab00, )∈ ∗∼ ,( S) . It follows that ∗∼, ∗∼,,∗∼ ∗∼, ∗∼ ,, ∗∼ (ab, )∈ ( S) and consequently, since LSa ( ) ⊆ J, we have LSaa( ) = LJ( ) . A similar argument ∗∼,∗∼ ,, ∗∼ shows that RSRJaa( ) = ( ) . ∗∼,, ∗∼ ∗∼ , ∗∼, From the last paragraph, we have HJaa( ) = HS( ) so that J is a r -ample semigroup. Furthermore, if ab, ∈ J∗∼, , then by Proposition 6, (ab, )∈ ∗∼, ( S) so that, by Corollary 3, there is an element ∗∼,∗∼ , ∗∼ ,, ∗∼ ∗∼ ,, ∗∼ ∗∼, ∗∼, ∗∼, c in LSRab( ) ( S) = LJ a( ) R b( J) . Thus, ab, are -related in J so that J is a ∗∼, -simple semigroup. We need the following crucial lemma.
Lemma 10 Let S= ( YS; α ) be a r -ample semigroup.
1) Let aS∈ α and αβ≥ . Then, there exists bS∈ β with ab≥ ; 2) Let abc,,∈ S, bc∗∼, and a≥ bc, . Then, bc= ; 3) Let a∈ ES( ) and bS∈ be such that ab≥ . Then, b∈ ES( ) .
00 0 ∗∼, Proof. 1) Let bS∈ β . Then, by Lemma 1, a( aba) ,( aba) a and (aba) are in the same -class and 0000 0 so a( aba) = ( aba) a( aba) = ( aba) a . Let b= a( aba) . Then bS∈ β and ab≥ . 2) By the definition of “ ≥ ”, there exist ef,,, gh∈ E( S) such that b= ea = af , c= ga = ah . From eb= b and bb∗∼,0, we have eb00= b . Similarly, ch00= c . Thus, ec= ecc0 = ebc 00 = bc = c . Similarly, bh= b and so b= bh = eah = ec = c, as required. 3) We have b= ea = af for some ef, ∈ ES( ) whence
OPEN ACCESS APM 66 H. S. YUAN, X. Z. KONG
b22=( ea)( af) = ea f = b. Following Proposition 7, we can easily prove the following lemma Lemma 11 Let φ be a homomorphism from a completely ∗∼, -simple semigroup S into another 0 completely ∗∼, -simple semigroup T . Then (aaφφ) = 0 . If φ is a homomorphism between two completely ∗∼, -simple semigroups. Then the Green (∗∼, ) -relations ∗∼, , ∗∼, are preserved, so that ∗∼, is preserved. We call a homomorphism preserving ∗∼, , ∗∼, are good. By Proposition 7 and Lemma 10, we can show that a completely ∗∼, -simple semigroup is primitive.
3. G-Strong Semilattice Structure of r-Ample Semigroups In this section, we introduce the G -strong semilattice of semigroups which is a generalization of the well known strong semilattice of semigroups.
Definition 12 Let S= ( YS; α ) be a semilattice Y decomposition of semigroup S into subsemigroups
SYα (α ∈ ) . Suppose that the following conditions hold in the semigroup S .
(C1) for any αβ, ∈Y , there is a band congruence ραβ, on Sβ with congruence classes αβ∈ αβ αβ α ∈ ρ {Sdd (αβ, ) :,( ) D( ,)} , where D( , ) is the index set and for Y , αα, is the universal relation ω Sα ;
(C2) for αβ≥ on Y and any dD(αβ,,)∈ ( αβ) , there is a homomorphism φd (αβ, ) from Sα into Φ=φ αβ∈ αβ Sd (αβ, ) . Let αβ, { d (αβ, ) :,dD( ) ( ,)} . Then
1) for α ∈Y , the homomorphism φD(αα, ) : SSαα→ is the identity automorphism of the semigroup Sα .
2) for αβγ≥≥ on Y , Φαβ,, Φ βγ ⊆Φ αγ ,, where ΦΦαβ,, βγ is the set φ φ αβ∈∈ αβ βγ βγ { dd(αβ,,) ( βγ) :d( ,) Dd( ,,) ( ,) D( ,)} . b 3) for bS∈ β , there exists φd (α, αβ ) ∈Φα, αβ , for all aS∈ α ,
= φφba ab( add(α,, αβ) )( b ( β αβ ) ). If the semigroup S satisfies the above conditions, then we call S a G -strong semilattice of subsemi- α ∈ = Φ groups SYα ( ) and write S GYS;,α αβ, . One can easily see that a G -strong semilattice = Φ αβ = αβ≥ S GYS;,α αβ, is the ural strong semilattice if and only if all D( ,1) for all on Y . Following Theorem 9, we can easily see that a r -ample semigroup S is a semilattice of completely ∗∼, -
simple semigroups SYα (α ∈ ) . In this section, we introduce the band congruence ραβ, on a regular r -ample
semigroup S= ( YS; α ) and the structure homomorphisms set Φαβ, . Finally, we will show the main result of the paper, that is, a r -ample semigroup is a regular r -ample semigroup if and only if it is a G -strong semilattice of completely ∗∼, -simple semigroups.
Lemma 13 Let S= ( YS; α ) be a regular r -ample semigroup, that is, S is a r -ample semigroup with the Green (∗∼, ) -relation ∗∼, as a regular band congruence on S . Then, for any element αβ, ∈Y , we define
ραβ, on Sβ as the following: 00 ( x,, y∈ Sβ)( x y) ∈⇔ραβ, (axa) =( aya)
for some aS∈ α . Then
1) ραβ, is a band congruence on Sβ and for xy, ∈ Sβ , ( xy, )∈ ραβ, if and only if for any bS∈ α , 00 (bxb) = ( byb) . αβγ≥≥ ρρ⊆ ρ ω 2) for on Y , αγ,, βγ and αα, is the universal relation Sα on Sα .
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3) for αβ≥ on Y and aS∈ α , bS∈ β , abρραβ,, b αβ ba .
Proof. We only prove 1), 2) and 3) can be proved similarly. Let xy, ∈ Sβ with ( xy, )∈ ραβ, , then there 00 00 exists aS∈ α such that (axa) = ( aya) . For any element bS∈ α , we have b( axa) b= b( aya) b . Thus
0000 (b( axa) b) = ( b( aya) b) . By the property of regular bands and Lemma 1 and Lemma 8, we easily have 00 (bxb) = ( byb) . Now the proof is completed. ρ αβ∈ αβ αα We denote the αβ, -congruence classes by {Sdd (αβ, ) :,( ) D( ,)} , following Lemma 13, D( , ) is a singleton.
Lemma 14 Let S= ( YS; α ) be a regular r -ample semigroup.
1) For any αβ≥ on Y and dD(αβ,,)∈ ( αβ) . Let aS∈ α , there exists a unique element
aSdd(αβ,,) ∈ ( αβ) such that aa≥ d (αβ, ) .
0 2) For any αβ≥ on Y and aS∈ α , xS∈ d (αβ, ) , if ae≥ for some idempotent eS∈ d (αβ, ) , then 0 eax= ax, xae = xa , ea= ae and (ea) = e .
Proof. 1) By Lemma 13 2) and Lemma 10 1), for any cS∈ d (αβ, ) , the element
00 0 0 add(αβ,,) = a( aca) =( aca) a ∈ S (αβ) such that aa≥ d (αβ, ) . Easily see ad (αβ, ) = ( aca) . If there is another
bS∈ d (αβ, ) such that ab≥ , then there are idempotents gh, ∈ E( S) such that b= ga = ah and so ba00= b = a b , thus ba00= b 0 = ab 00 since bb∗∼,0, which implies ba00≤ and hence 0 000 000 ∗∼, b= a b a =( aba) =( aca) = ad (αβ, ) , that is, ba d (αβ, ) . Thus by Lemma 10 (ii), abd (αβ, ) = is required. 2) Since
000 (aaxaa00000000( ) ) = aaxa( ) = aaaxa( ( ) )
000 000000 and a00,00( ax) a∗∼( a( ax) a ) , we have (aaxa00000000( ) ) a= ( aaxa( ) ) = aaaxa( ( ) ) , that is, 0 0 00≥ 0 0 ∈ ∈ ∈ = 000 a( a( ax) a ) . Also, since aSα and xSd (αβ, ) , we have ax Sd (αβ, ) and so that e( a( ax) a )
000 00 by (i). Thereby, we have eax= ( a00( ax) a) ax= ( a 0000( ax) a) a( ax) a ax= ax . Similarly, we have
xae= xa . Since x is arbitrarily chosen element in Sd (αβ, ) , we can particularly choose xe= . In this way, we 0 0 obtain that ea= ae and consequently, by Lemma 1, we have (ea) =( ea0 ) = e .
Lemma 15 Let S= ( YS; α ) be a regular r -ample semigroup. For any αβ≥ on Y and
dD(αβ,,)∈ ( αβ) , define a mapping φd (αβ, ) from Sα into Sd (αβ, ) with aaφdd(αβ,,) = ( αβ) , where ad (αβ, ) Φ=φ αβ∈ αβ is defined in Lemma 14. Write αβ, { d (αβ, ) :,dD( ) ( ,)} . Then
1) φd (αβ, ) is a homomorphism.
2) for α ∈Y , φD(αα, ) is the identity homomorphism of Sα .
3) for αβγ≥≥ on Y , Φαβ,, Φ βγ ⊆Φ αγ ,. b 4) for bS∈ β , there exists φd (α, αβ ) ∈Φα, αβ , for all aS∈ α ,
= φφba ab( add(α,, αβ) )( b ( β αβ ) ).
Proof. 1) Following Lemma 14, φd (αβ, ) is well defined. For ab, ∈ Sα and cS∈ d (αβ, ) , by Lemma 14 again,
OPEN ACCESS APM 68 H. S. YUAN, X. Z. KONG
φφ= 0 00= = 0 ≤ (add(αβ,,) )( b( αβ) ) ( aca) ab( bcb) ( aca) ab ab( bcb) ab and so
φ= φφ (ab) d(αβ,) ( a dd(αβ ,,) )( b ( αβ) ).
2) It follows easily since Sα is primitive.
3) We only need to show that for any dD(αβ,,)∈ ( αβ) dD(βγ,,)∈ ( αγ) , SSdd(αβ,,)φ ( βγ) ⊆ d( αγ ,) for 00 some dD(αγ,,)∈ ( αγ) . Let aS∈ α , bb12, ∈ Sd (αβ, ) and cS∈ γ , we have (ab12 a) = ( ab a) and 0 0 b1φd (βγ, ) = b 1( b 11 cb ) , b2φd (βγ, ) = b 2( b 22 cb ) and so
0000 φφ= 00= = (abaabbcbaabbcbaaba( 1dd(βγ,,) ) ) ( 1( 11) ) ( 2( 2 2) ) ( ( 2(βγ) ) ) ,
which implies SSdd(αβ,,)φ ( βγ) ⊆ d( αβ ,) for some dD(αγ,,)∈ ( αγ) . ∈ Φ=φ ∀ β αβ ∈ β αβ ⊆ 4) For bSβ , we need to prove that bbdβ, αβ { dd(β,, αβ ) ( ,,) D( )} S(α αβ ) . In fact, it φ φ ∈Φ φφ∈ ρ suffices to show that for any d (β, αβ ) and d′(β, αβ ) β, αβ , we have (bbdd(β,, αβ) , ′( β αβ ) ) α, αβ . For this ′ 0 purpose, we let xS∈ d (β, αβ ) and xS∈ d′(β, αβ ) . Then, by (i), we have bφd (β, αβ ) = b( bxb) and ′ 0 ∗∼, bφd′(β, αβ ) = b( bx b) . Let aS∈ α , then, because Sαβ is a completely -simple semigroup, and aba , φ φ φ ∈ ∗∼, b d (β, αβ ) , b d′(β, αβ ) are elements in Sαβ . We obtain that (aba,( aba)( bd (β, αβ ) )( aba)) and φ ∈ (aba,( aba)( bd′(β, αβ ) )( aba)) . By Lemma 1, we conclude that
00 φφ= ((aba)( bdd(β,, αβ ) )( aba)) (( aba)( b′(β αβ ) )( aba)) .
00 00 In other words, ((aba)( b( bxb) )( aba)) = (( aba)( b( bx′ b) )( aba)) . Thus, by the regularity of the band 00 00 S ∗∼, , we can further simplify the above equality to (a( b( bxb) ) a) = ( a( b( bx′ b) ) a) , that is, 00 φφ= ρ (ab( dd(β,, αβ ) ) a) ( ab( ′(β αβ ) ) a) . It hence follows, by the definition of α, αβ , that is φφ∈ ρ (bbdd(β,, αβ) , ′( β αβ ) ) α, αβ . 0 Now let cS1 ∈ d′(α, αβ ) , cS2 ∈ d′(β, αβ ) . Then (ac1 a) ∈ Sd′(α, αβ ) because Sd′(α, αβ ) is a ρα, αβ -equivalence 0 0 class of Sαβ . Now, by (i), aφd′(α, αβ ) = ( ac1 a) a and bφd′(β, αβ ) = b( bc2 b) for φd′(α, αβ ) ∈Φα, αβ and 0 φd′(β, αβ ) ∈Φβ, αβ . Since we assume that aSΦ⊆α, αβ d′(β, αβ ) , we have aφdd′′(α,, αβ ) =( ac1 a) a ∈ S (β αβ ) . Similarly,
we have bφddd′′′(β,,, αβ) ∈ SS( α αβ) ( β αβ ) . Thus, we have φφ= 00= 0 (add′′(α,, αβ ) )( b(β αβ ) ) ( ac12 a) ( ab( bc b) ) ab( bc 2 b) and also
φφ= 0 00= (add′′(α,, αβ ) )( b (β αβ ) ) ((ac1 a) ab)( bc 21 b) ( ac a) ab. However, by the definition of the natural partial order “ ≤ ” on semigroup S , we have ≥ φφ ∗∼, ab( add′′(α,, αβ ) )( b (β αβ ) ) . On the other hand, because every Sαβ is a completely -simple semigroup, = φφ Sαβ is a primitive semigroup. Hence, we obtain that ab( add′′(α,, αβ ) )( b (β αβ ) ) .
Finally, we can easily see that for any aa12, ∈ Sα and β ∈Y , if a1,Φα αβ and a2,Φα αβ are all subsets of
OPEN ACCESS APM H. S. YUAN, X. Z. KONG 69
the same ρβ, αβ -class Sd′(β, αβ ) , then a1 and a2 determine the same mapping φd′(β, αβ ) ∈Φβ, αβ and hence for ∈ = φφbb= φφ any bSβ , we have aba11( dd(α,, αβ ) )( b′′(β αβ ) ) and aba2( 2dd(α,, αβ ) )( b(β αβ ) ). Theorem 16 A r -ample semigroup S is a regular r -ample semigroup if and only if it is a G -strong semilattice of completely ∗∼, -simple semigroups. Proof. We have already proved the necessity from Lemma 14 and Lemma 15. We now prove the sufficiency part of the theorem. We first show that the Green’s (∗∼, ) -relation ∗∼, is a congruence on S . In fact, if ∗∼, aS∈ α , bS∈ β then by the definition of G -strong semilattice and that each Sα is a completely
simple semigroup, we see that there exist φd (α, αβ ) ∈Φα, αβ and φd (β, αβ ) ∈Φβ, αβ satisfying the following equa- lities = φφ 00= 0φ0 φ ab( add(α,, αβ) )( b ( β αβ ) ) and a b( ad(α,, αβ ) )( b d(β αβ ) )
since ραβ, is a band congruence on Sβ . Hence, we deduce that
0 0 0 00 ab= aφφ b = a φ b φ ( ) ( dd(α,, αβ) )( ( β αβ ) ) ( d(α , αβ ) ) ( d(β , αβ ) ) 0 0 = a0φφ b0 = ab00 . ( dd(α,, αβ ) )( (β αβ ) ) ( )
∗∼, Now by Lemma 1, is a congruence on S= ( YS; α ) . To see that S ∗∼, is a regular band, by a result of [18], we only need to show that the Green’s relations and are both congruences on S ∗∼, . We only show that is a congruence in S ∗∼, as is a ∗∼, congruence in S which can be proved in a similar fashion. Since S= ( YS; α ) is a r -ample semigroup, we ∗∼,, ∗∼ ∗∼,, ∗∼ can let ef, and gS∈ , where ef, ∈ Sα ES( ) , g∈ Sβ ES( ) with (ef, )∈ . Then, we have ef= e and fe= f . By the definition of G -strong semilattice, we can find homomorphisms ef φd (β, αβ ) f g and φd (β, αβ ) ∈Φβ, αβ , φd (α, αβ ) ∈Φα, αβ such that
∗∼,,∗∼ ( gegf) = { g( ef) ( gf )}
= gφef ef φg ⋅ g φφf f g ∗∼, {( d(β, αβ ) )(( ) d(α , αβ ) )( dd(β ,, αβ ) )( (α αβ ) )} = gfφφef g ∗∼, ( dd(β,, αβ ) )( (α αβ ) ) and
ge∗∼,,= g ef∗∼ = gφef ef φg ∗∼,= g φφef f g ∗∼,. ( ) ( ) ( d(β,, αβ ) )(( ) d(α αβ ) )( dd(β,, αβ ) )( (α αβ ) ) Thereby, ( gegf)∗∼,,= ( ge) ∗∼. Analogously, we can also prove that ( gfge)∗∼,,= ( gf ) ∗∼. This proves that is left compatible on S ∗∼, . Since is always right compatible, we see that is a congruence on S ∗∼, , as required. Dually, is also a congruence on S ∗∼, . Thus by [18] (see II. 3.6 Proposition), S ∗∼, is a regular band and hence S is a regular cryptic r -ample semigroup. Our proof is completed.
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OPEN ACCESS APM 70 H. S. YUAN, X. Z. KONG
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