EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 1, No. 1, 2008, (46-59) ISSN 1307-5543 – www.ejpam.com Honorary Invited Paper On the structure of regular H̃-cryptogroups Xiangzhi Kong 1 , Yue Ding, K.P. Shum2,∗,† 1 School of Science, Jiangnan University, Wuxi, Jiangsu, 214122, China 2 Department of Mathematics,The University of Hong Kong, Pokfulam Road, Hong Kong (SAR), China Abstract. We introduce the concepts of Green ∼-relations on H̃-abundant semigroups. By using the gen- eralized strong semilattice of semigroups, we show that an H̃-cryptogroup is a regular H̃-cryptogroup if and only if it is an H̃G-strong semilattice of completely J̃ -simple semigroups. This result not only extends a known result of Petrich from the class of completely regular semigroups to the class of semiabundant semigroups but also generalizes a well known result of Fountain on superabundant semigroups from the class of abundant semigroups to the class of semiabundant semigroups. AMS subject classifications: 20M10 Key words: The Green ∼-relations; Homomorphisms of H̃-abundant semigroups; H̃-cryptogroups. 1. Introduction It was proved by Clifford [1] that a regular semigroup is a union of groups if and only if it is a semilattice of completely simple semigroups. It is also known that if the set of all idempotents of a completely regular semigroup S is the center of S, then S can be expressed by a strong semilattice of groups (see [1]). Thus, we usually regard the completely regular semigroups as generalized groups. Moreover, by Petrich and Reilly, we call a completely regular semigroup S a normal cryptogroup if the Green relation H on S is a normal band congruence on S. In particular, a completely regular semigroup S is a normal cryptogroup if and only if S can be expressed by a strong semilattice of completely simple semigroups (see [12] and [13]). This result was further generalized by Fountain by proving that an abundant semigroup S is a superabundant semigroup if and only if S is a semilattice of completely J ∗ -simple semigroups [4]. The structure of superabundant semigroups whose set of idempotents forms a subsemigroup have been recently extensively investigated by Ren and Shum in [15] and [16]. The Green ∗-relations on a semigroup S were first defined by Pastijn [11] which can be re- garded as the Green relations in some oversemigroups of S. These relations were formulated by ∗Corresponding author. Email addresses: xiangzhikong@163.com (X. Kong), kpshum@maths.hku.edu.hk (K.P. Shum) †The research of K.P. Shum is partially supported by a Wu Jiehyee Charitable foundation grant no. 7103084, 2006-07 http://www.ejpam.com 46 c© 2007 EJPAM All rights reserved. baris Stamp X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 47 Fountain [4] as follows: L∗ = {(a, b) ∈ S × S : (∀x, y ∈ S1)ax = ay ⇔ bx = by}, R∗ = {(a, b) ∈ S × S : (∀x, y ∈ S1)xa = ya ⇔ xb = yb}, H∗ = L∗ ∩R∗, D∗ = L∗ ∨R∗. Later on, El-Qallali further generalized the Green ∗-relations to Green ∼-relations [3] as follows: L̃ = {(a, b) ∈ S × S : (∀e ∈ E(S))ae = a ⇔ be = b}, R̃ = {(a, b) ∈ S × S : (∀e ∈ E(S))ea = a ⇔ eb = b}, H̃ = L̃ ∩ R̃, D̃ = L̃ ∨ R̃. We can easily see that L̃ and R̃ are equivalent relations on S, however, the L̃ relation is not neces- sary to be right compatible with the semigroup multiplication and the R̃ relation is not necessary to be left compatible with the semigroup multiplication. We now denote the L̃ -class containing the element a of the semigroup S by L̃a and we observe thatL ⊆ L∗ ⊆ L̃. Among the usual Green relations or the above relations, L- or the generalized L-relations are duals of the corresponding R -relations or generalized R-relations. In what follows, we only discuss the properties which are related to the L- relation and the generalized L-relation, respectively. One can easily see that there is at most one idempotent of the semigroup S in each H̃ -class. If e ∈ H̃a ∩E(S), for some a ∈ S, then we simply denote the idempotent e by x0, for any x ∈ H̃a. Clearly, for any x ∈ H̃a with a ∈ S, we have x = xx0 = x0x. If a semigroup S is regular, then every L-class of S contains at least one idempotent, and so does every R-class of S. If S is a completely regular semigroup, then every H-class of S contains an idempotent. According to Fountain [4], a semigroup is abundant if every L∗- and R∗-class of S contains some idempotents. In other words, the term “abundant” means that the semigroup has plenty of idempotents. Clearly, we have L∗ = L on the set of all regular elements of a semigroup. Thus, regular semigroups are obviously special abundant semigroups. Thus, Fountain called such semigroup superabundant [4] if its everyH∗-classes contains an idempotent. Obviously, completely regular semigroups are special superabundant semigroups. Following El- Qallali [3], we call a semigroup S a semiabundant semigroup if every L̃-class and every R̃-class of S contain at least one idempotent. A semigroup S is called H̃-abundant if every H̃-class contains an idempotent of S. Clearly, the H̃-abundant semigroups are generalizations of superabundant semigroups in the class of semiabundant semigroups. One can easily see that L̃ = L on the set of regular elements in any H̃-abundant semigroup. Throughout this paper, we call a band B a regular band (right quasi normal band) if B satisfies the identity axya = axaya(xya = xaya). According to Petrich and Reilly [12], a completely regular semigroup S was called a regular cryptogroup if the Green relation H on S is a regular band congruence on S. The structure of regular cryptogroup was investigated by Kong-Shum in [8] and [9]. In the class of abundant semigroups, Guo and Shum [5] called an abundant semigroup whose set of idempotents forms a regular band a cyber group. The semilattice structure of regular cyber groups have been recently investigated in [9]. Naturally, one would ask : can we establish an analogous result of superabundant semigroups [4] in the class of semiabundant semigroups or an analogous result of cryptogroups [12] in the X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 48 class of H̃ -abundant semigroups? In this paper, we will establish a theorem for H̃-cryptogroups by using the Green∼-relations and theKG-strong semilattice of semigroups, as described in [10]. We will show that an H̃-cryptogroup is a regular H̃-cryptogroup if and only if it is an H̃G-strong semilattice of completely J̃ -simple semigroups. Our results in this paper also generalize and enrich the corresponding results given in [1], [4], [7], [8] and [13]. 2. KG-strong semilattices We now restate the concept of G-strong semilattice decomposition of semigroup S given by Kong and Shum in [8] and [9]. Let S = (Y ;Sα) be a semilattice of the semigroups Sα, where each Sα is a subsemigroup of the semigroup S and Y is a semilattice. We define the G-strong semilattice of semigroups by generalizing the well known strong semilattice of semigroups ( see [9]). Definition 2.1 Let S = (Y ; Sα) be a semigroup. Suppose that the following conditions S are satisfied: (i) (∀α, β ∈ Y, α > β), there exists a family of homomorphisms ϕd(α,β) : Sα −→ Sβ , where d(α, β) ∈ D(α, β) and D(α, β) is a non-empty index set. (ii) (∀α ∈ Y ), D(α, α) is a singleton. Denote the element in D(α, α) by d(α, α). In this case, the homomorphism ϕd(α,α) : Sα −→ Sα is the identity automorphism of the semigroup Sα. (iii) (∀α, β, γ ∈ Y, α > β > γ), if we write ϕα,β = {ϕd(α,β) : d(α, β) ∈ D(α, β)} then ϕα,βϕβ,γ ⊆ ϕα,γ , where ϕα,βϕβ,γ = {ϕd(α,β)ϕd(β,γ) : ∀d(α, β) ∈ D(α, β), d(β, γ) ∈ D(β, γ)}. (iv) for each α, β ∈ Y , there is a mapping from Sα into the set ϕβ,αβ whose value at any given element a ∈ Sα is denoted by ϕa d(β,αβ) such that for all b ∈ Sβ , ab = (aϕb d(α,αβ))(bϕ a d(β,αβ)). Then the above semilatttice of semigroups is called the generalized strong semilattice of semi- groups Sα and in brevity , the “G-strong semilattice” of semigroups Sα and denoted it by S = G[Y ; Sα, ϕα,β]. The following definition is a more general version of G-strong semilattices. Definition 2.2 Let K be any equivalent relation on a G-strong semilattice of semigroups S = G[Y ; Sα, ϕα,β]. Then, we call S a “KG-strong semilattice of semigroups Sα” if for every α, β ∈ Y, the mapping a 7−→ ϕa α(β,αβ) has the property that ϕa d(β,αβ) = ϕb d(β,αβ) whenever the elements a, b ∈ Sα are in the same K-class of S. Thus, it is clear that the G-strong semilattice of semigroups S can be determined by an equivalent X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 49 relation K. We therefore call the above generalized strong semilattice of semigroups Sα a “KG -strong semilattice of semigroups Sα” and is denoted by S = KG[Y ; Sα, ϕα,β], where K is any one of the Green relations L, R, D and H, respectively. Remark 2.3 It is clear that the KG-strong semilattice is stronger than the G-strong semilattice but it is weaker than the usual strong semilattice. In fact, if ρ and δ are equivalent relations on the semigroup S = (Y ; Sα) with ρ ⊆ δ, then one can observe that δG[Y ; Sα, ϕα,β] is “stronger” than ρG[Y ; Sα, ϕα,β]. As special cases, 1SG[Y ; Sα, ϕα,β] is the “weakest”KG-strong semilattice of semigroups since 1S is the “smallest” equivalent relation on S and also ηG[Y ; Sα, ϕα,β] is the strongest KG-strong semilattice of semigroups since η is the “greatest” equivalent relation on S, where 1S is the identity relation on S and η is the semilattice congruence on S which partitions the semigroup S into disjoint subsemigroups Sα(α ∈ Y ) of S. Hence, we can easily see that ηG[Y ;Sα, ϕα,β] is the usual strong semilattice of semigroups since in this case, every index set D(α, β) is a singleton for α > β on Y and hence there exists one and only one structure homo- morphism in the set of structure homomorphisms ϕα,β . We have already defined the Green ∼-relations L̃, R̃, H̃ and D̃ on a semigroup S. In order to define the Green ∼-relation J̃ on S, we consider the left ∼-ideal L of a semigroup S. Definition 2.4 A left (right) ideal L (R) of a semigroup S is called a left ∼-ideal of S if L̃a ⊆ L(R̃a ⊆ R) holds, for all a ∈ L(a ∈ R). We call a subset I of a semigroup S a ∼-ideal of S if it is both a left ∼-ideal and a right ∼-ideal. It is noteworthy that if S is a regular semigroup, then every left (right, two-sided) ideal of S is a left (right, two-sided) ∼-ideal. We also observe that for any idempotent e in a semigroup S , the left (right) ideal Se(eS) is a left(right) ∼-ideal. For if a ∈ Se, then a = ae , and hence for any element b in L̃a, we have b = be ∈ Se. By Definition 2.4, we see that the semigroup S is always a∼-ideal of itself, and we denote the smallest ∼-ideal containing the element a of S by J̃(a). Now, we define J̃ = {(a, b) ∈ S × S : J̃(a) = J̃(b)}. Definition 2.5 An H̃-abundant semigroup S is called completely J̃ -simple if S does not contain any non-trivial proper ∼-ideal of S. We now give some properties of the H̃-abundant semigroups. Some of the properties may have already been known or can be easily derived, however, for the sake of completeness, we provide here the proofs. Lemma 2.6 Let S be an H̃-abundant semigroup. Then the following properties hold: (i) The Green ∼-relation H̃ is a congruence on S if and only if for any a, b ∈ S, (ab)0 = (a0b0)0. (ii) If e, f are D̃-related idempotents of S, then eDf. (iii) D̃ = L̃ ◦ R̃ = R̃ ◦ L̃. X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 50 (iv) If e, f are idempotents in S such that eJ f , then eDf . Proof. (i) (Necessity). For any a, b ∈ S, we have aH̃a0 and bH̃b0. Since H̃ is a congruence on S, abH̃a0b0. But abH̃(ab)0, and so (ab)0 = (a0b0)0 since every H̃-class contains a unique idempotent. (Sufficiency). We only need to show that H̃ is compatible with the semigroup multipli- cation of S since H̃ is an equivalent relation on S. Let (a, b) ∈ H̃ and c ∈ S. Then (ca)0 = (c0a0)0 = (c0b0)0 = (cb)0 and hence, H̃ is left compatible to the semigroup mul- tiplication. Dually, H̃ is right compatible with the semigroup multiplication and thus H̃ is a congruence on S. (ii) Since eD̃f , there exist elements a1, · · · , ak of S such that eL̃a1R̃a2 · · · ak L̃ f . Since S is an H̃-abundant semigroup, eLa0 1Ra0 2 · · · a0 kLf . Thus eDf . (iii) If a, b ∈ S and aD̃b, then by (ii), a0Db0. Hence there exist elements c, d in S with a0LcRb0 and a0RdLb0, and consequently, aL̃cR̃b and aR̃dL̃b. Thus the result is proved. (iv) Since SeS = SfS, there exist elements x, y, s, t in S such that f = set and e = xfy. Let h = (fy)0 and k = (se)0. Then hfy = fy = ffy and so h = h2 = fh and sek = se = see, and thereby, k = k2 = ke. Hence, hf, ek are the idempotents satis- fying the relations hfRh and ekLk . These imply that ehfReh and ekfLkf . Now by eh = xfyh = xfy = e and kf = kset = set = f , we have eRefLf . This shows that eDf . Similar to the definition of cyber group given by Guo and Shum [5], we formulate the following definition. Definition 2.7 An H̃-abundant semigroup S is called an H̃-cryptogroup if the Green ∼-relation H̃ is a congruence on S. Also, we call an H̃-abundant semigroup S a regular H̃-cryptogroup if H̃ is a congruence on S such that S/H̃ is a regular band. Thus, H̃-cryptogroups are analogy of cryptogroups in the class of H̃-abundant semigroups. Also, we see in [5] that an H̃-cryptogroup is a generalized cyber groups. The H̃-cryptogroup S has the following properties: Lemma 2.8 (i) For any element a of the H̃-cryptogroup S , J̃(a) = Sa0S. (ii) For the H̃-cryptogroup S, J̃ = D̃. (iii) If the H̃-cryptogroup S is completely J̃ -simple, then the idempotents of S are primitive. X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 51 (iv) If the H̃-cryptogroup S is completely J̃ -simple, then the regular elements of S generate a regular subsemigroup of S. Proof. (i) Obviously, we have a0 ∈ J̃(a) and so Sa0S ⊆ J̃(a). We need to show that the ideal Sa0S is in fact a ∼-ideal and since a = aa0a0 ∈ Sa0S, J̃(a) ⊆ Sa0S. Let b = xa0y ∈ Sa0S(x, y ∈ S) and k = (a0y)0. Then a0a0y = a0y = ka0y so that a0(a0y)0 = k2 = k. Also since H̃ is a congruence, xa0yH̃xk. Now let h = (xk)0 = (xa0y)0. Then xkh = xk = xkk so that h = h2 = hk = ha0k ∈ Sa0S. Hence if c ∈ L̃b, d ∈ R̃b, then c = ch, d = hd ∈ Sa0S and hence, Sa0S is a ∼-ideal, as required. (ii) Suppose that (a, b) ∈ S with aJ̃ b. Then by (i), we have Sa0S = Sb0S. Now, by Lemma 2.6 (iv), a0Db0 and so aH̃a0Db0H̃b. This implies that aD̃b and hence J̃ ⊆ D̃. Conversely, let a, b ∈ S with aD̃b. Then by Lemma 2.6 (iii), there exists an element c ∈ S such that aL̃cR̃b. This leads to a0Lc0Rb0 and so Sa0S = Sc0S = Sb0S. Now, by (i), (a, b) ∈ J̃ and hence D̃ ⊆ J̃ . Therefore, J̃ = D̃. (iii) Let e, f be idempotents in S with e 6 f . Since S is completely J̃ -simple, f ∈ SeS. Now by the first part of Exercise 3 in [14][§8.4], there exists an idempotent g of S such that fDg and g 6 e. Let a ∈ S be such that fLaRg. Then fLa0Rg and since g 6 f, we have a0 = ga0(gf)a0 = g(fa0) = gf = g. Now by noting that g 6 f and gLf, we have f = fg = g. However, since g 6 e , we obtain e = f and hence all idempotents of S are primitive. (iv) Let a, b be regular elements of S. Since S consists of a single D̃-class, by (ii) and by Lemma 2.6 (iii), there exists an element c ∈ S such that aL̃cR̃b. Hence aL̃c0R̃b. This leads to c0b = b and aLc0 since a is regular. Now we have abLb and so the regularity of ab follows from the regularity of b. We now establish the following theorem for H̃ -cryptogroups. Theorem 2.9 Let S be an H̃ -cryptogroup. Then S is a semilattice Y of completely J̃ -simple semigroups Sα(α ∈ Y ) such that for every α ∈ Y and a ∈ Sα, we have L̃a(S) = L̃a(Sα) and R̃a(S) = L̃a(Sα). Proof. If a ∈ S, then aH̃a2 and so, J̃(a) = J̃(a2). Now for a, b ∈ S, we have (ab)2 ∈ SbaS, and hence, it follows that J̃(ab) = J̃((ab)2) ⊆ J̃(ba). Now, by symmetry, we obtain J̃(ab) = J̃(ba). Since, by Lemma 2.8 (i), we have J̃(a) = Sa0S and J̃(b) = Sb0S so that if c ∈ J̃(a) ∩ J̃(b), then c = xa0y = zb0t for some x, y, z, t ∈ S. Now c2 = zb0txa0y ∈ Sb0txa0S ⊆ J̃(b0txa0) and hence, J̃(b0txa0) = J̃(a0b0tx) by using previous arguments. Thus, c2 ∈ J̃(a0b0) and since cH̃c2, we have c ∈ J̃(a0b0). Since aH̃a0, X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 52 bH̃b0 and H̃ is a congruence on S, we have abH̃a0b0 . Consequently, c ∈ J̃(ab), and thereby J̃(a) ∩ J̃(b) ⊆ J̃(ab). The converse containment is clear so that J̃(a) ∩ J̃(b) = J̃(ab). We can easily see that the set Y of all∼-ideals J̃(a)(a ∈ S) forms a semilattice under set intersection and that the mapping a 7→ J̃(a) is a homomorphism from S onto Y . The inverse image of J̃(a) is just the J̃ -class J̃a which is a subsemigroup of S. Hence S is a semilattice Y of the semigroups J̃a. Now let a, b be elements of J̃ -class J̃ and suppose that (a, b) ∈ L̃(J̃). Then, a0, b0 ∈ J̃ so that (a0, b0) ∈ L̃(J̃), that is, a0b0 = a0, b0a0 = b0 and (a0, b0) ∈ L̃(S). It follows that (a, b) ∈ L̃(S) and consequently, by L̃a(S) ⊆ J̃ , we have L̃a(S) = L̃a(J̃). By using a similar argument, we can show that R̃a(S) = R̃a(J̃). From the above discussion, we can deduce that H̃a(J̃) = H̃a(S) and so J̃ is indeed an H̃-abundant semigroup. Furthermore, if a, b ∈ J̃ , then by Lemma 2.8 (i), (a, b) ∈ D̃(S) and hence, by Lemma 2.6 (iii), there exists an element c in L̃a(S) ∩ R̃b(S) = L̃a(J̃) ∩ R̃b(J̃). Thus a, b are D̃-related in J̃ and so J̃ is J̃ -simple. For the H̃-cryptogroups, we have the following theorem. Theorem 2.10 Let S be an H̃ -cryptogroup which is expressed by the semilattice of semigroups S = (Y ; Sα). Then the following statements hold: (i) For α, and β in the semilattice Y with α > β, if a ∈ Sα then there exists b ∈ Sβ with a > b; (ii) For a, b, c ∈ S with bH̃c, if a > b, a > c then b = c; (iii) For a ∈ E(S) and b ∈ S, if a > b then b ∈ E(S) . Proof. (i) Let c ∈ Sβ . Then, by Lemma 2.6 (i), we see that a(aca)0, (aca)0a and (aca)0 are all in the same H̃-class of the semigroup S and hence, a(aca)0 = (aca)0a(aca)0 = (aca)0a. Write b = a(aca)0. Then b ∈ Sβ and a > b. (ii) By the definition of “>”, there exist e, f, g, h ∈ E(S) such that b = ea = af , c = ga = ah. From eb = b and bH̃b0, we have eb0 = b0. Similarly, c0h = c0. Thus ec = ec0c = eb0c = b0c = c. By using similar arguments, we have bh = b and so, b = bh = eah = ec = c, as required. (iii) We have b = ea = af for some e, f ∈ E(S), and whence b2 = (ea)(af) = ea2f = b. The following fact can be easily observed: Fact 2.11 Let ϕ be a homomorphism which maps an H̃-cryptogroup S into another H̃ -cryptogroup T . Then (aϕ)0 = a0ϕ. 3. Properties of regular H̃-cryptogroups Lemma 3.1 Let S be a regular H̃ -cryptogroup(that is, H̃ is a congruence on the H̃-abundant semigroup S such that S/H̃ is a regular band). For every a ∈ S, we define a relation ρa on S by (b1, b2) ∈ ρa if and only if (ab1a)0 = (ab2a)0, (b1, b2 ∈ S). Then the following properties hold on S: X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 53 (i) ρa is a band congruence on S; (ii) (∀a, a1 ∈ Sα), ρa = ρa1 , that is, ρa depends only on the component Sα containing the element a rather than on the element itself, hence we can write ρα = ρa, for all a ∈ Sα. (iii) (∀α, β ∈ Y with α > β), ρα ⊆ ρβ and ρβ|Sα = ωSα , where ωSα is the universal relation on Sα. Proof. (i) It is easy to see that ρa is an equivalent relation on S, for all a ∈ S. We now prove that ρa is left compatible with the semigroup multiplication. For this purpose, we let (x, y) ∈ ρa and c ∈ S. Then, by the definition of ρa, we have (axa)0 = (aya)0. Since S is a regular H̃-cryptogroup, by Lemma 2.6 (i) and the regularity of the band S/H̃, we obtain that (acxa)0 = (ac(axa))0 = ((ac)0(axa)0)0 = ((ac)0(aya)0)0 = (acya)0. Hence, (cx, cy) ∈ ρa. Dually, we can prove that ρa is right compatible with the semigroup multi- plication. Thus ρa is a congruence on S. Obviously, H̃ ⊆ ρa and so ρa is a band congruence on S. (ii) Let (x, y) ∈ ρa. Then, by the definition of ρa, we have (axa)0 = (aya)0 and so a0 1(axa)0a0 1 = a0 1(aya)0a0 1. This leads to (a0 1(axa)0a0 1) 0 = (a0 1(aya)0a0 1) 0. Since S/H̃ = (Y ; Sα/H̃) is a reg- ular band and by Lemma 2.6 (i), we obtain (a1aa1xa1aa1)0 = (a1aa1ya1aa1)0. However, since a, a1 are elements of the completely J̃ -simple semigroup Sα, (a1aa1)0 = a0 1. Thereby, by Lemma 2.6 (i) again, we have (a1xa1)0 = (a1ya1)0, that is, (x, y) ∈ ρa1 . This shows that ρa ⊆ ρa1 . Sim- ilarly, we also have ρa1 ⊆ ρa. Thus, ρa = ρa1 . Since this relation holds for all a ∈ Sα, we usually write ρa = ρα. (iii) Let a ∈ Sα, b ∈ Sβ and α > β. We need to prove that ρα ⊆ ρβ . For this purpose, we let (x, y) ∈ ρα = ρa, by (ii). Then, by the definition of ρa, we have (axa)0 = (aya)0 and hence b(axa)0b = b(aya)0b. By Lemma 2.6 (i) and the regularity of the band, we have (babxbab)0 = (babybab)0. Since α > β in Y and a ∈ Sα, b ∈ Sβ , we have (bab)0 = b0. By using Lemma 2.6 (i) again, we can show that (bxb)0 = (byb)0, that is, (x, y) ∈ ρb = ρβ. Thus, ρα ⊆ ρβ as required. Furthermore, it is trivial that ρβ|Sα = ωSα , which is the universal relation on the semigroup Sα. We now use the band congruence ρα defined in Lemma 3.1 to describe the structural homomor- phisms for the H̃ -cryptogroup S = (Y ; Sα), where each Sα is a completely J̃ -simple semigroup. We first consider the congruence ρα,β = ρα|Sβ for α, β ∈ Y , which is a band congruence on the semigroup Sβ . Now, we denote all the ρα,β-classes of Sβ by {Sd(α,β) : d(α, β) ∈ D(α, β)}, where D(α, β) is a non-empty index set. In particular, the set D(α, α) is a singleton and we can therefore write d(α, α) = D(α, α). We have the following lemma. Lemma 3.2 Let S = (Y ;Sα) be a regular H̃-cryptogroup. Then, for all α, β ∈ Y with α > β, the following statements hold for all d(α, β) ∈ D(α, β). (i) For all a ∈ Sα, there exists a unique ad(α,β) ∈ Sd(α,β) satisfying a > ad(α,β) ; (ii) For all a ∈ Sα and x ∈ Sd(α,β), if a0 > e for some idempotent e ∈ Sd(α,β) then eax = ax, xae = xa, ea = ae and (ea)0 = e; X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 54 (iii) Let a ∈ Sα. Define ϕd(α,β) : Sα −→ Sd(α,β) by aϕd(α,β) = ad(α,β), where ad(α,β) ∈ Sd(α,β) and a > ad(α,β). Then ϕd(α,β) is a homomorphism and ad(α,β) = a(aba)0 = (aba)0a for any b ∈ Sd(α,β). Proof. (i) We first show that for any a ∈ Sα and b ∈ Sd(α,β), we have ab ∈ Sd(α,β), that is, (ab, b) ∈ ρα,β . In fact, since S = (Y, Sα) is an H̃ -cryptogroup, each Sα is a completely J̃ -simple semigroup. Hence, we have (xax)0 = x0, for all x ∈ Sα. This leads to (xabx)0 = (xaxbx)0 = (xbx)0 by the regularity of the band S/H̃ and Lemma 2.6 (i). Thereby, (ab, b) ∈ ρα,β . Similarly, we also have ba ∈ Sd(α,β). Invoking the above results, we have aba ∈ Sd(α,β) for any b ∈ Sd(α,β). Since H̃ is a band congruence on S, by Lemma 2.6 (i) again, we see that a(aba)0, (aba)0 and (aba)0a are in the same H̃-class of S so that a(aba)0 = (aba)0a(aba)0 = (aba)0a. Let a(aba)0 = ad(α,β). Then by the natural partial order imposed on S, we have a > ad(α,β). In order to show the uniqueness of ad(α,β), we assume that there is another a∗d(α,β) ∈ Sd(α,β) satisfying a > a∗d(α,β). Then, by the definition of “6”, we can write a∗d(α,β) = ea = af for some e, f ∈ E(S) and so a∗d(α,β)a 0 = a∗d(α,β) = a0a∗d(α,β). By the fact a∗d(α,β)H̃a0, we have (a∗d(α,β)) 0a0 = (a∗d(α,β)) 0 and a0(a∗d(α,β)) 0 = (a∗d(α,β)) 0. Consequently, by the definition of “ 6 ”, we have a0 > (a∗d(α,β)) 0. By Lemma 2.6 (i) again, we deduce that (a∗d(α,β)) 0 = (a0(a∗d(α,β)) 0a0)0 = (aa∗d(α,β)a)0 = (aba)0. Hence, (a∗d(α,β), ad(α,β)) ∈ H̃, and consequently, by Theorem 2.10 (ii), a∗d(α,β) = ad(α,β). This shows the uniqueness of ad(α,β). (ii) It is easy to see that, by the definition of “6”, a0 > (a0(ax)0a0)0. Also, since a ∈ Sα and x ∈ Sd(α,β), we have ax ∈ Sd(α,β) by (i). Moreover, since Sd(α,β) is a ρα,β -congruence class, (ax)0 ∈ Sd(α,β). Thus, by (i) again, we have (a0(ax)0a0)0 ∈ Sd(α,β) and e = (a0(ax)0a0)0. Thereby, we have eax = (a0(ax)0a0)0a0(ax)0a0ax = ax. Sim- ilarly, we have xae = xa. Since x is arbitrarily chosen element in Sd(α,β), we can particularly choose x = e. In this way, we obtain ea = ae and consequently, by Lemma 2.6 (i), we have (ea)0 = (ea0)0 = e. (iii) By using the result in (i), we can define ϕd(α,β) : Sα −→ Sd(α,β) by aϕd(α,β) = ad(α,β) = a(aca)0 = (aca)0a, for any a ∈ Sα and c ∈ Sd(α,β). Then, for any a, b ∈ Sα, we have, by (ii), (aϕd(α,β))(bϕd(α,β)) = ad(α,β)bd(α,β) = (aca)0ab(bcb)0 = (aca)0(ab(bcb)0) = ab(bcb)0. Similarly, we can show that (aϕd(α,β))(bϕd(α,β)) = (aca)0ab. Hence, ab > (aϕd(α,β))(bϕd(α,β)). Thus (ab)ϕd(α,β) = (aϕd(α,β))(bϕd(α,β)), by the definition of ϕd(α,β). This shows that ϕd(α,β) is indeed a homomorphism. We now proceed to show that the homomorphisms given in Lemma 3.2 (iii) are the structural homomorphisms for the G-strong semilattice G[Y ; Sα , ϕα,β] induced by the semigroup S = (Y ;Sα) under the band congruence ρα on the semigroup Sα. X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 55 Lemma 3.3 Let S = (Y ; Sα) be an H̃ -cryptogroup and ϕα,β = {ϕd(α,β) | d(α, β) ∈ D(α, β)} for α > β on Y , where D(α, β) is a non-empty index set. Then (i) ϕα,βϕβ,γ ⊆ ϕα,γ for α > β > γ on Y . (ii) For a ∈ Sα and β ∈ Y , aϕα,αβ = {aϕd(α,αβ)|∀d(α, αβ) ∈ D(α, αβ)} ⊆ Sd(β,αβ), for some ρβ,αβ-class Sd(β,αβ). Proof. (i) Clearly, ϕd(α,α) is an identity automorphism of Sα. We now prove that ϕα,βϕβ,γ ⊆ ϕα,γ for α > β > γ on Y . Pick ϕd(α,β) : Sα −→ Sd(α,β) ⊆ Sβ and ϕd(β,γ) : Sβ −→ Sd(β,γ) ⊆ Sγ . We show that ϕd(α,β)ϕd(β,γ) = ϕd(α,γ) for some ϕd(α,γ) : Sα −→ Sd(α,γ) ⊆ Sγ . For this purpose, we let a ∈ Sα, b1, b2 ∈ Sd(α,β) and c ∈ Sd(β,γ). Then, because S/H̃ is a band, by Lemma 3.2, we have b1ϕd(β,γ) = b1(b1cb1)0, b2ϕd(β,γ) = b2(b2cb2)0. Since b1, b2 ∈ Sd(α,β), by the definition of ρα,β , (b1, b2) ∈ ρα,β . This leads to (ab1a)0 = (ab2a)0. Now, by the regularity of the band S/H̃, we can easily deduce that (a(b1ϕd(β,γ))a)0 = (ab1(b1cb1)0a)0 = ((ab1a)0c(ab1a)0)0 = ((ab2a)0c(ab2a)0)0 = (a(b2(b2cb2)0)a)0 = (a(b2ϕd(β,γ))a)0. Thus, by the definition of ρα,γ , we have (b1ϕd(β,γ), b2ϕd(β,γ)) ∈ ρα,γ . In other words, there ex- ists a ρα,γ-class Sd(α,γ) satisfying Sd(α,β)ϕd(β,γ) ⊆ Sd(α,γ). Also, ϕd(α,β)ϕd(β,γ) clearly maps Sα into Sd(α,γ) by the transitivity of “6 ”, and hence ϕd(α,β)ϕd(β,γ) = ϕd(α,γ). This proves that ϕα,βϕβ,γ ⊆ ϕα,γ . (ii) It suffices to show that for any ϕd(α,αβ) and ϕd′(α,αβ) ∈ ϕα,αβ , we have (aϕd(α,αβ), aϕd′(α,αβ)) ∈ ρβ,αβ . For this purpose, we let x ∈ Sd(α,αβ) and x′ ∈ Sd′(α,αβ). Then, by Lemma 3.2 (iii), we have aϕd(α,αβ) = a(axa)0 and aϕd′(α,αβ) = a(ax′0. Let b ∈ Sβ . Since Sαβ is a completely J̃ -simple semigroup, and bab, aϕd(α,αβ), aϕd′(α,αβ) are elements in Sαβ , we obtain that (bab, (bab)(aϕd(α,αβ))(bab)) ∈ H̃ and (bab, (bab)(aϕd′(α,αβ))(bab)) ∈ H̃. Since every H̃-class of Sαβ contains a unique idempotent, ((bab)(aϕd(α,αβ))(bab))0 = ((bab)(aϕd′(α,αβ))(bab))0. In other words, we have ((bab)(a(axa)0)(bab))0 = ((bab)(a(ax′0)(bab))0. Thus, by the regularity of the band S/H̃, we can further simplify the above equality to (b(a(axa)0)b)0 = (b(a(ax′0)b)0, that is, (b(aϕd(α,αβ))b)0 = (b(aϕd′(α,αβ))b)0. By the definition of ρβ,αβ , we see that (aϕd(α,αβ), aϕd′(α,αβ)) ∈ ρβ,αβ . Finally we show that S = (Y ; Sα) equipped with the above structural homomorphisms acting on the ρα,β-equivalence class of S forms a G-strong semilattice of semigroups Sα. We need the following lemma. Lemma 3.4 Let S = (Y ; Sα) be a regular H̃-cryptogroup. For any a ∈ Sα, b ∈ Sβ , suppose that aϕα,αβ ⊆ Sd(β,αβ), bϕβ,αβ ⊆ Sd(α,αβ), where ϕα,αβ and ϕβ,αβ are the structural homomorphisms defined in Lemma 3.3. Then we have ab = (aϕd(α,αβ))(bϕd(β,αβ)). X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 56 Proof. Let c1 ∈ Sd(α,αβ), c2 ∈ Sd(β,αβ). Then (ac1a)0 ∈ Sd(α,αβ) because Sd(α,αβ) is a ρα,αβ- equivalence class of Sαβ . Now, by Lemma 3.2, aϕd(α,αβ) = (ac1a)0a and bϕd(β,αβ) = b(bc2b)0 for ϕd(α,αβ) ∈ ϕα,αβ and ϕd(β,αβ) ∈ ϕβ,αβ . Since we assume that aϕα,αβ ⊆ Sd(β,αβ), we have aϕd(α,αβ) = (ac1a)0a ∈ Sd(β,αβ). Similarly, we have bϕd(β,αβ) ∈ Sd(α,αβ) ∩ Sd(β,αβ). Thus, by Lemma 3.2 (ii), we have (aϕd(α,αβ))(bϕd(β,αβ)) = (ac1a)0(ab(bc2b)0) = ab(bc2b)0 and also (aϕd(α,αβ))(bϕd(β,αβ)) = ((ac1a)0ab)(bc2b)0 = (ac1a)0ab. However, by the definition of the natural partial order “6”, we have ab > (aϕd(α,αβ))(bϕd(β,αβ)). On the other hands, since every semigroup Sαβ is primitive, we obtain ab = (aϕd(α,αβ))(bϕd(β,αβ)). 4. Structure of regular H̃-cryptogroups In this section, we use the KG-strong semilattice to characterize regular H̃-cryptogroups. Also, we consider the question when will the Green ∼-relation H̃ to be a right quasi-normal band congruence? By using the KG-strong semilattice, we are able to give a description for the normal H̃ -cryptogroups. We note here that the orthodox regular H̃ -cryptogroups with KG-strong semi- lattices have been studies in [10]. A construction theorem of orthodox regular H̃ -cryptogroups was also given in [8]. Theorem 4.1 An H̃-cryptogroup S is a regular H̃-cryptogroup if and only if S is an H̃G-strong semilattice of completely J̃ -simple semigroups, that is, S = H̃G[Y ; Sα, ϕα,β]. Proof. By the definition of the KG-strong semilattice and the results obtained in §3, we have already proved the necessity part of Theorem 4.1 since it is obvious that H̃|Sβ ⊆ ρα,β for α > β on Y . We now prove the sufficiency part of the theorem. To prove that S/H̃ is a regular band, we use a result in [14]. What we need is to prove that the usual Green relations L and R are congruences on S/H̃. In fact, we only need to verify that L is a left congruence on S/H̃ since R is a right congruence on S/H̃ can be proved in a similar fashion. Since S = (Y ; Sα) is an H̃-cryptogroup, we can let eH̃, fH̃ and gH̃ ∈ S/H̃, where e, f ∈ Sα ∩ E(S), g ∈ Sβ ∩ E(S) with (e, f) ∈ L̃. Then, we have ef = e and fe = f . By the definition of H̃G-strong semilattice H̃G[Y ;Sα, ϕα,β], we can find the homomorphisms ϕef d(β,αβ) and ϕf d(β,αβ) ∈ ϕβ,αβ , ϕg d(α,αβ) ∈ ϕα,αβ such that (gegf)H̃ = {[g(ef)](gf)}H̃ = {[(gϕef d(β,αβ))((ef)ϕg d(α,αβ))][(gϕf d(β,αβ))(fϕg d(α,αβ))]}H̃ = [(gϕef d(β,αβ))(fϕg d(α,αβ))]H̃ X. Kong, Y.Ding, K.P.Shum / Eur. J. Pure Appl. Math, 1 (2008), (46-59) 57 and (ge)H̃ = [g(ef)]H̃ = [(gϕef d(β,αβ))((ef)ϕg d(α,αβ))]H̃ = [(gϕef d(β,αβ))(fϕg d(α,αβ))]H̃. Thereby, (gegf)H̃ = (ge)H̃. Analogously, we can also prove that (gfge)H̃ = (gf)H̃ . This proves that L is left compatible with the multiplication of S/H̃. Since L is always right congru- ence, L is a congruence on S/H̃, as required. Dually, R is also a congruence on S/H̃. Thus by [14] (see II. 3.6 Proposition ), S/H̃ forms a regular band and hence S is indeed a regular H̃-cryptogroup. Our proof is completed. Recall that a right quasi-normal band is a band satisfying the identity yxa = yaxa [6]. Also, a left quasi-normal band is a band satisfying the identity axy = axay. Thus, we can easily observe that both the right quasi-normal bands and the left quasi-normal bands are special cases of the regular bands. Also, a normal band (that is, a band satisfies the identity axya = ayxa) is a special right quasi-normal band and a left quasi-normal band. Based on the above observation, we are able to establish the following theorem for right quasi-normal H̃-cryptogroups. Theorem 4.2 An H̃-abundant semigroup S is a right quasi-normal H̃-cryptogroup if and only if S is an L̃G-strong semilattice of completely J̃ -simple semigroups, that is, S = L̃G[Y ;Sα, ϕα,β]. Proof. (Necessity) Let S be a right quasi-normal H̃ -cryptogroup. Then S/H̃ is a right quasi-normal band. To show that S is an L̃G-strong semilattice, by invoking Lemma 3.3 and its proof, we only need to show that for any δ > γ on Y , L̃|Sγ ⊆ ρδ,γ . In fact, for a ∈ Sδ, x, y ∈ Sγ with (x, y) ∈ L̃, we have (axa)H̃ = ((axy)a)H̃ = (ayxya)H̃ = (aya)H̃ by the right quasi-normality of the band S/H̃. Thus, by the definition of ρδ,γ , we have L̃|Sγ ⊆ ρδ,γ as re- quired. This shows that S = L̃G[Y ; Sα, ϕα,β]. (Sufficiency) Let a ∈ Sα, x ∈ Sβ , and y ∈ Sγ . Then, since S = L̃G[Y ;Sα, ϕα,β] is an H̃G-strong semilattice of Sα and by Theo- rem 4.1, H̃ is a congruence on S. Moreover, we have xa = (xϕa d(β,αβ))(aϕx d(α,αβ)) and thereby, axa = (aϕx d(α,αβ))(xϕa d(β,αβ))(aϕx d(α,αβ)). By the fact ((xa)0, (axa)0) ∈ L, we can easily see that (xa, axa) ∈ L̃|Sαβ , and so, by our hypothesis, S = L̃G[Y ;Sα, ϕα,β]. This implies that there exist some homomorphisms ϕ eL d(αβ,αβγ) ∈ ϕαβ,αβγ and ϕ eL d(γ,αβγ) ∈ ϕγ,αβγ satisfying the con- ditions y(xa) = (yϕ eL d(γ,αβγ))((xa)ϕ eLd(αβ,αβγ)) and y(axa) = (yϕ eL d(γ,αβγ))((axa)ϕ eLd(αβ,αβγ)). Hence, it follows that (y(xa))H̃ = [(yϕ eL d(γ,αβγ))((xa)ϕ eLd(αβ,αβγ))]H̃ ={(yϕ eL d(γ,αβγ)){[(xϕa d(β,αβ))(aϕx d(α,αβ))]ϕ eL d(αβ,αβγ)}}H̃ =[(yϕ eL d(γ,αβγ))((aϕx d(α,αβ))ϕ eL d(αβ,αβγ))]H̃ REFERENCES 58 and (y(axa))H̃ = {(yϕ eL d(γ,αβγ)){[(aϕx d(α,αβ))(xϕa d(β,αβ))(aϕx d(α,αβ))]ϕ eL d(αβ,αβγ)}}H̃ = [(yϕ eL d(γ,αβγ))((aϕx d(α,αβ))ϕ eL d(αβ,αβγ))]H̃. This leads to (yxa)H̃ = (yaxa)H̃ and so S/H̃ is a right quasi-normal band. Thus, S is indeed a right quasi-normal H̃-cryptogroup. Since we have already mentioned that a band B is a normal band if for all elements e, f, g in B, the identity efge = egfe holds in B( see [6]). In closing this paper, we characterize the normal H̃ -cryptogroups. In fact, this result gives a modified version of the theorem of Petrich and Reilly in [11] on normal cryptogroups, in particular, the theorem on normal cryptogroups in [11] and also the theorem of Fountain on superabundant semigroups in [4] is now refined and amplified in the class of quasiabundant semigroups. Theorem 4.3 An H̃-abundant semigroup S is a normal H̃-cryptogroup if and only if S is a D̃G- strong semilattice of completely J̃ -simple semigroups, that is, S = D̃G[Y ; Sα, ϕα,β]. Proof. (Necessity) The proof is similar to the necessity part given in Theorem 4.2, that is, we only need to prove that D̃|Sβ ⊆ ρα,β for all α, β ∈ Y with α > β. Since every semigroup Sα can be regarded as a D̃-class of S, we can just let a ∈ Sα, x, y ∈ Sβ . Recall that S = (Y ; Sα) is a normal H̃-cryptogroup, S/H̃ is a normal band. Now, by the normality of the band S/H̃, we have (axa)H̃ = (a(xyx)a)H̃ = (ayxya)H̃ = (aya)H̃. Thus, by Lemma 3.1, we see that (x, y) ∈ ρα,β and whence D̃|Sβ ⊆ ρα,β . This proves that S = D̃G[Y ;Sα, ϕα,β]. (Sufficency) Let S = D̃G[Y ; Sα, ϕα,β], where each Sα is a completely J̃ - simple semigroup, for all α ∈ Y . Then by definition, S is an L̃G-strong semilattice of semigroups Sα and also S is an R̃G-strong semilattice of semigroups Sα . By applying Theorem 4.2 and its dual, we immediately deduce that H̃ is a congruence on S and for all a, x, y ∈ S, we have [(axy)a]H̃ = [ay(xya)]H̃ = (ayxyxa)H̃ = (ayxa)H̃. This shows that S/H̃ is a normal band. Moreover, since each Sα is a D̃-class of S, for every α, β ∈ Y with α > β, the set D(α, β) is just a singleton. This means that S is a strong semilattice of completely J̃ -simple semigroups Sα. Our proof is completed. References [1] A. H. Clifford , Semigroups admitting relative inverses, Ann of Math. 42, 1037-1049, (1941) [2] A. H. Clifford and G. B. Preston , The Algebraic Theory of Semigroups, Mathematical Surveys 7, Vols 1 and 2, American Mathematical Society, Providence, R.I., (1967) [3] A. El-Qallali , Structure Theory for Abundant and Related Semigroups, PhD Thesis, York University, England, (1980) REFERENCES 59 [4] J. B. Fountain , Abundant semigroups, Proc London Math Soc 43 (3) 103-129, (1982) [5] X. J. Guo and K. P. Shum , On left cyber groups. Int. Math. J, 5 705–717, (2004) [6] J. M. Howie , Fundamental of Semigroup Theory, Clarendon Press, Oxford, (1995) [7] X. Z. Kong and K. P. Shum , Completely regular semigroups with generalized strong semilattice decompositions, Algebra Colloqium , 12 (2) 269-280, (2005) [8] X. Z. Kong and K. P. Shum , On the structure of regular crypto semigroups, Comm. in Algebra, 29 (6), 2461-2479 ,(2001) [9] X. Z. Kong and K. P. Shum , Semilattice structure of regular cyber groups, Pragmatic Algebra, 1 1-12, (2006) [10] X. Z. Kong and Z. L. Yuan , KG-strong semilattice decomposition of regular orthocryptosemigroups, Semigroup Forum, 73, 95-108, (2006) [11] F. Pastijn , A representation of a semigroup by a semigroup of matrices over a group with zero, Semigroup Forum, 10, 238-249,(1975) [12] M. Petrich and N. R. Reilly , Completely Regular Semigroups, John Wiley & Sons, 162-242, (1999) [13] M. Petrich , The structure of completely regular semigroups, Trans Amer Math Soc, 189, 211-236, (1974) [14] M. Petrich , Lectures in Semigroups, Wiley & Sons Inc. London, (1976) [15] X. M. Ren and K. P. Shum , The structure of superabundant semigroups, Sci in China, Ser. A, 47 (5), 756-771, (2004) [16] X. M. and K. P. Shum , On superabundant semigroups whose set of idempotnets forms a subsemi- group, Algebra Colloqium, 14 (2), 215-228, (2007)