KG -Strong Semilattice Structure of Regular Orthocryptosemigroups

Semigroup Forum Vol. 73 (2006) 95–108

c 2006 Springer


DOI: 10.1007/s00233-006-0614-5

RESEARCH ARTICLE

KG-Strong Semilattice Structure of Regular Orthocryptosemigroups ∗ Xiang-zhi Kong and Zhi-ling Yuan Communicated by Francis J. Pastijn

Abstract The concept of KG -strong semilattice of semigroups is introduced. By using this concept, we study properties of regular orthocryptosemigroups. Necessary and sufficient conditions for a semisuperabundant orthosemigroup to be a regular orthocryptosemigroup or a right quasi-normal orthocryptosemigroup and a normal orthocryptosemigroup are obtained. Important results of [11], Theorem 2.7, Chapter 4 on normal orthocryptogroups are generalized and enriched. Keywords: Regular orthocryptosemigroup, KG -strong semilattice, Natural partial order. AMS Classification: 20M10.

1. Introduction In 2001, Zhang L., Shum K.P. and Zhang R.H. introduced the concept of a refined semilattice of semigroups [14], which is a natural generalization of the notion of a strong semilattice of semigroups [5], [11]. They proved that a band is a regular band (i.e., a band that satisfies the identity axya = axaya) if and only if it is a refined semilattice of rectangular bands. Thus, a number of results in the literature concerning strong semilattice decomposition can be further developed [12], [15]. In [6] and [7], we gave the concept of KG -strong semilattice of semigroups, which is another natural generalization of a strong semilattice of semigroups and it is more “natural” and simpler than the refined semilattice of semigroups. By using KG -strong semilattice of semigroups, we proved that a band is a regular band if and only if it is a G -strong semilattice of rectangular bands [6] and a semigroup is a regular cryptogroup (i.e., a completely regular semigroup with its Green’s relation H is a regular band congruence) if and only if it is an HG -strong semilattice of completely simple semigroups. These results generalized the following remarkable results of [13]: a band is a normal band (i.e., a band that satisfies the identity axya = ayxa) if and only if it is a strong semilattice of rectangular bands; and at the same time, also generalized the famous results of [1]: a semigroup is a Clifford semigroup if and only if it is a ∗ The research is supported by a NSF of Shandong Province, People’s Republic of China (No. Y2005A11).

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strong semilattice of groups, and of [9]: a semigroup is a normal crytpogroup if and only if it is a strong semilattice of completely simple semigroups respectively. Let X be a nonempty set. In the rest of the paper, we always denote the universal relation on X by ωX and the identity relation on X by εX . We also denote the set of idempotents of a semigroup S by E(S) . For further notation and terminology, the reader is referred to [5], [11]. For some concepts that have appeared in the literature, we occasionally use alternative, though equivalent, definitions. First we recall two families of generalized Green’s relations used to study certain classes of semigroups. Let S be an arbitrary semigroup. The following Green’s ∗ -relations are due to Pastijn and appear for example in Fountain [3]: L∗ = {(a, b) ∈ S × S: (∀x, y ∈ S 1 )ax = ay ⇔ bx = by}, R∗ = {(a, b) ∈ S × S: (∀x, y ∈ S 1 )xa = ya ⇔ xb = yb}, H∗ = L∗ ∩ R∗ . Later, El-Qallali generalized Green’s ∗ -relations to Green’s ∼-relations [2]: L
= {(a, b) ∈ S × S: (∀e ∈ E(S))ae = a ⇔ be = b},
= {(a, b) ∈ S × S: (∀e ∈ E(S))ea = a ⇔ eb = b}, R
= L
∩ R.
H Among the usual Green’s relation or the above relations, L- or the generalized L-relations are the dual of the corresponding R-relations. So, in what follows, most of the times we only discuss properties related to L- or to the generalized L-relations. One can see that L ⊆ L∗ ⊆ L
 and that there is at most
-class. If e ∈ H
a E(S) , for some a ∈ S , one idempotent contained in each H
a . Clearly, for any x ∈ H
a with a ∈ S , then we write e as x0 , for any x ∈ H we have x = xx0 = x0 x. A semigroup with a unique idempotent, which is its identity, is called a unipotent semigroup(briefly u-semigroup). Groups (left or right), cancellative monoids, and a semigroup S 1 such that the semigroup S has
= L
= R
no idempotent are all examples of u-semigroups. It is evident that H on a u-semigroup. 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. A semigroup S is rpp [8] if each of its L∗ -classes contains at least one idempotent. An lpp semigroup is the dual of an rpp semigroup. A semigroup is abundant [3] if it is both rpp and lpp. One can see that L∗ = L on regular elements of a semigroup. So regular semigroups are obviously abundant semigroups. A semigroup S is called superabundant if each of its H∗ -classes contains an idempotent. A

semigroup S is semiabundant [2] if both each L-class and each R-class of S
contains at least one idempotent. In an rpp semigroup, we have L = L∗ [4]. A
-classes contains an semigroup S is called semisuperabundant if each of its H idempotent.

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A semigroup S is called a rectangular u-semigroup if S is the direct product of a u-semigroup and a rectangular band. A semigroup S is called
is a congruence and S is called an orthosemigroup if a cryptosemigroup if H E(S) is a subsemigroup of S . An orthocryptosemigroup is a semisuperabundant cryptosemigroup if its idempotents set forms a band and is said to be a regular (respectively, right quasi-normal, normal) orthocryptosemigroup if its set of idempotents form a regular(respectively, right quasi-normal, normal) band. Recall that a band is said to be right quasi-normal if it satisfies the identity xya = xaya. 2. KG -strong semilattices In this section, we introduce the concept of KG -strong semilattice decomposition of a semigroup S = (Y ; Sα ) , where Y is a semilattice and each Sα is a subsemigroup of S . In this paper, we always denote a semigroup S which is a semilattice Y of subsemigroups Sα (α ∈ Y ) by S = (Y ; Sα ) . Definition 2.1. Let S = (Y ; Sα ) be a semigroup. Suppose that the following conditions hold on the semigroup S . (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 which runs between the interval [α, β] . (ii) (∀α ∈ Y ) , D(α, α) is a singleton set. Denote the element in D(α, α) by d(α, α). In this case, the homomorphisms 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,β Φβ,γ ⊆ Φα,γ , where Φα,β Φβ,γ = {φd(α,β) φd(β,γ) : ∀d(α, β) ∈ D(α, β), d(β, γ) ∈ D(β, γ)}. (iv) (∀a ∈ Sα )(∀β ∈ Y ), there exists a unique φad(β,αβ) ∈ Φβ,αβ such that for any b ∈ Sβ , ab = (aφbd(α,αβ) )(bφad(β,αβ) ). We call the above semilattice of semigroups Sα , with a set of structure homomorphisms Φα,β , a generalized strong semilattice if S = (Y ; Sα ) satisfies the above conditions (i)–(iv). For the sake of brevity, we call the above generalized strong semilattice of Sα the “G -strong semilattice” of Sα and is denoted by S = G[Y ; Sα , Φα,β ] . The following definition is a stronger version of G -strong semilattices. Definition 2.2. Let K be an equivalence relation on a G -strong semilattice of semigroups S = G[Y ; Sα , Φα,β ] . We say that S is a KG -strong semilattice of semigroups if for each α, β ∈ Y , the mapping a −→ φad(β,αβ) has the property

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that φad(β,αβ) = φbd(β,αβ) whenever the elements a, b ∈ Sα belong to the same K -class. Hence under the above multiplication, we obtain a G -strong semilattice of semigroups S determined by the equivalence relation K . We therefore call S a KG -strong semilattice of Sα and denote it by S = KG[Y ; Sα , Φα,β ] . Note. It is clear that the KG -strong semilattice is stronger than the G strong semilattice but it is weaker than the strong semilattice. In fact, if ρ and δ are equivalences on the semigroup S = (Y ; Sα ) with ρ ⊆ δ , then we observe that, δG[Y ; Sα , Φα,β ] is “stronger” than ρG[Y ; Sα , Φα,β ]. As the special case, 1S G[Y ; Sα , Φα,β ] is the “weakest” KG -strong semilattice since 1S is the “smallest” equivalence relation on S and ηG[Y ; Sα , Φα,β ] is the strongest KG strong semilattice since η is the “biggest” equivalence relation on S , where 1S is the identity equivalence relation on S and η is the semilattice congruence on S which divides S into Sα (α ∈ Y ). Hence, it can be easily seen that ηG[Y ; Sα , Φα,β ] is the usual strong semilattice 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 homomorphism in the set of structure homomorphisms Φα,β . We observe that if the set D(α, β) for α
β is just a singleton then the G -strong semilattices, KG -strong semilattices and strong semilattices are the same since there is no distinction between their respective multiplications. 3. Properties
is a Lemma 3.1. Let S be a semisuperabundant orthosemigroup. Then H 0 0 0 congruence if and only if for any a, b ∈ S , (ab) = a b .
0 . Since H
is a
0 and bHb Proof. Necessity. For any a, b ∈ S , then aHa 0 0 0 0 0 0

congruence, we have abHa b . But abH(ab) , and so (ab) = a b since every
-class contains at most one idempotent and E(S) is a band. H
is an equivalence, we only need to show that H
is Sufficiency. Since H 0 0 0 0 0 0
compatible. Let (a, b) ∈ H and c ∈ S . Then (ca) = c a = c b = (cb) and
is left compatible. Dually, H
is right compatible and so H
is a congruence. so H Lemma 3.2. Let S be an orthocryptosemigroup. Then the relation J
de
fined by J = {(x, y) ∈ S × S: x0 J (E(S))y 0 } is a semilattice congruence and each J
-class is a rectangular u-semigroup. Proof. It is easy to see that J
is an equivalence on S . Suppose that (a, b) ∈ J
, let c ∈ S . Then a0 J (E(S))b0 . By Lemma 3.1, (ca)0 = c0 a0 J (E(S))c0 b0 = (cb)0 since J (E(S)) is a congruence on E(S). Thus we show that J
is left compatible. Dually, J
is right compatible and so J
is a congruence. Since
⊆ J
and each H
-class of S is a u-semigroup, J (E(S)) is a semilattice H congruence on E(S) , we easily see that J
is a semilattice congruence on S .

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To show every J
-class is a rectangular u-semigroup, denote the set of idempotents of J
a by Ea for any a ∈ S . Recalling the definition of an orthocryptosemigroup, we know, from the definition of J
, that Ea = Ea0 is a J
(E(S)) -class of E(S) , so that it is a rectangular band. Easily see that
a is a u-semigroup. We now show that J
a is isomorphic to Ea × H
a . Define H


a mapping φ from Ja to Ea × Ha . For any x ∈ aJ , let xφ = (x0 , a0 xa0 ) .
a and so φ is well defined. For any x, y ∈ J
a , we have Obviously, a0 xa0 ∈ H (xφ)(yφ) = (x0 , a0 xa0 )(y 0 , a0 ya0 ) = (x0 y 0 , a0 xa0 a0 ya0 ) = (x0 y 0 , a0 xx0 a0 y 0 ya0 ) = (x0 y 0 , a0 xya0 ) = (xy)φ, so that φ is a homomorphism. If (x0 , a0 xa0 ) = (y 0 , a0 ya0 ) , then we obtain that x0 = y 0 and a0 xa0 = a0 ya0 , hence x0 a0 xa0 x0 = y 0 a0 ya0 y 0 which leads
a , let x = ebe, to x = y . So φ is injective. Furthermore, for any (e, b) ∈ Ea × H 0 0 0 then we have xφ = ((ebe) , a (ebe)a ) = (e, b) , which means that φ is surjective. So J
a is isomorphic to a rectangular u-semigroup. Thus, we can regard an orthocryptosemigroup S as S = (Y ; Sα ) , where Y is a semilattice and Sα is a rectangular u-semigroup for any α ∈ Y . On the basis of this semilattice decomposition, we can describe a regular orthocrytposemigroup in terms of KG -strong semilattice of semigroups. Through straightforward calculations, we can obtain the following result. Lemma 3.3. Let S = I × U × Λ be a rectangular u-semigroup, where I, U and Λ are left zero semigroup, u-semigroup and right zero semigroup respec
v, µ) if and only if tively. Then for any (i, u, λ), (j, v, µ) ∈ S , (i, u, λ)L(j, λ = µ. Lemma 3.4.

Let S = (Y ; Sα ) be an orthocryptosemigroup.

(i) Let a ∈ Sα , b ∈ Sβ , and a
b . Then there exist e, f ∈ E(Sβ ) with b = ea = af ; (ii) Let a ∈ Sα and α
β . Then there exists b ∈ Sβ with a
b ;
and a
b, c. Then b = c; (iii) Let a, b, c ∈ S , bHc, (iv) Let a ∈ E(S) and b ∈ S be such that a
b . Then b ∈ E(S). Proof. (i) Let g, h ∈ E(S) be such that b = ga = ah. Then e = (ga)0 g, f = 0 h(ah) ∈ E(Sβ ) and ea = (ga)0 ga = ga = b = ah = ah(ah)0 = af. (ii) Let b ∈ Sβ , by Lemma 3.1, a(aba)0 , (aba)0 a and (aba)0 are in the
-class and so a(aba)0 = (aba)0 a(aba)0 = (aba)0 a. Take b = a(aba)0 , same H then b ∈ Sβ and a
b .

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(iii) Suppose that a ∈ Sα , b, c ∈ Sβ . By (i), there exist e, f, g, h ∈ E(Sβ )
0 , we have eb0 = b0 . such that b = ea = af , c = ga = ah. From eb = b and bHb 0 0 0 0 0 0 Similarly, c h = c . Thus ec = ec c = eb c = b c = c c = c. Similarly, bh = b so that b = bh = eah = ec = c as required. (iv) We have b = ea = af for some e, f ∈ E(S) whence b2 = (ea)(af ) = ea2 f = b.
is a conLemma 3.5. Let S be a regular orthocryptosemigroup(that is, H gruence on the semisuperabundant semigroup S such that E(S) is a regular band). For every a ∈ S , we define a relation ρa on S by (b1 , b2 ) ∈ ρa if and only if (ab1 a)0 = (ab2 a)0 , (b1 , b2 ∈ S). Then the following properties hold on S : (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. (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 equivalence relation on S , for all a ∈ S . We now prove that ρa is left compatible. 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 orthocryptosemigroup, by using Lemma 3.1 and the regularity of the band E(S), we obtain that (acxa)0 = a0 c0 x0 a0 = a0 c0 a0 x0 a0 = a0 c0 a0 y 0 a0 = (acya)0 . This shows that (cx, cy) ∈ ρa . Dually, we can prove that ρa is right compatible.
⊆ ρa , so ρa is a band congruence. Obviously, H (ii) Let (x, y) ∈ ρa . Then, by the definition of ρa , we have (axa)0 = 0 (aya) and so a01 (axa)0 a01 = a01 (aya)0 a01 . Since E(S) is a regular band and by Lemma 3.1, we obtain a01 (axa)0 a01 = a01 a0 x0 a0 a01 = a01 a0 a01 x0 a01 a0 a01 = a01 x0 a01 = (a1 xa1 )0 , similarly, a01 (aya)0 a01 = (a1 ya1 )0 . Thus, (a1 xa1 )0 = (a1 ya1 )0 , which means ρa ⊆ ρa1 . Similarly, ρa1 ⊆ ρa . Hence ρa = ρa1 . Since this relation holds for all a ∈ Sα , we write ρa = ρα . (iii) Let a ∈ Sα , b ∈ Sβ and α
β . To show that ρα ⊆ ρβ , we let (x, y) ∈ ρα = ρa , by (ii). Then, by the definition of ρa , we have (axa)0 = (aya)0 and hence b0 (axa)0 b0 = b0 (aya)0 b0 . Since α
β in Y and a ∈ Sα , b ∈ Sβ , we have (bab)0 = b0 . By using Lemma 3.1 and the regularity of the band, we can show that (bxb)0 = (byb)0 , that is, (x, y) ∈ ρb = ρβ . Thus, ρα ⊆ ρβ as required. Furthermore, it is trivial to see that ρβ |Sα = ωSα , the universal relation on Sα .

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We now use the congruence ρα defined in Lemma 3.5 to describe the structural homomorphisms for the semigroup S = (Y ; Sα ) , where each Sα is a rectangular u-semigroup. Our idea is to extend the method of [11], Theorem IV.1.6 for the homomorphisms acting on Sα to the homomorphisms acting on the equivalence classes of S determined by ρα . We first let ρα,β = ρα |Sβ for α, β ∈ Y , which is a band congruence on Sβ . Then, we denote all the ρα,β -classes by {Sd(α,β) : d(α, β) ∈ D(α, β)} , where D(α, β) is a non-empty index set for [α, β] . In particular, D(α, α) is a singleton and we write d(α, α) = D(α, α) . We now have the following lemma. Lemma 3.6. Let S = (Y ; Sα ) be a regular orthocryptosemigroup. 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; (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)0 a for any b ∈ Sd(α,β) . Proof. (i) We first claim 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 a regular orhtocryptosemigroup, each Sα is a rectangular u-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 E(S) and Lemma 3.1. Thereby, (ab, b) ∈ ρα,β . Similarly, we also have ba ∈ Sd(α,β) . Invoking the above results, we have aba ∈ Sd(α,β)
is a band congruence on S , by Lemma 3.1 for any b ∈ Sd(α,β) . Since H 0
-class of S again, we see that a(aba) , (aba)0 and (aba)0 a are in the same H 0 0 0 0 0 so that a(aba) = (aba) a(aba) = (aba) a. Let a(aba) = 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 “”, we can write a∗d(α,β) = ea = af for some e, f ∈ E(S) . Thus by Lemma 3.1, (a∗d(α,β) )0 = ea0 = a0 f , that is, a0
(a∗d(α,β) )0 . Now by Lemma 3.1 and 3.5, we have (a∗d(α,β) )0 = (a0 (a∗d(α,β) )0 a0 )0 = (aa∗d(α,β) a)0 = (aba)0 ,
and so by Lemma 3.4 (iii), a∗ which means that (a∗d(α,β) , ad(α,β) ) ∈ H 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 “”, a0
a0 (ax)0 a0 . 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 the proof of (i), a0 (ax)0 a0 ∈ Sd(α,β) and e = a0 (ax)0 a0 . Thereby, we have eax =

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a0 (ax)0 a0 ax = ax. Similarly, 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 that ea = ae and consequently, by Lemma 3.1 again, we have (ea)0 = (ea0 )0 = e. (iii) By the result in (i), we now define φd(α,β) : Sα −→ Sd(α,β) by aφd(α,β) = ad(α,β) = a(aca)0 = (aca)0 a for any a ∈ Sα , and c ∈ Sd(α,β) . Then, for any a, b ∈ Sα , by the proof of (i), we have b(bcb)0 , (aca)0 ∈ Sd(α,β) and a0 ≥ (aca)0 , thus, (aφd(α,β) )(bφd(α,β) ) = ad(α,β) bd(α,β) = (aca)0 ab(bcb)0 = (aca)0 a(b(bcb)0 ) = ab(bcb)0

(by (ii)).

Similarly, we have (aφd(α,β) )(bφd(α,β) ) = (aca)0 ab. Hence, ab
(aφd(α,β) ) × (bφd(α,β) ) . By the definition of φd(α,β) , we have (ab)φd(α,β) = (aφd(α,β) ) × (bφd(α,β) ) . This shows that φd(α,β) is indeed a homomorphism. The proof is completed. Lemma 3.7. Let S = (Y ; Sα ) be a regular orthocryptosemigroup and Φα,β = {φd(α,β) |d(α, β) ∈ D(α, β)} when α
β on Y , where D(α, β) is the non-empty index set induced by Lemma 3.5. Then (i) φd(α,α) is the identity automorphism on Sα for each α ∈ Y ; (ii) (∀α, β, γ ∈ Y , α
β
γ), Φα,β Φβ,γ ⊆ Φα,γ ; (iii) (∀α, β ∈ Y )(a ∈ Sα ), there exists a ρβ,αβ -class Sd(β,αβ) such that aΦα,αβ ⊆ Sd(β,αβ) . Proof. (i) Let a ∈ Sα and aφd(α,α) = b . By the definition of φd(α,α) , b = ea = af for some idempotents e, f ∈ E(Sα ) . According to Lemma 3.3,
and so a0 = b0 . Thus b = a0 b = a0 ea = a0 ea0 a = a. So φd(α,α) is (a, b) ∈ H the identity automorphism of Sα . (ii) 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, by Lemma 3.6, we have b1 φd(β,γ) = b1 (b1 cb1 )0 , b2 φd(β,γ) = b2 (b2 cb2 )0 . Since b1 , b2 ∈ Sd(α,β) , by the definition of ρα,β , we have (b1 , b2 ) ∈ ρα,β . This leads to (ab1 a)0 = (ab2 a)0 . Now, by the regularity of the band E(S) , we can deduce that (a(b1 φd(β,γ) )a)0 = (ab1 (b1 cb1 )0 a)0 = a0 b01 c0 b01 a0 = a0 b01 a0 c0 a0 b01 a0 = a0 b02 a0 c0 a0 b02 a0 = (ab2 (b2 cb2 )0 a)0 = (a(b2 φd(β,γ) )a)0 .

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Thus, by the definition of ρα,γ , we have (b1 φd(β,γ) , b2 φd(β,γ) ) ∈ ρα,γ . In other words, there exists a ρα,γ -class Sd(α,γ) satisfying Sd(α,β) φd(β,γ) ⊆ Sd(α,γ) . Also, φd(α,β) φd(β,γ) is clearly a mapping which maps Sα into Sd(α,γ) , by the transitivity of “”, we obtain φd(α,β) φd(β,γ) = φd(α,γ) . This proves that Φα,β Φβ,γ ⊆ Φα,γ . (iii) In fact, 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 3.6, (iii), we have aφd(α,αβ) = a(axa)0 and aφd
(α,αβ) = a(ax a)0 . Let b ∈ Sβ . Then, because Sαβ is a rectangular u-semigroup, and bab , aφd(α,αβ) , aφd
(α,αβ) are elements in Sαβ , we ob
and (bab, (bab)(aφd
(α,αβ) )(bab)) ∈ tain that (bab, (bab)(aφd(α,αβ) )(bab)) ∈ H

H. Since every H -class of Sαβ is a u-semigroup, we conclude that ((bab)(aφd(α,αβ) )(bab))0 = ((bab)(aφd
(α,αβ) )(bab))0 . In other words, ((bab)(a(axa)0 )(bab))0 = ((bab)(a(ax a)0 )(bab))0 . Thus, by the regularity of the band E(S) , we can further simplify the above equality to (b(a(axa)0 )b)0 = (b(a(ax a)0 )b)0 , that is, (b(aφd(α,αβ) )b)0 = (b(aφd
(α,αβ) )b)0 . It hence follows, by the definition of ρβ,αβ , that (aφd(α,αβ) , aφd
(α,αβ) ) ∈ ρβ,αβ . The proof is completed. Lemma 3.8. Let S = (Y ; Sα ) be a regular orthocryptosemigroup. For any a ∈ Sα , b ∈ Sβ , suppose that aΦα,αβ ⊆ Sd(β,αβ) , bΦβ,αβ ⊆ Sd(α,αβ) . Then we have ab = (aφd(α,αβ) )(bφd(β,αβ) ). Proof. Let c1 ∈ Sd(α,αβ) , c2 ∈ Sd(β,αβ) . Then (ac1 a)0 ∈ Sd(α,αβ) because Sd(α,αβ) is a ρα,αβ -equivalence class of Sαβ . Now, by Lemma 3.6, aφd(α,αβ) = (ac1 a)0 a and bφd(β,αβ) = b(bc2 b)0 for φd(α,αβ) ∈ Φα,αβ and φd(β,αβ) ∈ Φβ,αβ . 0 Since we assume that aΦα,αβ ⊆ Sd(β,αβ) , we have  aφd(α,αβ) = (ac1 a) a ∈ Sd(β,αβ) . Similarly, we have bφd(β,αβ) ∈ Sd(α,αβ) Sd(β,αβ) . Thus, by Lemma 3.6 (ii), we have (aφd(α,αβ) )(bφd(β,αβ) ) = (ac1 a)0 (ab(bc2 b)0 ) = ab(bc2 b)0 and also (aφd(α,αβ) )(bφd(β,αβ) ) = ((ac1 a)0 ab)(bc2 b)0 = (ac1 a)0 ab. However, by the definition of the natural partial order “”, we have ab
(aφd(α,αβ) )(bφd(β,αβ) ) . By the same method with the proof of Lemma 3.6 (i),we can obtain that ab = (aφd(α,αβ) )(bφd(β,αβ) ). Note. By Lemma 3.7, we can easily see that for any a1 , a2 ∈ Sα and β ∈ Y , if a1 Φα,αβ and a2 Φα,αβ are all subsets of the same ρβ,αβ -class Sd(β,αβ) , then a1 and a2 determine the same mapping φd(β,αβ) ∈ Φβ,αβ and hence for any

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b ∈ Sβ , we have a1 b = (a1 φbd(α,αβ) )(bφd(β,αβ) ) and a2 b = (a2 φbd(α,αβ) )(bφd(β,αβ) ), where φbd(α,αβ) varies when b varies in Sβ , but on the other hand, we observe that φd(β,αβ) does not vary with b , that is, φd(β,αβ) is fixed for all b ∈ Sβ . Thus, if we let δ be equivalence on S and if δ|Sβ ⊆ ρα,β for all α
β on Y , we can easily deduce that for any a1 , a2 ∈ Sα and b1 , b2 ∈ Sβ with (a1 , a2 ) ∈ δ , (b1 , b2 ) ∈ δ , there exist some homomorphisms φδd(α,αβ) ∈ Φα,αβ and φδd(β,αβ) ∈ Φβ,αβ such that a1 b1 = (a1 φδd(α,αβ) )(b1 φδd(β,αβ) ) and a2 b2 = (a2 φδd(α,αβ) )(b2 φδd(β,αβ) ). In other words, if δ|Sβ ⊆ ρα,β for any α
β then S must be a δG[Y ; Sα , Φα,β ] . 4. Structure In this section, we use the KG -strong semilattice to characterize regular or
thocryptosemigroups. Also, we consider the question when will the Green H relation be a right quasi-normal or normal band congruence. In the literature, [11] only showed that if S is a completely regular semigroup with Green H relation is a normal band congruence, then S is a strong semilattice of completely simple semigroups. Our result given here generalizes their result substantially. Theorem 4.1. Let S be a semisuperabundant orthosemigroup. Then S is
-strong semilattice of a regular orthcryptosemigroup if and only if S is an HG
rectangular u-semigroups, that is, S = HG[Y ; Sα , Φα,β ]. Proof. By the definition of KG -strong semilattice and the results obtained in § 3, we have already proved the necessity part of the theorem since it is obvious
S ⊆ ρα,β for α
β on Y . We now prove the sufficiency part of the that H| β
is a congruence on S . In theorem. We first show that the Green’s ∼-relation H
fact, if a ∈ Sα , b ∈ Sβ then by the definition of HG -strong semilattice and that
each S is a rectangular u-semigroup, we see that there exist φH ∈Φ α

and


φH d(β,αβ)

d(α,αβ)

∈ Φβ,αβ satisfying the following equalities





H 0 0 0 H 0 H ab = (aφH d(α,αβ) )(bφd(β,αβ) ) and a b = (a φd(α,αβ) )(b φd(β,αβ) ). Hence, we deduce that

H 0 (ab)0 = [(aφH d(α,αβ) )(bφd(β,αβ) )]

0 H 0 0 = [(aφH d(α,αβ) ) (bφd(β,αβ) ) ]

0 H 0 = [(a0 φH d(α,αβ) )(b φd(β,αβ) )] = a0 b0 .

α,αβ

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H 0
since easily see that a0 φH d(α,αβ) = (aφd(α,αβ) ) and H is obviously a congruence
is a congruence on on a rectangular u-semigroup. Now by Lemma 3.1, H S = (Y ; Sα ) . To see that E(S) is a regular band, by a result of [10], we only need to show that the Green’s relations L and R are both congruences on E(S) . We only show that L is a congruence in E(S) as R is a congruence in E(S) can be proved in a similar fashion. Since E(S) = (Y ; E(Sα )) , we can let e, f ∈ Sα ∩ E(S) , g ∈ Sβ ∩ E(S) with (e, f ) ∈ L(E(S)). Then, we have ef = e
-strong semilattice HG[Y
and f e = f . By the definition of HG ; Sα , Φα,β ] , we ef f g can find homomorphisms φd(β,αβ) and φd(β,αβ) ∈ Φβ,αβ , φd(α,αβ) ∈ Φα,αβ such that gegf = [g(ef )](gf ) g f g = [(gφef d(β,αβ) )((ef )φd(α,αβ) )][(gφd(β,αβ) )(f φd(α,αβ) )] g = (gφef d(β,αβ) )(f φd(α,αβ) )

and ge = g(ef ) g = (gφef d(β,αβ) )((ef )φd(α,αβ) ) g = (gφef d(β,αβ) )(f φd(α,αβ) ).

Thereby, gegf = ge. Analogously, we can also prove that gf ge = gf . This proves that L is left compatible on E(S) . Since L is always right compatible, we see that L is a congruence on E(S) , as required. Dually, R is also a congruence on E(S) . Thus by [10] (see II.3.6 Proposition), E(S) is a regular band and hence S is a regular orthcrypto semigroup. Our proof is completed.

Recall that a right quasi-normal band is a band satisfying the identity yxa = yaxa. 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 all special cases of the regular bands. Also, normal bands are special case of both right quasi-normal bands and left quasi-normal bands. Based on the above observations, we establish the following theorem. Theorem 4.2. A semisuperabundant orthosemigroup S is a right quasi
-strong semilattice of normal orthocryptosemigroup if and only if S is an LG
rectangular u-semigroups, that is, S = LG[Y ; Sα , Φα,β ]. Proof. =⇒) Let S be a right quasi-normal orthocryptosemigroup. Then
E(S) is a right quasi-normal band. To show that S = LG[Y ; Sα , Φα,β ] , by

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-strong semilattice and invoking Lemma 3.8 and its proof, the definition of LG
S ⊆ ρα,β . In fact, for we only need to show that for any α
β on Y , L| β 0
a ∈ Sα , x, y ∈ Sβ with (x, y) ∈ L, then x L(E(S))y 0 and so by Lemma 3.1 and the right quasi-normality of the band E(S) , we have (axa)0 = ((axy)a)0 =
S ⊆ ρα,β as (ayxya)0 = (aya)0 . Thus, by the definition of ρα,β , we have L| α
required. This shows that S = LG[Y ; Sα , Φα,β ] .
⇐=) Let a ∈ Sα , x ∈ Sβ , and y ∈ Sγ . Then, since S = LG[Y ; Sα , Φα,β ] ,

it is an HG -strong semilattice of Sα and by Theorem 4.1, H is a congruence on S . Moreover, we have xa = (xφad(β,αβ) )(aφxd(α,αβ) ) and thereby, axa =
S , )(xφa )(aφx ). It is now easy to see that (xa, axa) ∈ L| (aφx d(α,αβ)

d(β,αβ)

d(α,αβ)

αβ


and so, by our hypothesis, S = LG[Y ; Sα , Φα,β ] , this implies that there exist

L some structural homomorphisms φL d(αβ,αβγ) ∈ Φαβ,αβγ and φd(γ,αβγ) ∈ Φγ,αβγ satisfying y(xa) = (yφL
)((xa)φL
) and y(axa) = (yφL
)
× ((axa)φL d(αβ,αβγ) ).

d(γ,αβγ)

d(αβ,αβγ)

d(γ,αβγ)

Hence, it follows that



L 0 (y(xa))0 = [(yφL d(γ,αβγ) )((xa)φd(αβ,αβγ) )]

a x L 0 = {(yφL d(γ,αβγ) ){[(xφd(β,αβ) )(aφd(α,αβ) )]φd(αβ,αβγ) }}

x L 0 = [(yφL d(γ,αβγ) )((aφd(α,αβ) )φd(αβ,αβγ) )] and

x a x L 0 (y(axa))0 = {(yφL d(γ,αβγ) ){[(aφd(α,αβ) )(xφd(β,αβ) )(aφd(α,αβ) )]φd(αβ,αβγ) }}

x L 0 = [(yφL d(γ,αβγ) )((aφd(α,αβ) )φd(αβ,αβγ) )] . This leads to y 0 x0 a0 = (yxa)0 = (yaxa)0 = y 0 a0 x0 a0 and so E(S) is a right quasi-normal band. Thus, S is a right quasi-normal orthocryptosemigroup. The proof is completed. Finally, we characterize the normal orthocryptosemigroup. This result is a modified version of Theorem IV.1.6 in [11] on normal cryptogroups, in particular, their result is amplified and enriched. Theorem 4.3. A semisuperabundant orthosemigroup S is a normal orthocryptosemigroup if and only if S is a J
G -strong semilattice of rectangular u-semigroups, that is, S = J
G[Y ; Sα , Φα,β ]. Proof. =⇒) The proof is similar to the necessity part given in Theorem 4.2, that is, we only need to prove that J
|Sβ ⊆ ρα,β for all α, β ∈ Y with α
β . Since every Sα can be regarded as a J
-class of S , we can just let a ∈ Sα , x, y ∈ Sβ and so x0 J (E(S))y 0 , which means that x0 y 0 x0 = x0 , y 0 x0 y 0 = y 0 .

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Recall that S = (Y ; Sα ) is a normal orthocryptosemigroup, E(S) is a normal band. Now, by the normality of the band E(S) , we have (axa)0 = (a(xyx)a)0 = (ayxya)0 = (aya)0 . By Lemma 3.5, we see that (x, y) ∈ ρα,β and whence J
|Sβ ⊆ ρα,β . This proves that S = J
G[Y ; Sα , Φα,β ] . ⇐=) Let S = J
G[Y ; Sα , Φα,β ] , where each Sα is a rectangular u
-strong semilattice of semigroup, for all α ∈ Y . Then by definition, S is an LG
-strong semilattice of Sα . By applying Theorem 4.2 Sα and also S is an RG
is a congruence on S and for all and its dual, we can immediately see that H a, x, y ∈ S , we have a0 x0 y 0 a0 = a0 y 0 x0 y 0 a0 = a0 y 0 x0 y 0 x0 a0 = a0 y 0 x0 a0 . This shows that E(S) is a normal band. Our proof is completed. Moreover, since each Sα is a J
-class of S , for every α, β ∈ Y with α
β , D(α, β) is just a set of singleton. This means that S is a strong semilattice of rectangular u-semigroups Sα . Acknowledgments The authors would like to thank their advisor, Professor Yuqi Guo, Professor Francis Pastijn and the referee for their valuable comments and suggestions. References [1] Clifford, A. H., Semigroups admitting relative inverses, Ann. of Math. 42 (1941), 1037–1049. [2] El-Qallali, A., “Structure Theory for Abundant and Related Semigroups”, PhD thesis, York, 1980. [3] Fountain, J. B., Abundant semigroups, Proc. London Math. Soc. 44(3) (1982), 103–129. [4] He, Y., “Some Studies on Regular Semigroups and Generalized Regular Semigroups”, PhD thesis, Kunming, 2002. [5] Howie, J. M., “Fundamental of Semigroup Theory”, Clarendon Press, Oxford, 1995. [6] Kong, X. Z. and Z. L. Yuan, On the semilattice structure of regular bands, Adv. Math. (in Chinese) 31(5) (2002), 476–482. [7] Kong, X. Z. and Z. L. Yuan, KG -strong semilattice decomposition of regular cryptogroups, Adv. Math. (in Chinese) 33(6) (2004), 697–702.

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[8] Pastijn, F. and M. Petrich, Congruences on regular semigroups, Trans. Amer. Math. Soc. 295(2) (1986), 607–633. [9] Petrich, M., The structure of completely regular semigroups, Trans. Amer. Math. Soc. 189 (1974), 211–236. [10] Petrich, M., “Lectures in Semigroups”, Wiley & Sons, London, 1976. [11] Petrich, M. and N. R. Reilly, “Completely Regular Semigroups”, John Wiley & Sons, 1999. [12] Wang, Z. P., R. H. Zhang and M. Xie, Regular orhtocryptou semigroups, Semigroup Forum 69(2) (2004), 281–302. [13] Yamada, M. and N. Kimura, A note on idempotent semigroups, II, Proc. Japan Acad. 34 (1958), 110–112. [14] Zhang, L., K. P. Shum and R. H. Zhang, On refined semilattices, Algebra Colloquium 8(1) (2001), 93–108. [15] Zhang, R. H., On the refine semilattice decomposition structure of left C-rpp semigroups, Southeast Asian Bulletin of Math. 24 (2000), 137–145.

Department of Mathematics Qufu Normal University Qufu, Shandong, 273165 People’s Republic of China [email protected] [email protected]

Received June 1, 2005 and in final form March 9, 2006 Online publication October 30, 2006