Orthodox Super Rpp Semigroups 1

International Journal of Algebra, Vol. 8, 2014, no. 10, 485-494 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.4546 Φ, Ψ Product Structure of Orthodox Super Rpp Semigroups 1 Yang Li and Zhenlin Gao College of Science University of Shanghai for Science and Technology Shanghai 200093, P. R. China Copyright c 2014 Yang Li and Zhenlin Gao. 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. Abstract The aim of this paper is to give the structure of orthodox super rpp semigroups using the method of Φ, Ψ product which extends both left Δ product and right Δ product. Our result not only generalizes the concept of Δ product, but also enriches the methods to construct semigroups. Mathematics Subject Classification: 20M10 Keywords: Orthodox super rpp semigroups, Bands, C-rpp semigroups, Φ, Ψ product 1 Introduction As a class of generalized regular semigroups, rpp semigroups have been considered by many authors. Fountain [1] has first proved that a C-rpp semigroup can be expressed as a strong semilattice of left cancellative monoids. Based on Fountain s work, Guo et al. [3] defined the concept of strongly rpp semigroups which is the generalization of completely regular semigroups in the range of rpp semigroups. By using this concept, the definition of left C-rpp semigroups 1 This work is supported by the National Natural Science Foundation of China No.11171220

486 Yang Li and Zhenlin Gao was introduced and the semi-spined product structure of left C-rpp semigroups was obtained. Two years later, the right dual of left C-rpp semigroups was investigated by Guo [4]. He et al. [7] studied the orthodox super rpp semigroups and proved that an orthodox super rpp semigroup is isomorphic to a semilattice of left cancellative planks, which extends some main results in [3] and [4]. After then, many methods to construct some kinds of rpp or wrpp semigroups were discussed, such as semi-spined product, Δ product, wreath product, left cross product and band-like extension See [3], [5]-[7], [10]-[12]. In this paper, we first construct the Φ, Ψ product of semigroups which generalizes the concepts of left Δ product and right Δ product defined in [12] and [11] respectively, and then we will establish a structure theorem for orthodox super rpp semigroups in terms of Φ, Ψ product. Our result not only gives a new description for orthodox super rpp semigroups, but also enriches the methods to construct semigroups. 2 Preliminaries In this section, some basic definitions and known results used in the sequel will be recalled. Lemma 2.1 [7] Let S be a semigroup. Then the following statements hold: 1 L = {a, b S S x, y S 1,ax= ay bx = by} is a right congruence on S; 2 For a S, e E S the set of all idempotents in S, a, e L if and only if a = ae, and for any x, y S 1, ax = ay implies ex = ey. Obviously, a left cancellative monoid is a semigroup with a unique idempotent which is called a unipotent semigroup in [9]. Lemma 2.2 [9] Let S = S α be a semilattice of unipotent semigroups S α, then S is a strong semilattice of S α. By Fountain [2], a semigroup S is rpp if and only if for all a S, L a contains at least an idempotent of S, where L a is the L equivalence class of S containing a. Definition 2.3 [3],[7]We call a rpp semigroup S a strongly rpp semigroup if for all a S, there exists a unique idempotent a + satisfying al a + and a = a + a. A strongly rpp semigroup S is called orthodox strongly rpp if E S forms a band. We call an orthodox strongly rpp semigroup S an orthodox super rpp if R + is a left congruence on S, where R + = {a, b S S a +,b + R}, and R is the usual Green s R relation on S.

Φ, Ψ product structure of orthodox super Rpp semigroups 487 Lemma 2.4 [7] Let S be an orthodox strongly rpp semigroup. Then S is an orthodox super rpp semigroup if and only if S is isomorphic to a semilattice Y of S α = E α M α, where each E α is a rectangular band, and M α is a left cancellative monoid. Definition 2.5 The component x i i =1, 2,..., n of the element x 1,x 2,...,x n in the Cartesian product X = X 1 X 2 X n is called the projection of the element x 1,x 2,...,x n onto the set X i i =1, 2,..., n. We denote it by x 1,x 2,...,x n P Xi = x i 1 i n. Lemma 2.6 [7] Let S be a semilattice Y of S α = E α M α, where each E α is a rectangular band, and M α is a left cancellative monoid. Then the following statements hold: 1 For i, a, i,a S α, j, b, j,b S β, we have [i, aj, b] P Mαβ =[i,aj,b] P Mαβ 2 The set union M = M α forms a C-rpp semigroup with respect to the following operation defined by : For any a M α and b M β, a b = c there exist i, a S α and j, b S β such that [i, aj, b] P Mαβ = c. 3 The Φ, Ψ Product Structure of Orthodox Super Rpp Semigroups In this section, we first introduce the concept of Φ, Ψ product of semigroups, and then point out this concept indeed generalizes both left Δ product and right Δ product. Finally, we shall use this concept to give a structure theorem for orthodox super rpp semigroups. Let Y be a semilattice and M =Y ; M α a semilattice decomposition of a semigroup M into subsemigroups M α. Let a band E =Y ; E α be a semilattice Y of rectangular bands E α. Denote the direct product of E α and M α for every α Y by S α = E α M α. Then for any α, γ Y with α γ, assume that there exist the following two mappings: Φ α,γ : S α T l E γ, i, a ϕ i,a α,γ, i, a S α Ψ α,γ : S α T r E γ, i, a ψ i,a α,γ, i, a S α where T l E γ and T r E γ are the left transformation semigroup and right transformation semigroup on E γ respectively. For any ϕ T l E γ and ψ T r E γ, we define a mapping on E γ as follows: ϕ ψ : E γ E γ, ϕ ψλ =ϕ λ λ ψ,λ E γ.

488 Yang Li and Zhenlin Gao We now list some notations used in the sequel: for every α Y, ϕ 1,ϕ 2 T l E α, we let ϕ 1,ϕ 2 R E α stands for ϕ 1 λ,ϕ 2 λ R E α, λ E α ; dually, ψ 1,ψ 2 T r E α, ψ 1,ψ 2 L E α stands for λ ψ 1, λ ψ 2 L E α, λ E α, where R E α and L E α are Green s R relation and L relation on E α respectively; finally, for a constant mapping σ : E α E α, denote [σ] by the value of σ. Under the above notations, we have the following lemma: Lemma 3.1 For i, a S α, j, b S β, suppose that the following conditions hold: C1 For i 1,a 1, i 2,a 2 S α,λ 1,λ 2 E α, we have the equality ϕ i 1,a 1 α,α λ 1 λ 2 ψ i 2,a 2 α,α = i1 i 2 ; C2 ϕ i,a α,αβ ϕj,b β,αβ ψ i,a α,αβ ψj,b β,αβ is a constant mapping on E αβ ; C3 For α, β, δ Y with αβ δ, put [ ] ν = ϕ i,a α,αβ ϕj,b β,αβ ψ i,a α,αβ ψj,b β,αβ, then ϕ ν,ab αβ,δ,ϕi,a α,δ ϕj,b β,δ R E δ C3.1, and ψ ν,ab αβ,δ,ψi,a α,δ ψj,b β,δ L E δ C3.2; C4 For α, γ Y with α γ, k, l E γ,ifk, l L E γ, then k ψ α,γ i,a, l ψ α,γ i,a L E γ C4.1; if k, l R E γ, then ϕ α,γ i,a k,ϕ α,γ i,a l R E γ C4.2. Denote the union of S α by S = S α, and define a binary operation on S : [ ] i, a j, b = ϕ i,a α,αβ ϕj,b β,αβ ψ i,a α,αβ ψj,b β,αβ,ab Then S, forms a semigroup, and S = S α is a semilattice of S α = E α M α with respect to the binary operation. Proof. For i, a S α, j, b S β, k, c S γ,wehave [ ] i, a j, b k, c = ϕ i,a α,αβ ϕj,b β,αβ ψ i,a α,αβ ψj,b β,αβ [ = ϕ ν,ab αβ,αβγ ϕk,c γ,αβγ ψ ν,ab αβ,αβγ ψk,c γ,αβγ,ab k, c ], abc We denote ϕ i,a α,αβγ ϕj,b β,αβγ ϕk,c γ,αβγ and ψi,a α,αβγ ψj,b β,αβγ ψk,c γ,αβγ by θ 1 and θ 2 respectively. It follows by C3.1 that ϕ ν,ab αβ,αβγ ϕk,c γ,αβγ,θ 1 R E αβγ 3.1; on the other hand, by C3.2 and C4.1, we have ψ ν,ab αβ,αβγ ψk,c γ,αβγ,θ 2 L E αβγ 3.2. Consequently, by 3.1 and 3.2, it follows that [ ] ϕ ν,ab αβ,αβγ ϕk,c γ,αβγ ψ ν,ab αβ,αβγ ψk,c γ,αβγ =[θ 1 θ 2 ]

Φ, Ψ product structure of orthodox super Rpp semigroups 489 Similarly, by C3.2, C3.1 and C4.2, we also have i, a j, b k, c=[θ 1 θ 2 ], abc Hence i, a j, b k, c =i, a j, b k, c, and this implies that S, forms a semigroup. For every α Y,i 1,a 1, i 2,a 2 E α M α, and λ E α, the following equality holds by C1: ϕ i 1,a 1 α,α ϕ i 2,a 2 α,α ϕ i 2,a 2 α,α λ ψ i 1,a 1 α,α ψ i 2,a 2 α,α λ = ϕ i 1,a 1 α,α λ ψ i 1,a 1 α,α ψ i 2,a 2 α,α = i 1 i 2. Thus we have i 1,a 1 i 2,a 2 =i 1 i 2,a 1 a 2. Consequently S = S α is indeed a semilattice of S α = E α M α with respect to the binary operation. The proof is completed. The above constructed semigroup S, is called Φ, Ψ product of a band E and a semigroup M, and we denote it by S = E Φ,Ψ M. We now consider that what the above conditions C1-C4 change into when each E α is a left zero band. First, if each E α is a left zero band, then ϕ ψ = ϕ, and the equality in C1 is equivalent to ϕ i 1,a 1 α,α λ 1 =i 1 ; the constant mapping in C2 is equivalent to ϕ i,a α,αβ ϕj,b β,αβ ; the formula C3.1 in C3 is equivalent [ ] to ϕ ν,ab αβ,δ = ϕ i,a α,δ ϕj,b β,δ, where ϕ i,a α,αβ ϕj,b β,αβ = ν, however the formula C3.2 and the condition [ C4 always ] hold; finally, the formula is equivalent to i, a j, b = ϕ i,a α,αβ ϕj,b β,αβ,ab. Summarizing the above arguments, we have the following corollary: Corollary 3.2 [12] In each direct product S α = E α M α,ife α is a left zero band, and for i, a S α, j, b S β, Φ α,γ γ Y,α γ, the following conditions are satisfied: C1 ϕ i,a α,α λ =i, λ E α ; C2 ϕ i,a value by [ ϕ i,a α,αβ ϕj,b β,αβ α,αβ ϕj,b β,αβ is a constant mapping on E αβ, and denote this constant ] ; [ ] C3 For δ Y with αβ δ, put ϕ i,a α,αβ ϕj,b β,αβ ϕ i,a α,δ ϕj,b β,δ. Define a binary operation ons = S α : i, a j, b = Then S, forms a semigroup, and S = E α M α with respect to the binary operation. = ν, then ϕ ν,ab αβ,δ = [ ] ϕ i,a α,αβ ϕj,b β,αβ,ab. S α is a semilattice of S α = In corollary 3.2, we have S = E Φ M which is called left Δ product of a left regular band E and a semigroup M in [12]. Dually, when each E α is a right zero

490 Yang Li and Zhenlin Gao band, the Φ, Ψ product can also changes into the right Δ product defined in [11]. Hence the Φ, Ψ product indeed generalizes both left Δ product and right Δ product. We now give some properties of the Φ, Ψ product. Lemma 3.3 For S = E Φ,Ψ M defined as above, the following statements hold: 1 E S = {i α,e α i α E α,e α E M α }, and E S is a band if and only if E M is a band; 2 If each M α is a left cancellative monoid, then S is strongly rpp if and only if P: For every i, a S α, j, 1 β S β, k, 1 γ S γ, ϕ i,a α,αβ ϕ j,1 β β,αβ,ϕi,a α,αγϕ k,1 γ γ,αγ R E αβ with αβ = αγ implies ϕ i,1 α α,αβ ϕ j,1 β β,αβ,ϕi,1 α α,αγ ϕ k,1 γ γ,αγ R E αβ, where 1α is the identity of M α for every α Y. Proof. 1 This part can be easily verified by routine checking. So we omit the details. 2 Suppose that the condition P holds, and i, a x = i, a y, for i, a S α, x, y S 1. We first put x = j, b S β,y = k, c S γ, namely i, a j, 1 β j, b = i, a k, 1 γ k, c 3.3. Then we can easily deduce that ϕ i,a α,αβ ϕ j,1 β β,αβ,ϕi,a α,αγϕ k,1 γ γ,αγ R E αβ with αβ = αγ from formula 3.3, so we have ϕ i,1 α α,αβ ϕ j,1 β β,αβ,ϕi,1 α α,αγ ϕ k,1 γ γ,αγ R E αβ 3.4 by the condition P. However, by the equality i, 1 α j, b = i, 1 α j, 1 β j, b, it follows that ϕ i,1 α α,αβ ϕ j,1 β β,αβ,ϕi,1 α α,αβ ϕj,b β,αβ R E αβ 3.5, and similarly, ϕ i,1 α α,αγ ϕ k,1 γ γ,αγ,ϕ i,1 α α,αγ ϕ k,c γ,αγ R E αβ 3.6 also holds. Combining 3.4,3.5 with 3.6, we have ϕ i,1 α α,αβ ϕj,b β,αβ,ϕi,1 α α,αγ ϕ k,c γ,αγ R E αβ 3.7. On the other hand, by the equality i, a i, 1 α j, b =i, a i, 1 α k, c, it can be easily verified that ψ i,1 α α,αβ ψj,b β,αβ,ψi,1 α α,αγ ψ γ,αγ k,c L E αβ 3.8. Consequently, by 3.7 and 3.8, we have [ ] ϕ i,1 α α,αβ ϕj,b β,αβ ψ i,1 α α,αβ ψj,b β,αβ = [ ϕ i,1 α α,αγ ϕ k,c γ,αγ ψ i,1 α α,αγ ψ γ,αγ] k,c 3.9 By Lemma 2.2, M is in fact a strong semilattice of M α since each M α is a left cancellative monoid, namely M =Y ; M α,φ α,τ, where φ α,τ is a homomorphism from M α into M τ with α τ. So ab = ac can be expressed by aφ α,αβ bφ β,αβ =aφ α,αγ cφ γ,αγ, and we have 1 α φ α,αβ bφ β,αβ =1 α φ α,αγ cφ γ,αγ,

Φ, Ψ product structure of orthodox super Rpp semigroups 491 namely 1 α b =1 α c 3.10, since M αβ is a left cancellative monoid. By 3.9 and 3.10, it follows that i, 1 α j, b =i, 1 α k, c. If one of x and y is 1 S, then it is trivial to verified that i, a x =i, a y implies i, 1 α x =i, 1 α y. Hence by Lemma 2.12 we have i, a L i, 1 α. We can also easily observe that i, a + =i, 1 α. So S is strongly rpp. Conversely, if S is strongly rpp, then obviously i, a L i, 1 α. Suppose that for every i, a S α, j, 1 β S β, k, 1 γ S γ, ϕ i,a α,αβ ϕ j,1 β β,αβ,ϕi,a α,αγϕ k,1 γ γ,αγ R E αβ with αβ = αγ. Denote the constant mappings ϕ i,a α,αβ ϕ j,1 β β,αβ ψ i,a α,αβ ψ j,1 β β,αβ and ϕ i,a α,αγϕ k,1 γ γ,αγ ψ α,αγψ i,a k,1 γ γ,αγ by ξ and η respectively. Then we can clearly see that ϕ i,a α,αβ ϕ j,1 β β,αβ,ξ R E αβ and ϕ α,αγϕ i,a k,1 γ γ,αγ,η R E αβ. Hence [ξ], [η] R E αβ, and then there exist s, t E 1 αβ such that [ξ] =[η] s, [η] = [ξ] t. Consequently, we have i, a j, 1 β = i, a k, 1 γ s, 1 αβ and i, a k, 1 γ =i, a j, 1 β t, 1 αβ, then this implies that i, 1 α j, 1 β = i, 1 α k, 1 γ s, 1 αβ 3.11 and i, 1 α k, 1 γ =i, 1 α j, 1 β t, 1 αβ 3.12 by i, a L i, 1 α. So we can easily see that ϕ i,1 α α,αβ ϕ j,1 β β,αβ,ϕi,1 α α,αγ ϕ k,1 γ γ,αγ by the formula 3.11 and 3.12. This completes the proof. R E αβ If a C-rpp semigroup M is written as M =Y ; M α, then we always mean that every M α is a left cancellative monoid. So by Lemma 2.4 and Lemma 3.3, we can deduce that the Φ, Ψ product of a band E and a C-rpp semigroup M satisfying the condition P is an orthodox super rpp semigroup. We now proceed to prove that every orthodox super rpp semigroup is isomorphic to aφ, Ψ product of a band and a C-rpp semigroup satisfying the condition P. Lemma 3.4 Let S be an orthodox super rpp semigroup. Then there exist a band E and a C-rpp semigroup M such that S = E Φ,Ψ M. Proof. According to Lemma 2.4, we can write S = S α, where S α = E α M α, each E α is a rectangular band, and M α is a left cancellative monoid. We form the set unions E = E α and M = M α. In Lemma 2.6, it have been known that M = M α is a C-rpp semigroup with respect to the multiplication defined there. Then we define a multiplication on E = E α : For every i E α,j E β, ij = k i, 1 α j, 1 β =k, 1 αβ. It can be also seen that f : E S E,i, 1 α i is an isomorphism from E S ontoe. So E = E α forms a band with respect to the multiplication defined above.

492 Yang Li and Zhenlin Gao For α, γ Y with α γ, we construct the following mappings: defined by Φ α,γ : S α T l E γ, i, a ϕ i,a α,γ i, aλ, 1 γ = ϕ i,a α,γ λ, a1 γ, and Ψ α,γ : S α T r E γ, i, a ψ α,γ i,a defined by λ, 1 γ i, a = λψ α,γ i,a, 1 γa. We now prove that the conditions C1-C4 in Lemma 3.1 are satisfied by Φ α,γ and Ψ α,γ defined above. For i 1,a 1, i 2,a 2 S α and λ 1,λ 2 E α, obviously, we have i 1,a 1 λ 1, 1 α = i 1 λ 1,a 1, λ 2, 1 α i 2,a 2 =λ 2 i 2,a 2, and so ϕ i 1,a 1 α,α λ 1 λ 2 ψ i 2,a 2 α,α =i1 λ 1 λ 2 i 2 =i 1 i 2. Hence the condition C1 is satisfied. In order to prove that C2 is satisfied, for i, a S α, j, b S β and λ E αβ, we first have the following equality: i, aj, bλ, 1 αβ i, aj, b =i, a ϕ j,b β,αβ λ, b1 αβ λψ i,a α,αβ, 1 αβa j, b =i, a ϕ j,b β,αβ λ, 1 αβ ϕ j,b β,αβ λ, b1 αβ λψ i,a α,αβ, 1 αβa λψ i,a α,αβ, 1 αβ j, b = ϕ i,a α,αβ ϕj,b β,αβ λ, a1 αβ ϕ j,b β,αβ λ, b1 αβ λψ i,a α,αβ, 1 αβa λψ i,a α,αβ ψj,b β,αβ, 1 αβb = ϕ i,a α,αβ ϕj,b β,αβ λ λψ i,a α,αβ ψj,b β,αβ, ab 2 3.13 On the other hand, if we write i, aj, b =ν, ab, then i, aj, bλ, 1 αβ i, aj, b = ν, ab 2 3.14, and comparing 3.14 with 3.13, we have ϕ i,a α,αβ ϕj,b β,αβ λ λψ i,a α,αβ ψj,b β,αβ =ν for every λ E αβ. This indicates that ϕ i,a α,αβ ϕj,b β,αβ ψ i,a α,αβ ψj,b β,αβ is a constant mapping on E αβ. Consequently, the condition C2 is satisfied. We denote the above constant mapping by θ. Then by the above arguments, we also can obtain i, aj, b =[θ],ab=ν, ab 3.15. For δ Y with αβ δ and λ E δ, it first follows that i, aj, bλ, 1 δ =i, a ϕ j,b β,δ λ, 1 δ ϕ j,b β,δ λ, b1 δ = ϕ i,a α,δ ϕj,b β,δ λ, a1 δ ϕ j,b β,δ λ, b1 δ = ϕ i,a α,δ ϕj,b β,δ λ ϕ j,b β,δ λ, ab1 δ 3.16.

Φ, Ψ product structure of orthodox super Rpp semigroups 493 On the other hand, we also have i, aj, bλ, 1 δ =ν, abλ, 1 δ = ϕ ν,ab αβ,δ λ, ab1 δ 3.17. Comparing 3.17 with 3.16, ϕ ν,ab αβ,δ λ = ϕ i,a α,δ ϕj,b β,δ λ Moreover, we have ϕ i,a α,δ ϕj,b β,δ λ = ϕν,ab αβ,δ λ ϕ i,a α,δ ϕj,b ϕ j,b β,δ λ 3.18 holds. β,δ λ 3.19 since E δ is a rectangular band. Consequently, by 3.18 and 3.19, the formula C3.1 holds. Dually, using similar method, we can also verify that the formula C3.2 holds. Hence the condition C3 is satisfied. For γ Y with α γ and k, l E γ, if k, l R E γ, then there exist m, n E 1 γ such that k = lm, l = kn, namely, k, 1 γ =l, 1 γ m, 1 γ, l, 1 γ = k, 1 γ n, 1 γ. Furthermore, we have i, ak, 1 γ =i, al, 1 γ m, 1 γ, i, al, 1 γ =i, ak, 1 γ n, 1 γ, and by calculating the above two equalities, we can obtain ϕ i,a α,γ k, a1 γ = ϕ i,a α,γ l, a1 γ m, 1γ = ϕ α,γ i,a l m, a1 γ, and ϕ i,a α,γ l, a1 γ = ϕ i,a α,γ k, a1 γ n, 1γ = ϕ i,a α,γ k n, a1 γ. Consequently, it follows that ϕ i,a α,γ k = ϕ i,a α,γ l m, ϕ i,a α,γ l = ϕ α,γ i,a k n, and ϕ i,a α,γ k,ϕ i,a α,γ l R E γ. So the formula C4.2 holds. Dually, we can also prove that the formula C4.1 holds if k, l L E γ. Hence the condition C4 is satisfied. By the above conclusions, it is clear to see that the band E = E α and the C-rpp semigroup M = M α can form a Φ, Ψ product E Φ,Ψ M. Moreover, by the formula 3.15, we can also see that the multiplication on S coincides with that on the Φ, Ψ product E Φ,Ψ M. Hence S = E Φ,Ψ M. The proof is completed. By Lemma 3.32, we can easily deduce that the Φ, Ψ product E Φ,Ψ M constructed above satisfies the condition P. So summarizing the results of Lemma 3.1, 3.3 and 3.4, we can obtain the following structure theorem: Theorem 3.5 Let a band E =Y ; E α be a semilattice of rectangular bands E α. Let a C-rpp semigroup M =Y ; M α be a semilattice of left cancellative monoids M α. Then the Φ, Ψ product E Φ,Ψ M satisfying the condition P is an orthodox super rpp semigroup. Conversely, every orthodox super rpp semigroup is isomorphic to a Φ, Ψ product of a band and a C-rpp semigroup satisfying the condition P.

494 Yang Li and Zhenlin Gao References [1] J. B. Fountain, Right pp monoids with central idempotents, Semigroup Forum, 13 1977, 229-237. [2] J. B. Fountain, Abundant semigroups, Proc. London Math. Soc., 44 1982, 103-129. [3] Yuqi Guo, K. P. Shum, Pinyu Zhu, The structure of left C-rpp semigroups, Semigroup Forum, 50 1995, 9-23. [4] Yuqi Guo, The right dual of left C-rpp semigroups, Chinese Science Bulletin, 4219 1997, 1599-1603. [5] Xiaojiang Guo, Ming Zhao, K. P. Shum, Wreath product structure of left C-rpp semigroups, Algebra Colloquium, 15 2008, 101-108. [6] Xiaojiang Guo, Young Bae Jun, Ming Zhao, Pseudo-C-rpp semigroups, Acta Mathematica Sinica English Series, 26 2010, 629-646. [7] Yong He, Yuqi Guo, K. P. Shum, The construction of orthodox super rpp semigroups, Sciences in China Ser. A, Mathematics, 474 2004, 552-565. [8] J. M. Howie, An Introduction to Semigroup Theory, Academic Press, London, 1976. [9] Jing Leng, Huilan Pan, Zhengpan Wang, On semilattices of unipotent semigroups, Journal of Southwest China Normal University Natural Science Edition, 3710 2012, 35-37. [10] X. M. Ren, X. L. Ding, K. P. Shum, A new structure theorem of left C-wrpp semigroups, International Journal of Algebra, 11 2007, 41-49. [11] K. P. Shum, X. M. Ren, The structure of right C-rpp semigroups, Semigroup Forum, 68 2004, 280-292. [12] Xiaomin Zhang, Left Δ product structure of left C-wrpp semigroups, Journal of Mathematical Research and Exposition, 292 2009, 327-334. Received: May 5, 2014