Another Structure of Weakly Left C-wrpp Semigroups

It is known that a left C-wrpp semigroup can be described as curler structure of a left band and a C-wrpp semigroup. In this paper, we introduce the class of weakly left C-wrpp semigroups which includes the class of weakly left C-rpp semigroups as a subclass. We shall particularly show that the spined product of a left C-wrpp semigroup and a right normal band is a weakly left C-wrpp semifroup. Some equivalent characterizations of weakly left C-wrpp semigroups are obtained. Our results extend that of left C-wrpp semigroups. Keywords—Left C-wrpp semigroup,left quasi normal regular band, weakly left C-wrpp semigroup. THROUGHOUT this paper, we adopt the notation and terminologies given by Howei[1] and Du[2]. Tang[3] considered a Green-like right congruence relation L∗∗ on a semigroup S : for a, b ∈ S, aL∗∗b if and only if axRay ⇔ bxRby for all x, y ∈ S. Moreover, Tang pointed out in [3] that a semigroup S is a wrpp semigroup if and only if each L∗∗-class of S contains at least one idempotent. Recall that a wrpp semigroup S is a C-wrpp semigroup if the idempotents of S are central. It is well known that a semigroup S is a C-wrpp semigroup if and only if S is a strong semilattice of left-R cancellative monoids(see[3]). Because a Clifford semigroup can be expressed as a strong semilattice of groups and a C-rpp semigroup can be expressed as a strong semilattice of left cancellative monoids(see[4-9]), we see immediately that the concept of C-wrpp semigroups is a common generalization of Clifford semigroups and C-rpp semigroups. For wrpp semigroups, Du-Shum [2] first introduced the concept of left C-wrpp semigroups, that is, a left C-wrpp semigroup whose satisfy the following conditions: (i) for all e ∈ E(L∗∗ a ), a = ae, where E(L∗∗ a ) is the set of idempotents in L∗∗ a ; (ii) for all a ∈ S, there exists a unique idempotent a satisfying aL∗∗a+ and a = aa ; (iii) for all a ∈ S, aS ⊆ L∗∗(a) ,where L∗∗(a) is the smallest left **-ideal of S generated by a. For such semigroups, Du-Shum[2] gave a method of construction. Zhang[10] showed that the spined product of a left C-wrpp semigroup and a right normal band which is a weakly left C-wrpp semigroup by virtue of left C-full Ehremann cybergroups. In this paper, we first define the concept of E. X. Yuan is with the School of Yishui, Linyi University, Shandong 276400 P.R.China (corresponding phone: 86-539-2251004; fax: 86-539-2251004; (e-mail: lygxxm1992@ 126..com). X.M. Zhang is with the School of Logistics, Linyi University, Shandong 276005 P.R.China. weakly left C-wrpp semigroups. A equivalent descriptions of weakly left C-wrpp semigroups is therefore obtained and our results generalize that of Cao on weakly left C-rpp in[5]. In view of the theorems given in this paper, one can easily observe that the results of weakly left C-wrrp semigroups are a common generalizations of weakly left C-semigroups and left C-wrpp semigroups in range of wrpp semigroups. We first recall some known results used in the sequel. To start with, we introduce the concept of simi-spined product. Let T = ∪α∈Y Tα and I = ∪α∈Y Iα be the semilattice decomposition of the semigroups T and I with respect to semilattice Y respectively. For all α ∈ Y , we denote the direct product Iα × Tα by Sα. Let S = ∪α∈Y Sα. we define the mapping η by the following rules: η : S → Tl(I), (i, a) → η(i, a), η(i, a) : I → I, j → (i, a)j, where Tl(I) is a left transformation semigroup on I . Suppose that the mapping η satisfies the following conditions: (Q1)If (i, a) ∈ Sα, j ∈ Iβ , then (i, a)j ∈ Iαβ ; (Q2)If (i, a) ∈ Sα, (j, b) ∈ Sβ with α ≤ β, then (i, a)j = ij, where ij is the semigroup product in the semigroup I = ∪α∈Y Iα; (Q3)If (i, a) ∈ Sα, (j, b) ∈ Sβ , then η(i, a)η(j, b) = η((i, a)j, ab), where ab is the semigroup product in the semigroup T = ∪α∈Y Tα. Then we define a multiplication ” ◦” on S = ∪α∈Y Sα by (i, a) ◦ (j, b) = ((i, a)j, ab). By a straightforward verification, we can prove that the multiplication ” ◦” satisfies the associative law and hence (S, ◦) becomes a semigroup, denoted by S = I ×η T . We call this semigroup the semi-spined product of I and T with respect to the structure mapping η. Lemma 1[2] Let I be a left regular band which is expressed as a semilattice of left zero bands Iα (that is, I = ∪α∈Y Iα ) and let T = ∪α∈Y Tα be a C-wrpp semigroup(that is, T is a strong semilattice of left-R cancellative monoids [Y ;Tα, φα,β ])(see[3]). If the structure mapping η satisfies the following condition: (Q): kerη(i, a) = kerη(j, b) for every (i, a), (j, b) ∈ Sα. Then S is a left C-wrpp semigroup. Conversely, every left C-wrpp semigroup S can be constructed in terms of above method. Lemma 2[5] A semigroup S is a weakly left C-semigroup, that is, S is a regular semigroup and (∀e ∈ E(S))η′ e : S → eS, x → ex Enxiao Yuan, Xiaomin Zhang