Wreath product structure of left C-rpp semigroups

Wreath product structure of left C-rpp semigroups K.P. Shum∗ Faculty of Science, The Chinese University of Hong Kong Shatin, Hong Kong (SAR), China e-mail: [email protected]

Abstract The concept of “ wreath product” of semigroups was first initiated by B. H. Neumann in 1960 and later on, his concept was used by G. B. Preston to investigate the structure of some inverse semigroups.However,in the study of the structure theory of generalized regular semigroups,people usually use spined product and semi-spined product to construct semigroups and surprisingly,the concept of “ wreath product” has seldom been mentioned.In this paper,we modify the “ wreath product” given by Neumann and Preston to study the structure of some generalized Clifford semigroups. In particular,we prove that a semigroup is a left C-rpp semigroup if and only if it is the wreath product of a left regular band and a C-rpp semigroup. Our theorem provides new insight to the structure of left C−rpp semigroups. Key words: Left C-rpp semigroup; Left regular band,; Left cancellative monoid; Wreath product MR (1991) Subject Classification: 20M10

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Introduction

Throughout this paper, we follow the notations and terminologies given by Fountain [2]. Similar to rpp rings, a semigroup S is called an rpp semigroup if for all a ∈ S, aS 1 , regarded as a right S 1 -system, is projective. In studying the structure of rpp semigroup, Fountain [2] considered a Green-like right congruence relation L∗ on a semigroup S defined by: (∀a, b ∈ S) aL∗ b if and only if ax = ay ⇔ bx = by for all x, y ∈ S 1 . Dually, we can define the ∗

The research is partially supported by a UGC (HK) grant # 2160210/04-05

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left congruence relation R∗ on a semigroup S. It can be observed that for a, b ∈ S, aL∗ b if and only if aLb when S is a regular semigroup. Also, we can easily see that a semigroup S is an rpp semigroup if and only if each L∗ -class of S contains at least one idempotent. Later on, Fountain [3] called a semigroup S an abundant semigroup if each L∗ -class and each R∗ -class of S contains at least one idempotent. An important subclass of the class of rpp semigroups is the class of C-rpp semigroups. We call an rpp semigroup S a C-rpp semigroup if the idempotents of S are central. It is well known that a semigroup S is a C-rpp semigroup if and only if S is a strong semilattice of left cancellative monoids [2]. Because a Clifford semigroup can always be expressed as a strong semilattice of groups, we see immediately that the C-rpp semigroups are proper generalizations of Clifford semigroups. In order to further generalize the C-rpp semigroups, Guo-Shum-Zhu [12] introduced the concept of “strongly rpp semigroups”. They called a rpp semigroup S a strongly rpp semigroup if every L∗a contains a unique idemT potent a† ∈ L∗a E(S) such that a† a = a holds, where E(S) is the set of all idempotents of S. They then call a strongly rpp semigroup S a left C-rpp semigroup if L∗ is a congruence on S and eS ⊆ Se holds for all e ∈ E(S). It is noted that the set E(S) of a left C-rpp semigroup S forms a left regular band, that is, ef = ef e for e, f ∈ E(S). Because of this crucial observation, we can describe the left C-rpp semigroup by using the left regular band and C-rpp semigroup. The structures of left C-rpp semigroups have been investigated by many authors (see [4-13], for example).In particular,it was proved in [12] that if S is a strongly rpp semigroup whose set of idempotents E(S) forms a left regular band,then S is a left C-rpp semigroup if and only if S is a semilattice of the direct products of a left zero band and a left cnacellative monoid, that is, it is a semilattice of left cancellative strips. Also, if we S let M = α∈Y Mα be the semilattice of the left cancellative monoids and S I = α∈Y Iα a semilattice of left zero bands, then S is a left C-rpp semigroup if and only if S is the semi-spined product of the C-rpp semigroup M and the left regular band I (see [12]). Recently, Li-Shum [15] have further shown that a weakly left C-rpp semigroup can be described as the spined product of a left C-rpp semigroups and a right normal band. However, in all the investigations of left C-rpp semigroups and their generalizations, we notice that the concept “semi-direct product” of semigroups, firstly developed by B. H. Neumann [1] in 1960, has surprisingly not been used. The hurdle may be perhaps that it is not an easy task to link up all the ingredients by a structure mapping which is also compatible with the semigroup multiplication. In fact, Neumann did use his “semidirect prod2

uct” to define a “wreath product” of semigroups. The wreath product was then adopted by W. R. Nico [16] to study product of monoids and G. B. Preston used “wreath product” to study the structure of inverse semigroups. In this paper, we first define the wreath structure mapping and then following the ideas of Neumann [1] and Preston [18], we define the wreath product of a C-rpp semigroup M and a left regular band I, S = M ωr I, by using the newly defined wreath mappings.We establish the following nice theorem: Theorem 1.1 A semigroup S is a left C-rpp semigroup if and only if S is the wreath product of a left regular band and a C-rpp semigroup with respect to some wreath structure mapping.

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Preliminaries

We first define the wreath structure mapping of the C-rpp semigroup M and the left regular band I. Definition 2.1 Let M = (Y ; Mα ) be the semilattice decomposition of the C-rpp semigroup M into the left cancellative monoids Mα and I = (Y ; Iα ) the semilattice decomposition of the left regular band I into the left zero bands Iα . Define a mapping ϕ : M → End(I) by ϕ(m) = ϕm satisfying the following conditions: (WR1) for all x ∈ Iα and m ∈ Mβ , we have ϕm x ∈ Iαβ ; (WR2) for all u ∈ Mα and v ∈ Mβ , there exists g ∈ Iαβ such that ϕu ϕv = λg ϕuv , where λg is the inner left translation of I determined by g; (WR3) for all u ∈ Mα , x ∈ Iα , y ∈ Iβ and z ∈ Iγ , if x(ϕu y) = x(ϕu z), then x(ϕ1α y) = x(ϕ1α z), where 1α is the identity of the monoid Mα . We call the mapping ϕ the “wreath structure mapping” of M and L. Definition 2.2 In the above definition, we form the set S = {(x, u) ∈ I × M : (∃α ∈ Y )x ∈ Tα and u ∈ Mα }. Define the multiplication ◦ on S by (x, u) ◦ (y, v) = (x(ϕu y), uv).

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Then, it can be easily seen that the “ ◦ ” is well defined and is associative. Thus (S, ◦) forms a semigroup. We call this semigroup (S, ◦) the wreath product of I and M with respect to the wreath structure mapping ϕ, denoted by Iωrϕ M . Definition 2.3 Let S be a strongly rpp semigroup whose the set of idempotents E(S) forms a left regular band. Define a relation ηS on S by ηS = {(x, y) ∈ S × S : (∃e ∈ E(y † ))x = ey}, where E(y † ) is the D-class of E(S) containing the idempotent y † ∈ L∗y such that y † y = y. Then, we call ηS the left C-rpp relation.

T

E(S)

We give an important property of left C-rpp semigroups. Lemma 2.4 If S is a left C-rpp semigroup, then for all e ∈ E(S) and a ∈ S, we have ae = (ae)† a. Proof: Let S be a left C-rpp semigroup. Then, by the main result in [12], S can be expressed as the semilattice of the direct product of a left zero band Iα and a left cancellative monoid Mα . Hence, we can write S = (Y ; Iα × Mα ). Now,pick any (x, 1α ) ∈ E(S) and (y, m) ∈ Iβ × Mβ .Since L∗ is a right congruence on S,we have (y, m)(x, 1α ) ∈ Iαβ × Mαβ . On the other hand, because S is particularly an rpp semigroup, we have g ∈ E(Iαβ × Mαβ ) such that gL∗ ((y, m)(x, 1α ))† (y, m), by Proposition 6.9 in Fountain [3]. This result leads to ((y, m)(x, 1α ))† (y, m)g = ((y, m)(x, 1α ))† (y, m). Since g and g(x, 1α ) are in Iαβ × Mαβ , we can derive that (y, m)(x, 1α ) = = = = =

((y, m)(x, 1α ))† ((y, m)(x, 1α )) ((y, m)(x, 1α ))† (y, m)g(x, 1α ) ((y, m)(x, 1α ))† (y, m) ◦ g ◦ g(x, 1α ) ((y, m)(x, 1α ))† (y, m)g ((y, m)(x, 1α ))† (y, m).

Now, write a = (y, m) ∈ S, e ∈ E(S). Then we see immediately that ae = (ae)† a. Thus, the proof is completed. 2 The following is a crucial lemma of this paper. Lemma 2.5 ( left C-rpp congruence Lemma) Let S be a strongly rpp semigroup whose set of idempotents E(S) forms a left regular band. Then S is a left C-rpp semigroup if and only if the left C-rpp relation ηS on S is a congruence on S such that S/ηS is a C-rpp semigroup. 4

Proof: ⇒) Suppose that S is a left C-rpp semigroup. Then,by Guo-GuoShum in [7], we can express S as the semi-spined product M ∞ξ I, where S M is a C-rpp semigroup and I a left regular band. Write M = α∈Y Mα , S where each Mα is a left cancellative monoid, and I = α∈Y Tα , where each S Iα is a left zero band. Then S = α∈Y Iα × Mα is the semispined product of I and M . By the definition of L∗ on S, we can verify that ηS ⊆ L∗ . On the other hand, by Proposition 6.9 in Fountain [3], every L∗ -class of S is of the S form Iα × Mα . Thus, in particular, we can write ηS = α∈Y ηIα ×Mα . Now, it is easy to check that for every (iα , aα ), (jα , bα , ) ∈ Iα × Mα , we have (aα , iα )ηIα ×Mα (jα , bα ) ⇔ aα = bα . Consequently, we can verify that ηS is an equivalent relation on S. Also, for all (iα , aα ) ∈ Iα × Mα , (iβ , aβ ) ∈ Iβ × Mβ , we have (aα , iα )ηS (aβ , iβ ) ⇔ α = β and aα = aβ . Thus,by the definition of semi-spined product [12], we can see immediately that ηS is a congruence on S so that the semigroup S/ηS is isomorphic to M . This shows that S/ηS is indeed a C-rpp semigroup. ⇐) Assume that ηS is a congruence on S such that S/ηS is a C-rpp semigroup. Then, we can easily verify that if xηS is an idempotent in S/ηS , then x is an idempotent of S by the definition of ηS . This means that ηS is an idempotent-pure congruence on S. Thus, we deduce that by the well known Lallament lemma, E(S)/ηS = E(S)/L∗ = E(S)/L = E(S/ηS ) which is a semilattice. In other words, L=L∗ is a semilattice congruence on S and S thereby, E(S) is a left regular band. Hence, we can write E(S) = α∈Y Eα , where Eα is a left zero band. Moreover, it is obvious to see that Y is isomorphic to the semilattice E(S)/L. We now identify E(S)/L by Y . In S fact, by our hypothesis, we can let S/ηS = α∈Y Mα , where each Mα is a S left cancellative monoid. By putting T = α∈Y Iα × Mα and consider the mapping θ : S → T defined by s 7→ (s† , sηS ) for all s ∈ S. Then it can be easily seen that θ is well defined. Now let (x, t) ∈ T . Then there exists s ∈ S such that sηS = t and xL∗ sL∗ s† . It is also easy to see that (xs)ηS = sηS = t. On the other hand, if u, v ∈ S 1 with xsu = xsv, then we have su = s† xs† ◦ su = s† ◦ (xsu) = s† ◦ (xsv) = sv. This leads to s† u = s† v since sL∗ s† . Thus we have xu = xs† u = xs† v = xv. 5

Hence, we have shown that xL∗ xs. Because x ◦ xs = xs and since S is still a strongly rpp semigroup, we see that (xs)† = x. Thus, θ(xs) = (x, t) and consequently θ is a surjective mapping. To prove that θ is an injective mapping, we let (s† , sηS ) = (t† , tηS ). In this case, we have s† = t† and t = es for some e ∈ E(s† ), and thereby, we deduce that t = t† t = s† es = s† es† ◦ s = s† s = s. This shows that θ is injective. S Finally, we define a multiplication # on T = α∈Y Iα × Mα by (s† , sηS )#(t† , tηS ) = ((st)† , (st)ηS ). Then we can easily check that # is associative and hence (T, #) is a semigroup. This shows that θ is an isomorphism of S onto (T, #). Moreover, by the definition of #, (T, #) is a semilattice of the semigroups Iα × Mα , where α ∈ Y . Thus, by the main result in [12], S is a left C-rpp semigroup. The proof is completed. 2

3

Main theorem

In this section, we establish a new structure theorem for left C-rpp semigroups. Theorem 3.1 A semigroup S is a left C-rpp semigroup if and only if S is a wreath product of a left regular band and a C-rpp semigroup with respect to some wreath structure mapping. Proof: We first proceed to prove the sufficiency part. Let M be a C-rpp semigroup so that M can be written as a semilattice of left cancellative monoids Mα , that is, M = (Y ; Mα ). Also, let L be a left regular band so that L = (Y ; Lα ), where Lα is a left zero band. Then, according to our Definition 2.1, we can define the wreath structure mapping ϕ of M and L by ϕ : M → End(L) such that ϕ(m) = ϕm for each m ∈ M. By using the wreath structure mapping, we form the wreath product S = Lωr ϕ M under the multiplication ◦ defined by (x, u) ◦ (y, v) = (x(ϕu y), uv). We have checked that (S, ◦) is a semigroup. We now want to show that (S, ◦) is a left C-rpp semigroup. We prove this part via the following steps: 6

Step 1. We first prove that E(S) = {(x, 1α ) ∈ S : α ∈ Y } and E(S) ∼ = L. For this purpose, we let (x, m) ∈ E(S). Then we have (x, m) = (x, m)2 = (x(ϕm x), m2 ) and so, we have m = m2 . This leads to m = 1α . Conversely, since L is a left regular band, each Lα is a left zero band. We have (x, 1α )2 = (x(ϕ1α x), 1α ) = (x, 1α ) ∈ E(S). The other parts are clear. Step 2. Let (x, u) ∈ Lα × Mα and (y, v) ∈ Lβ × Mβ . Then, we need to prove that (x, u)L∗ (y, v) if and only if α = β. However, by the Definition of L∗ , we only need to prove that (x, u)L∗ (x, 1α ), where 1α is the identity of Mα . In fact, by the wreath mapping condition (WR1), we have (x, u)(x, 1α ) = (x, u). On the other hand, for (y, v), (z, w) ∈ S 1 satisfying (x, u)(y, v) = (x, u)(z, w), we have x(ϕu y) = x(ϕu z) and uv = uw. Thus, by the condition (WR3) and uL∗ 1α , we have x(ϕ1α y) = x(ϕ1α z) and 1α v = 1α w. Consequently, we deduce that (x, 1α )(y, v) = (x(ϕ1α y), 1α v) = (x(ϕ1α z), 1α w) = (x, 1α )(z, w). This shows that (x, u)L∗ (x, 1α ), as required. Step 3. We now prove that S is a left C-rpp semigroup. Because S = Lωrϕ M is a wreath product of L and M with respect to ϕ, by the condition (WR3) and the wreath multiplication “◦” on S, we can see immediately that S is a semilattice of the semigroups Lα ×Mα for α ∈ Y . In view of the result in [12], we need only prove that S is a strongly rpp semigroup. In fact, in Step 2, we have already proved that S is an rpp semigroup and (x, 1α )L∗ (x, u). T Clearly, we have (x, 1α )(x, u) = (x, u). Now, we let (y, 1α ) ∈ L∗(x,u) E(S) with (y, 1α )(x, u) = (x, u). Then, we have (x, u) = (y(ϕ1α x), u). This leads to x = y(ϕ1α x), and thereby x = y. This shows that (x, 1α ) = (y, 1α ). In other words, the idempotent e in the L∗ − class L∗(x,u) is unique and satisfying the condition e(x, u) = (x, u). This proves that S is indeed a strongly rpp semigroup. Hence the wreath product S = Lωr ϕ M is a left C-rpp semigroup. To prove the necessity part, we first let S be a left C-rpp semigroup. Then, by the definition of left C-rpp semigroup, the set E(S) of all idempotents of S forms a left regular band. Hence, we can write E(S) = (Y ; Eα ), where each Eα is a left zero band. By the Definition of the left C-rpp congruence ηS , we see immediately that ηS ⊆ L∗ and the restriction ηS |E(S) of ηS 7

to E(S) is just LE(S) . Thus, we obtain that E(S/ηS ) = E(S)/L ∼ = Y . Now, by applying the left C-rpp congruence Lemma (Lemma 2.5), we may assume that M = S/ηS = (Y ; Mα ) is a semialattice of left cancellative monoids Mα with an identity 1α inMα for α ∈ Y . Denote by T the set of all representative elements of the ηS -classes of the left C-rpp semigroup S. Define an operation “ ? ” on T by x ? y = the representative element of the ηS − class containing xy, f or all x, y ∈ S. Then, it is routine to verify that “?” is well defined and (T, ?) is a semigroup. Also, the mapping θ : T → M ; x 7→ xηS is an isomorphism of T onto M . Hence we can identify T with M . Now, for u ∈ S, we define φu : E(S) → E(S) by φu (x) = (ux)† . Then it is obvious that φu maps E(S) into E(S). We now proceed to finish the proof of the necessity part of our Theorem 3.1 via the following steps: Step A. φu defined above is an endomorphism of E(S). Proof: Let x ∈ Eα , y ∈ Eβ and u ∈ Mγ . Let S be a left C-rpp semigroup.Then, we observe that L∗ is a congruence on S. This implies that (ux)† ∈ Eαγ , (uy)† ∈ Eβγ . Moreover, we also have (ux)† (uy)† , (uxy)† ∈ Eαβγ . Since every Eα is a left zero band, it follows that (ux)† (uy)† L(uxy)† and (ux)† (uy)† L∗ uxy. By Theorem 2.4 in [12], L∗ is a semilattice congruence on S. Hence, we have (ux)† (uy)† uL∗ uxy, and thereby g = ((ux)† (uy)† u)† ∈ Eαβγ . Since Eαβγ is a left zero band, we have g ◦ xy = g ◦ gxy = g and thus, by Lemma 2.4, we deduce that (ux)† (uy)† (uxy) = = = = = =

((ux)† (uy)† u) ◦ g ◦ xy (ux)† (uy)† u ◦ g (ux)† (uy)† u (ux)† uy uxy (uxy)† (uxy).

Now, by the uniqueness of the idempotent a† , we have (ux)† (uy)† = (uxy)† . This proves that φu (xy) = φu (x)φu (y). Hence φu is indeed an endomorphism of E(S). 2 Step B. Define φ : M → End(E(S) by m 7→ φm . We need to verify that φ is a wreath structure mapping, that is, we need to verify that φ satisfies conditions (WR1-3). 8

Since S is a left C-rpp semigroup, by Definition, L∗ is a congruence on S and so condition (WR1) holds. To prove that condition (WR2) holds, we let e ∈ E(S) and u, v ∈ M . Since (uv)ηS (u ? v), there exists gL∗ u ? v such that uv = g(u ? v). Also, by the property of left C-rpp semigroup (see Lemma 2.4), we have (u(vx)† )† uvx u(vx)† ◦ vx uvx g(u ? v)x λg φu?v (x) ◦ uvx.

φu φv (x)uvx = = = = =

Since L∗ is a congruence on S, we have φu φv (x)Lλg φu?v (x)L∗ uvx. Now,by the uniqueness of the idempotent a† , it follows that φu φv (x) = λg φu?v (x). This verifies that φ satisfies condition (WR2). To verify that φ satisfies condition (WR3), we let (x, u) ∈ Eα × Mα , y ∈ Eβ and z ∈ Eγ . Suppose that x(φu y) = x(φu z). Then, by Lemma 2.4, we have xuy = x(φu y)u = x(φu z)u = xuz. Since x and u† are in Eα , we have uy = u† xuy = u† xuz = uz. However, because M is, in particular, a C-rpp semigroup, we have 1α L∗ u and hence we deduce from uy = uz that 1α y = 1α z. Obviously, x(1α y) = x(1α z) and consequently x(φ1α y) = x(φ1α z). This verifies that φ satisfies condition (WR3). Hence φ is indeed a wreath structure mapping of E(S) and M . Step C. Our last step is to prove that S ∼ = E(S)ωr φ M . We first define ψ : S → E(S)ωr φ M ; T

s 7→ (s† , u),

where u ∈ M aηS . Clearly, ψ is injective. To see that ψ is surjective, we let (x, u) ∈ E(S)ωrφ M . Then, we have xL∗ u and by using the proof of the converse part of Lemma 2.5, we have (xu)† = x. This leads to ψ(xu) = (x, u). Hence, ψ is surjective. To show that ψ is a semigroup momorphism, we let s, t ∈ S. Then, by using similar arguments as in the proof of Lemma 2.5, we can find u ∈ T T Mα sηS and v ∈ Mβ tηS such that s = s† u and t = t† v. Because S is a

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left C-rpp semigroup, L∗ is a congruence on S. Hence, s† (st† )† L∗ st. On the other hand, because E(S) is a left regular band, we have s† (ut† )st = s† (ut† )† s† ◦ ut† ◦ t = s† ◦ (ut† )† ut† ◦ t = s† u ◦ t† t = st. Thus, by the uniqueness of a† , we have s† (ut† )† = (st)† , and hence s† (φu t† ) = (st)† . Therefore we deduce that ψ(s)ψ(t) = (s† , u)(t† , v) = (s† (ut† )† , u ? v) = ((st)† , u ? v) = ψ(st). This proves that ψ is indeed a semigroup momorphism. Thus the proof is completed. 2

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