The construction of orthodox super rpp semigroups

552 Science in China Ser. A Mathematics 2004 Vol. 47 No. 4 552565

The construction of orthodox super rpp semi-groups

HE Yong1,2,4, GUO Yuqi 2 & Kar Ping Shum3

1. School of Computer Science, Hunan University of Science & Technology, Xiangtan 411201, China;

2. Department of Mathematics, Southwest Normal University, Chongqing 400715, China;

3. Faculty of Science, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong, China;

4. Department of Mathematics, Soochow University, Suzhou 215000, China

Correspondence should be addressed to He Yong (email: [email protected])

Received December 29, 2002

Abstract We define orthodox super rpp semigroups and study their semilattice decompo-sitions. Standard representation theorem of orthodox super rpp semigroups whose subband ofidempotents is in the varieties of bands described by an identity with at most three variables areobtained.

Keywords: orthodox super rpp semigroup, band-like extension, left cancellative plank,

semilattice decomposition, standard representation.

DOI: 10.1360/02ys0365

Throughout this paper, for any element a in a semigroup S, the set of [left,right] idempotent identities of a in S is denoted by Ia [I la, I

ra ]. For terminology

and notation not given in this paper the reader is referred to the refs. [13].

It was mentioned by Petrich in ref. [1] that there are exactly 18 varietiesof bands which can be described by an identity in at most three variablesand he gave the standard representations for such bands. The authors of refs.[2,47] have characterized the structure of some classes of orthogroups. How-ever, standard representations of bands have not been completely generalizedto orthogroups. We observe that characterization theorems for left seminormaland left semiregular orthogroups as well as their right dual have not yet beenobtained.

The powerful tools in studying the generalized regular semigroups are the-Greens equivalences[3], namely L, R, H, D and J on a semigroup S. Wecall a semigroup S a rpp, lpp and superabundant semigroup, respectively, if everyL, R and H-class in S contains at least one idempotent[3,8]. Some authorshave concentrated on these classes of semigroups. For example, the structures ofsuperabundant semigroups have been described by Fountain, Kong, and Shum(see refs. [3,9,10]). But for orthodox superabundant semigroups (that is, or-thodox superabundant semigroups whose idempotents form a subsemigroup),their structures have not been investigated. In the following lemma, every resultrelated with L also holds for its dual R.

Lemma 1[3,8]. Let S be a semigroup. Then the following statements hold:

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The construction of orthodox super rpp semigroups 553

(1) L = {(a, b) S S|(x, y S1) ax = ay bx = by}, which is a rightcongruence on S;

(2) for any a S and e ES, (a, e) L if and only if e Ira and, for anyx, y S1, ax = ay implies ex = ey;

(3) for any e, f ES, (e, f) L if and only if (e, f) L;(4) each H-class in S contains at most one idempotent; (if it exists, the

unique idempotent in Ha (a S) is denoted by a.);(5) if S is a left cancellative monoid, then it is a unipotent semigroup (that

is, a monoid with a unique idempotent);

(6) S is a C-rpp semigroup (i.e. a rpp semigroup with central idempotents)[8]if and only if it is a semilattice of left cancellative monoids, and if and only if itis a strongly semilattice of left cancellative monoids. (In this paper, by a C-rppsemigroup [Y ;T] we always mean that the semigroups T on the semilatticeY are left cancellative monoids.)

A rpp semigroup S is said to be a strongly rpp semigroup[11] if, for anya S, the set La I la contains a unique element a. In fact, by Lemma 1(2), La I la = La Ia. We naturally call a lpp semigroup S strongly lpp if forany a S, the set Ra Ira contains a unique element a+. Several classes ofspecial strongly rpp semigroups have already been discussed in refs. [8,1113].Clearly, the relation R = {(a, b) S S|I la = I lb} on a semigroup S is anequivalent relation [14]; moreover, if S is a strongly rpp semigroup, then therelation R+ = {(a, b) S S|a R b} dened by Guo, Guo and Shum1),2) onS is also an equivalence.

Definition 1. A strongly rpp semigroup is said to be orthodox if itsidempotents form a subsemigroup. Orthodox super rpp semigroups are ortho-dox strongly rpp semigroups on which R+ is a left congruence. We call anorthodox supper rpp semigroup an A super rpp semigroup if its idempotentsform an A band, where A stands for an adjective such as left semiregular,left seminormal, etc. A semigroup S is said to be L-compatible if L is acongruence on S.

1 The structural characterizations for orthodox super rpp semi-groups

We denote the set of transformations on a non-empty set X by T (X) andthe identical transformation on X by X . For any T (X) and i X, we use< > and < i >, respectively, to denote the constant transformations on Xwith values < > and i. By taking the transformation composition on the leftby ( )(x) = ((x)) for all x X, the set T (X) becomes a semigroup which

1) Guo, X., Guo, Y. Q., Shum, K. P., Super-rpp semigroups, To be submitted.2) Guo, X., Guo, Y. Q., Shum, K. P., Matrix representation of strongly rpp semigroups, To be

submitted.

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is called the left transformation semigroup on X. We denote this semigroupby Tl(X). Similarly, if we take the transformation composition on the rightby x() = (x) for all x X, then the set T (X) also forms a semigroupwhich is called the right transformation semigroup on X, in notation Tr(X).The elements in Tl(X) and Tr(X) are called left and right transformations onX, respectively.

Let T be a semigroup with semilattice decomposition [Y ;T]. For everyelement Y , we associate with two non-empty sets I and such thatI I = = whenever = in Y . Denote S = IT. Then,for any in Y , we dene the mappings by

, : S Tl(I ), a a, , , : S Tr( ), a a,such that the following statements hold for any (i, x, ) S and (j, y, ) S:(C.1) (1) (i,x,), =< i >,

(i,x,), =< >;

(2) ((k, ) I ) (i,x,), (j,y,), =< k >, (i,x,), (j,y,), =< >;(3) ( ) (k,xy,), = (i,x,), (j,y,), , (k,xy,), = (i,x,), (j,y,), .

Then, by condition (C.1) (2), we can see that the following operation

(i, x, )(j, y, ) = (< (i,x,), (j,y,), >,xy,< (i,x,), (j,y,), >)

on the set S =

Y S is well-dened. In this case, condition (C.1) (3) isevidently equivalent to the following condition

(C.1) (3) ( ) (i,x,)(j,y,), = (i,x,), (j,y,), , (i,x,)(j,y,), = (i,x,), (j,y,), .By direct calculations, we can easily show that, with respect to the operationdened above, S forms a semigroup, of which [Y ;S] is a semilattice decomposi-tion. We call S a band-like extension of T and denote it by B[Y ;T; I,; ,, ,].If all I are just singletons, then we usually omit the rst component of the ele-ments in S and call S a right band-like extension of T , in notation B[Y ;T; ; ,].Left band-like extensions B[Y ;T; I; ,] of T can be dually dened.

The direct product of a left cancellative monoid T and a rectangular bandI is called a left cancellative plank [12] and is denoted by I T . Theprojection of a subdirect product A of sets Ai (i I) onto some Ai is denotedby PAAi or simply PAi if no ambiguity arises.

Summing up the above information, we can formulate the following Theo-rem.

Theorem 1. Let S be a semigroup. Then the following statements areequivalent:

(1) S is an orthodox super rpp semigroup;

(2) S can be expressed as a semilattice Y of left cancellative planks S =



I T which satises the condition(C.2) for any in Y, (i, x, ) S and (j, 1T , ), (k, 1T , ) S,

((i, x, )(j, 1T , ))PI = ((i, x, )(k, 1T , ))PI= ((i, 1T , )(j, 1T , ))PI = ((i, 1T , )(k, 1T , ))PI ;

(3) S can be described as a band-like extension B[Y ;T; I,; ,, ,] ofa C-rpp semigroup [Y ;T] which satises the condition

(C.3) for any in Y and (i, x, ) IT, there exists (i,x,), Tl(I)such that (i,1T ,), =

(i,x,), (i,x,), .

Before proving Theorem 1, we need the following Denition 2 and Lemmas25.

Definition 2. Let S be an orthodox super rpp semigroup. Then the semi-lattice Y of left cancellative planks S and the band-like extension B[Y ;T; I,; ,, ,] of the C-rpp semigroup [Y ;T] as described in Theorem 1 arecalled the structural semilattice decomposition and the standard representationof S, respectively. The C-rpp semigroup [Y ;T] is called the C-rpp branch of S.In what follows, we always denote the structural semilattice decomposition ofS by [Y ;S] and the standard representation of S by B[Y ;T; I,; ,, ,].

Lemma 2. If a semigroup S is a semilattice Y of left cancellative planksS = I T , then ES is a subsemigroup of S.

Proof. Let (i, 1T , ) ES and (j, 1T , ) ES be two arbitrary ele-ments in ES such that (i, 1T , )(j, 1T , ) = (l, x, ). We rst consider the case . In this case, we observe that

(l, x, ) = (i, 1T , )(j, 1T , ) = (i, 1T , )(i, 1T , )(j, 1T , ) = (i, x, ).We now claim that l = i. In fact, by setting (i, 1T , )(j, 1T , ) = (i, y, ), weget

(i, x, ) = (i, x, )(j, 1T , ) = (i, x, )(i, 1T , )(j, 1T , ) = (i, xy, )so that = and yx = x. Since T is a left cancellative monoid, y = 1T and wecan immediately deduce that (i, 1T , ) l (j, 1T , ). Thus (j, 1T , )(i, 1T , ) ES which is L-equivalent to (i, 1T , ). As a consequence, we further deducethat

(l, x, )2 = (i, 1T , )(j, 1T , )(i, 1T , )(i, x, )= (i, 1T , )(i, x, ) = (i, x, ) = (l, x, ).

For the general case, since , in Y , in virtue of the above result, we cansimilarly derive that

(l, x, )2 = (l, x, )(l, 1T , )(i, 1T , )(j, 1T , )(l, 1T , )= (l, x, )(l, 1T , ) = (l, x, ).

Thus ES is indeed a subsemigroup of S.

Lemma 3. Let S be a semilattice Y of left cancellative planks S =I T . Then the set T =

Y T forms a C-rpp semigroup, which is a

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homomorphic image of S, with respect to the following operation dened byxy = z ((i, x, ), (j, y, ) S)((i, x, )(j, y, ))PT = z.(x T, y T)

Proof. By Lemma 2 we have ES = [Y ;ES = I {1T} ]. Let(i, x, ) S and (j, y, ) S be two arbitrary elements in S. For any(k, ), (k

,

) I, put (i, x, )(k, 1T , ) = (l, u, ) and (l, 1T , )(i, x, ) =

(l, u

,

). Then, we can deduce that

u= ((l, 1T , )(i, x, )(k, 1T , ))PT= ((l, 1T , )(i, x, )(l

, 1T ,

)(k

, 1T ,

)(k, 1T , ))PT

= ((i, x, )(k, 1T ,

))PT .

This shows that u = ((i, x, )(k, 1T , ))PT is independent of (k, ). Du-ally, we can also deduce that ((k, 1T , )(j, y, ))PT is also independent of(k, ). Now we choose (i

,

) I , (j , ) I and suppose that

(i, x, )(j, y, ) = (k, z, ). Then, by calculations, we can derive the followingequalities

z= ((i, 1T ,

)(k, 1T , )(i, x, )(j, y, )(k, 1T , )(j

, 1T ,

))PT

= ((i, 1T ,

)(i, x, )(j, y, )(j

, 1T ,

))PT

= ((i, x,

)(i, 1T , )(j, 1T , )(j

, y,

))PT

= ((i, x,

)(i, 1T , )(j, 1T , ))PT ((i, 1T , )(j, 1T , ))(j

, y,

))PT

= ((i, x,

)(i

, 1T ,

)(j

, 1T ,

))PT ((i

, 1T ,

)(j

, 1T ,

)(j

, y,

))PT

= ((i, x,

)(i

, 1T ,

)(j

, 1T ,

)(j

, y,

))PT

= ((i, x,

)(j

, y,

))PT .

Thereby ((i, x, )(j, y, ))PT is independent of both (i, ) and (j, ) so that(T, ) becomes a groupoid. Moreover, it is clearly that PST is a surjective ho-momorphism of S onto (T, ) and hence (T, ) is a semigroup. Thus, it followsfrom Lemma 1 (6) that T = [Y ;T] is a C-rpp semigroup.

Lemma 4. The following statements are equivalent for an orthodoxstrongly rpp semigroup S:

(1) S is an orthodox super rpp semigroup;

(2) for any a S, I la I la ;(3) R+ = R;(4) the relation = {(a, b) S S | (a, b) DES} on S is a semilattice

congruence;

(5) S is a semilattice of left cancellative planks.

Proof. (1)(2). Let S be an orthodox super rpp semigroup. Then, forany a S and e I la, we have ea = (ea) R (ea) = a. Consequently, wehave ea = eaa = a.

(2)(3). Assume that (2) holds. Let a, b S. Then, by I la I la, we have(a, a) R. Also, by R|ES = RES , we can see that (a, b) R if and only ifCopyright by Science in China Press 2004


(a, b) R. Similarly, we can see that (a, b) R if and only if (a, b) R+.This shows that R+ = R.

(3)(4). Assume that (3) holds. Then, for any a S, by (a, a) R+,we get (a, a) R whence (2) holds. Let [Y ;E] be the greatest semilatticedecomposition of ES. Then, for any Y , put S = {x S | x E}.Clearly, is an equivalence on S containing L such that S/ = {S | Y }.Now, we claim that S is a subsemigroup. This is because for any a, b Sand x, y S1

abx = aby= abx = aby= bx = bbx = babx = baby = bby = by= abx = aby.

Hence ab S. In other words, S is a subsemigroup of S and our claim holds.Suppose that a S, b S such that ab S and ba S . Then

because we have a I lab I l(ab). Also, by (ab) L ab = abb L (ab)b, wehave (ab) L (ab)b. Thereby and bab = b(ab)(ab) S. Dually,we can show that also. By ba L ba, we obtain that ba S and(ba) E so that b(ba) E . In virtue of b(ba) L b(ba), we have(bab)(ba) = (ba)(b(ba)) S whence . Dually, we also have .Thus ab ba. As a consequence, we can easily show that is a semilatticecongruence.

(4)(5). Assume that is a semilattice congruence on S and the semilatticedecomposition of S induced by is S = [Y ;S]. Then ES = [Y ;ES ]. Forany Y , choose e ES and put Se = {a S | a = e}. Then, forany a, b Se, by ab L eb = b and e I lab we have (ab) = e and whenceSe is a subsemigroup of S with e acting as its the identity. Let c Se besuch that ca = cb. Then it follows by Lemma 1 (1) that a = ea = eb = b.Thus Se is a left cancellative monoid. For any d S, it is obvious thate ede = ede = ede e L (ede)e L e. Now, by Lemma 1 (3) we also have(ede) = e so that ede Se. Since d(ede)d = deddded = ddd = d,we can show by routine checks that that the mapping : d (ede, d) is anisomorphism of S onto Se ES .

(5)(1). Let S be a semilattice Y of left cancellative planks S = IT, a = (i, x, ) S, b = (j, y, ) S and c S. If a S , then aa = a sothat ; and by a(i, 1T , ) = a we have a(i, 1T , ) = a and hence also. This shows that = . Moreover, since a Ia but (i, 1T , ) is theunique element in Ia S, we claim that a = (i, 1T , ). By direct calculationwe can easily show that (ca)PI = (caa)PI = (ca(ca)a)PI = (ca)PI .Therefore (a, b) R+ if and only if = and i = j. Thus, for this case, wecan show that (a, b) R and ca, cb S. Since R is a left congruence on S,we have (ca, cb) R also and whence (ca)PI = (cb)PI . Consequently,we have (ca)PI = (ca)PI = (cb)PI = (cb)PI so that (ca, cb) R+.The proof is completed.

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Lemma 5. A semigroup S is a semilattice of left cancellative planks ifand only if it is a band-like extension of a C-rpp semigroups.

Proof. We only need to consider the necessity part. Suppose that S is asemilattice Y of left cancellative planks S = IT. Then, by Lemma 2and Lemma 3, ES is a subsemigroup of S and T = [Y ;T] is a C-rpp semigroup.Thus, for any in Y , we have (i, x, ) S, j I and , , since

((i, x, )(j, 1T , ))PI = ((i, x, )(j, 1T , )(j, 1T , ))PI= ((i, x, )(j, 1T , ))PI .

We now dene a left transformation on the set I as follows:

(i,x,), : I I, j ((i, x, )(j, 1T , ))PI ( ).

The right transformation (i,x,), on can always be dually dened. It isnow routine to check that these transformations satisfy condition (C.1) and theoperation on B[Y ;T; I, ; ,, ,] coincides with the operation dened onS. This prove the lemma.

We are now ready to prove Theorem 1.

Proof. (1)(2). Let S be an orthodox super rpp semigroup. Then, byLemma 4, S is a semilattice Y of some left cancellative planks S = I T . For any in Y , (i, x, ) S and (j, 1T , ), (k, 1T , ) S, if((i, x, )(j, 1T , ))PI = ((i, x, )(k, 1T , ))PI , then by the following equation

((i, x, )(j, 1T , ))PI = ((i, x, )(j, 1T , )(j, 1T , ))PI= ((i, x, )(j, 1T , ))PI

and Lemma 3 we can deduce that(i, x, )(j, 1T , )= (((i, x, )(j, 1T , ))PI , x 1T , )

= (((i, x, )(k, 1T , ))PI , x 1T , )= (i, x, )(k, 1T , ).

Since (i, x, ) L (i, 1T , ), we have (i, 1T , )(j, 1T , ) = (i, 1T , )(k, 1T , )so that((i, 1T , )(j, 1T , ))PI = ((i, 1T , )(j, 1T , ))PI = ((i, 1T , )(k, 1T , ))PI .

(2)(3). Assume that (2) holds and let S = [Y ;S]. Then, by Lemmas25, we see that ES is a subsemigroup of S and S is a band-like extensionB[Y ;T; I,; ,, ,] of C-rpp semigroup T = [Y ;T]. For any in Yand (i, x, ) I T , we dene (i,x,), Tl(I) by

(j I) (i,x,), (j) ={

(i,1T ,), (j) if j ran(i,x,), ,

j otherwise.

Then it is routine to check that (i,1T ,), = (i,x,), (i,x,), .

(3)(1). Assume that (3) holds and let S = B[Y ;T; I,; ,, ,].Then S can be expressed as a semilattice Y of left cancellative planks S =I T . Now, it is obvious that (i, 1T , ) I(i,x,) for any (i, a, ) S.Suppose that a, b S1 such that (i, x, )a = (i, x, )b. If a = (j, y, ) S andCopyright by Science in China Press 2004


b = (k, z, ) S , then = of course. Applying conditions (C.1) (2) and(3), we deduce that

(i,x,), (j,y,), = (i,x,)(i,1T ,), (j,y,), = (i,x,), (i,1T ,), (j,y,), = (i,x,), (i,1T ,), (j,y,), = (i,1T ,), (j,y,), .

Thus, we further deduce that

((i,x,), (j,y,), , (1Tx)(1Ty), (i,x,), (j,y,), )= (i, x, )(i, 1T , )(j, y, )= (i, x, )(j, y, )= (i, x, )(k, z, )= (i, x, )(i, 1T , )(k, z, )= ((i,x,), (k,z,), , (1T x)(1Tz), (i,x,), (k,z,), )

and whence(i, 1T , )(j, y, ) = ((i,1T ,), (j,y,), , 1Ty, (i,1T ,), (j,y,), )

= ((i,x,), (i,x,), (j,y,), , 1Ty, (i,x,), (j,y,), )= ((i,x,), (i,x,), (k,z,), , 1Tz, (i,x,), (k,z,), )= ((i,1T ,), (k,z,), , 1Tz, (i,1T ,), (k,z,), )= (i, 1T , )(k, z, ).

If at least one of a and b is 1S , then it is trivial to see that (i, x, )a = (i, x, )bimplies that (i, 1T , )a = (i, 1T , )b also. Therefore, by Lemma 1 (2), we have(i, x, ) L (i, 1T , ) and hence (i, 1T , ) I(i,x,) L(i,x,). If (l, 1T , ) I(i,x,) L(i,x,) also, then by Lemma 1 (3) we have (l, 1T , ) R (i, 1T , ) sothat = . Since (i, 1T , ) is the unique element in I(i,x,) S, we can easilysee that (i, 1T , ) = (l, 1T , ) and so (i, 1T , ) = (i, x, ). Thus S is indeeda strongly rpp semigroup. Furthermore, by Lemma 2 and Lemma 4, S is anorthodox super rpp semigroup. The proof is now completed.

2 Some special cases

We now characterize some special orthodox supper rpp semigroups.

Theorem 2. Let S be a semigroup. Then the following statements hold:

(1) S is a rectangular super rpp semigroup if and only if it is a left can-cellative plank; in particular, S is a left [right] zero super rpp semigroup if andonly if it is the direct product of a left [right] zero band and a left cancellativemonoid;

(2) S is an orthodox super rpp semigroup in which ES is a semilattice ifand only if it is a C-rpp semigroup;

(3) S is a right regular super rpp semigroup if and only if it is a rightband-like extension of a C-rpp semigroup;

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(4) S is a left regular super rpp semigroup if and only if it is a left band-likeextension of a C-rpp semigroup which satises condition (C.3);

(5) S is a regular super rpp semigroup if and only if there exist a left regularsuper rpp semigroup S1 and a right regular super rpp semigroup S2, which havethe same C-rpp branch T , such that S = S1 T S2;

(6) S is a L-compatible right regular super rpp semigroup if and only ifthere are a C-rpp semigroup T = [Y ;T] and a right regular band B = [Y ;B]such that S = T Y B;

(7) S is a left [right] quasi-normal super rpp semigroup if and only if thereare a left [right] regular super rpp semigroup T = [Y ;T] and a right [left]normal band B = [Y ; ] such that S = T Y B;

(8) S is a [left, right] normal super rpp semigroup if and only if there are aC-rpp semigroup T = [Y ;T] and a [left, right] normal band B = [Y ;B] suchthat S = T Y B;

(9) S is a right semiregular [right seminormal] super rpp semigroup if andonly if it is a right band-like extension of a left [right] regular super rpp semi-group;

(10) S is a left semiregular [L-compatible orthodox, left semi-normal] superrpp semigroup if and only if it is a left band-like extension B[Y ;T; I; ,] ofa right regular [L-compatible right regular, right normal] super rpp semigroupT = [Y ;T], which satises the condition

(C.4) for any in Y and (i, x) I T, (i,x)

, = (i,x), (i,x), for some

(i,x), Tl(I).

In order to establish Theorem 2, we need the following Lemmas 68.

Lemma 6. The L-compatible orthodox strongly rpp semigroups are allorthodox super rpp semigroups.

Proof. Let S be a L-compatible orthodox strongly rpp semigroup. Forany a S and e I la, we have ea I la and ea L ea = a so that ea = a.Thus, by Lemma 4, S is orthodox super rpp.

The possible right dual of the results in the following lemma hold. We omitthe details.

Lemma 7. Let S = B[Y ;T; I,; ,, ,] be a band-like extension ofa C-rpp semigroup [Y ;T]. Then the following statements hold:

(1) ES is a right semiregular band if and only if (i,x,), =

(i,x), (that is, for

any in Y and (i, x, ) S, (i,x,), is independent of );(2) ES is a right seminormal band if and only if

(i,x,), = i,;

(3) ES is a regular band if and only if (i,x,), =

(i,x), and

(i,x,), =

(x,), ;

(4) ES is a right quasinormal band if and only if (i,x,), = i, and (i,x,), =



(x,), ;

(5) ES is a normal band if and only if (i,x,), = i, and (i,x,), = ,;

(6) ES is a left regular band if and only if each is a singleton;

(7) ES is a left normal band if and only if each is a singleton and(i,x,), = i,;

(8) ES is a semilattice if and only if every and I are singletons.

Proof. By Lemma 2, ES is a subsemigroup of S. Moreover, if we identifythe element (i, 1T , ) of ES with (i, ) and set ,|I = ,, ,|I =,, then (Y ; I, ,; , ,) is a standard representation of ES. In virtue ofthe relations between the varieties of bands and by using the notion of standardrepresentations of the bands involved in this result, we only need to check thenecessities of (1) and (2).

(1) Let ES be a right semiregular band. Then, for any in Y and(i, x, ), (i, x, ) IT, by the standard representations of right semireg-ular bands we have (i,), =

(i,), and thereby we have

(i,x,), =

(i,x,), (i,), =

(i,x,), (i,), = (i,x,), .

(2) Similar to (1), if ES is a right seminormal bands, then(i,), = (i,), = (i,), = (i,),

and hence(i,x,), =

(i,), (i,x,), = (i,), = (i,), = (i,), (i,y,), = (i,y,), .

Lemma 8. Let S = B[Y ;T; I,; ,, ,] be a band-like extension ofa C-rpp semigroup [Y ;T]. Then the following statements hold:

(1) if S is right semiregular super rpp, then teh condition (C.3) holds forsome (i,x,), with

(i,x,), =

(i,x), ;

(2) if ES is a right seminormal band, then S naturally satises condition(C.3);

(3) if S is an orthodox super rpp semigroup, then S is L-compatible if andonly if (i,x,), = ,;

(4) if S is a left seminormal super rpp semigroup, then S is L-compatible.Proof. (1) This part follows from Lemma 7 (1) and the denition of

(i,x,), given in the proof of Theorem 1.

(2) If ES is a right seminormal band, then, by Lemma 7 (2), we see that(i,x,), =

(i,1T ,), for any in Y and (i, x, ) S. Hence (i,1T ,), =

I (i,x,), .(3) Let S be an orthodox super rpp semigroup. Then, it is routine to show

that (i, x, ) L (j, y, ) in S if and only if = . Thus the suciency partholds.

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Conversely, if S is L-compatible, then (i,x,), = (j,y,), for any in Yand (i, x, ), (j, y, ) I T , since for any (k, ) I , we have(k, 1Tx,

(i,x,), ) = (k, 1T , )(i, x, ) L (k, 1T , )(j, y, ) = (k, 1Tx, (j,y,), ).

(4) This part follows by Lemma 7 (2) and the above statement (3). Weomit the details.

We now prove Theorem 2.

Proof. In fact, Theorem 2 (1)(4) follows from Theorem 1, Lemma 7and Lemma 8 directly. We now only need to prove (5) and (6). The other partscan be shown similarly.

(5) Let S = B[Y ;T; I,; ,, ,] be a regular super rpp semigroup.Then, for any in Y and (i, x) IT, it follows, by Lemma 7 (3), that(i,x), =

(i,x,), for any is a well-dened left translation on I. Moreover,

by (4) and Lemma 8 (1), S1 = B[Y ;T; I; ,] is a left regular super rppsemigroup, where , is a mapping which maps I T into Tl(I) dened by(i, x) (i,x), . Dually, we can also construct a right regular super rpp semigroupS2 = B[Y ;T; ; ,]. Thus, by using Lemma 3, one may easily show thatS = S1 [Y ;T] S2.

Conversely, assume that S1 = B[Y ;T; I; ,] and S2 = B[Y ;T; ; ,]are a left and a right regular super rpp semigroup, respectively. Then, for any in Y and (i, x, ) I T , we can dene (i,x,), = (i,x), and

(i,x,), =

(x,), . It follows by the above statement (4), Theorem 1 and Lemma 7

(3), we can see that S1 [Y ;T] S2 = B[Y ;T; I,; ,, ,] is a regular superrpp semigroup.

(6) The necessity part follows from (3) and Lemma 8 (3). Let S = B[Y ;T; ; ,]be a L-compatible right regular super rpp semigroup. Now, for any in Yand , dene , = (1T ,), so that (Y ; ; ,) is a right regular band.Then, for any (x, ) T and (y, ) T , by Lemma 8 (3) we alsohave

(x, )(y, ) = (xy,< (x,),(y,), >) = (xy,<

(1T ,),

(1T ,)

, >) = (xy, ).

Hence, we have S = [Y ;T]Y [Y ; ]. The proof completed.

3 Some remarks

Remark 1. Let A be an innite cyclic semigroup with generator a andB an innite cyclic monoid with generator b and identity e = b0. Let S =AB {1} and dene an operation on S which extends those operations on Aand B and has 1 as its identity by putting ambn = bm+n and bnam = an+m forintegers m > 0 and n 0. Then S/L = {A{1}, B} and S/ R = {AB, {1}}.Clearly, S is an orthodox strongly rpp semigroup having R as a left congruence.Since (a, 1) R+ but (a, e) = (ea, e1) R+, S is not an orthodox super rppsemigroup. This illustrates that an orthodox strongly rpp semigroup S is not



necessarily an orthodox super rpp semigroup even if the equivalence R on S isa left congruence.

Remark 2. By using Theorem 1 and Lemma 7, we may obtain the char-acterization theorems of structural semilattice decompositions and standardrepresentations of various special orthodox super rpp semigroups consideredabove. For example, we can obtain that a semigroup S is a normal super rppsemigroup if and only if S is a strong semilattice of left cancellative planks, andif and only if S is a band-like extension B[Y ;T; I,; ,, ,] of a C-rppsemigroup [Y ;T] such that

(i,x,), = i, and (i,x,), = ,. Recall that a

L-compatible strongly rpp semigroup is a perfect strongly rpp semigroup[12] ifits idempotents form a normal band. Then, by Lemma 6 and the above result,we can assert that the perfect rpp semigroups are exactly the normal super rppsemigroups.

Remark 3. By using our Lemma 6 and Lemma 8, we can give a criterionfor a strongly rpp semigroup to be a left regular supper rpp semigroup. Let Sbe a strongly rpp semigroup. Then the following statements are equivalent:

(1) S is a left regular super rpp semigroup;

(2) S is L-compatible and ES is a left regular band;(3) eS Se for any e ES.Proof. It follows by Lemma 8 (3) and Lemma 6 that (1) and (2) are

equivalent. Assume that (1) holds and let S = [Y ;S]. Then, for any a Sand e ES , by (ea) ES we get ea = ea(ea)e = eae whence (3) holds.Conversely, if (3) holds, then it can be easily shown that ES is a left regularband. Moreover, for any a S and e I la, there is u S such that a = ea =ue = uee = eae = ae. Thus, by Lemma 1 (1), ae = a and so a L ea. Thisimplies that ea L a. Since ea I la, we also have ea = a, whence (1) holdsas well.

Remark 4. A L-compatible strongly rpp semigroup S is called a leftC-rpp semigroup[11] if eS Se for all e ES. It follows by Remark 3 that theaxioms in the denition of left C-rpp semigroup are not mutually independentand the left C-rpp semigroups are exactly the left regular super rpp semigroups.By Theorem 2.4 in ref. [11], the authors have considered the semispined productof a C-rpp semigroup and a left regular band. In fact, it turns out to be a left C-rpp semigroup. Since the concept of semispined product of a C-rpp semigroupand a left regular band is essential the same as that of left band-like extensionof a C-rpp semigroup, by the following example, we can see that Theorem2.4 in ref. [11] has a gap in the proof and an additional condition (C.3) [oralternatively, (C.2)] is needed. Let T = [Y ;N ], where N is the usual additionalsemigroup of nonnegative integers. Let Y = {1, 2 | 1 > 2} be a semilattice.Set I1 = {e} and I2 = {f, g, h}. For any (i, a) S = I N ( Y ), dene

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(i,a), =< i > and put

(e,0)1,2 =

(

f g h

h g h

)

, (e,a)1,2 =

(

f g h

h h h

)

(a N\{0}).Then, by direct calculation, we can see that these transformations satisfy con-dition (C.1) and so S = B[Y ;N ; I; ,] is a left band-like extension of a C-rppsemigroup. Since (e, 1)(f, 0) = (h, 1) = (e, 1)(g, 0) but (e, 0)(f, 0) = (h, 0) =(g, 0) = (e, 0)(g, 0), we have (e, 1) L(e,1). Clearly, (f, 0), (g, 0), (h, 0) L(e,1).This shows that the semigroup S is not rpp.

Remark 5. Orthodox superabundant semigroups have been recentlystudied by Ren and Shum [10]. By using Lemma 1 (2) and Lemma 1 (4), we canalso give a description for orthodox superabundant semigroups. A semigroupS is superabundant if and only if S is both strongly rpp and strongly lpp. Inthis case, a = a = a+ for any a S.

Proof. Let S be a semigroup and a S. If S is superabundant, then, byLemma 1 (4), a I la La. Since a R a, by Lemma 1 (2) we conclude thatI la I la whence a = a. Dually, we have a = a+ also. Conversely, assumethat S is both strongly rpp and strongly lpp. Since a L a R a+, it followsby Lemma 1 (2) and its dual that a = aa+ = a+. Thus a = a+ = a.

Remark 6. Monoids, which are left cancellative but not right cancella-tive, are orthodox super rpp but not orthodox superabundant. If S is an or-thodox superabundant semigroup, then a = a+ for any a S. It followsby Lemma 1 (2) and its dual that I la I la . Therefore orthodox superabun-dant semigroups are orthodox super rpp. Adding the symmetric requirementsof (C.2) and (C.3) in Theorem 1 (2) and (3), respectively, we can obtain thestructural semilattice decomposition and standard representation of orthodoxsuperabundant semigroups. Similar to Theorem 2, we can also obtain the char-acterizations for various orthodox superabundant semigroups.

Remark 7. Clearly, orthogroups are orthodox super rpp semigroups anda band-like extensions. The semigroup S = B[Y ;G; I,; ,, ,] of a C-rpp semigroup T = [Y ;G] is regular if and only if T is a Cliord semigroup.In this case, we have (i,1G ,), =

(i,x1,), (i,x,), for any in Y and

(i, x, ) I G . Thus a semigroup S is an orthogroup if and only if Sis a band-like extension of a Cliord semigroup. By using Theorem 1, Lemma7 and Theorem 2, we can obtain the structural semilattice decomposition andstandard representation of various classes of special orthogroups. The followingresults are new.

Theorem 3. (1) A semigroup S is a left [right] quasinormal orthogroupif and only if S is a spined product of a left [right] orthogroup [Y ;T] and aright [left] normal band [Y ; ] with respect to the semilattice Y ;

(2) A semigroup S is a left semiregular [L-compatible, left seminormal]orthogroup if and only if S is a left band-like extension of a right regular [L-compatible right regular, right normal] orthogroup.



Acknowledgements This work was supported by the National Natural Science Foundation ofChina (Grant No. 10071068), a Youth Scientific Foundation grant of Hunan Education Department(Grant No. 02B024) and a UGC (HK) (Grant No. 2060187 (02/04)).

References

1. Petrich, M., Lectures in Semigroups, New York: Pitman, 1976.2. Petrich, M., Reilly, N., Completely Regular Semigroups, New York: John Wiley & Sons, 1998.3. Fountain, J. B., Abundant semigroups, Proc. London Math. Soc. 1982, 44: 103129.4. Guo, Y. Q., Ren, X. M., Shum, K. P., Another structure of left C-semigroups, Adv. Math., 1995,

24: 3943.5. Guo, Y. Q., The structure of weakly left C-semigroups, Chinese Science Bulletin, 1995, 40:

17441747.6. Guo, Y. Q., Shum, K. P., Zhu, P. Y., On quasi-C-semigroups and some special subclasses, Algebra

Colloquium, 1999, 6: 105120.7. Zhu, P. Y., Guo, Y. Q., Shum, K. P., The characterization and structure of left C-semigroups,

Sci. China, Ser. A, 1991, (6): 582590.8. Fountain, J. B., Right pp monoids with central idempotents, Semigroup Forum, 1977, 13: 229

237.9. Kong, X. Z., Shum, K. P., On the structure of regular crypto semigroups, Comm. Algebre, 2001,

29(6): 24612479.10. Ren, X. M., Shum, K. P., On superabundant semigroups, Sci. China, Ser. A. To appear.11. Guo, Y. Q., Shum, K. P., Zhu, P. Y., The structure of left C-rpp semigroups, Semigroup Forum,

1995, 50: 923.12. Guo, X. J., Shum, K. P., Guo, Y. Q., Perfect rpp semigroups, Comm. Algebra, 2001, 29 (6):

24472459.13. Guo, X. J., Guo, Y. Q., The translation hull of strongly right type-A semigroups, Sci. China,

Ser. A, 1999, 29: 10021008.14. Lawson, M., Semigroups and ordered categories, I. The reduced case, J. Algebra, 1991, 141 (2):

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The construction of orthodox super rpp semigroups