On constructions of semigroups Текст научной статьи по специальности «Математика»

Tom 154, kh. 2

УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА

Физико-математические пауки

2012

UDK 512.53

ON CONSTRUCTIONS OF SEMIGROUPS K.P. Shum, X.M. Ren, C.M. Gong

Abstract

This paper gives a brief survey of constructions of semigroups by using structures of some semigroups belonging to the class of regular semigroups, quasi-regular semigroups, and abundant semigroups. In particular, we show some basic notation and structure theorems for some semigroups, for example, Rees matrix semigroups over the 0-group G0 and their generalizations, bands, £-ideal quasi-regular semigroups, c*-quasiregular semigroups, l* -inverse semigroups, q* -inverse semigroups, and regular ortho-lc-monoids.

Key words: regular semigroups, quasi-regular semigroups, abundant semigroups, constructions.

1. Rees matrix semigroups and their generalizations

For notation and terminology not given in this paper, the reader is referred to fl 13]. A semigroup is called completely 0-simple if it is 0-simple and has a primitive idempo-tent. In 1940. Rees provided the following recipe for manufacturing completely 0-simple semigroups.

Let G be a group with identity element e, and let I, A be non-empty sets. Let P = (px,i) be a A x I matrix with entries in the 0-group G0(= G U {0}), and suppose that P is regular, in the sense that no row or no column of P consists entirely of zeros. Formally.

(Vi e I)(3A e A) pXi =0,

(VA e I)(3i e I) PAi = 0. (l)

Let S = (I x G x A) U {0}, and define a multiplication on S by

(i, a, A) • (b,j,M) = (0i'aPAjPA =0'

[0 if pAj = 0,

(i, a, A) • 0 = 0 • (i, a, A) = 0 • 0 = 0. (2)

The semigroup constructed with this recipe will be denoted by M0[G; I, A; P] and will I x A G0

P

G0 I, A

empty sets, and, let P = (pAi^ n A x I matrix with entries in G0. Suppose that P is regular in the sense of (1). Let S = (I x G x A) U {0}, and, define a multiplication on S by (2). Then S is a completely 0-simple semigroup.

Conversely, every completely 0-simple semigroup is isomorphic to one constructed, in this way.

In 1990. M.V. Lawson in [15] gave another abstract characterization of Rees matrix semigroups as follows.

Let S be a monoid with identity 1 and zero element 0, having group of units G(S). Let A Mid J be non-empty sets, and let P be a A x J matrix over S with entries pAi where (A, i) G A x J. The matrix semigroup M = M0(S; J, A; P) is the set of triples J x S x A with a zero 0 adjoined and where we identify all the elements of the form (i, 0, A) with 0, under a multiplication given by

(i,x,A) • (j,y,M)40i,XPAj^ ^r =0' 0

Theorem 2 [15]. Let S be a Rees semigroup with e G U \ {0}. Then

(i) S is abundant if and, only if eSe is abundant;

(ii) S eSe

(iii) S is inverse if and only if S is reduced, eSe is inverse and Regu(S) is a subsemigroup (for details, see [15]).

To further generalize the Rees matrix semigroup constructed above, we have recently established in [16] the following construction of semigroups by using semigronpoids.

A semigroupoid is a pair (S, S0) consisting of a set S of morphisms and a set S0 of objects, together with the functions t : S ^ S0 and w : S ^ S0, and a function ^ which is so called "multiplication" from the set S * S = {(x, y) G S x S | t(x) = w(y)} to S; we usually write xy instead of ^(x, y^d if (x, y) G S * S, then we write 3xy; in addition, the following two axioms hold:

(CI) If 3xy, then t(xy) = t(y^d w(xy) = w(x); (C2) x(yz) = (xy)z whenever the products are defined.

Let A, B G S0. Then, in this rase, the set Mor (A, B) = {x G S | t(x) = A and w(x) = B} is called the Morseifrom A to B. A semigroupoid S is said to be strongly

(A, B)

J A S

Define two surjective functions F : J ^ S0 and G : A ^ S0. Now,let p : A x J ^ S be a function such that

P(Aji) GMor(F(i), G(A)).

We simply write P(x,i) = P\i so that the entries of the A x J matrix P = (pAi) are p\i. Let M = M(S, F, G; P) be the following set

M = {(i, x, A) G J x S x A | x GMor(G(A),F(i))},

equipped with the multiplication given by (i, x, A)(j, y, = (i, xpAjy, .

Then it is easy to check that the set M = {(i, x, A) G J xS x A | x GMor (G(A), F (i))}

M

oofir a semigroupoid (for details, S66 [161).

2. Presentations of bands and their generalizations

SS Petrich in [17] gave the general structure theorem for bands.

Theorem 3 [14, Theorem 4.4.5]. Let Y be a semilattiee, and let {Ea | a G Y}

be a family of pairwise disjoint rectangular bands indexed by Y. For each a, let Ea = Ja x Aa, and for each pair a, p of elements of Y such that a > p, let : Ea ^ Tip * x TAp be a morphism, where

a$a„a = (¿p) (a G Ea).

Suppose also that

(i) if a = (i,p) G Ea, then ¿aa and 1aa are constant maps, and

<¿2^ > = i,

= m;

(ii) if a G Sa, b G Sp and ap = 7, then ¿a¿^ and are constant maps;

(iii) if <^a¿^> is denoted by j and > % then for all S < 7,

¿j'v) = ^, =

Let B = |J{Ea | a G Y }, and define the product of a in Ea and b G Ep by

a * b=«¿a¿y>, <iaiY>),

w/iere 7 = ap. Tften (B, *) is a band, whose J-classes are the rectangular bands Ea.

Y

rectangular bands Ea = Ia x Aa indexed by Y, and a family of morphisms $a,p : Ea — 7}^* x Ta^ (a, p G Y, a > p) satisfying (i), (ii) mid (iii).

An element a of a semigroup S is called regular if there exists x G S such that a = axa; an element a of S is called quasi-regular if there exists a natural number n such that a" is regular. A semigroup S is called regular (quasi-regular) if every S

are generalizations of regular semigroups. As a generalization of bands, in 1989 Ren and

E

S E S

E(S) S

E

constructions.

The set Q with a partial operation is called a partial power breaking semigroup if there is a partial binary operation on the set Q such that for any p, q, r G Q, (pq)r G Q (well-defined) if and only if p(qr) G Q; in this case, (pq)r = p(qr) holds, and for every a G Q, there exists n G N such that a" / Q.

Let Y be a semilattice, and let {Ea = Ia x Aa | a G Y} be a family of pairwise disjoint rectangular bands. Let Q be a partial power breaking semigroup together with the mapping ^ : Q — U Ea satisfying the following properties.

a£Y

(i) For any a, b G Q, if y>(a) G Ea, y(b) G Ep Mid ap = 7, then ab G Q implies y(ab) G E7. For every pair a, p G Y with a > p, we can construct two mappings:

: ^-1(Ea) —► 7}^* x ,

a - (¿p,1a)

: Ea -► 7}^ * x ,

e - (^p,1p)

that satisfy the following properties:

(ii) If e = (i,j) G Ea, then ¿a, 1a are constant transformations on Ia and Aa, respectively, and <^a> = i, <ia> = j. Here we denote the values of the constant transformations by <^> and <ia> j respectively.

(iii) 1° If e G Ea, f G Ep, and S < 7 = ap, then ¿^¿jrnd if are transformations on I7 and A7, respectively. Let (¿^¿Y> = i, if > = j, we have

¿P = ¿^f = mf.

2° If e G Ea, a G Q, ^(a) G E^^d S < 7 = a^, then ¿Y¿a, ¿a¿Y ^^ are constant transformations on J^d AY respectively. Let (¿Y¿a} =

k, (^e ^a} = i> (¿a ¿y } = k, an d (^a^e} = i',we have

¿f0 = ¿e = ^a, ¿f -1') = ¿a^f' •'') =

3Mf a, b G Q, ab / Q, y>(a) G Ea, y(b) G E,g, and S < 7 = a^, then ¿a¿Y, ^Y

Ly, respectively. Let (¿a¿Y} = u, (^a^Y}

are constant transformations on J^d AY, respectively. Let (¿a¿Y} = u, } = v,

we have

(iv) If a, b G Q, ab G Q, y(a) G Ea, <^>(b) G E^^d S < 7 = a^, then We now write J2 = QU U Ea and define an operation * on £ as follows:

a£Y

a) If a, b G Q and ab G Q, then a * b = ab. If a, b G Q, y(a) G Ea, y(b) G E^ and a^ = 7, tat ab / Q, then

a * b=¿y}, ^Y}).

b) If e G Ea, a G Q, y(a) G E^^d a^ = 7, then

a * e = (^¿Y}, (^l}),

e * a = ((¿Y^a}, (V>|^

c) If e G Ea, f G E^^d a^ = 7, then

e * f = ((¿Y¿Y}, (V>|^}).

We denote the above system consisting of £ and the operation * on £ by I] = ZXQ, U Ea, . It is easy to show that J] = £(Q, U Ea, $,<£>) is

a£Y a£Y

*

Theorem 4 [18]. £ei S be a semigroup. Then S is an E-ideal quasi-regular

semigroup if and only if S is isomorphic to some semigroup of type £ =

= e(q, ee Ea,

a£Y

3. A-products and generalized A-products

A regular semigroup S is called a left C-semigroup (in short, LC-semigroup) if for any a G S, aS C Sa. In 1991, Zhu, Guo and Shum in [19] gave the following

C

SE S

(i) S is a left C -semigroup;

(ii) (Ve G E) eS C Se;

(iii) (Ve G E)(Va G S) eae = ea;

(iv) DS n (E x E) = ;

(v) S is a semilattice of left groups;

(vi) L = J is a semilattice congruence on S.

C

Ae

Let Y be a semilattice. Let T = |J Ta be a semilattice of semigroups Ta and

a£Y

J = U J« a semilattice partition of the set J on the semilattice Y. For each

a£Y

a G Y, write Sa = Ta x Ja; for any a, p G Y, a > ft, define the mapping

^a„8 : Sa -► 7/^ ,

a ^

satisfying the following conditions:

(Pi) If (u, i) G Sa, i' G Ja,then ^O^i' = i. (P2) If (u,i) G Sa, (v j) G S,, then

(a) ^"a, ^ia, are constant value mappings on Ja,, denote the value by

(b) if ap > S, } = k, we have = •

S

(u,i) * (v ,j) = (uv }), (u,i) G Sa, (v,j) G S,.

where uv is the product of u and v in the semigroup T.

S = Sa

a£Y

The semigroup S constructed above is called a A-product of a semi group T and a set J with respect to semilattice Y, denoted by S = TAY,^J.

Theorem 6 [21]. Lei T = [Y; Ga, ya,,] &e a strong semilattice of group Ga, and lei J = UaeY Ja a semilattice decomposition of a left regular band J for left zero bands Ja. Then the A-product S = TAY,^J o/ T mid J with respect to Y is a LC-semigroup.

S

According to [22], a quasi-regular semigroup S is called a C* -quasiregular semigroup if for any e G E(S), the mapping : S1 ^ eS 1e defined by x ^ exe is a semigroup

homomorphism and RegS is an Weal of S.

C*

and Guo in [22].

S

(i) S C*

(ii) S is a quasi-completely regular semigroup in which RegS is an ideal of S and E(S)

(iii) S is a quasi-completely regular semigroup such that eS U Se C RegS and the mapping : E(S) ^ eE(S)e defined by f ^ efe ¿s a semigroup homomorphism for all e G E(S);

(iv) S

(Va G S)(3m G N) amS U Sam C RegS

E(S)

(v) S C

It is well-known that the structure of completely regular semigroups can be described by the translational hull of semigroups (see M. Petrich in [23]). Inspired by the

work of M. Petrich, wo can also construct qnasi-coriiplotoly regular semigroups by nsing translations on semigroups.

To obtain structure of C*-quasiregular semigroups, we consider a more general construction for semigroups rather than the A-product structure. We call this new structure the generalized A-product structure. We first cite the following concepts.

A mapping 0 from a power breaking partial semigroup Q to another one is called a partial homomorphism if (ab)0 = a0b0,whenever a, b, ab G Q.

Write T(X )(T*(X)) for the semigroup of all left (right) transformations on a set X. Also, we use the symbol <y> to denote the value of a constant mapping y acting on X

A

(I) Let t be a partial homomorphism from a power breaking partial semigroup Q to a semilattice Y; write Qa = t-1(a), for any a G Y.

(II) Let T = [Y, Ta, £ap] be a strodg semilattice of demigroups Ta, where £ap is the

I = Ia A = Aa

a£Y a£Y

I A Y

if Ta are groups, then the strong semilattice T = [Y, Ta,£ap] is a Clifford semigroup. For every a G Y, form the following three sets, namely, the sets

Sa = Qa U Ta,

Sa = Qa U (Ia x Ta^

Sr = Qa U (Ta x Aa).

(III) For any a, p G Y with a > p, define the mapping

: S0 — Tp by a — a0a,p,

and we require that 0a p satisfies the following condition. (PI) (i) 0a,p k = L,p.

(ii) if a G Qa and a > p > 7, then a0a,p0p7 = a0 a,7.

(iii) if a G Q a, b G Qp and ab G Qap with ap > S, then (ab)0ap,5 = a0ab0p,5.

(IV) For a, p G Y with a > p, define the following two mappings y a,p and 1 a,p:

ya,p : Sl — T(Ip) by a — ya,p; 1a,p : Sr — T* (Ap) by a — V£,p.

Let ya,p and 1a,p satisfy the following conditions (PI), (P2), (P2)* and (P3)*, respectively.

(P2) If (i, g) G Ia x Ta and j G Ia, then y 2;g)j = i. (P2)* If (g, A) G Ta x A a and m G A a, then m1 i3^ = A.

For the sake of convenience, we write (i,g)0a,p = g0a(g, A)0a,p = g0a,p for (i, g) I x T (g, A) T x A (P3) Let a, p and S G Y with ap > S.

(i) If a G S£, b G S^, and ab G Qap, then y^ = y a,5yp,^

(ii) If a G S^, b G Sp and ab / Qap, then ya,apyp, ap is a constant mapping acting

I p

Let k = <ya,apyp, ap> be the constant value of ya,apyp, ap and g = a0a,apb0p, ap-Then

y afcp$ = y a,5 yp,5.

(P3)* Let a, p and S G Y with ap > S.

(i) If a G Sr, b G Sp, and ab G Qap, then 12p,5 = 12,51p,5.

(ii) If a G Sr, b G Sp, and ab / Qap, then 12,ap 1p,ap a constant mapping

A p

Let u = <1 2,ap 1p, ap> be the constant value of 12,ap 1p,ap and v = a0a,apb0p, ap-Then

S = S = (Q U (I x T x A ))

a£Y a£Y

operation "*" on S satisfying the following conditions. [Ml] If a G Qa, b G Qp, and ab G Qap, then a * b = ab. [M2] If a G Qa, b G Qp, and ab / Qap, then

a * b = (<y a,a pyp,a p> a0a,apb0p, ap, <12,ap 1p, ap>).

[M3] If a G Qa, (i, g, A) G Ip x Tp x Ap, then

a * (i,g,A) = (<y a,a p ypiap >,a0a ,ap g0p, ap, <12,ap V^p >),

(i, g, A) * a = (<yp,apya,ap>,g0p, apa0a,ap, ^^12,ap>). [M4] If (i, g, A) G Ia x Ta x A a, (j, h, m) G Ip x Tp x Ap, then

(i,g, A) * (j h,M) = (<yi*apypj^>,g^a,a phCp,ap, ^^>).

( S, * )

Now, we write E = {ya,p, 1mp, 0a,p | a, p G Y, a > p} and call it the structure mapping of the semigroup S = |J (Qa U (Ia x Ta x Aa)).

a£Y

Snmmarizing all the above steps, we give the following definition.

S

A-product of the power breaking partial semigroup Q, the semigroup T, the sets I and A with respect to the semilattice Y and the structure mapping E. Denote this semigroup by S = AY,s(Q, I, T, A).

C*

Theorem 8 [22]. Let Y be a semilattiee, Q be a power dreaking partial semigroup, G = [Y, Ga, £a,p] &e a strong semilattiee of groups Ga, I = |J Iat and A = |J Aa 6e

a£Y a£Y

A

Ayjs(Q, I, G, A) ¿s a C* -quasiregular semigroup. *

A-product Ay,e(Q,I, G, A).

4. Left wreath products and wreath products

In 1982, an abundant semigroup was first introduced and studied by J.B. Fountain [5]. To show the definition of an abundant semigroup, we first cite a set of relations called Green's star relations on a semigroup S:

C* = {(a, b) £ S x S | (Vx, y G S1)ax = ay ^ bx = by}, R* = {(a, b) G S x S | (Vx, y G S1)xa = ya ^ xa = yb},

H* = L* A R*, D* = L* VR*,

J * = {(a, 6) G S x S | J* (a) = J * (6)},

where J* (a) denote the principal *-ideal generated by the element a in S (see [5] and [24]).

Clearly, on any semigroup S we have L C L* and R C R*. It is easy to see that for regular elements a, 6 G S, (a, 6) G L* if and only if (a, 6) G L. Moreover, we can easily see that L* is a right congruence and R* is a left congruence on S, respectively.

An abundant semigroup S is a semigroup in which each L*-class and each R*-class contains an idempotent. It is clear that a regular semigroup is abundant. In fact, abundant semigroups can be regarded as natural generalizations of regular semigroups.

An abundant semigroup S is called an L* -inverse semigroup if S is an JC semigroup whose idempotents form a left regular band (for details, see [25]).

To obtain structure of L* -inverse semigroups, the concept of left wreath product of semigroups was introduced by Ren and Shuni in [25].

Let r be a type-A semigroup with semilattice Y of idempotents. Let B = UaeYBa be a semilattice decomposition of a left regular band B into left zero bands Ba.

Because the type-A semigro up r is abundant, we can always identify the element Y G r by its corresponding idempotent yt G R^(r) n E or by y* G L^(r) n E, respectively. Moreover, since the type-A semigro up r is also an JC abundant semigroup, there is a connecting isomorphism n : (w^ ^ (w*} such that aw = w(an) for any a G (w^ and w G r.

Now, we form the set B ix r = {(e, y)| e G BYt,y G r}. In order to make this set B ix r a semigroup, we need to introduce a multiplication "*" defined on the set B [X r % the following mapping. Firstly, we define a mapping ^ : r ^ End(B) by y ^ > where a7 G End(B), which is the endomorphism semigroup on B. This mapping satisfies the following properties.

(P1) Absorbing: for each y G r and a G Y, we have Baa7 C B,Ya)t. In particular, if Y G Y, then a7 is an inner endomorphism on B such that = f e for some f G BY and all e G B.

(P2) Focusing: for a, p G r and f G B(a,)t, we have a,aaSf = aa,Sf, where Sf is an inner endomorphism induced by f on B satisfying hf = f h for all h G B.

(P3) Homogenizing: for e G Bwt, g G BT^^d h G B^t, if wt = w^d egCT^ = = ehff- , then fgff-* = f , for any f G Bw*.

(P4) Idempotent connecting: assume that for any w G r, n is the connecting isomorphism which maps (w^ to (w*} by a ^ an- If (e,wt^d (f, w*) G B ix r, then there IS 3. bijection 0 : (e} ^ (f} such that

(i) e0 = ^dj = e(g0)ff- , for g G (e};

(ii) for g G (e} and a G (wt}, (g, a) G B ix r if and only if (g0, an) G B ix r.

Equipped with the above mapping ^, we now define a multiplication " *" on B ix r

by

(e, w) * (f,T) = (ef,wt)

for any (e, w), (f, t) G B ix r, where f= faw .

It can be verified that the multiplication "*" defined above for the set B ix r is

B

a type-A semigro up r under a map ping denoted by B ixv r.

We are now going to establish a structure theorem for L*-inverse semigroups.

Theorem 9 [25, Theorem 4.1]. A semigroup S is an L* -inverse semigroup if and, only if S is a left wreath product of a left regular band B and a type-A semigro up r.

In [26], we call an IC abundant semigroup S a Q* -inverse semigroup if the set of its idempotents E forms a regular band, i.e. E satisfies the identity e/ege = e/ge, for all e, / and g in E.

S Q* E

ular band. Denote the ^-class containing the element e G E by E(e). We first have the following result.

SS

by aSb if and, only if b = ea/ and a = gbh for some e G E(a+), / G E(a*), g G E(b+) and h G E(b*), then the equivalence relation S is the smallest type-A good congruence S

S Q* E

relations Mi and Mr on S as follows:

(a, b) G Mi ^ (xa, xb) G L* (x G E),

(a, b) G Mr ^ (ax, bx) G R* (x G E).

Put pi = S n Mr and p2 = S n Mi on S (see [26]). We are now able to establish the Q*

Q*

wreath product of semigroups was introduced by Ren and Shum in [26] as follows. In the wreath product of semigroups, we need the following ingredients:

(a) Y: a semilattice.

(b) r a typ e-A semigroup whose s et of idmmpotents is the semi lattice Y.

(c) I: a left regular band such that I = |J Ia, where Ia is a left zero band for

a£Y

all a G Y.

(d) A: a right regular band such that A = |J Aa, where Aa is a right zero band

a£Y

for all a G Y.

We now form the following sets:

I M r = {(e,w)| w G r, e G Iw+ },

r 04 A = {(w,i)| wG r, iG Aw*},

I 04 r 04 A = {(e,w, i)| w G r, e G Iw+ andiG Aw*}.

Since w G r and r is a type-A semigroup, there are some idempotents wt g R* (r) n E(r^d w* G L*(r) n E(r). Also since the set of idempotents of r forms a semilattice, w^d w* are in Y. This illustrates that the sets I 04 r, r 04 A^d I 04 r 04 A are well-defined. We only need to define an associative multiplication on the set I 04 r 04 A so that the set I 04 r 04 A under the multiplication turns out to be a semigroup.

Before we define a multiplication on I 04 r 04 A, we need to give a description for the structure mappings.

Define a mapping y : r ^ End(I) by 7 ^ a7 for 7 G r and a7 G End(I) satisfying the following conditions.

(P1) For each 7 G r and a G Y, we have IaaY C I(Ya)t • In particular, if 7 G Y, then is an inner endomorphism on I such that there exists g G IY with = ge, for all e G I, where ^raotes eaY.

(P2) For a, p G r and / G I(ap)t, we have apaaSf = aapSf, where Sf is an inner endomorphism induced by / on I satisfying h5f = /h = /h/, for all h G I.

(P3) For e G Jwt, g G JTt and h G J^t, if wt = w^d egCT^ = ehCT^, then = fh^.. , for all f G Jw..

(P4) Assume that for any w G r, n is the connecting isomorphism which maps (w^ to (w*} by a ^ an- If (e, wt) and (f, w*) G J n r, then there is a bijection 0 : (e} ^ (f} such that

(i) e0 = f and geCTa = e(g0)CT^ for any g G (e} and a G (wt};

(ii) for any g G (e} and a G (wt}, (g, a) G J n r if and only if (g0, an) G J n r.

Similarly, define a mapping ^ : r ^ End(A) by y ^ pY for y G ^d pY G End(A)

satisfying the following conditions.

(P1)' For each y G ^d a G Y, we have AapY C A(aY).. In particular, if y G Y, then pY is an inner endomorphism on A such that there exists i G AY with jPY = ji for all j G A, where jPY denotes jp7.

(P2)' For a, p G r and i G A(a,)., we have pap,e = pa,£j, where e is an inner endomorphism induced by i on A such that j£i = ji = iji for any j G A.

(P3)' For i G Aw., j G ATand k G A?., if tw = ^^d ji = i, then jp.t m = kp.t m for all m G Awt •

(P4)' Assume that for any w G r, n is the connecting isomorphism which maps (w^ to (w*} by a ^ an- If (wt,j^d (w*,i) G r u A, then there is a bijection 0' : (i} ^ (j} such that the following conditions hold:

(i) j0' = i, i = iPan (k0'), for any k G (j} Mid a G (wt};

(ii) for any k G (i} and a G (wt}, (a, k) G r u A if and only if (an, k0') G r u A.

After gluing up the above components J, r and A together with the mappings ^

and ^, we now define a multiplication on the set Jm r m A by

(e, w, i) * (f,T, j) = (ef, wt, ipT j), (3)

for any (e, w, i), (f, t, j) G Jm r u A, where f= f<rw and iPT = ipT.

Using the properties (P1), (P2), (P1)' Mid (P2)', we can easily verify that the above multiplication " *" on Jm r m A is associative. We now call the above constructed semigroup a wreath product of J, r and A with respect to <p and ^, and denote it by Q = J r ixty A.

Theorem 11 [26, Theorem 4.4]. The wreath product J r A of a left regular band I, a type-A semigroup r and a right regular band A with respect to the mappings <p and ^ is a Q* -inverse semigroup.

Conversely, every Q* -inverse semigroup S can be expressed by a wreath product of J r ixty A.

Remark 1. The class of Q*-inverse semigroups contains several interesting classes of semigroups as its special subclasses. We only discuss some of these special subclasses as follows.

(a) L*-inverse semigroups and R*-inverse semigroups

By Theorem 11, a Q*-inverse semigroup S can be expressed as a wreath product J r A of J, r and A with respect to the mappings <p and where r is a type-A semigroup, J Mid A are respectively a left regular band and a right regular band. In Theorem 11, if A = 0, then S = J r, which is an L*-inverse semigroup. Similarly, if we let J = 0, then r Kty A becomes an R*-inverse semigroup. Thus, the class of L*-inverse semigroups and the class of R*-inverse semigroups are two special subclasses of the class of Q* -inverse semigroups. In this case, we can easily re-obtain Theorem 10 for structure of L* -inverse semigroups, as a corollary of Theorem 11.

(b) Quasi-inverse semigroups

Wo know that a quasi-inverse semigroup is a regular semigroup whose set of idem-

potent forms a regular band. It is clear that a qnasi-inverse semigroup is a special Q*

S S S aSb

only if b = ea/ for some e G E(aa') and / G E(a'a), where a' is an inverse element of a.

S

on S, and so r = S/S is the greatest inverse semigroup homomorphism image of S. Obviously, the inverse semigroup r = S/S must be a type-A semigroup whose set of idempotents forms a semilattice. As a result, a wreath product I 04v r 04^ A of S, regarded as a Q* -inverse semigroup, can be simplified by using the so-called half-direct product (in short, H.D.-product) of a quasi-inverse semigroup given by M. Yamada in [27] as described in the following Theorem 12.

S

E S S

such that r = S/S is the greatest inverse semigroup induced by S, and let Y be the semilattice of r. Define the congruences n1, n2 on E by e^/ if and only if eR/; e^2/ if and only if eL/, respectively.

For X C E, write XX = {e | e G X} and X = {e | e G X}, where e and e are the n1 -class and the n2 -class containing e G X, respectively. Then the following statements hold:

(i) E/n1 = E is a left regular band such that E = |J Ea, where every Ea is a left

a£Y

zero band; E/n2 = E is a right regular band such that E = |J Ea, where each Ea is

a£Y

a right zero band, for every a G Y.

(ii) S is isomorphic to an H.D.-product of E, r and E with respect to the mappings y' and i>', respectively. Conversely, any H.D. -product of a left regular band I = |J Aa, an inverse semi group T and aright regular band A = |J Aa with respect

a£Y a£Y

to the mappings y' and i>' is a quasi-inverse semigroup S, where r is the greatest

SY

ofT.

S

and T are two semigroups having a common homomorphic image H, and if ^ : S ^ H and ^ : T ^ H ^re ^^^^^OTphisms onto H, then the spined product of S and T with respect to H, ^d ^ is defined by

Y = {(s,t) G S x T | s^ = t^}.

Q*

S Q*

if and only if S is a spined product of an L* -inverse semigroup S1 = I 04v r and an R* -inverse semig roup S2 = r 04^ A with respect to a type-A semigro up T.

5. Semi-spined product of semigroups

In generalizing regular semigroups, apart from weakening the definition of regularity, one of the most suitable approach is to modify the usual Green's relations on semigroups. During the recent 40 years, a series of generalized Green's relations have been established, such as (*)-Green's relations, (l)-Green's relations, (*, ~)-Green's relations, -Green's relations and )-Green's relations (see [28]). In this section, we only introduce (*, -Green's relations.

S

aSx(a G S), regarded as the right S 1-systems, are projective (see [29] and [24]). It was shown in [24] that a semigroup S is rpp if and only if for any a G S, the set

Ma = {e G E(S)|S 1a C Se & Kera; C Kere;}

E(S) S

to be strongly rpp [20] if

(Va G S) (3! e G Ma) ea = a.

In [20] and [30]. (l)-Green's relations have been applied to the study of rpp semigroups. especially strongly rpp semigroups. However, strongly rpp semigroups do not form a satisfactory generalization of completely regular semigroups in the class of rpp semigroups.

In order to get a satisfactory generalization of completely regular semigroups in the class of rpp semigroups, Guo, Shum and Gong [31] introduced the so-called (*,

S

L*,~ = l*

H*

d l*

D*

a J *'~6

where, for any a, 6 G S,

aRb

R* ~ : " A R*' J d L*'

_d. ; i j *

nR*

R,

= L*'' VR*'~,

"(a) = J *'~(6)"

(Ve G E(S))"ea = a <-► e6 = 6" [15],

and J*'~(a) is the smallest ideal containing and saturated by L*'~ and R*'~.

Let S be a semigroup and E(S) the lattice of all equivalences on S. For any a G E(S), call A C S a subset saturated, by a if A is a union of some a-classes of S; call S a -abundant if every a-class of S contains idempotents of S.

A semigroup S is called r-wide [31] if S is L*'~-abundant and R*'~-abundant. An r-wide semigroup is called a super-r-wide semigroup [31] if S is *, ^-abundant. Call a semigroup S an ortho-le-monoid if S is a super-r-wide semigroup with E(S) < S [31]. An ortho-lc-monoid S is called a regular ortho-le-monoid if E(S) forms a regular band. It is clear that an ortho-lc-monoid is strongly rpp, and each H*'~-class is a left cancellative monoid (in short, lc-monoid).

For (*, -Green's relations, we have the following results.

S,

(i) R*'~ is usually not a left congruence even if S is an R*'~ -abundant semigroup.

(ii) In general, we have L*'~ o R*'~ = R*'~ o L*'~.

(iii) If S is super-r-wide, then R*'~ is a left congruence and D*'~ = L*'~ o R*'~ (= R*~ oL*'~, of course).

Thus,

(iv) When S is super-r-wide, the corresponding Green's Lemma holds for the (*, -Green's relations.

By using (*, -Green's relations, we can give some characterizations of super-r-wide semigroups and ortho-lc-monoids.

SS

group if and only if S is a strongly rpp semigroup on which *, ~ = L*'~ o R*'~ = = R*-~ o L* ~ holds.

Call that a semigroup is a rectangular Ic-monoid if it is isomorphic to the direct product of a rectangular band and a left cancellative monoid.

SS

and, a semilattice of rectangular Ic-monoids.

Based on the semilattice decomposition of ortho-lc-monoids, the semi-spined product structure of regular ortho-lc-monoids was provided in [31].

Let M = [Y; Ma, ya,p] be an la-Clifford semigroup, i.e. a strong semilattice Y of left cancellative monoids Mas, I = |J Ia a semilattice decomposition of the left regular

a£Y

band I into left zero bands Ia's, and A = |J Aa a semilattice decomposition of the

a£Y

right regular band A into right zero bands Aa' s. We define the following mappings

S : A = J Ia x Ma Ti(I)

a£Y

and

£ : U (Ma x Aa) -^Tr (A)

a£Y

satisfying the following conditions: (PI) If (i, a) G Ia x Ma and j G Ip, then S(i, a)j G Iap; (Ql) If (b, m) G Ma x Aa and A G Ap, then Ae(b, m) G Aap; (P2) If a < p holds in (PI) for a, p G Y, then S(i, a)j = i; (Q2) If a < p holds in (Ql) for a, p G Y, then Ae(b, m) = m; (P3) If (i, a) G Ia x M^d (j, b) G Ip x Mp, then

S(i, a)S(j, b) = S(S(i, a)j, aya,apbyp,ap);

(Q3) If (a, A) G Ma x Aa and (b, m) G Mp x Ap, then

£(a, A)e(b, m) = e(aya,apbyp,ap, Ae(b, m));

(P4) Let (i, a) G Ia x Ma, j G Ip rnd k G IY. If

S(i, a)j = S(i, a)k,

then

S(i, 1a)j = S(i, 1a)k.

o

S = y (Ia x Ma x Aa)

a£Y

by

(i, a, A)(j, b, m) = (S(i, a)j, aya,apbyp,ap, Ae(b, m)). (4)

S

Definition 2 [31]. The semigroup S constructed above is called the semi-spined product of the lc-Clifford semigroup M = [Y; К the left regular band

I = U I« the right regular band Л = |J Ла with respect to the semilattice

Y and structure mappings S and e.

By using the above definition of semi-spined product of semigroups, we obtain the following structure theorem for regular ortho-lc-monoids.

Theorem 17 [31]. The semi-spined product of semigroups described in Definition 2 is a regular ortho-lc-monoid. Conversely, every regular ortho-lc-monoid can be constructed in this manner.

The research of the first corresponding author is partially supported by the grant of Wu Jiehyee Charitable Foundation, Hong Kong 2007/10. The research of the second anthos is supported by the grant of the National Natural Science Foundation of China (Grant No. 10971160).

Резюме

К.П. Шум, G.M. Реи, Ч.М. Гун. О методах построения полугрупп.

В статье представлен краткий обзор методов построения полугрупп с использованием структур некоторых полугрупп, относящихся к классам регулярных, квазирегулярпых и избыточных полугрупп. В частности, приведены основные обозначения и структурные теоремы для некоторых полугрупп, таких как рисовские полугруппы матричного типа над 0-группой G и их обобщения, связки, В-вде^^пые квазирегулярные полугруппы, C* -квазирегулярные полугруппы, £*-^версные и Q* -инверсные полугруппы и регулярные орто-1с-мопоиды.

Ключевые слова: регулярные полугруппы, квазирегулярпые полугруппы, избыточные полугруппы, конструкции.

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Поступила в редакцию 16.01.12

Shum, Kar-Ping PliD, Professor of Mathematics, Institute of Mathematics, Yunnan University, Kunming, China.

Шум, Кар-Пин доктор паук, профессор математики Института математики Юпь-папьского университета, г. Купьмип, Китай.

E-mail: kpshumQynu.edu.сп

Ren, Xue-Ming PhD, Full Professor, Department of Mathematics, Xi'an University of Architecture and Technology, Xi'an, China.

Рен, Сюэ-Мин доктор паук, профессор отделения математики Сиапьского университета архитектуры и технологий, г. Сиань, Китай.

E-mail: xmrenQxauat.edu.сп

Gong, Chun-Mei PhD, Associate Professor, Department of Mathematics, Xi'an University of Architecture and Technology, Xi'an, China.

Гун, Чун-Мей доктор паук, адъюнкт-профессор отделения математики Сиапьского университета архитектуры и технологий, г. Сиапь, Китай.