Structure of Regular Semigroups with a Regular Idempotent

Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 12, 557 - 570

Structure of Regular Semigroups with a Regular Idempotent K. Indhira and V. M. Chandrasekaran School of Advanced Sciences VIT University, Vellore – 632 014, India [email protected] Abstract Let S be a regular semigroup and let E(S) be the set of idempotents of S. In [6], we introduced the concept of regular semigroups with a regular idempotent and study the properties of regular semigroups with regular idempotent. An idempotent u of S is called regular if fu R f L uf for each f ∈ E(S). The main theorem of this paper describes the structure of regular semigroups with regular idempotent which generalizes the structure theorem of regular semigroups with a medial idempotent. Mathematics Subject Classification: 20M17 Keywords: regular semigroup, regular idempotent, medial idempotent, midunit

Let S be a regular semigroup and let E(S) be the set of idempotents of S. In [9], McAlister and McFadden have shown that a regular semigroup S contains a quasi-ideal inverse transversal if and only if S can be imbedded as an ideal in a locally inverse semigroup T which contains an idempotent u for which

fu R f L uf for each f = f in S. Amongst the semigroups which contain such an idempotent one those (locally inverse) semigroup which can be naturally ordered in such a way as to have a maximum idempotent u. Then u can be characterized as an idempotent satisfying e = eue for each e ∈ 〈 E (T )〉 , the idempotent generated part of T (see [1]). Such an idempotent was called medial 2

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K. Indhira and V. M. Chandrasekaran

normal by Blyth and McFadden [1], who obtained a structure theorem for all regular semigroups containing a maximum idempotent in some-imposed natural order. An idempotent u of S is called medial if x = xux for each, x ∈ 〈 E (S )〉 the idempotent generated subsemigroup of S. In [7], Loganathan described the structure of regular semigroups that contain a medial idempotent. In this paper we introduce a new class of regular semigroups that contains an idempotent u for which fu R f L uf for each f ∈ E (S ) .We call u a regular idempotent of S. In [6], we introduced the concept of regular semigroups with a regular idempotent and study the properties of regular semigroups with regular idempotent. We start in section 1 by recalling the concept of regular semigroups with a regular idempotent and we investigate some elementary properties of regular semigroups with a regular idempotent. In section 2 we prove our main result. It generalizes the structure theorem for regular semigroups with a medial idempotent [7].

1. Regular semigroups with a regular idempotent Definition 1.1. Let S be a regular semigroup. An idempotent u of S is called regular if fu R f L uf for each f ∈ E (S ) . The following lemma contains some elementary properties of regular semigroups with a regular idempotent which we shall find useful in this paper.

Lemma 1.2. Let u be a regular idempotent of a regular semigroup S. Then (1) V (uxu ) ∩ uSu = V ( x) ∩ uSu for all x ∈ S . (2) For all x, y ∈ S , x * ∈ V (uxu ) ∩ uSu , y * ∈ V (uyu ) ∩ uSu , we have

uxyu = uxu ( x * xyy * ) uyu (3) For all x, y ∈ S , (i) x L y ⇒ uxu L uyu ; (ii) x R y ⇒ uxu R uyu . Proof. (1). In particular for any x ∈ S , we prove uV ( x)u ⊆ V (uxu ) ∩ uSu.

⊆ V ( x) ∩ uSu . Let x * ∈ V (uxu ) ∩ uSu . Then for any x ** ∈ V (uxu ) ∩ uSu , we have xx ** x = x.x * xx ** xx * x = xx * (uxux**uxu ) x * x = xx* (uxu ) x * x

Structure of regular semigroups

559

= xx* xx* x = x and x ** xx ** = x ** (uxu ) x **

= x ** Hence x ** ∈ V ( x) ∩ uSu. (2). The proof is obvious. (3). This is immediate from (2). Let us give some examples of regular semigroups with a regular idempotent. Example 1.3. Let S be a regular semigroup containing a medial idempotent u. Then x = x u x for all x ∈ 〈 E ( S )〉 , the idempotent generated part of S. In particular, f u f = f . f = f implies fu R f L uf for each f ∈ E ( S ) . Therefore u is a regular idempotent of S. Example 1.4. Let S be a regular semigroup with midunit u[5]. Then x u y = xy for any x, y ∈ S . In particular, f u f = f . f = f implies fu R f L uf for each f ∈ E (S ) . Therefore u is a regular idempotent of S. Example 1.5. Let S be a regular semigroup. An element u ∈ S is said to be a

weak

middle

unit

[2]

if,

for

all

x∈S

and

all x ' ∈ V( x) ,

x u x ' = xx ' and x 'u x = x ' x . If u is a weak middle unit then u is necessarily idempotent, for u = uu 'u for all u ' ∈ V (u ) = uuu ' .u since uu ' = uuu ' = u.u Let f ∈ E (S ) . Since f ∈ V ( f ), f u f = f implies that fu R f L uf . Therefore, u is a regular idempotent. ⎛1 1 ⎞ ⎟⎟ is the Example 1.6. The semigroup S = M 0 ({1}, {1,2}, {1,2}; p ) where p = ⎜⎜ ⎝1 0 ⎠ smallest non-orthodox regular semigroup[3]. Then it is easy to see that (1,1,1) is a regular idempotent of S. Let S be a regular semigroup. For any u ∈ E (S ) , let S u = {x ∈ S :V ( x) ∩ uSu ≠ φ } .

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Clearly uSu ⊆ S u and every element of S u is regular. If x, y ∈ S u with x ∗ ∈

V ( x) ∩ uSu and y ∗ ∈ V ( y ) ∩ uS then by Theorem 1.2 of [10], y ∗ gx ∗ ∈ V ( xy ) ∩ uSu, where g ∈ S ( x∗ x, yy ∗ ) so that xy ∈ S u . Hence S u is a regular subsemigroup of S. Moreover, by Lemma 1.2, V (uxu ) ∩ uSu ⊆ V ( x) for any x ∈ S u. We have the following useful characteristics of S u .

Proposition 1.7. Su is the largest regular subsemigroup of S such that u is a

regular idempotent in S u . Proof. Let f ∈ E ( S u ) .Choose f ∗ ∈ V (ufu ) ∩ uSu ⊆ V ( f ) . Then

f = ff ∗ f

∗

= fuf uf implies fu R f L uf .Hence u is regular idempotent in S u . Now suppose that K is a regular subsemigroup of S such that u is regular in K. We first show that E ( K ) ⊆ S u . Take any f ∈ E (K ) . Then fu R f L uf implies f = fux = ∗

∗

∗

yuf for some x, y ∈ S . Pick f ∗ ∈ V (ufu ) ∩ uSu. Then f ff = f (ufu ) f and

∗

= f∗

(ufu ) f ∗ (ufu ) = ufu ⇒ uff ∗ fu = ufu ⇒ yuff ∗ fux = yufux ⇒ ff ∗ f = f

So V ( f ) ∩ uSu ≠ φ and hence f ∈ S u .Next take any x ∈ K and x ∗ ∈ V ( x) ∩ K . Then xx ∗ , x ∗ x ∈ E ( K ) ⊆ S u . Pick a a ∈ V ( xx ∗ ) ∩ uSu and b ∈ V ( x ∗ x) ∩ uSu . From xbx ∗ = xx ∗ and x ∈ S u and K ⊆ S u .

x ∗ ax = x ∗ x

we

get

bx ∗ a ∈ V ( x) ∩ uSu. Hence

Corollary 1.8. Let S be a regular semigroup. Then the following statements are equivalent. (i) u is a regular idempotent of S. (ii) V ( f ) ∩ uSu ≠ φ for any f ∈ E (S ). (iii) V ( x) ∩ uSu ≠ φ for any x ∈ S . In particular, the idempotent u of S is regular if and only if S u = S. Theorem 1.9. Let S be a locally inverse semigroup. Then the idempotent u of S is regular if and only if uSu is an inverse transversal of S.


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Proof. Suppose that u is a regular idempotent of S. Given S is locally inverse and hence uSu is an inverse subsemigroup of S. Since V ( x) ∩ uSu ≠ φ for each x ∈ S , it is enough to prove that V ( x) ∩ uSu contains exactly one element. Let

x ∗ , x ∗∗ ∈ V ( x) ∩ uSu. Then x = xx ∗ x , x ∗ = x ∗ xx ∗ and x∗∗ = x∗∗ xx∗∗ , x = xx ∗∗ x , . But since

x ∗ , x ∗∗ in V (uxu ) ∩ uSu and ∗

uSu is an inverse

∗∗

subsemigroup of S implies that x = x , as required. Conversely, suppose that uSu is an inverse transversal of S. Then for any x ∈ S , V ( x) ∩ uSu = 1. By Proposition 1.7, u is a regular idempotent. Since uSu is also a quasi-ideal of S, Theorem 2.2 of [3] can be stated as follows.

Corollary 1.10. A regular semigroup S contains a quasi ideal transversal if and only if S can be embedded as an ideal in a locally inverse semigroup which contains a regular idempotent.

2. Description of regular semigroups with a regular idempotent Let u be a regular idempotent of a regular semigroup S. Following [3,4], we define I = {i ∈ E ( S ) : uiu ∈ E (uSu ) and i L uiu} Λ = {λ ∈ E ( S ) : uλu ∈ E (uSu ) and λ R uλu} . It is easy to see that I=E (Su), Λ=E(uS) and E(uSu) ⊂ I ∩ Λ . Needless to say that these two sets play am important role in our theory. In the next few lemmas we obtain some basic properties of these sets.

Lemma 2.1. (i). If i ∈ I , then i R e for some e ∈ E (uSu ) implies i ∈ E (uSu ) ; If λ ∈ Λ, then λ L e for some e ∈ E (uSu ) implies λ ∈ E (uSu ) .

(ii). *

If

x∈S ,

then

for

any

x * ∈ V (uxu ) ∩ uSu ,

we

have

*

xx ∈ I and x x ∈ Λ. (iii) If i R x L λ , where i ∈ I , λ ∈ Λ, x ∈ S, then there exists an inverse x * ∈ V (uxu ) ∩ uSu such that xx* = i and x * x = λ. Proof. (i).If i R e , then from Lemma 1.2(3), we have uiu R ueu = e R i , while i ∈ I implies i L uiu . Thus i and uiu are H -related idempotents. Hence i = uiu ∈ E (uSu ). (ii). This is immediate from Lemma 1.2(2).

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(iii).


From

Lemma

1.2(3),

we

have

x * ∈ V (uxu ) ∩ uSu such that uxux* = uiu and

uiu R uxu L uλu ;

choose

x *uxu = uλu . Then by Lemma

1.2(1), x * ∈ V ( x) so that xx* R x R i and x * x L x L λ . Also xx* L i and x * x R λ . Thus xx * H i and x * x H λ ; so xx * = i and x * x = λ. . For each e ∈ E (S ), let I e = {i ∈ I : ui Re} and Λ e = {λ ∈ Λ : λu L e} . Then Lemma 2.2. (i). I e and Λ e are rectangular bands. (ii). For all e, e'∈ E (uSu ), i ∈ I with e' L e R uiu L i , we have e' i = e' uiu ∈ V (e). Dually, for all e, e'∈ E (uSu ), λ ∈ Λ with e' R e L uλu L λ , we have λe' = uλue'∈ V (e). Proof. (i). Let i, i '∈ I e . Then by Lemma 1.2(1), uiu, ui' u ∈ V (i ' ) and hence, by Green’s lemma, implies uiui ' = ui ' u. This ii ' = iuiui ' = iui ' u ∈ I e , since uiui' u = ui' u R e . Thus, I e is a band and, by Lemma 1.2(1), I e ⊂ V (e) . Hence I e is a rectangular band. Similarly Λ e is also a rectangular band. (ii). Since ei = uiu, e' i = e' ei = e' uiu. Again, e' L e R uiu, it follows that e' uiu ∈ V (e) . The second statement can be proved dually. Lemma 2.3. (i). If i R i' in I, and λ L λ ' in Λ , then uλiu = uλu (λ ' i' )uiu . (ii) If x ∈ S, i ∈ I , λ ∈ Λ are such that uiu R x L uλu , then i R ixλ L λ and uixu = uxλu = uixλu = x . Proof. (i) Immediate from Lemma 1.2(2) and Lemma 2.2(i). follows from Green’s lemma while (ii). That i R ixλ L λ uixu = uxλu = uixλu = x is an immediate consequence of Lemma 1.2(2). Lemma 2.4. For all i, i '∈ I , uiui' = i ' implies ii '∈ I and ii ' L i ' ; further , for any i ' '∈ I , with i ' ' R i , we have i ' ' i ' = ii '. Dually, for all λ , λ '∈ Λ, uλuλ ' = λ ' implies λλ '∈ Λ and λλ ' R λ ' ; further , for any λ ' '∈ Λ, with λ ' ' L λ , we have λ ' ' λ ' = λ λ ' . Proof. Since i L uiu, i ' L ui' u, and uiui ' = i ' , by Lemma1.2(2), uii ' u = uiu (uiuii' ui' u )ui' u = uiu (uiui' )ui' u = uiui' u = uiui' u = ui' u. In particular, ui' u ∈ V (ui' u ) ∩ uSu = V (uii ' u ) ⊂ V (ii ' ). This implies ii ' = ii ' ui' uii' = ii ' ii ' , since i ' L ui' u. Further, ii ' L uii ' u, since uii ' u = uiu Lii ' ui' u = ii '. Therefore, ii '∈ I and ii ' L ui' u L i '. If i ' '∈ I with i ' ' R i , then by Green’s lemma i ' ' uiu = i , and hence i ' ' i ' = i ' ' uiui ' = ii '. The second statement can be proved dually.

Before proceeding further let us fix some notations.


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Let S o be a regular semigroup and E( S o ) be the set of idempotents of S o . For each x∈ S o , let

{

}

{

}

r ( x) = R x ∩ E ( S o ) = e ∈ E ( S o ) : eRx and

l ( x) = L x ∩ E ( S o ) = e ∈ E ( S o ) : eLx

in particular, if e∈E( S o ), then r(e) (resp. l(e)) is the R-class (resp. L –Class) of e in E( S o ). Let E( S o )/R be the partially ordered set of R-Classes of E( S o ) and

E( S o ) / L be the partially ordered set of L-classes of E( S o ). In the following we shall regard E( S o )/R and E( S o )/L as small categories. Thus, for example, the objects of E( S o )/R are the R-classes of E( S o ) and, for any two objects r(e), r(f), there is exactly one morphism, denoted (r(e),r(f)), from r(e) to r(f) if r(e) ≥ r(f); otherwise there are no morphisms from r(e) to r(f). We denote by P the category of pointed sets and base point preserving maps. Given a functor F:C→P from a category C to P, we always assume that Fe ∩ Ff = φ whenever e and f are distinct objects of C. We denote the base point of Fe by e itself. Let u be a regular idempotent of a regular semigroup S. Let I/R be the set of R – classes of I and Λ/L the set of L-classes of Λ. For each i∈I, let i denote the R-classes of I containing i. Similarly, for each λ ∈ Λ, let λ denote the Lclasses of Λ containing λ . Observe that by Lemma 2.1(i), e =r(e) and e =l(e), whenever e∈E(uSu). With this notation, the following is an immediate consequence of Lemma 2.4. Theorem 2.5. The association r (e) a Ar (e) , (r (e), r ( f )) a A(r (e), r ( f )) where

Ar (e) = I e / R = { i ∈ I / R : uiu R e} with e = r (e) as base point, and where the map A(r (e), r ( f )) : Ar (e) → Ar ( f ) is given by i A(r (e), r ( f )) = uf , defines a functor A : E (uSu ) / R → P from E (uSu ) / R to P , the category of pointed sets and base point preserving maps. Dually, The association l (e) a Bl (e) , ( l (e), l ( f )) a B(l (e), l ( f )) where Bl (e) = Λ e / L = { λ ∈ Λ / R : uλu L e} with e = l (e) as base point, and where the

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map B(l (e), l ( f )) : Bl (e) → Bl ( f ) is given by λ A(r (e), r ( f )) = fλ , defines a functor B : E (uSu ) / L → P from E (uSu ) / L to P , the category of pointed sets and base point preserving maps. Definition 2.6. Let S o be a regular semigroup. An S o -pair (A,B) is a pair of functors

A : E (S o ) / R ⎯ ⎯→ P, B : E ( S o ) / L ⎯ ⎯→ P . Given an S o -pair (A,B), a BxA matrix over S o is a function ∗ : (b, a) a b ∗ a : Bl (e) x Ar ( f ) → S o . U U l ( e )∈E ( S ) / L

r ( f )∈E ( S ) / R

Theorem 2.7. Let S o be a regular semigroup and let (A,B) be an S o -pair with identity 1. Let ∗ be a BxA matrix over S o satisfying

(1) if b ∈ Bl (e) and a ∈ Ar ( f ) , then b ∗ a∈ l(e) S o r(f) ; (i) e (b ∗ aA(r(f),r(f’)))f’ = e(b ∗ a)f’ (ii) e’(bB(l(e),l(e’) ∗ a)f = e’(b ∗ a)f (3) for any b ∈ Bl (e), a ∈ Ar ( f ) , b ∗ r ( f ), l (e) ∗ a ∈ l (e)r ( f ) , (2)

for all e, e’, f, f’ ∈ E( S o ) with l(e) ≥ l(e’), r(f) ≥ r(f’). Then

{

W=W ( S o ;A,B, ∗ ) = (a, x, b ) : x ∈ S o , a ∈ Ar ( x ) , b ∈ Bl ( x )

is a regular semigroup under the multiplication (a,x,b) (c,y,d) = (aA(r ( x), r ( z ) ), z , dB(l ( y ), l ( z ) )) where z=x(b ∗ c)y.

} (2.1)

The map η : S o →W, xη = (r(x),x,l(x)) is an injective

homomorphism of S o to W. If we identify S o with S o η, via η, then S o (= S o η) = uWu, where u = (r(1),1,l(1)) is a regular idempotent of W. Conversely, every regular semigroup with a regular idempotent can be constructed in this way.

Proof. The associativity of the multiplication follows from (2). We first show

that η is an injective homomorphism. Clearly η is one-to-one. If x, y ∈ S o , then x(l ( x) ∗ r ( y )) y = xy because l ( x) ∗ r ( y ) ∈ l ( x)r ( y ) . This implies xη ⋅ yη = (r ( x), x, l ( x)) ⋅ (r ( y ), y, l ( y )) = (r ( xy ), xy, l ( xy )) = ( xy)η .


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Hence η is an injective homomorphism. We next show that W is a regular semigroup. Take any (a, x, b) ∈ W and let x * ∈ V ( x) ∩ S o . Then using (3) we get (a, x, b)(r ( x * ), x * , l ( x * ))(a, x, b) = (a, xx * , l ( x * ))(a, x, b) = (a, x, b) and (r ( x * ), x * , l ( x * ))(a, x, b)(r ( x * ), x * , l ( x* )) = (r ( x * ), x * x, b)(r ( x* ), x * , l ( x * )) = (r ( x* ), x* , l ( x* )) so that (r ( x * ), x * , l ( x * )) ∈ V (a, x, b). Hence W is a regular semigroup. Next we prove that u = (r(1),1,l(1)) is a regular idempotent of W. But this follows, since for each (a,x,b) ∈W, V ((r ( x), x, l ( x))) ⊆ V ((a, x, b)) ∩ S o , where Using (1) to (3), it is easy to verify that S o (= S o η) = uWu. ∗ ∗ ∗ (r ( x ), x , l ( x ))∈V ((a, x, b)) ∩ uWu. By Corollary 1.8. it follows that u = (r(1),1,l(1)) is a regular idempotent of W. Conversely, if u is a regular idempotent of S. Maintaining the notation of Theorem 2.5, let ( A, B) be the S o = uSu -pair. We define a B × A matrix * over

S o = uSu as follows. Fix an R-invariant map α : I → I so that α is constant on each R-class of I. Similarly, fix an L-invariant map β : Λ → Λ so that β is constant on each L-class of Λ . For each

λ ∈ Bl (e) , i ∈ Ar ( f ) define

λ * i = u ( β (λ )α (i ))u . Clearly * is well defined and it satisfies (1). We now show that * satisfies (2). Take any λ ∈ Bl (e) , i ∈ Ar ( f ) and let r ( f ) ≥ r ( f ' ) . Then i A(r ( f ), r ( f ' )) = if ' and λ * i = u ( β (λ )α (i ))u = ( β (λ )α (i )). Therefore e(λ * i A(r ( f ), r ( f ' )) f ' = e(λ * if ' ) f ' = e( β (λ )α (if ' )) f ' = e( β (λ )α (i )) f '

= e(λ * i ) f ' . Hence 2(i) is satisfied. A dual argument proves 2(ii). Now for any e, f ∈ E ( S o ) = E (uSu ), λ ∈ Bl (e) , i ∈ Ar ( f ) , l (e) * i = u ( β (e)α (i ))u = ( β (e)α (i )) = β (e)α (i )uα (i )u = β (e) uα (i )u ∈ l (e)r ( f ) .

Similarly, λ * r ( f ) ∈ l (e)r ( f ). Hence (3) holds. Now define θ : S → W by sθ = ( ss * , usu , s * s ) , s ∈ S , s * ∈ V (usu ) ∩ uSu. Then θ is bijective map with inverse χ : W → S given

by (i , x, λ ) χ = ixλ. The map θ is a homomorphism, since sθ ⋅ tθ = ( ss * , usu , s * s )(tt * , utu , t *t ) = ( ss * h, usu ( s * s * tt * )utu , kt *t )

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= ( ss * h, usu ( us * stt *u )utu , kt *t ) by Lemma 2.3 = ( ss * h, ustu , kt *t ) = ( st )θ where s * ∈ V (usu ) ∩ uSu , t * ∈ V (utu ) ∩ uSu , h ∈ I , k ∈ Λ with h R usu ( us * stt *u )utu L k The last step follows since h R usu ( us * stt *u )utu L k

implies ss * h Rss * usu ( us * stt *u )utu = stt *utu R st and ktt * L usu ( us * stt *u )utu t *t = usus * st L st . This implies that θ : S → W is an isomorphism. This completes the proof of the theorem. Note that an idempotent u is a medial idempotent of S if and only if uSu is an orthodox semigroup and for any g ∈< E ( S ) >, ugu is an idempotent[7]. In this case Theorem 2.7 reduces to the following Corollary 2.8. Let S o be an orthodox semigroup with identity 1 and let (A,B) be an S o -pair. Let ∗ be a BxA matrix over S o satisfying (1)

if b ∈ Bl (e) and a ∈ Ar ( f ) , then b ∗ a∈ l(e) S o r(f) ;

(i) e (b ∗ aA(r(f),r(f’)))f’ = e(b ∗ a)f’ (ii) e’(bB(l(e),l(e’) ∗ a)f = e’(b ∗ a)f (3) for any b ∈ Bl (e), a ∈ Ar ( f ) , b ∗ r ( f ), l (e) ∗ a ∈ l (e)r ( f ) , (2)

for all e, e’, f, f’ ∈ E( S o ) with l(e) ≥ l(e’), r(f) ≥ r(f’). Then

{

W=W ( S o ;A,B, ∗ ) = (a, x, b ) : x ∈ S o , a ∈ Ar ( x ) , b ∈ Bl ( x )

}

is a regular semigroup under the multiplication (a,x,b) (c,y,d) = (aA(r ( x), r ( z ) ), z , dB(l ( y ), l ( z ) )) The map η : S o →W, xη = (r(x),x,l(x)) is an injective homomorphism of S o to W. If we identify S o with S o η, via η, then S o (= S o η) = uWu, where u = (r(1),1,l(1)) is a medial idempotent of W.

where z=x(b ∗ c)y.

Conversely, every regular semigroup with a medial idempotent can be constructed in this way.

Remark 2.9. In the above Corollary 2.8, if we define the multiplication (a,x,b) (c,y,d) = (ae, xbcy, fd)


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when e R xbcy L f, then it reduces to Theorem 3.3. of [7]. We conclude this section with the following result. Theorem 2.10. Let S be a regular semigroup with identity 1 and let be the idempotent - generated part of S. Let M be an idempotent – generated regular semigroup with regular idempotent u such that ≅ uMu (assume E(S) ≠ E(M)). Then there exists a regular semigroup T with a regular idempotent u and S ≅ uTu such that the diagram E (S )

∩ M

⊆

S

⊆

∩ T

is a push-out in the category of regular semigroups. Proof: For x∈S, r(x) = E(S) ∩ Rx ⊆ E ( S ) , l(x) = E(S) ∩ Lx ⊆ E ( S ) and since M is an idempotent – generated regular semigroup with regular idempotent u such that ≅ uMu , by Theorem 2.7., take M=M ( E ( S ) , A,B, ∗ ) where ∗

is a B × A matrix over E ( S ) M=M

satisfying (1) – (3). So

( E (S ) ; A, B;∗) = {(i, x, λ ) : x ∈

}

E ( S ) , i ∈ Ar ( x ) , λ ∈ Bl ( x ) .

Let

(

)

⎫. T = T (S;A,B; ∗ ) = ⎧⎨ i ', x ′, λ ' : x ′ ∈ S , i ' ∈ Ar ( x ′) , λ ' ∈ B ⎬ l (x ′) ⎭ ⎩ Then T is a regular semigroup under the multiplication (2.1). The map η : S→T defined by xη= (r ( x), x, l ( x) ) is an injective homomorphism. If we identify S with S η, via η, then S (=Sη) = uTu, where u = (r(1),1,l(1)) is a regular idempotent of T. x∈ E ( S ) , x i1 η = xi2φ = (r(x),x,l(x)) and the diagram . For i1 E ( S ) ⎯⎯→ S

↓η

i2 ↓

M ⎯⎯→ T

φ

commutes. Here i1, i2 and φ denote the inclusion maps.

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Now let W be a regular semigroup and φ′ : M → W, η:S→ W be two homomorphisms such that i1η′=i2φ′. Define χ: T(S;A,B; ∗ ) → W by

(i, x, λ )χ = (i, xx*, l ( xx*))φ ′.(r ( x), x, l ( x))η ′.(r ( x * x), x * x, λ )φ ′ where x*∈V(x)∩S. Clearly the map is well defined. Now we show that the map χ is a homomorphism, ϕχ=ϕ′ and ηχ=η′. Take (i1 , x1 , λ1 ), (i2 , x 2 , λ 2 ) ∈ T . Then ((i1, x1, λ1 )(i2 , x2 , λ2 ))χ = (i1 , A(r ( x1 ), r ( x1.(λ1 ∗ i2 ).x 2 )), x1 (λ1 ∗ i2 ) x 2 , λ2 B (l ( x 2 ), l (x1 (λ1 ∗ i2 ) x 2 ))χ = (i1 , A(r ( x1 ), r ( z ), zz * l ( zz*) ))φ ′(r ( z ), z , l ( z ) )η ′(r ( z * z ), z * z , λ 2 B(l ( x 2 ), l ( z ) ))φ ′ = (i1 , x1 x1 *, l ( x1 x1*))φ ′(r ( zz*), zz*, l ( zz*))φ ′(r ( z ), z , l ( z ) )η ′ (r ( zz*), z * z, l ( z * z ) )φ ′(r ( x2 * x2 ), x2 * x2 , λ2 )φ ′ (2.2) = (i1 , x1 x1 *, l ( x1 x1*) )φ ′(r ( z ), z , l ( z ) )η ′(r ( x 2 * x 2 ), x 2 * x 2 , λ 2 )φ ′ and

(i1, x1, λ1 )χ (i2 , x2 , λ2 )χ = (i1 , x1 x1 *, l ( x1 x1*))φ ′(r ( x1 ), x1 , l ( x1 ) )η ′(r ( x1 * x1 ), x1 * x1 , λ1 )φ ′ (i2 , x2 x2 *, l ( x2 x2 *))φ ′(r ( x2 ), x2 , l ( x2 ))η ′(r ( x2 * x2 ), x2 * x2 , λ2 )φ ′ = (i1 , x1 x1 *, l ( x1 x1*))φ ′(r ( x1 ), x1 , l ( x1 ) )η ′(r ( x1 * zx 2 *), x1 * zx 2 *, l ( x1 * zx 2 *))φ ′ (r ( x2 ), x2 , l ( x2 ) )η ′(r ( x 2 * x 2 ), x 2 * x2 , λ2 )φ ′ = (i1 , x1 x1 *, l ( x1 x1*) )φ ′(r ( z ), z , l ( z ) )η ′(r ( x 2 * x 2 ), x 2 * x 2 , λ 2 ) )φ ′ (2.3) where z = x1 (λ1 ∗ i2 )x 2 , x* ∈ V ( x) ∩ S ≅ uWu , z* ∈ V ( z ) ∩ S ≅ uWu . By (2.2) and (2.3), it follows that, χ is a homomorphism. Take (i, x, λ )∈ M , where x ∈ E ( S ) . Then

(i, x, λ )φχ = (i, x, λ )χ


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( ) ( ) = (i, xx*, l ( xx*) )φ ′(r ( x), x, l ( x) )φ ′(r ( x ), x, l ( x ) )φ ′(r ( x * x), x * x, λ )φ ′ = (i, x, λ )φ ′

= i, xx*, l ( xx*) φ ′(r ( x), x, l ( x) )η ′ r ( x * x), x * x, λ φ ′

implies that φχ = φ ′ , where x*∈V(x)∩S. Now take (r ( x), x, l ( x) ) in S(=Sη) ≅ uWu and x*∈V(x)∩S ≅ uWu then

(r ( x), x, l ( x) )ηχ = (r ( x), x, l ( x) )χ = (r ( x), xx*, l ( xx*) )φ ′(r ( x), x, l ( x) )η ′(r ( x * x ), x * x, l ( x ) )φ ′ = (r ( x), xx*, l ( xx*))η ′(r ( x), x, l ( x) )η ′(r ( x * x), x * x, l ( x) )η ′ = (r ( x), x, l ( x) )η ′ implies that ηχ=η′. Hence the diagram is a push out in the category of regular semigroups. Acknowledgement. The authors wish to thank Professor M. Loganathan for his encouragement and help during the preparation of this paper.

References [1] T.S.Blyth and R.B.McFadden, Naturally ordered regular semigroups with a greatest idempotent, Proc. Roy. Soc. Edinburgh, Vol.91A (1981),107-122. [2] T.S.Blyth and M.H.Almeida Santos, Naturally ordered regular semigroups with inverse monoid transversal, Semigroup Forum, Vol.76 (2008),71-86. [3] T.S.Blyth and R.B.McFadden, On the construction of a class of regular semigroups, Journal of Algebra, Vol.81 (1983), 1-22. [4] V. M. Chandrasekaran and M.Loganathan, Split Map and Idempotent Separating Congruence, Int. J. App. Math and Comp.,Vol.12 (2005), 351-360. [5] A.H.Clifford and G,B,Preston, The algebraic theory of semigroups, Math. Surveys no.7, Amer. Math. Soc., Providence, R.I., Vol.1, 1961, Vol.2, 1967. [6] K. Indhira and V. M.Chandrasekaran, Regular semigroups with a regular idempotent, International Conference on Mathematical Methods and Computation Tiruchi, 1-5, 2009

[7] M. Loganathan, Regular semigroups with a medial idempotent, Semigroup Forum, Vol.36 (1987), 69-74. [8] M. Loganathan and V.M. Chandrasekaran, Regular semigroups with a split map, Semigroup Forum, Vol. 44 (1992),199-212.

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[9] D.B. McAlister and R.B. McFadden, Regular semigroups with inverse transversals, Quart.J.Math. Oxford (2), Vol.34 (1983),459-479. [10] K.S.S.Nambooribad, Structure of regular semigroups I, Mem. Amer. Math. Soc., Vol.22, No.224, 1979. Received: September, 2010