# Transitive Compatible Pair Congruence

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Vol. 9, No. 1, 2015, 119-124

ISSN: 2279-087X (P), 2279-0888(online) Published on 30 January 2015

www.researchmathsci.org

119

### Transitive Compatible Pair Congruence

Pei Yang1, Zhenlin Gao2 Hai-zhen Liu3 and Jing Zhang4 1

College of Science, University of Shanghai for Science and Technology Shanghai-200093, China. Email: [email protected]

2

College of Science, University of Shanghai for Science and Technology Shanghai-200093, China. Email: [email protected]

3

College of Science, University of Shanghai for Science and Technology Shanghai-200093, China. Email: [email protected]

4

College of Science, Dalian University of Technology Dalian-116024, China. Email: [email protected]

Corresponding author: Pei Yang

Received 19 January 2015; accepted 28 January 2015

Abstract. In this paper, we introduce the concepts of transitive compatible pair congruence

### ρ

(H K, ). Prove that any congruence on a semigroup S can be expressed as a transitive compatible pair congruence.

Keywords: Semigroup ; Transitive compatible pair; congruence AMS Mathematics Subject Classification (2010): 20M10 1. Introduction and preliminaries

The Green's star relations L∗， ， ， ，R∗ D∗ H,∗ J ∗on semigroup S are defined as follows.

( , )a b ∈L∗⇔ 1

(∀x y, ∈S ) ax=aybx=by;

( , )a b ∈R∗⇔ (∀x y, ∈S1) xa= yaxb =yb;

D∗=L ∗∨ R∗ ，H ∗=L∗∧ R∗，a J ∗bJ∗( )a =J b∗( ).

whereJ∗( )(x xS)represents the minimalJ ∗-ideal generated by x.

Definition 1.1. Assume that P represents a kind property about semigroups, if the semigroup S has P property, then S is called a P semigroup. A congruence ρ on the semigroup S is called P congruence, if S ρ is a P semigroup.

We shall call a semigroup S a rpp semigroup, if every L∗-class of S contains idempotents of S and for any aS, *

( a )

(2)

The following basic conclusions are needed throughout this paper.

Lemma 1.1. Let S be a semigroup, for anyaS,eE S( )the following statements are equivalent:

(1) aL ∗e(a R∗e)；

(2) ae=a ea( =a), and for anys t, ∈S1,as=at sa( =ta)implieses=et se( =te).

Lemma 1.2. (1)LandL∗both are right congruences on semigroupS;

(2)RR∗，LL∗，DD H∗, ⊆H J∗, ⊆J∗； (3)R |Re ( )g S R L, |Re ( )g S L

==

.

Lemma 1.3. (1) Band B has a semilattice decomposition Y

B Jα

α∈

= ∪ which rest with the

semilattice Y , whereJα

∀ ∈α Y

### )

areJ-classes of B.

(2) Left regular bandIis a semilattice of left zero bandsLα, whereLαareL-classes of I.

2. Transitive compatible pair congruences

From this section, we always assume that semigroup S mentioned below contains zero element, unless otherwise stated.

Definition 2.1. Let H and K be two subsemigroups of semigroup S ,such that

S=HKH andρK

be two relations on H and K respectively. We define a relation on S by

Hand

### ρ

Kas follows:

( , ) { ( , ) | ( , ) or ( , ) }

H K

H K

H K a b S S a b a b

ρ =ρ ∪ρ = ∈ × ∈ρ ∈ρ . (2.1)

If

Hand

### ρ

Ksatisfy the following conditions:

( , )a b ∈ρH, ( , )b c ∈ρK ⇒( , )a c ∈ρ(H K, ) ; (2.2)

( )( , ) ( , ( , ) )

( , ) or ( , ) ,

(2.3) ( )( , ) , ( , ) ;

H K

H K

H K

x S a b resp a b

xa xb xa xb

or c H K xa c c xb

ρ ρ

ρ ρ

ρ ρ

∀ ∈ ∈ ∈

 ⇒ 

∃ ∈ ∩ ∈ ∈



( , ) or ( , ) ,

( )( , ) , ( , )

### ,

(2.4)

H K

H K

ax bx ax bx

or c H K ax c c bx

ρ ρ

ρ ρ

 

∃ ∈ ∩ ∈ ∈



then,(H K, )is called a transitive compatible pair of

### ρ

(H K, ).

Theorem 2.1. Let ( ,H K be a transitive compatible pair of)

### ρ

(H K, )defined as Definition 2.1. Then the following conclusions hold :

(3)

121 ( , )|

H

H K H

ρ =ρ or ρK(H K, )|K； (2.5) (2)

### ρ

(H K, ) is a congruence on S if and only if

Hand

### ρ

Kare congruences on H and

K respectively.

Proof: (1)⇒. Assume that the relation

### ρ

(H K, ) is an equivalent relation on S, for any aH , ( , )a a ∈ρ(H K, ) implies ( , ) H

a a ∈ρ , thus the relation ρH

is reflexive.

Let ( , )a b ∈ρH, by formula (2.1) we have( , )a b

### ρ

(H K, ), thus( , )b a ∈ρ(H K, ), namely

( , )b a ∈ρHby ( , )b a ∈ ×H Hand formula (2.1), this shows that the relation ρH is symmetric. Similarly, we have

( , ) ( , )

( , ), ( , )a b b c ∈ρH ⇒( , ),( , )a b b c ∈ρH K ⇒( , )a c ∈ρH K ⇒( , )a c ∈ρH. So we conclude that ρH

is an equivalent relation on H . Also, using the same argument as above, we obtain that ρK

is an equivalent relation on K . ⇐. Let ρ ρH, K

be equivalent relations on H and K respectively, for anyaS,

then aH or aK . Hence ( , )a a ∈ρH or ( , )a a ∈ρK , this implies ( , )a a ∈ρ(H K, )by

formula (2.1), thus the relation ρ(H K, ) is reflexive. Let( , )a b ∈ρ(H K, ), then ( , )a bH

ρ or ( , ) K

a b ∈ρ by formula (2.1). Hence, we can infer that ( , ) H

b a ∈ρ or ( , ) K b a ∈ρ , this fact shows( , )b a ∈ρ(H K, ).Namely,ρ(H K, )is symmetric. Next, we shall illustrate that

(H K, )

### ρ

istransitive. Let( , ), ( , )a b b c ∈ρ(H K, ), according to formula (2.1), there exist four

cases as follows:

(i) ( , ),( , )a b b c ∈ρH; (ii)( , )a b ,( , )b c ∈ρK; (iii) ( , )a b ∈ρH,( , )b c ∈ρK; (iv) ( , )a b ∈ρK, ( , )b c ∈ρH.

For case (i) (resp, case(ii)),we know ( , )a c ∈ρH(resp, ( , )a c ∈ρK), it allows that ( , )a c

(H K, )

ρ

∈ . Now we considers the last two cases. Since ( , )H K is a transitive compatible pair of

### ρ

(H K, ), by formula (2.2) we have( , )a c ∈ρ(H K, )by formula(2.1). Summing up the

above cases, we know that the relation

### ρ

(H K, )is transitive. Consequently, the relation

(H K, )

### ρ

is an equivalent relation onS. Finally, we shall prove formula (2.5). Clearly,

( , )|

H

H K H

ρ ⊆ρ , ( , )|

K

H K K

ρ ⊆ρ . We assume that ( , )|

K

H K K

ρ ⊂ρ , then we will illustrate ( , )|

H

H K H

ρ =ρ . Since

( , )| ( ( , )|( ))

K

H K K H K H K

ρ =ρ ∪ ρ ∩ , thus according to Definition 2.1 we have that

( , ) ( ( , )| ) ( ( , )| ) ( ( , )| ) ( , )

K H K

H K H K H H K K H K H H K

ρ = ρ ∪ ρ ⊇ ρ ∪ρ ⊇ρ ∪ρ =ρ . So, we can include that ( , )|

H

H K H

ρ =ρ , and ( , )|( ) H

H K H K

ρ ⊇ρ ∩ .

(2)⇐.

### ρ

(H K, )is an equivalent relation onS. Let( , )a b ∈ρ(H K, ), then ( , )a b

H

or

( , )a b

### ρ

K. Now we consider the situation ( , ) H

(4)

( , )a b ∈ρH and

### ρ

His a congruence on H , thus we have (xa xb, )∈ρH, (ax bx, )∈

### ρ

H. If xK, by formula (2.3) given in Definition 2.1, we have

( , ) H or ( , ) K, or ( )( , ) H,( , ) K xa xb ∈ρ xa xb ∈ρ ∃ ∈ ∩c H K xa c ∈ρ c xb ∈ρ . Hence, we obtain (xa xb, )∈

### ρ

(H K, ). Similarly, we have(ax bx, )∈

### ρ

(H K, ).

For the situation( , )a b

### ρ

K, similarly, we also have(xa xb, ), (ax bx, )∈

(H K, ).

Thus

### ρ

(H K, ) is a congruence onS.

⇒. Let ( , ) H

a b ∈ρ , then( , )a b ∈ρ(H K, ). ForhH, since H is a subsemigroup of S , thus (ha hb, ), (ah bh, )∈ ×H H.By the given condition,

### ρ

(H K, ) is a congruence onS, we

immediately have (ha hb, ), (ah bh, )∈

### ρ

(H K, ) , therefore ( , ), ( , ) H ha hb ah bh ∈ρ by

formula(2.1) and (ha hb, ), (ah bh, )∈ ×H H. This shows that ρHis a congruence onH .

In a similar way, we also can obtain that ρK

is a congruence onK.

Definition 2.2. If the relation ( , )

H K

H K

ρ =ρ ∪ρ defined by formula (2.1) is a congruence on semigroupS,then we call

### ρ

(H K, )a transitive compatible pair congruence

onS.For the transitive compatible pair congruence ( , )

H K

H K

ρ =ρ ∪ρ onS, if at least one

of H and K is a proper subsemigroup ofSand H 1 K 1

Hor K

ρ ≠ ρ ≠ ,then we call

### ρ

(H K, )a

proper transitive compatible pair congruence onS,and (H K, )a proper transitive compatible pair of

### ρ

(H K, ).Otherwise, we call

### ρ

(H K, )trivial.

Obviously, by Definition 2.2, the Rees congruence =(ρ H×H)∪1Sdetermined by a proper ideal is the proper transitive compatible pair congruence

### ρ

(H S, )on S and ( , )H S is a proper transitive compatible pair ofρ.In order to study a nontrivial congruenceρ, we need to obtain some proper transitive compatible pair congruence representation of

ρ. So, we need to prove the following theorem.

Theorem 2.2. For any nontrivial congruenceρon semigroup S can be expressed as a

proper transitive compatible pair congruence

### ρ

(H K, )onS.

Proof: Let 1S ≠ ⊂ ×ρ S S.Now, we shall prove that ρcan be expressed as a proper transitive compatible pair congruence

### ρ

(H K, )onS.

Denote the zero element of S

### ρ

by

__

0 , let

__

0 ={ρ aS a| ∈0 },

__ __

={ | 0 } { 0 } K xS ≠ ∈x S ρ ∪ , __

0 { | is not a zero divisor of }

K = xK x S ρ , H=0ρ∪(K K\ 0).

IfK0≠∅, then for anyx y, ∈K0,

_ _ ___ __

0

(5)

123

0

K is a proper subsemigroup ofSwithout zero divisors. Since for anya∈0ρ,xS,

__ _ _ ___

0= ⋅ =a x ax, therefore we have ax∈0ρ. Similarly, we obtainxa∈0ρ. Hence we can deduce that 0ρis a proper ideal of S . Next, we shall prove that H is a subsemigroup

ofS.For anya b, ∈H, we distinguish two cases as follows. Case 1. At least one of elements a and b belong to 0ρ;

Case 2. Elements a and bboth are inK K\ 0, i.e. a b, ∈K K\ 0.

For the former, since 0ρis an ideal ofS, thenab∈0ρH. For the latter, leta b, ∈K

0

\K , if

___ __

0 ba= (or

___ __

0

ab= ), thenba(orab)∈K K\ 0. If

___ __ ___ __

0, 0 baab≠ , since

__ __

0 a≠ is a

zero divisor in S ρ, then there existsxK K\ 0such that

_ _ ___ __

0 a x=ax= (or

_ _ __

0 x a= ).

Therefore

___ __ __ ___ __

0 ba x =b ax= (or

__ ___ ___ __ __

0

x ab=xa b = ), and consequently

___ __

0 ba≠ (or

___ __

0 ab≠ ) is a zero divisor in S ρ. We immediately have ba (or ab )K K\ 0. Hence, we obtain

thatHis a subsemigroup of S . In particular, letK0= ∅, then K=K K\ 0 is a proper subsemigroup of S.

Now we distinguish the following two situations.

(i) IfK0= ∅, then H=0ρ∪K=S. Since 0ρ is a proper ideal of S , it is obvious that

(0ρ,K) ( |0ρ) ( |K)

ρ = ρ ∪ ρ is a proper transitive compatible pair congruence on S and

(0 ,Kρ )

ρ ρ= .

(ii) If K0≠ ∅ , we shall prove that

0

(H K, 0) ( | )H ( |K )

ρ = ρ ∪ ρ is a proper transitive compatible pair congruence on S . Clearly,

|Hand

0 |K

### ρ

are congruences on H and

0

K respectively. SinceHK0= ∅, thus formula (2.2) naturally holds.

Let ( , )a b ∈ρ|H, for any xS, then xH orxK0. If xH, we have (xa xb, ),(ax,

) |H

bx ∈ρ . IfxK0, since ( , )a b ∈ρ|H, then ( , )a b ∈ρ, and therefore (xa xb, )∈ρ, (ax bx, )∈ρ. For the case (xa xb, )∈ρ, we will illustrate that either (xa xb, )∈ ×H H or

0 0

(xa,xb)∈K ×K . IfxaK0, letϑ:SS ρ ϑ,x =xρbe the natural homomorphism.

We have 1

0

(( ) )

xbxa ρ ϑ− ⊆K by (xa)ρ=(xb)ρ, hencexbK0. If xaH, we also can get xbHusing the same means as above. Thus, we deduce that either (xa xb, )∈ ×H H or(xa xb, )∈KK0. Since (xa xb, )∈ρ, then either (xa xb, )∈ρ|Hor (xa xb, )∈ρ|K0

by (xa xb, )∈ ×H Hor(xa,xb)∈K0×K0. Consider the other case (ax bx, )∈ρ, xS,we also obtain that either (ax bx, )∈ρ|H or

0

(ax bx, )∈ρ|K using the similar arguments. Similarly, for the situation

0

(6)

Finally, we will prove (

0

, )

H K

ρ ρ= .Suppose that( , )a b ∈ρ, if ( , )a b ∉ρ|K0, since

0

HK = ∅ , then ( , )a b ∈ ×H H and consequently ( , )a b ∈ρ|H . If ( , )a b ∉ρ|H , similarly we can get

0

( , )a b ∈ρ|K . This fact implies ( ) 0 ,

H K

### ρ ρ

⊆ . On the contrary, let

( , 0)

( , )a b

### ρ

H K , then either ( , )a b ∈ρ|H or

0

( , )a b ∈ρ|K , namely ( , )a b ∈ρ . To

sum up, we know ( ) 0 , = H K

### ρ ρ

.

By Theorem 2.2, we know that transitive compatible pair congruence on semigroup

S is a kind of universal representation method fitting to any congruence onS . Concretely speaking, let H and K be subsemigroups of semigroup S , for a congruence

ρon S , so long as (H K, ) is a transitive compatible pair of ρ, then ρ can be represented as ρ ρ= (H K, )=( | )ρ H ∪( | )ρ K .

REFERENCES

1. Howie J. M. An introducation to semigroup theory, London. Academic Press, 1976.

2. ZhenLin Gao, On the Lest Property of the Semilattice Congruences on po- Semigroups, Semigroup Forum, 56 (1998) 323-333.

3. Zhu Pinyu, Guo Yuqi, Structure and characterizations of left Clifford semigroups, Sci. China, 21(6) (1991) 582- 590.

4. J.B.Fountain, Right pp monoids with central idempotents, Semigroup Forum, 13 (1977) 229-237.

5. K.P.Shum and X.M.Ren, The structure of right C-rpp semigroups, Semigroup Forum, 68 (2004) 280-292.

6. T.Vasanthi and M.Amala, Some special classes of semirings and ordered semirings, Annals of Pure and Applied Mathematics, 4(2) (2013) 145-159.

7. T.Vasanthi and N.Sulochana, On the additive and multiplicative structure of semirings, Annals of Pure and Applied Mathematics, 3(1) (2013) 78-84.

References

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