*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 Yang*1

*2*

**, Zhenlin Gao***3*

**Hai-zhen Liu***4 1*

**and Jing Zhang**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*=*ay*⇔*bx*=*by*;

( , )*a b* ∈R∗⇔ (∀*x y*, ∈*S*1) *xa*= *ya*⇔ *xb* =*yb*;

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

where*J*∗( )(*x* *x*∈*S*)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 f*or any a*∈*S*, *

( *a* )

The following basic conclusions are needed throughout this paper.

**Lemma 1.1. Let ** *S* be a semigroup, for any*a*∈*S*,*e*∈*E S*( )the following statements
are equivalent:

(1) *a*L ∗*e*(*a *R∗*e*)；

(2) *ae*=*a ea*( =*a*), and for any*s t*, ∈*S*1,*as*=*at sa*( =*ta*)implies*es*=*et se*( =*te*).

**Lemma 1.2. (1)***L*and*L*∗both are right congruences on semigroup*S*;

(2)*R*⊆*R*∗，*L*⊆*L*∗，*D*⊆*D 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* *J*_{α}

α∈

= ∪ which rest with the

*semilattice Y , whereJ*_{α}

### (

∀ ∈α*Y*

### )

are*J*-classes of B.

(2) Left regular band*I*is a semilattice of left zero bands*L*_{α}, where*L*_{α}are*L*-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*=*H*∪*K*,ρ*H*
andρ*K*

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

### ρ

*H*and

### ρ

*K*as follows:

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

*H* *K*

*H K*

*H* *K* _{a b}_{S}_{S}_{a b}_{a b}

ρ =ρ ∪ρ = ∈ × ∈ρ ∈ρ . (2.1)

If

### ρ

*H*and

### ρ

*K*satisfy the following conditions:

( , )*a b* ∈ρ*H*, ( , )*b c* ∈ρ*K* ⇒( , )*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 :

121 ( , )|

*H*

*H K* *H*

ρ =ρ or ρ*K* =ρ_{(}_{H K}_{,} _{)}|* _{K}*； (2.5)
(2)

### ρ

_{(}

_{H K}_{,}

_{)}is a congruence on

*S*if and only if

### ρ

*H*and

### 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

*a*∈

*H*, ( , )

*a a*∈ρ

_{(}

_{H K}_{,}

_{)}implies ( , )

*H*

*a a* ∈ρ , thus the relation ρ*H*

is reflexive.

Let ( , )*a b* ∈ρ*H*, by formula (2.1) we have( , )*a b* ∈

### ρ

_{(}

*H K*

_{,}

_{)}, thus( , )

*b a*∈ρ(

*H K*, ), namely

( , )*b a* ∈ρ*H*by ( , )*b a* ∈ ×*H* *H*and formula (2.1), this shows that the relation ρ*H* is
symmetric. Similarly, we have

( , ) ( , )

( , ), ( , )*a b* *b c* ∈ρ*H* ⇒( , ),( , )*a b* *b c* ∈ρ*H K* ⇒( , )*a c* ∈ρ*H K* ⇒( , )*a c* ∈ρ*H*.
So we conclude that ρ*H*

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

*is an equivalent relation on K . *
⇐. Let ρ ρ*H*, *K*

be equivalent relations on *H* and *K* respectively, for any*a*∈*S*,

*then a*∈*H* *or a*∈*K* . Hence ( , )*a a* ∈ρ*H* or ( , )*a a* ∈ρ*K* , 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 b* ∈
*H*

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* ∈ρ*H*; (ii)( , )*a b* ,( , )*b c* ∈ρ*K*;
(iii) ( , )*a b* ∈ρ*H*,( , )*b c* ∈ρ*K*; (iv) ( , )*a b* ∈ρ*K*, ( , )*b c* ∈ρ*H*.

For case (i) (resp, case(ii)),we know ( , )*a c* ∈ρ*H*(resp, ( , )*a c* ∈ρ*K*), 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 on*S*. 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 on

*S*. Let( , )

*a b*∈ρ(

*H K*, )

**, then ( , )**

*a b*∈

*H*

### ρ

**or**

( , )*a b* ∈

### ρ

*K*. Now we consider the situation ( , )

*H*

( , )*a b* ∈ρ*H* and

### ρ

*His a congruence on H , thus we have (xa xb*, )∈ρ

*H*, (

*ax bx*, )∈

### ρ

*H*. If

*x*∈

*K*, 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*, )∈

### ρ

(*H K*, ).

Thus

### ρ

_{(}

_{H K}_{,}

_{)}is a congruence on

*S*.

⇒. Let ( , ) *H*

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

### ρ

_{(}

_{H K}_{,}

_{)}is a congruence on

*S*, 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 ρ*H*is a congruence on*H* .

In a similar way, we also can obtain that ρ*K*

is a congruence on*K*.

**Definition 2.2. ** If the relation ( , )

*H* *K*

*H K*

ρ =ρ ∪ρ defined by formula (2.1) is a
congruence on semigroup*S*,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 ofS*and *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*)∪1* _{S}*determined 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 congruence**ρon semigroup *S* can be expressed as a

proper transitive compatible pair congruence

### ρ

_{(}

_{H K}_{,}

_{)}on

*S*.

* Proof: Let 1S* ≠ ⊂ ×ρ

*S*

*S*.Now, we shall prove that ρcan be expressed as a proper transitive compatible pair congruence

### ρ

_{(}

_{H K}_{,}

_{)}on

*S*.

Denote the zero element of *S*

### ρ

by__

0 , let

__

0 ={_{ρ} *a*∈*S a*| ∈0 },

__ __

={ | 0 } { 0 }
*K* *x*∈*S* ≠ ∈*x* *S* ρ ∪ ,
__

0 { | is not a zero divisor of }

*K* = *x*∈*K x* *S* ρ , *H*=0_{ρ}∪(*K K*\ _{0}).

If*K*0≠∅, then for any*x y*, ∈*K*0,

_ _ ___ __

0

123

0

*K is a proper subsemigroup ofS*without zero divisors. Since for any*a*∈0_{ρ},*x*∈*S*,

__ _ _ ___

0= ⋅ =*a x* *ax*, therefore we have *ax*∈0_{ρ}. Similarly, we obtain*xa*∈0_{ρ}. Hence we can
deduce that 0_{ρ}*is a proper ideal of S .* *Next, we shall prove that H* is a subsemigroup

of*S*.For any*a b*, ∈*H*, we distinguish two cases as follows.
**Case 1. *** At least one of elements a and * *b* belong to 0_{ρ};

**Case 2. Elements ** *a* and *b*both are in*K K*\ 0, i.e. *a b*, ∈*K K*\ 0.

For the former, since 0_{ρ}is an ideal of*S*, then*ab*∈0_{ρ} ⊆*H*. For the latter, let*a b*, ∈*K*

0

*\K* , if

___ __

0
*ba*= (or

___ __

0

*ab*= ), then*ba*(or*ab*)∈*K K*\ _{0}. If

___ __ ___ __

0, 0
*ba*≠ *ab*≠ , since

__ __

0
*a*≠ is a

*zero divisor in S* ρ, then there exists*x*∈*K K*\ _{0}such 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

that*His a subsemigroup of S . In particular, letK*_{0}= ∅, then *K*=*K K*\ _{0} is a proper
*subsemigroup of S*.

Now we distinguish the following two situations.

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

(0_{ρ},*K*) ( |0_{ρ}) ( |*K*)

ρ = ρ ∪ ρ * _{is a proper transitive compatible pair congruence on S}* and

(*0 ,K*_{ρ} )

ρ ρ= _{. }

(ii) If *K*_{0}≠ ∅ , we shall prove that

0

(*H K*_{, 0}) ( | )*H* ( |*K* )

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

### ρ

|*and*

_{H}0
|*K*

### ρ

*are congruences on H*and

0

*K* respectively. Since*H*∩*K*0= ∅, thus formula (2.2) naturally holds.

Let ( , )*a b* ∈ρ|* _{H}, for any x*∈

*S, then x*∈

*H*or

*x*∈

*K*0

*. If x*∈

*H*, we have (

*xa xb*, ),(

*ax*,

) |_{H}

*bx* ∈ρ . If*x*∈*K*0, 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* . If*xa*∈*K*_{0}, letϑ:*S*→*S* ρ ϑ,*x* =*x*ρbe the natural homomorphism.

We have 1

0

(( ) )

*xb*∈ *xa* ρ ϑ− ⊆*K* by (*xa*)ρ=(*xb*)ρ, hence*xb*∈*K*_{0}*. If xa*∈*H*, we also can
*get xb*∈*H*using the same means as above. Thus, we deduce that either (*xa xb*, )∈ ×*H* *H*
or(*xa xb*, )∈*K*0×*K*0. Since (*xa xb*, )∈ρ, then either (*xa xb*, )∈ρ|*H*or (*xa xb*, )∈ρ|*K*_{0}

by (*xa xb*, )∈ ×*H* *H*or(*xa*,*xb*)∈*K*_{0}×*K*_{0}. Consider the other case (*ax bx*, )∈ρ*, x*∈*S*,we
also obtain that either (*ax bx*, )∈ρ|* _{H}* or

0

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

0

Finally, we will prove _{(}

0

, )

*H K*

ρ ρ= .Suppose that( , )*a b* ∈ρ, if ( , )*a b* ∉ρ|_{K}_{0}, since

0

*H*∩*K* = ∅ , then ( , )*a b* ∈ ×*H* *H* and consequently ( , )*a b* ∈ρ|* _{H}* . If ( , )

*a b*∉ρ|

*, similarly we can get*

_{H}0

( , )*a b* ∈ρ|* _{K}* . This fact implies

_{(}

_{)}0 ,

*H K*

### ρ ρ

⊆ . On the contrary, let( , _{0})

( , )*a b* ∈

### ρ

*, then either ( , )*

_{H K}*a b*∈ρ|

*or*

_{H}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 on*S* .
*Concretely speaking, let H and K be subsemigroups of semigroup S , for a congruence *

ρ* on S , so long as * (

*H K*, ) is a transitive compatible pair of ρ, then ρ can be represented as ρ ρ=

_{(}

_{H K}_{,}

_{)}=( | )ρ

*∪( | )ρ*

_{H}*.*

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