01_08.pdf - Khulna University of Engineering & Technology

STANDARD IDEALS AND HOMOMORPHISM KERNEL OF A LATTICE Md. Abul Kalam Azad¹, Md. Mizanur Rahman² and Md. Zaidur Rahman¹

1 Department of Mathematics, Khulna University of Engineering & Technology

2 Department of Mathematics, Rajshahi Govt. City College, Rajshahi

Received: 20 November 2010 Accepted: 15 December 2010 ABSTRACT

A meet semilattice together with the property that any two elements possessing a common upper bound have a suprimum. Standard ideal had extended to convex sub lattice by [7] & [11]. Here we generalize the characterization of standard ideals and also generalize the standard ideals, Homomorphism kernels and Isomorphism theorems. For any two lattice L₁ and L₂, a map :L₁L₂ is called an isotone, if for any

L y

x,  , with xy implies 

   

x  y . Also the above mapping is called a meet homomorphism if for all,



x y

    

 x  y

    . Therefore meet homomorphism is an isotone. And since  is isotone.

 

x 

  

y  x y



   . Therefore 

 

x 

 

y exist by upper bound property of L₂ [2] & [3] have characterize those lattice whose all congruence are standard and neutral. Here we generalize characterization of this lattice whose all congruence are standard.

1. INTRODUCTION

This Paper studies extensively standard ideal and homomorphism kernels. The idea of standard ideals in lattice was first introduced by [8], [10]. It had extended the ideal to convex sub lattices and proved many result on homomorphism by [7], [11] also [4] and [5].

A congruence  of a lattice L is called standard if _{ s} for some standard ideal S of L. For any two lattice L1and L₂, a map :L₁L₂ is called an isotone if for any x,yL, with xy implies 

   

x  y , also the above mapping is called a meet homomorphism if for all x,yL₁, 



xy

    

 x  y . Therefore meet homomorphism is an isotone. And since  is an isotone.

    

x  y  xy



   . Therefore 

   

x  y exsit by upper bound property of L₂.

Latif in his thesis have introduce the concept of standard n-ideals of a lattice. We conclude this paper with some more properties of standard and neutral ideals. We also extended the result of Cornish and A.S.A Noor [1] , we also show that If I is an arbitrary ideal and S is standard ideal then IS and IS are principal, then I itself principal.

Secondly we have discussed Homomorphism, kernel and stadard ideals. Gratzer and Schmitd in [8] were translate several thorem of Group theory to lattice theory. Here we have generalized so of their result, we have shown that if S is a standard ideal of a lattice L, then _s the extension of 

 

S to I

 

L and 

 

S is the restriction of _S to the lattice L. Then we have shown that in a sectionally complemented lattice all congruences are standard. We also show that in a relatively complemented lattice L with 0, if every standard ideal of L is generated by a finite number of standard elements, then the congruence lattice C

 

L is Boolean.

Finally we have generalized two results of [2] and [3] regarding lattices all of whose congruence are standard.

We know that the set of all standard ideals of a lattice L is a sub lattice of I

 

L . Also the congruence _S where S is standard form a sub lattice of 



I

 

L



, and S_S is an isomorphism. Suppose  is a congruence relation of L.  defines in the natural way a homomorphism of I

 

L under which I J



I,JI

 

L



if and only if to any xI there exists a,bJ such that x y

 

 and conversely. We call this congruence relation the extension of  to I

 

L .

On the otherhand any congruence relation  of I

 

L induces a congruenc relation of L under which xy if and only if



x







y

  

 . This is called the restriction of  to L.

Thirdly in [8] Gratzer and Schmidt have proved Isomorphism theorem for standard ideal in lattices. In their paper they have translated several theorems of group theory to lattice theory using ideal, standard ideal, factor group and group operation. Here we generalze isomorphism theorem.

an International Journal

* Corresponding author: azadmath.azad8@gmail.com KUET@JES, ISSN 2075-4914/01(2), 2010

We refer the reader to [3] [4] [5] [6] for a necessary background on this paper.

Theorem 1.1: Let S be an ideal in a lattice L. Then the following conditions are equivalent.

(i) S is a standard ideal.

(ii) The binary relation _{ S} , defined by xy_{ S} if and only if

x



x y

 

 xa



y



x y

 

 yb



for some a,bS is a congruence relation.

iii) The binary relation  defined by xy

 

 , if and only if for all tL,



xt

 

 tc

 

 yt

 

 tc



for some cS is a congruence .

iv) For each ideal K,

S  K  s  k

: sk exists, and sS and kK. Moreover, (ii) and (iii) represent the same congruence, viz. 

 

S , the smallest congruence of L having S as congruence class.

Proof:

(i) implies (ii). If (i) holds, then the relation JK

 

_S



J,KI

 

L



if and only if J 



JK

 

 JS



and K 



JK

 

 KS



is a congruence on I

 

L .

Then _S/L restriction to L is a congruence on L and x y

 

_S /L if and only if

  

x  x y

 

 xS



and



y







xy

 





y



S



. Thus to prove (ii), it is sufficient to prove that



x







xy





 

x



S



implies



x y

 

x a



x    for some aS. This is proved by induction. By the property of the supremum of two ideals,

      

^

0









x Ln

S x y

x Where L₀



xy





 

x



S



and L_n



tL:t pq;pq exists and

1



,qA_n

p for n=1,2,...n. Indeed, we show by induction that



xy





 

x



S

 

 t:t



x y







xa



for some aS



. If tL then t



x y



or t



x



S. In the first instance, t xy



xy

 

 xs



for any sS and in the second instance



x y

 

x t



x t

t      and tS. Thus the result holds for n0. Suppose the result hold for n-1 for some n1. Let tL, then tpq with p,qL_n1 so p



xy

 

 xs₁



and



x y

 

x s₂



q    for some s₁,s₂S.

Then t



x y

 

 xs₁

 

 xs₂

 

 xy

 

 xs



.

For some sS (Since



xs₁

 

 xs₂



s and is in S, it is of the form



xs



for some sS. Thus, we have



xy





 

x



s







t:t



xy

 

 xs



for some sS; in effect , x



xy

 

 xa



for some

S

a and so x



xy

 

 xa



; as required.

ii) Implies (iii) , Let xy





 

S



. Since 

 

S is a congruence, xt yt





 

S



for any tL, and so



x y t

 

x t a



t

x       and yt



xyt

 

 xtb



for some a,bS.Then Observe that



ta

 

 tb



S. Thus, x y

 



Conversely, if x y

 

 then for any tL,



xt

 

 tc

 

 yt

 

 tc



for some cS. Choosing x

t and ty, we have x



xy

 

 xc



and y



x y

 

 yc



respectively.

Thus xy





 

S



and  is congruence 

 

S

iii) Implies (iv). Let T 



sk:skexists and sS and kK



. Suppose xsk,sS,kK Clearly

 



S



k k

s   and so xx



sk

 

 xk

   

 S



. Hence for all tL,



xt

 

 tc

 

 xkt

 

 tc



for some cS. Choosing tx, we obtain



x k

 

x c



x    and so xT . But T is closed under existent finite suprima. It follows that T is an ideal of L and T SK.

iv) Implies (i). Let x J



SK



. Then xJ and x



sk



. So xsk for suitable S

s and kK. Then, x



xs

 

 xk



and so x



JS

 

 JK



. The reverse inclusion is obvious. Thus J



SK

 

 JS

 

 JK



; S is a standard ideal. The last part is clear from the proof of (ii) implies (iii). Now we give another characterization of standard ideals of a lattice. This is a generalization of [8, Theorem 2] which is also a nice improvement of the theorem 1.1. 

Theorem 1.2: For an ideal S of a Lattice L, the following conditions are equivalent;

(i) S is a standard ideal.

ii) The equality I



SK

 

 IS

 

 IK



holds if I and K are principal ideals.

(iii) If for the principal ideals I and J the inequality JSI holds , then J 



IS

 

 JI



iv) The relation 

 

S of L defined by x y





 

S



if and only if x



xy

 

 xa



,

 x y   y b 

y    

for some a,bS is a congruence relation.

Proof:

(i) Implies (ii) is obvious, from the definition of the standard ideal.

(ii) Implies (iii) is clear.

iii) Implies (iv). Obviously the relation is an equivalence relation. Let xy and x y

 

S then



y b



x

y   for some bS. Suppose for some tL, ytexists. Then xt exists. Hence,



x t

 

y b

 

x t

  

y t



b



y t t

y           

thus yt



xt

 





yt



b



So xt yt





 

S



. Now, ytx



yb

 

 x



S, So



yt







x



S.

Then by (iii)



yt







xyt







Syt



, 0



xt







S



yt

 

. Then a similar proof of (i) imlies (ii) of Theorem.1.1 shows that yt



xt

 

 yta



; for some aS.

Thus 

 

S is a congruence relation.

(iv) implies (i) holds by theorem 1.1.

We conclude this section with the following result, which is a generalization of a well known result of lattice theory of [8, Lemma 8]

Theorem 1.3: Let I be an arbitrary and be a standard ideal of the lattice L. If IS and IS are principal, then I itself is principal.

Proof: Let IS



a



and I S



b



. Then by theorem 1.1 axs for some xI and sS. Since a

b and xa so xb exists by the upper bound property of L.

We claim that I



xb



. Of course



xb



I. For the reverse inequality let tI . Since t,xba, so again by the upper bound property of L, wtxb exists and wL. Then



a



S



w



S



xb



S



x







a



i.e. S



w



S



xb



.

Further,



b



SIS



w



S



xb



S



b







b



, and so S



w



S



xb



. This two equalities imply that



w







xb



as S is standard and so wxb



xb



. Since tw,t



xb



and hence ^I



^x^b



, this complete the proof. 

Theorem 1.4: Homomorphic image of relatively complemented lattice is relatively complemented.

Proof: :L₁L₂ be an onto homomorphism and suppose L₁ is relatively complemented.

Let

 

^x1,y1 be any interval in L₂ since  is onto homomorphism,  images x and y for x¹,y¹ respectively such that (x) x¹ and x y(ax¹ y¹).

Thus

 

^x,y is an interval in L₁.

Let b

 

x¹,y¹ 





   

x, y



be any element then as before  a pre-image a of b. s, t 

 

a b and .

y a x 

Now L₁ relatively complemented implies that a has a complement a¹ relative to

 

^{x1,y .}

i.e.

1 1 1

1

1 1

1 1

) ( , ) (

) ( ) ( ) ( ), ( ) ( ) (

,

y a b x a b

y a a x a a

y a a x a a











































 ) (a¹



 is complement of b relative to

 

^{x1,y1 .}

Thus each element in any interval in L₂ has complement, going us the required result. 

Theorem 1.5: :L₁L₂ is an onto homomorphism where L₁, L₂ are lattices and 0¹ is least element of L₂ then kernel  is an ideal of L.

Proof: Since  is onto, 0¹L₂, thus ker   as pre image of 0¹ exists in L₁ . Now a,bker











(a)0¹ (b) (ab) (a) (b)0¹0¹ abker

Again aker , lL gives 

 

a 0¹ Also (al)(a)(l)0¹l0¹ alker Hence ker is an ideal of L 

Theorem 1.6: Let S be the a standard ideal. Then , is the extension of _{ S} to I

 

L and _{ S} is the restriction of _S to the lattice.

Proof: Let 

 

S be the extension of 

 

S to I

 

L and I J





 

S



.

We suppose IJ . Choosing a,yI . We can find an xI



yx



with x y

 

S and so there exists an

y

Sx with yx



ySxy



. The ideal S¹ generated by the yS_xy satisfies S¹S and IS¹J . Hence IJ

 

 . On the other hand, if IJ

 

S then IS¹J with a suitable S¹S. Then for any

J

y it follows that yIS and so for any yJ it follows that yI S and so



y s



x s x

y     for some sS as S is standard.

Thus ^x^y





 

^S



[ theorem 1.1] and hence 

 

S S.

To prove the 2nd assertion, suppose



a







b

  

_S . Then



a







a







b



_S 



ab

  

_S and hence



a







ab



S¹ for suitable S¹S. Then a



ab



S and since S is standard [theorem 1.1].

So a



ab

 

 as₁



for some s₁S.

Similarly, we can show that b



ab

 

 bs2



for some s₂S. Thus ab

 

S .

Hence _{ S} is the restriction of _S to L. 

Theorem 1.7: Let L be a sectionally complemented lattice. Then every homomorphism kernel of L is a standard ideal and every standard ideal is the kernel of precisely one congruence- relation.

Proof: Suppose the ideal I of L is homomorphism kernel induced by the congruence relation . Let ab

 

 , L

b

a,  then aba

 

 and 0aba. Since L is sectionally complemented, so there exits c; such that

0



b c

a and



ab



ca. This implies 0



ab



cacc

 

 . Since I is a homomorphism kernel, so cI. Moreover, a



ab



c



ab

 

 ac



; similarly, we can show that



a b

 

a d



b    for some d I. Therefore I is a standard ideal.

At the same time we have proved that if I is the kernel of the homomorphism induced by , then I

 

 . Hence every standard ideal is the kernel of Precisely one congruence relation.  Thus we have the following corollary;

Corollary 1.8: In a sectionally complemented lattice all congruencies are standard. 

We know, an ideal



s



is standard if and only if s is a standard element. Moreover, we can easily show that an ideal Generated by finite number of standard elements is standard.

Theorem 1.9: Let L be a relatively complemented lattice with 0. If every standard ideal of L is generated by a finite number of standard elements, then C

 

L , the congruence lattice is Boolean. Moreover, the converse of this is not true.

Proof: Let C

 

L with ww₁ where w and w₁ are the smallest and the largest congruence respectively. Since L is relatively complemented, So by above, 

 

S for some standard ideal S. Then



0



SL.Since every standard ideal is generated by a finite number of standard elements , so there exist standard elements a₁...a_m and b₁...b_n Such that S



a₁...a_m



and L



b₁...b_n



. Then



0







a₁...a_m







b₁... ...b_n



since



a₁...a_m







b₁...b_n



at least one of b



a₁...a_m



. Suppose b ,b ...b_Ir

2

1 1

1 are the only elements



b₁...b_n



. such that they do not belong to



a₁...a_m



. Then of course



a₁...a_m







b₁...b_n



L set C₁k 



a₁b₁



k ...



am b₁



_k for each k,

r k



1 , Then

IK K

I b

c

o  and each

IK

c is standard . Since L is sectionally complemented, there exist

IK

d such that c_IK d_IK 0 and c_IK d_IK b_IK. Since each d_iK is standard.

Now c_I (a₁...a_m]

K for each k.

Thus



^aI...^am







^dI_I...^dI_r







^c1_I...^cI_r







^dI_I...^d1_K



   



I



^.........



^...r



^.



^...I^.... _r



^,

r r I

I

I I I I

I I I

I

b b

b b

d c d

c



















and so

 

A a a

b b

d d

a a

m I

I I

I I

m r I

r I









] ...

( ] ...

...

(

] ...

...

( ....

..

..

1.

Also, as each

a

₁ is standard. So



aI...am







dI_K



   



ai   am







dI_K







bI_K



 ...

   



aI bI_K ^...... ambI_K

 

 dI_K



^,



   



aI bI_K   ambI_K







dI_K



 ...

 



 





⁰



^.

 c_IK d_IK Then using the standardness of each ,

IK

d we have



aI^......am



^



dI^......dI_K



^



⁰



Thus we obtain a standard ideal T



dI₁...dI_r



of L

such that 

 

T is the complement of . Therefore C

 

L is Boolean.

For the converse statement, consider the following lattice L Here it is easy to see that C

 

L is Boolean.



a d



L , , which is of course a standard ideal. But both a and d are not standard elements of L.  [5] and [6] have characterized those Lattices whose all congruence are standard and neutral.

our following theorems give characterizations to those lattice whose all congruence are standard and neutral.

These are certainly generalizations of above authors work.

Theorem 1.10: Let L be a lattice. Then the following conditions are equivalent.

(i) All congruence of L are standard.

(ii) L has a zero and for x,yL there exists aL such that, ^a⁰



^x^y,x



.

Proof: (i) implies (ii) since the smallest congruence w of L is standard .L must have a zero.

Let x,yL, then 



xy,x





 

I , for some standard ideal I.

i.e. x



xy

  

 I , where I is standard , hence x



xy

 

 xa



.for some aI Hence a0



xy,x



.

(ii) implies (i). Let  be a congruence and I 

 

0. Suppose x y

 

 . Then by (ii) there exists aL such that x



xy

 

 xa



and a0



x y,x



. Since 



xy,x



, so a0

 

 and Hence aI . Similarly y



xy

 

 yb



for some bI .

Thus I is a standard ideal and 

 

I , and so (i) holds. 

Theorem 1.11: Let L be a lattice. Then the following conditions are equivalent.

(i) All congruence of L are neutral.

(ii) L has a zero and satisfies the condition:

, ,

, ,

; ) (

)

(t y t z t x y z L

x      implies existence of

a  L

.Such that

) , (

), (

) (

) (

)

(t a a t y a t z x y a o x y x

x             

e

d

a b c

0

Figure 1.1

(iii) L has a zero and satisfies the condition

L z y x d t z t y

t

x  (  )  (  ) ; , , , ,  implies the existence of

a  L

such

that x  (t  a)  (t  a  y) (y  ((t  a) x)), a  0(x  y, x).

Proof: (i) implies (ii) L must have a zero of w

   

0 . Let x  (t  y)  (t  z);t, x, y, z  L .

Then 



x y,x





 

I for some neutral ideal I. since I is standard , by the above theorem there exists aI such that

x  ( x  y )  ( x  a

_I

)

Now

x  a

_I

 ( t  y )  ( t  z ) , a

_I

 I , ( x  a

_I

]  I ,

and



xa₁







(ty)(tz)]. Hence



xa₁



I

 

t y







tz

 





I



ty

 





I



tz

 

as I is neutral.

Therefore xa₁ pq, for somepI



ty



and qI



tz



.

Thus pty, qtz and p,qI. Hence ppty and qqtz. Let pqa , then xa₁a



pt y

 

 qtz

 

 at y

 

 atz



a. Hence a



at y

 

 atz



, aI.

But then a



xy



a



xa₁

 

 x y



a

 

xa₁

 

 xy

 

ax.

Thus



at



xaxa



x y

 

 at y

 

 atz

 

 x y



and a0







x y, x

 



 

I as aI.

(ii) implies (iii) . Let x,y,z,tL and x



t y

 

 tz



, then there exists aL such that a0



xy,x



and x



ta

 

 aty

 

 atz

 

 x y



.

Now x



ta

 

 aty

 

 atz







y



x



ta

   

 aty

 

 atz

 

 xy



x



ta



, hence x



ta

 

 atz







y



x



ta

  

. Thus (iii) holds.

(iii)implies (i). Let



be any congruence of L. Suppose xy and xy

 

 . Let I

 

0. Since xy x yx so by (iii) with tzx there exists an aL

such that x



xa

 

 axx







y



x



xa

   

 xa



 y i,e x



xa



y. Where, a0



xt,x



. [since theorem 1.1]

Hence I is a standard ideal and 

 

I . 

Now it suffices to show that all standard ideals of L are neutral. Let I be a standard ideal of L and



J K



I

x   for some ideals J and K. Then xI and xJK ThenxL_m for some









0,1,2,3

m where A₀ JK . L_m 



t pq:pq exist and p,qL_m1



Suppose xL₀. Then xI and xJ or xK , and so x



IJ

 

 IK



. Now we will use the induction. Suppose yL_m1, and yI implies that y



IJ

 

 IK



. Since xL_m, xpq for suitable p,qL_m1. set tpq. Then ^x



^t ^p

 

 ^t^q



. Then by (iii) there exists bL such that



t b

 

b t q

 

p

 

t b



x

 

x         , b0



x p,x



. Since x,x p,0I and I is a Homomorphism kernel, we get bI . Hence x



tb



I.

Further

 

x



tb

 

 p





 

x



tb

 

q





 

x



tb

 

 p

 

 btq



x



tb



. Putting



t b



x

a   , we get xa



a p

 

 aq



with aI

Now both a p,ar are members of I and L_m1. Thus both a p,aq belongs to



IJ

 

 IK



, and so x



I J

 

 IK



.

Hence I is neutral. 

Theorem 1.12: Let k an ideal in a lattice. Then the following condition are holds.

(i) k is a standard ideal.

(ii) The binary relation 

 

^k , defined by ^x^y





 

^x



if and only if ^x



^x^y

 

 ^x^a



,



x y

 

y b



y    for some a,bk is a lattice congruence.

(iii) The binary relation , define by x y

 

 if and only if for all tS



xt

 

 tc

 

 yt

 

 tc



for some ck, is a lattice congruence.

(iv) For each ideal H, kH 



kh: kh exists and kK,hH



. 

Theorem 1.13 : Let L be a lattice with the smallest element 0 in which each initial segment is a complemented lattice. Then the map K

 

K is a lattice isomorphism of the lattice of standard ideals of L on to the lattice congruence of L.

Proof: Let  be a lattice-congruence of L and J 



xL:x0

 





. Of course J is an ideal. Suppose

 

 b

a and let c and d be respective complements of ab in



a



and



b



. Then

 



0









c a c a b

c and ddb0

 

 . Also a



ab

 

 ac



and b



ab

 

 bd



with c,dJ.

Conversely, these last relations imply ab

 

 . Hence by the above theorem J is a standard ideal and

  

H J

 . 

The remainder follows from corollary: The standard ideals of a lattice L form a distributive sub lattice of the ideal-lattice J

 

L and the map K

 

K is a lattice embedding of this sub lattice into the distributive of all lattice congruence on L.

The situation is more complex when it comes to permutability. We close this section with some result in this direction.

A lower semi lattice



L:



is called medial if the supremum



xy

 

 ys

 

 zx



exists when the suptema of each pair exist. Thus the medial lower semi lattice is a lattice and so will be referred to as medial lattice.

Theorem 1.14: A mapping f:LM is an isomorphism if and only if f is isotone and has an isotone inverse.

Proof: Let f:LM be an isomorphism. Then f being one-one, onto f¹exists and is one-one , onto. Again by definition of isomorphism, f will be isotone. We show f¹:LMis also isotone.

Let y₁,y₂M where y₁y₂. Since f is onto, x₁,x₂L, s.t f

 

x₁  y₁, f

 

x₂ y₂

 

_{1 2} ¹

 

₂

1

1 f y ,x f y

x  ^  ^

 ^.

Now y₁ y₂  f

 

x₁  f

 

x₂ x₁x₂ [from the definition of isomorphism]

 

₁ ¹

 

₂ ¹

1 , ^{ }

  

 f y f y f is isotone.

Conversely, let f be isotone such that f¹ is also isotone , since f¹exists, f is one-one, onto . Again, as f is isotone x₁x₂ f

 

x₁  f

 

x₂ , x₁,x₂L.

Also f¹ is isotone implies f

   

x₁  x₁  f

 

x₂  f^1



f

 

x₁



 f^1



f

 

x₂



x₁x₂, thus

 

₁

 

₂

2

1 x f x f x

x    Hence f is an isomorphism. 

Theorem 1.15: L/ is a lattice.

Proof: Of course L/is a meet semilattice. We need only to show that it has the upper bound property. Let

     

a,b c then

      

a a c ac



,

      

b b c bc





Now,



ac

 

 bc



exists by the upper bound property of L. Hence



ac







bc





 

ac

 

 bc

 

 and so

   

a bexists. Therefore L/ is a lattice. If  is a congruence of a lattice L, then the map :LL/ defined by

 

a 

 

a

 is the natural homomorphism. This is known as the homomorphism induced by . For a standard ideal S of L we denote the quotient lattice L/

 

S , simply by L/S.

Now we give the homomorphism theorem for lattices which is a generalization of [Lattice Theory First concepts by Gratzer theorem 11 p-26].

Theorem 1.16: Every homographic image at a lattice L is isomorphic to a suitable quotient lattice L. In fact if M

L

: is a homomorphism of L on to M and if



is the congruence relation of L defined by xy if and only if 

   

x  y then L/L₁; an isomorphism is given by :

 

xq

 

x , xL.

Proof: Since



is a homomorphism then it is easy to check that



is a congruence relation . To prove that  is an isomorphism, we have to check that (i)  is well defined. Let

 

x 

 

y . Then x y

 

 ; thus

         



   



 x  p y  x   y  . i.e.  is well defined

(ii)  is one - one, 

  

x

 







 

y 

   

x  y then xy and so

 

x

 

 

 

y

 

 i.e.  is one - one .

(iii)  is on to : Let xL since



is onto. There is any yL with 

 

y x . Thus

  

y

 





 x i.e.  is onto .

(iv) Preserves the operations. i.e.  is homomorphism . Let

   

x , y L/, therefore

   



x  y 







x y







xy

    

 x  y 

  

x 





  

y 



 , and finally for , Suppose

 

x 

 

y  exists.

Then

 

x 

 

y 

 

t  for some tL. so

 

x

 

t  and

 

y 

 

t  . This implies

 

x 

 

x

 

t 



xt



,

similarly

 

y 



yt



.

Then 

  

x 

 

y 







xt







yt



 

   

 

 

   

 



x t

 

y t



t y t x

t y t x



































       

 



^



^

  

^

















y x

t y t

x









Hence  is a lattice homomorphism and so it is an isomorphism 

Theorem 1.17: [First isomorphism theorem for standard ideals]: Let L be a lattice, S be a standard ideal and I an arbitrary ideal of L Then SI is a standard ideal of I

and



IS



/SI/



IS



.

Proof: We can use the first isomorphism theorem for universal algebra [12, Theorem 1.2], Then it remains to prove that every congruence class of IS may be represent by an element of I . so let xIS. Then since theorem 1.1 xis for some iI, sS. Moreover xisi

 

S .

Hence congruence class that contains x may be represented by iI. That is

   

x  i 

 

S . Therefore



IS



/SI/



I S



. 

For the 2nd isomorphism theorem we need the following results. We omit the proofs as they are very trivial.

Lemma 1.18: Let the correspondence xx be lattice homomorphism of lattice L on to a lattice L. If s is standard element of L then s is a standard element of L. 

Corollary 1.19: Let xx be lattice homomorphism of lattice L onto I . Let S be an ideal of L and denote by S the homomorphic image of S under this homomorphism. If S is standard in L then S is standard in L.

Theorem 1.20: [Second isomorphism theorem for standard ideals].

Let L be a lattice, S be an ideal and T be a standard ideal of L . ST. Then S is a standard ideal of L if and only if S/T is a standard ideal in L/T and in this case L/S



L/T

 

/ S/T



.

Proof: First suppose that S is a standard ideal of L. let :LL/T be the natural mapping . Then xx is a lattice homomorphism and onto. So S is a standard ideal of L/T . Now SS/T . Hence S/T is a standard ideal of L/T .

Conversely, suppose that S/T is a standard ideal of L/T . We are to show that S is a standard ideal of L.

Let us define a relation 

 

S by 1.1 (ii), suppose xy with x y





 

S



. Then x ys for some sS. Thus for any

u L

. if xu exists,

then xu



yu



s. This implies xu yu





 

S



. To prove substitution property for , suppose a denotes the image of the element a under the homomorphism LL/T Suppose x y((S/T)).Since

T

S/ is standard in L/T , there is a suitable sS/T, such that xu (yu)s.

Further, since T is standard in L we can find aT such that xu



(yu)s



t. We put s₁st and get



y u



s s S u

x    ₁, ₁ .. Hence 

 

S is a congruence relation of L, and so by 1.1, S is standard.

In above proof we have also shown that the congruence classes of L/T under 



S/T



are the homomorphism image of those of L under 

 

S . Then the second isomorphism theorem for universal algebra [12 theorem 1.4 finishes the proof.

Thus in a lattice every standard ideal is a homomorphism kernel of at least one congruence relation and the congruence classes of L/T under

  

m S/T



are the homomorphism image of those of L under

  

H S . Also the theorem is the characterization of standard ideals of a lattice which I also the nice improvement and generalization of [8, theorem 2]

REFERENCE

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Chinthayamma Malliah and Parameshwara Bhatta-Lattices all of whose congruences are standard or distributive, manuscript.

Chinthayamma Malliah and Parameshwara Bhatta - Lattice all of whose congruences are neutral Amer Math.

Soc Vol 94 No. 1 May 1989.

Chinthayamma Malliah and Parameshwara Bhatta - A generalization of distributive ideals to convex sublatteces. Acta Math. Hunger. 48 (1986). No.1-2, Page 73-77

Chinthayamma Malliah and Parameshwara Bhatta.. Generalizations of dually distributive and neutral ideals to convex sublattices, manuscript.

E. Evans: Median lattices and convex aubalgebras. Manuscript[18], G.Gratzer, Generallattice theory, Birkhauser verlag, Basel(1978)

E.Fried and E.T. Schmidt: Standard sublattices Alg.Univ.5(1975), 203-211

G. Gratzer and E. T. Schmidt,: Standard ideals in lattices Acta Math, Acad. Sci Hung. 12(1961). 17-86.

G. Gratzer: General lattice theory, Birkhauser Verlag, Basel (1978)

M.F Jamowitz, A charecterization of standard ideals, Acta Math. Acad. Sci. Hung. 16(1965) 289-301.

J. Niemeinen: Distributive, standard and neutral convex sublattices of a lattice. Comment, Math. Universilatis, sancti, Pauli vol, 33 no-1 (1984) Japan.

Universal algebra, Springer-verlag, new york, Heidelbery Berlin, 2nd edition 1979.