( ) ( ) ( )s ( ) ( ) ) ( )s

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* Corresponding author : [email protected] KUET @ JES, ISSN 2075-4914 / 04(1), 2013

MODULAR IDEALS OF A NEARLATTICE

Md. Zaidur Rahman^* and Md. Bazlar Rahman

Department of Mathematics, Khulna University of Engineering & Technology. Khulna-9203, Bangladesh Received:24 October 2012 Accepted: 07 April 2013

ABSTRACT

In this paper we include several characterizations of modular ideals in a nearlattice. For a modular ideal M of a nearlattice S and a general ideal J, we have given a description of

M  J

. We prove that xm j for some

J j M

m ,  imply that x jm₁ j for somem₁M. We also prove that for a modular ideal M of a nearlattice S, if both IM and IM are principal for any ideal I, then I itself is a principal ideal.

Keywords: Nearlattice, Modular element, Modular ideal, Standard ideal.

AMS subject classification (2005): 06A12, 06A99, 06B10.

1. INTRODUCTION

A meet semilattice S together with the property that any two elements having a common upper bound have a supremum, is called a nearlattice. This property is known as the “Upper bound property”.

A lattice L is called modular if for all with zx, x



yz

 

 x y



z.

A nearlattice S is modular if for all with zx and if yz exists, then x



yz

 

 x y



z. By (Noor A.S.A. et al, 1997), this is equivalent to the condition that for all t,x,y,zS with zx,

   



t y t z

 

x t y

 

t z



x         .

(Talukdar M.R. et al, 1998) have defined a modular element in a join semilattice directed below.

By their definition, an element m in a lattice L is called a modular element if for all x,yL withyx,



m y

 

x m



y

x     .

Recently, (Rahman M.Z. et al, 2012) have given a nice description of modular element in a nearlattice.

Let S be a nearlattice. By (Rahman M.Z. et al, 2012), an element mS is called a modular element if for all S

y x

t, ,  withyx, x

 

tm

 

 ty

 





tmx

 

 ty



.

An ideal M of a nearlattice S is called a modular ideal if it is a modular element of the ideal lattice^I

 

^S . That is, M is modular if for all I,JI

 

S withJI , I



M J

 

 IM



J.

We already know that a lattice (nearlattice) is modular if and only if its every element is modular.

An element s of a lattice L is called a standard element if for all x

,

yL,

x   y  s    x  y    x  s . By (Cornish W.H. et al, 1982), an element s in a nearlattice S is called a standard element if for allt,x,yS,

   

 x y x s   t x y   t x s 

t          

.

An ideal I of a nearlattice S is called a standard ideal if it is standard element of the ideal lattice

I   S

^.

Of course, every standard ideal of a nearlattice (lattice) is modular, but the converse need not be true. In this paper we include some characterization of modular ideals of a nearlattice.

The following result is due to (Cornish et al., 1978)

Theorem 1. A nearlattice S is modular if and only if it does not contain a sublattice isomorphic to a pentagonal sublattice.●

It is well known that a lattice L is modular if and only if is modular. But by (Rahman, M.B., 1994), this is not true for nearlattices. We include the proof for the convenience of the reader.

L z y x, , 

S z y x, , 

 

L I

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Theorem 2. For a nearlattice S, if is modular, then S is modular, but the converse need not be true.

Proof: Suppose is modular. Let a,b,cS with ca and bc exists. Then



a





 

b







c

 



 

a







b

 





c



, and so



a



bc

 



 

ab



c



, which implies, . Therefore, S is modular.

To prove the converse, consider the nearlattice S in figure 1.

Here S does not contain a sublattice isomorphic to pentagonal lattice. So by theorem 1, S is modular. But in I

 

S ,

       

 0 , p , p , y , q , y , S 

is a pentagonal sublattice, and so I

 

S is not modular. ●

By (Rahman M.B., 1994) we know that the supremum of two ideals in a nearlattice is not very easy to handle. We have;

Theorem 3. For two ideals I and J of a nearlattice S, _n

n A

J

I ^

 



0 , where A₀ IJ and



^x ^S ^x ⁱ ^j^,

A_n    for some i,jA_n_₁



. ●

By (Cornish W.H. et al, 1982), we know that for a standard ideal K of a nearlattice S and for any JI

 

S ,



^k ^j ^k ^K ^j ^J



J

K    , 

But in case of a modular ideal M of a nearlattice, we are unable to give a simple description of MJ . Even J

M

x  does not imply xm j for some mM and jJ . For example, consider the following nearlattice of figure 2.

Here S is a modular nearlattice by Theorem1. InI

(S )

,

(b ]

is modular. Now

q   t    b . But qpq for any p   t  and q   b .

Theorem 4. Let L be a lattice and

m  L

, m is modular if and only if

(m ]

is modular inI

(L )

. Proof : Suppose m is modular in L. Suppose J I. Let

x  I    m   J .

Then

x  I

and

x   m   J

^.

This implies xm j for some jJ . So x jm j.

 

S

 

S I I



b c

 

a b



c

a    

x

p q y b

0 S

Figure 1

t x r

s p q y b

0 S

Figure 2

S

(t] (x]

(r]

(s] (y]

(p] (q] (b]

0 I(S)

Figure 3

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Now jJ I.

Thus x jI and

x  j   x  j    m  j     x  j   m   j

( as m is modular)

  I   m    J

^.

Therefore,

x   I   m    J

.

Since the reverse inequality is trivial, so

I    m   J    I   m    J

. Hence

(m ]

is modular in I

(L )

.

Conversely, let

(m ]

be modular in I

(L )

.

Suppose zx. Then

 x     m    z      x    m     z 

That is,

 x   m  z      x  m   z 

Therefore,

x   m  z    x  m   z

, and so m is modular. ●

Theorem 5. Let S be a nearlattice, I,JI

 

S and

I , J   a  for some aS. Then



^x ^S ^x ⁱ ^j

J

I     for some iI,jJ



Proof: Let xIJ. Then by theorem 3, xi j for some i,jA_n_₁, where A₀ IJ . Since i,jA_n_₁, so ii₁ j₁, ji₂ j₂ for some i₁,i₂,j₁,j₂A_n_₂.

Then xi₁i₂ j₁ j₂, the supremum exists by the upper bound property of S as i₁,i₂,j₁,j₂ a. Thus proceding in this way x



p₁ pn

 

 q₁qn



for some p_i

,

q_iA₀ IJ, and the supremum exists by the upper bound property again.

Therefore, xi j for some iI, jJ. ● But in a nearlattice S, Theorem 4 is not true.

Theorem 6. For an element m of a nearlattice S, if



m



is modular in I

 

S , then m is modular, but the converse may not be true.

Proof: Suppose



m



is a modular ideal in S. Let zx. Then for all

t  S

, tztxx implies

 t  z    x 

               

   



t x m

 

t z

 

t x

  

t m

 

t z

 

z t m x t z t m t x t































So

 t  x     t  m    t  z      t  x    t  m     t  z 

This implies

  t  x     t  m    t  z       t  x  m    t  z  

And so

x    t  m    t  z     t  x     t  m    t  z     x  t  m    t  z 

Therefore, m is modular in S.

To prove the converse

Consider the following nearlattice and its ideal lattice.

Here d is modular in S. But in I

 

S (figure 5),

  0  ,  d  ,  g  ,  g , e  , S 

is a pentagonal sublattice. Hence

 d 

^is

not a modular ideal.●

Theorem 7. Let M be a modular ideal of a nearlattice S and J be an ideal. If xm j for some mM,jJ, then x jm₁ j for some m₁M.

Proof: Let xm j, then x jm j.

Thus, x j



x j







M 



j

  





x j



M

 

 j



.

So by theorem 5, x jpq for some p



x j



M and q



j



. Since p



x j



M, so pM and px j.

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Thus x j pq p jx j implies x j p j, where pM. ●

Theorem 8. Let M be an ideal of a nearlattice S with the condition that for all ideals J of S,



x S x m j m j

J

M     ,  exists for some mM,jJ



. Then the following conditions are equivalent.

(i) M is modular.

(ii) xMJ implies x jm j for some mM, jJ.

Proof: (i)(ii) Suppose M is modular. Let xM J. Then by the given condition, xm j for some M

m ,jJ. Then by theorem 7,

j m j

x  

₁



and so (ii) holds.

) ( )

(ii  i Suppose (ii) holds.

Let I,JI

 

S with JI

Suppose xI



MJ



. Then xI and xMJ.

Thus by given condition, x jm j for some mM, jJ. Now, mx j implies mIM .

Therefore, x



IM



J, and so I



M J

 

 IM



J . Since the reverse inclusion is trivial. so I



M J

 

 IM



J. Hence M is modular. ●

In lattices, we know from (Rahman M.Z. et al, 2012)] that an element m is modular if and only if for all

b  a

with

a  m  b  m

&

a  m  b  m

imply

a  b

.

We conclude the paper with the following result which is proved by above characterization of modular elements.

Theorem 9. Let M be a modular ideal of a nearlattice S. If I M and IM are principal, then I is modular.

Proof: Let IM 



a



and IM



b



.

Then by theorem 5, aim for some iI,mM. Thus,



a



M IM 



bi



M 



i







a



. This implies M IM



bi



Also,



b



M IM 



bi







b



implies MIM



bi



Moreover,



bi



I.

Therefore, I 



bi



as M is modular.●

REFERENCES

Cornish W. H. and Hickman R. C., Weakly distributive semilattices, Acta Math Acad.Sci. Hungary, 32, 5- 16(1978)

(g,e]

(f] (g]

(e] (h]

(b] (c]

(a] (d]

0 I(S)

Figure 5

f g

e h

b c a d 0 S

Figure 4

(5)

Cornish W. H. and Noor A. S. A., Standard element in a nearlattice, Bull Austral. Math .Soc. 26(2), (1982); 185- 213.

Noor A. S. A. and Islam A. K. M. S., Relative annihilators in nearlattice, The Rajshahi University Studies (part- B) 25(1997); 117-120..

Rahman M. B., A study on distributive nearlattices, Ph.D Thesis, Rajshahi University, Bangladesh, (1994).

Rahman M. Z. and Noor A. S. A., Modular and Strongly Distributive elements in a nearlattice ,JMCMS, Vol- 7,No-1, July 2012, p: 988-997.

Talukdar M. R. and Noor A. S. A., Modular ideals of a join semilattice Directed below, SEA Bull. Math 22(1998); 215-218.