The Lattice of L-ideals of a ring is modular

IWM 2015--2-4 April, 2015Iffat Jahan

Ramjas College, University of Delhi

Fuzzy sets were introduced by Zadeh with a view to apply it in approximate reasoning.

If the closed unit interval [0,1] is replaced by a lattice L having least and greatest elements

Then is called an L-subset in X .

A fuzzy set in X is a mapping

: [0, 1].Xm ®: X Lm ®

m

The concept of L-fuzzy subset which is now known as L-subset is introduced by Goguen in the year 1967.

•Rosenfeld applied fuzzy set theory in algebra by introducing the notions of fuzzy subgroupoids and fuzzy subgroups in 1971.

•Wang Jin Liu initiated the studies of lattice valued ring theory by introducing the definitions of fuzzy L-subrings and fuzzy L- ideals in 1983.

We denote by R a commutative ring and by

a complete Heyting Algebra where ‘ ’ denotes the partial ordering of L, the join (sup ) and meet (inf ) of the elements of are denoted by ‘ ’ and ‘ ’ respectively.

, , , L L

L-subset

An L-subset in a set X is a function

The set of L-subsets of X is called the L-power set of X and is denoted by

: ,X L

.XL

' '

• For , We say that is contained in if for every

and is denoted by . .

, XL x X

( ) ( )x x

•For a family of L-subsets of X , where I is a nonempty index set, the union and the intersection of

are respectively defined by :

and

for each .

{ }i i I

ii I

ii I

{ }i i I

( ) { ( )}i ii I i I

x x

( ) { ( )}.i ii I i I

x x

x X

Let . Define as follows:

is called sum of and . By the definition of sum it follows that .

( ) { ( ) ( ) : ; , }.x y z x y z y z R

, RL RL

L-subring of R Let Then is said to be an L-

subring of R if for each

(i) (ii)

. RL , x y R

( ) ( ) ( ), x y x y

( ) ( ) ( ). xy x y

If is an L-subring of R, then and

Moreover, and

is known as the tip of

( ) (0)x

( ) ( ). x x sup ( ) (0)x R

x

.

(0)

L-Ideal of a ring R

Let be an L-subring of R. Then is said to be an L-( left, right ) ideal of R if for each

( )

The set of L-ideals of R is denoted by

, x y R

( ) ( ) ( ). xy x y

( ) ( ),xy y

( ) ( )xy x

L(R).

The intersection of any arbitrary family of L-ideals of R is an L-ideal of R.

Let . Then the L-ideal generated by is defined to be the least L-ideal of R which contains . It is denoted by . That is

{ : }.i

i iq q

q q qÍ

á ñ= ÎI L(R)

RLqÎ

•The set of L-ideals of a ring is a complete lattice under the ordering of L-set inclusion, where the infima and the suprema of a family of L-ideals of are defined as the intersection of the family and the ideal generated by their union respectively.

•In ring theory, the sum of two ideals I

and J of a ring R is again an ideal of R

which is the least ideal of R containing

both I and J. Thus

•The sum I+J provides the join of the

ideals I and J.

I + J = I JÈ .

•However in L-ring theory , the sum of two L-ideals of R is an L-ideal containing both if and only if they have identical tips.

and h q and h q

•Let denote the set of L-ideals of R, each having the same tip ‘t’ by

•Thus is a modular lattice with meet and join defined by

and .

tL (R).

tL (R)

h q h qÚ = +h q h qÙ = Ç

•However, if two L-ideals of R have different tips, then their sum fails to provide the join of given L-ideals.

Modularity of the lattice of L-ideals of a ring still remain an open problem.

•T. Head in his outstanding paper introduced the concept of a tip-extended pair of fuzzy subgroups and provided the join structure of a pair of fuzzy subgroups of the lattice of fuzzy normal subgroups.

A. Jain demonstrated the utility of this join structure to establish a direct proof of modularity of the lattice of fuzzy normal subgroups.

•We answer the question of modularity of the lattice of L-ideals of a ring in affirmative. In doing so, we extend the notion of tip-extended pair of fuzzy ideals to the L-setting for L-ideals of a ring and thus construct the join of two L-ideals in a very simple way. This join structure helps us to establish that the lattice of L-ideals of a ring is modular.

Here we introduce the idea of a tip-extended pair of L-ideals. We first prove the following proposition:

Let be an L-ideal of R and .

Then the L-subset of R defined by if x

0, t if x

= 0; is also an L-ideal of R.

t Lt

t (x) (x)

Let and be L-ideals of R . Define L-subsets

and of R as follows: and

for x 0,

( ) ( )x xhq q= ( ) ( )x xqh h=

(0) (0) (0) (0).

•It follows that

are L-ideals of R

and

For L-ideals and of R, from the above proposition it follows that and are also L-ideals of R. We call the pair , the tip-extended pair of L-ideals. We also notice that

,

(0) (0) (0)j qq j q j+ = Ú

If and are L-ideals of R, then is the least L-ideal of R containing

That is

j qq j+q j.q jÈ

.j qq j q j+ = È

The set of L-ideals of a ring is a lattice under the ordering of L-set inclusion , where the join ‘ ’ and the meet ‘ ’ in are defined as follows:

and

(R)L

Í

Ú Ù (R)L

,j qq j q jÚ = +

.q j q jÙ = Ç

The lattice of L-ideals of a ring R is modular.

L(R)

•Since modular inequality holds in every lattice, it is sufficient to establish that if

and and then

That is

, , L(R)q f c Îq fÊ

( ) ( ). q j c j q cÙ Ú Í Ú Ù

( ) ( ). q j c j q cÙ Ú Í Ú Ù

( ) ( )c j q c jq j c j q cÙÙ + Í + Ù

•I. Jahan, The lattice of L-ideals of a ring is modular, Fuzzy Sets and Systems 199 ( 2012) 121 – 129.

Reference

•N. Ajmal and K. V. Thomas, The lattices of fuzzy ideals of a ring, Fuzzy Sets and Systems 74 (1995) 371 – 379.

•J. A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145 – 174.

•G. Gratzer, General lattice theory, Academic Press, New York, 1978.

•T. Head, A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and Systems 73 (1995) 349 – 358.

•T. Head, Erratum to ‘‘A metatheorem for deriving fuzzy theorems from crisp versions ’’ Fuzzy Sets and Systems 79 (1996) 277 – 278.

•A. Jain, Tom Head’s join structure of fuzzy subgroups, Fuzzy Sets and Systems 125 (2002) 191 – 200.

•I. Jahan, The lattice of L-ideals of a ring is modular, Fuzzy Sets and Systems 199 ( 2012) 121 – 129.

•W. – J. Liu, Operations on fuzzy ideals, Fuzzy Sets and Systems 11 (1983) 31 – 41.

•J. N. Mordeson and D. S. Malik, Fuzzy commutative algebra, World Scientific Pub. Co. Pte. Ltd., 1999.

•A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512 – 517.

•L. A. Zadeh, Fuzzy Sets, Information and Control 8 (1965) 338 – 353.