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The lattice of L-ideals of a ring is modular
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Fuzzy Sets and Systems 199 (2012) 121 – 129 www.elsevier.com/locate/fss
The lattice of L-ideals of a ring is modular Iffat Jahan Department of Mathematics, Ramjas College, University of Delhi, Delhi 110007, India Received 30 January 2009; received in revised form 15 December 2011; accepted 20 December 2011 Available online 29 December 2011
Abstract In this paper, we extend the notion of a tip-extended pair of fuzzy subgroups to L-ideals of a ring. We prove that the sum of two tip-extended L-ideals of an arbitrary pair of L-ideals of a ring is the least L-ideal containing the union of the given L-ideals. Using this construction of join of L-ideals, we prove that the lattice of all L-ideals of a given ring is modular. © 2011 Elsevier B.V. All rights reserved. Keywords: Algebra; L-ideals; Ring; Sum of L-ideals; Lattice; Modularity
1. Introduction Rosenfeld initiated the studies of fuzzy algebraic substructures by introducing the notions of a fuzzy subgroupoid and a fuzzy subgroup of a group in his pioneering paper [13]. In the same paper, the idea of a least fuzzy subgroupoid containing a given fuzzy set was also introduced. Consequently, Rosenfeld constructed the lattice of all fuzzy subgroupoids of a given group. In a recent paper [9], this author has established that the lattice of all fuzzy ideals of a ring is modular. The technique of the proof in this paper is different from those of earlier authors. In fact, the proof of modularity is heavily based on the property of the unit interval that it is a dense chain. Ajmal and Thomas in 1994 initiated the construction of various types of lattices and sublattices of fuzzy subgroups [2,3]. Modularity of the lattice of all fuzzy normal subgroups was established in a systematic and stepwise manner in the papers [1–3,5]. However, not much attention was paid to the studies of the lattices of fuzzy ideals of a ring. The only notable attempts can be found in [4,9,16–18]. Head also discussed modularity of the lattice of all fuzzy normal subgroups in his outstanding work [6], wherein he formulated the well known metatheorem and the subdirect product theorem. In his erratum [7], the concept of a tip-extended pair of fuzzy subgroups was introduced in order to construct the join of two fuzzy normal subgroups of a group. In fact, modularity of the lattice of all fuzzy normal subgroups follows as a consequence of the subdirect product theorem in view of the fact that the Rep function commutes with the set product of two tip-extended fuzzy normal subgroups which is the join of those subgroups. Jain [8] employed this technique to provide the direct proof of modularity of the lattice of all fuzzy normal subgroups. Modularity of the lattices of L-normal subgroups of a group and L-ideals of a ring still remains an open question. The main feature of the techniques, so far employed in the proofs of modularity in the fuzzy setting, either involves E-mail address: [email protected]. 0165-0114/$ - see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.12.012
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I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
the applications of strong level subsets or subdirect product theorem introduced by Head. In the L-fuzzy setting, both of these above techniques fail in view of the fact that the properties of L-subgroups, L-normal subgroups and L-ideals cannot be characterized in terms of strong level subsets and the metatheorem or the subdirect product theorem formulated by Head are only applicable to fuzzy algebraic substructures. In this paper, 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. 2. Preliminaries Throughout this paper we denote by R a commutative ring and by L = L , ≤, ∨, ∧ a completely distributive lattice where ‘≤’ denotes the partial ordering of L, the join (sup) and meet (inf) of the elements of L are denoted by ‘∨’ and ‘∧’ respectively. We also write 1 and 0 for maximal and minimal elements of L respectively. In this section, we first introduce some basic definitions and results which are used in the sequel. An L-subset of X is a function from X into L. The set of all L-subsets of X is called the L-power set of X and is denoted by L X . We say that an L-subset of X is contained in an L-subset of X if (x) ≤ (x) for every x ∈ X and is denoted by ⊆ . In particular when L is [0,1], the L-subsets of X are called fuzzy subsets of X and the set [0, 1] X is referred as the fuzzy power set of X . Definition 2.1. Let , ∈ L R . Define + ∈ L R as follows: + (x) = ∨{(y) ∧ (z) : x = y + z; y, z ∈ R}. + is called sum of and . By the definition of sum it follows that + = + . Definition 2.2. Let ∈ L R . Then is said to be an L-subring of R if (i) (x − y) ≥ (x) ∧ (y), (ii) (x y) ≥ (x) ∧ (y); for all x, y ∈ R. It can be easily verified that if is an L-subring of R then (x) ≤ (0) and (x) = (−x) for all x ∈ R. We denote the set of L-subrings of a ring R by L(R). Definition 2.3 ([10]). Let ∈ L(R). Then is called an L-(left, right) ideal of R if ((x y) ≥ (y), (x y) ≥ (x)), (x y) ≥ (x) ∨ (y) for all x, y ∈ R. We denote the set of L-ideals of a ring R by L(R). Proposition 2.4. The intersection
i
i of any family {i } of L-ideals of a ring R is an L-ideal of R.
Definition 2.5. Let be an L-subset of a ring R. 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 ∈ L(R)}. = ⊆i
Proposition 2.6. The set of all L-ideals L(R) of a ring R is a complete lattice under the ordering of L-set inclusion, where the infima and the suprema of a family of L-ideals of R are defined as the intersection of the family and the ideal generated by their union respectively.
I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
123
3. The tip-extended pair of L-ideals and modularity Head in his erratum [7] of the paper [6] 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. Jain [8] demonstrated the utility of this join structure to establish a direct proof of modularity of the lattice of fuzzy normal subgroups. So far this approach of formation of join structure is not investigated in the studies of L-algebraic substructures. Here we introduce the idea of a tip-extended pair of L-ideals. We first prove the following proposition: Proposition 3.1. Let ∈ L(R) and t ∈ L. Then the L-subset t of R defined by (x) if x 0, t (x) = (x) ∨ t if x = 0 is also an L-ideal of R. Proof. Let x, y ∈ R. Case 1: x − y 0. Then either x 0 and y 0 or one of x or y is 0. In the first case we have t (x − y) = (x − y) ≥ (x) ∧ (y) = t (x) ∧ t (y). And in the later case either x − y = x or x − y = y, so that t (x − y) = t (x) or t (y). Hence t (x − y) ≥ t (x) ∧ t (y). Case 2: x − y = 0. Then t (0) = (0) ∨ t ≥ (0) ≥ (x) = t (x) for all 0 x ∈ R. Thus t (0) ≥ t (x) for all x ∈ R. Now, t (x − y) = t (0) ≥ t (x) and t (x − y) = t (0) ≥ t (y). This implies t (x − y) ≥ t (x) ∧ t (y). Next, we consider Case 1: x y 0 then x 0 and y 0. Thus t (x y) = (x y) ≥ (x) ∨ (y) = t (x) ∨ t (y). Case 2: x y = 0. Then as before t (x y) = t (0) ≥ t (x) and t (x y) = t (0) ≥ t (y). Therefore t (x y) ≥ t (x) ∨ t (y). Consequently t ∈ L(R). Definition 3.2. Let and ∈ L(R). Define L-subsets and of R as follows: (x) = (x) and (x) = (x) for all x 0, and (0) = (0) = (0) ∨ (0).
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I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
From the above Proposition 3.1 it follows that if , ∈ L(R), then and ∈ L(R). We call the pair , the tip-extended pair of L-ideals. In L-ideal theory, it is well known (see [11]) that for two L-ideals and , the sum + is the least L-ideal containing and provided their tips are identical. That is (0) = (0). However, the above assertion fails if and have different tips. In order to rectify the above situation, we use the concept of a tip-extended pair of L-ideals. We also notice that + (0) = (0) ∨ (0). For the sake of completeness, we prove that + is an L-ideal of R and also establish the following: Proposition 3.3. If , are L-ideals of R then + is the least L-ideal of R such that + = ∪ , where ∪ is the ideal generated by the union ∪ . Proof. First, we show that + is an L-ideal of R. Let x, y ∈ R. Define the following subsets of R × R: D(x) = {( p, q) ∈ R × R : x = p + q}, D(y) = {(r, s) ∈ R × R : y = r + s}, D(x + y) = {(u, v) ∈ R × R : x + y = u + v}, and D(x y) = {(u, v) ∈ R × R : x y = u + v}. Now we first take ( p, q) ∈ D(x) and (r, s) ∈ D(y), then x + y = ( p + q) + (r + s) = ( p + r ) + (q + s). This implies ( p + r, q + s) ∈ D(x + y). Therefore, we have + (x + y) = ( (u) ∧ (v)) (u,v)∈D(x+y)
≥
( ( p + r ) ∧ (q + s))
( p+r,q+s)∈D(x+y) ( p,q)∈D(x),(r,s)∈D(y)
≥
( ( p) ∧ (r ) ∧ (q) ∧ (s))
( p,q)∈D(x) (r,s)∈D(y)
=
( ( p) ∧ (q) ∧ (r ) ∧ (s)).
( p,q)∈D(x) (r,s)∈D(y)
Thus + (x + y) ≥
( ( p) ∧ (q) ∧ (r ) ∧ (s)).
( p,q)∈D(x) (r,s)∈D(y)
Now, we prove that the R.H.S. of the above inequality is + (x) ∨ + (y). As L is a completely distributive lattice, for a fixed ( p, q) ∈ D(x) we have ( ( p) ∧ (q) ∧ (r ) ∧ (s)) (r,s)∈D(y)
⎛
= ( ( p) ∧ (q)) ∧ ⎝
(r,s)∈D(y)
⎞ ( (r ) ∧ (s))⎠ .
I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
Consequently ( p,q)∈D(x)
⎧ ⎨ ⎩
(r,s)∈D(y)
⎧ ⎨
=
( p,q)∈D(x)
=
⎧ ⎨ ⎩
⎩
125
⎫ ⎬ { ( p) ∧ (q) ∧ (r ) ∧ (s)} ⎭ ⎛
( ( p) ∧ (q)) ∧ ⎝
( ( p) ∧ (q))
( p,q)∈D(x)
⎫ ⎬ ⎭
(r,s)∈D(y)
∧
⎧ ⎨
⎞⎫ ⎬ ( (r ) ∧ (s))⎠ ⎭
⎩
( (r ) ∧ (s))
(r,s)∈D(y)
⎫ ⎬ ⎭
= + (x) ∧ + (y). Thus + (x + y) ≥ + (x) ∧ + (y). Next, we observe that if (u, v) ∈ D(y) then x y = xu + xv. That is, (xu, xv) ∈ D(x y). Now + (y) = ( (u) ∧ (v)) (u,v)∈D(y)
≤
( (xu) ∧ (xv)) (as and ∈ L(R))
(u,v)∈D(y)
=
( (xu) ∧ (xv))
(xu,xv)∈D(x y)
≤
( (r ) ∧ (s))
(r,s)∈D(x y)
= + (x y). Similarly we can obtain + (x) ≤ + (x y). Consequently + is an L-ideal of R. Again, we consider + (x) = ( (y) ∧ (z)) ≥ (x) ∧ (0). (y,z)∈D(x)
Also, + (x) =
( (y) ∧ (z)) ≥ (0) ∧ (x).
(y,z)∈D(x)
Moreover, for all x ∈ R we have (0) = (0) = (0) ∨ (0) and this implies that (0) ≥ (x), (0) ≥ (x). Consequently it follows that (x) ∧ (0) = (x) and (0) ∧ (x) = (x). Thus + (x) ≥ (x) and + (x) ≥ (x). That is , ⊆ + . Also ⊆ and ⊆ . So ∪ ⊆ + . Thus + is an L-ideal containing ∪ . Now to prove that + is the smallest ideal containing ∪ , let ∈ L(R) be such that ∪ ⊆ and x ∈ R. If x = 0, then + (0) = (0) ∨ (0) ≤ (0).
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I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
So we let x 0 and consider ( ( p) ∧ (q)) + (x) = ( p,q)∈D(x)
=
⎧ ⎪ ⎨
⎪ ⎩ ( p,q)∈D(x)
( ( p) ∧ (q))
0 p,0 q
=
⎧ ⎪ ⎨
⎪ ⎩ ( p,q)∈D(x)
( ( p) ∧ (q))
0 p,0 q
=
⎧ ⎪ ⎨
⎪ ⎩ ( p,q)∈D(x) 0 p,0 q
≤
⎧ ⎪ ⎨
⎪ ⎩ ( p,q)∈D(x)
Hence
+ (x)
⎪ ⎭ ⎫ ⎪ ⎬ ⎪ ⎭
∨ {( (x) ∧ (0)) ∨ ( (0) ∧ (x))}
∨ { (x) ∨ (x)}
⎫ ⎪ ⎬
(( p) ∧ (q)) ∨{(x) ∨ (x)} ⎪ ⎭
(x)
0 p,0 q
⎫ ⎪ ⎬
⎫ ⎪ ⎬ ⎪ ⎭
∨ {(x) ∨ (x)} = (x). (As , ⊆ )
≤ (x) for all x ∈ R. That is + ⊆ . Consequently + = ∪ .
Proposition 3.4. The set of all L-ideals L(R) of a ring R is a lattice under the ordering of L-set inclusion, where the join ‘∨’ and the meet ‘∧’ in L(R) are defined as follows: ∨ = + and ∧ = ∩ . Theorem 3.5. The lattice of all L-ideals L(R) of a ring R is modular. Proof. Since modular inequality holds in every lattice, it is sufficient to establish that if , and ∈ L(R) and ⊇ then ∧ ( ∨ ) ⊆ ∨ ( ∧ ). In view of Proposition 3.3, we have ∨ = + and ∨ ( ∧ ) = ∧ + ( ∧ ) . Thus we shall prove ∧ ( + ) ⊆ ∧ + ( ∧ ) . Let x ∈ R. If x = 0, then ∧ ( + )(0) = (0) ∧ ((0) ∨ (0)) = (0) ∨ ((0) ∧ (0)) = (0) ∨ ( ∧ )(0) = (∧ + ( ∧ ) )(0). So we assume that x 0. Now as (0) = (0) = (0) ∨ (0), we have (x) ∧ (0) = (x) = (x)
(1)
(0) ∧ (x) = (x) = (x).
(2)
and
I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
127
Also as ⊇ , (x) ∧ (x) = (x).
(3)
Next, if x = y + z then (x) ∧ (y) ≤ (x − y) = (z).
(4)
Again as (x) ≤ (0) and ( ∧ )(x) ≤ ( ∧ )(0), we have (x) ∨ ( ∧ )(x) ≤ ( ∧ )(0) ∨ (0). By the definition of the tip-extended pair ∧ and ( ∧ ) , we have ∧ (0) = ( ∧ ) (0) = ( ∧ )(0) ∨ (0).
(5)
That is (x) ∨ ( ∧ )(x) ≤ ∧ (0) = ( ∧ ) (0), so that (x) ∨ ( ∧ )(x) = ∧ (0) ∧ ((x) ∨ ( ∧ )(x)). Define a subset of R × R as D(x) = {( p, q) ∈ R × R : x = p + q} and consider ( ∧ ( + ))(x) = (x) ∧ ( + )(x) ⎧ ⎫ ⎨ ⎬ = (x) ∧ ( (y) ∧ (z)) ⎩ ⎭ =
(y,z)∈D(x)
((x) ∧ (y) ∧ (z)) (as L is a completely distributive lattice)
(y,z)∈D(x)
=
⎧ ⎪ ⎨
⎪ ⎩ (y,z)∈D(x) 0 y,0 z
=
⎧ ⎪ ⎨
⎪ ⎩ (y,z)∈D(x)
⎫ ⎪ ⎬
((x) ∧ (y) ∧ (z)) ∨{{(x) ∧ (x) ∧ (0)} ∨ {(x) ∧ (0) ∧ (x)}} ⎪ ⎭ ⎫ ⎪ ⎬
((x) ∧ (y) ∧ (z))
⎪ ⎭
0 y,0 z
∨ {((x) ∧ (x)) ∨ ((x) ∧ (x))}
(using the definition of the tip-extended pair , , (1) and (2)) ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ ((x) ∧ (y) ∧ (y) ∧ (z)) ∨{(x) ∨ ((x) ∧ (x)} (using (3)) = ⎪ ⎪ ⎭ ⎩ (y,z)∈D(x) 0 y,0 z
≤
⎧ ⎪ ⎨
⎪ ⎩ (y,z)∈D(x) 0 y,0 z
((z) ∧ (y) ∧ (z))
⎫ ⎪ ⎬ ⎪ ⎭
∨ {(x) ∨ ( ∧ )(x)} (using (4))
(6)
128
I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
=
⎧ ⎪ ⎨ ⎪ ⎩ (y,z)∈D(x)
((y) ∧ ( ∧ )(z))
0 y,0 z
=
⎧ ⎪ ⎨
⎪ ⎭
∨ {∧ (0) ∧ ((x) ∨ (( ∧ )(x))} (using (6)) ⎫ ⎪ ⎬
⎪ ⎩ (y,z)∈D(x) 0 y,0 z
⎫ ⎪ ⎬
(∧ (y) ∧ ( ∧ ) (z) ∨{∧ (0) ∧ ∧ (x) ∨ (∧ (0) ∧ ( ∧ ) (x)} ⎪ ⎭
(as L is a completely distributive lattice) ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ ∧ ( (y) ∧ ( ∧ ) (z) ∨{(( ∧ ) (0) ∧ ∧ (x)) ∨ (∧ (0) ∧ ( ∧ ) (x))} (using (5)) = ⎪ ⎪ ⎭ ⎩ (y,z)∈D(x) 0 y,0 z
=
(∧ (y) ∧ ( ∧ ) (z)
(y,z)∈D(x) ∧
= (
+ ( ∧ ) )(x).
Hence ∧ ( + ) ⊆ ∧ + ( ∧ ) . This establishes of the modularity of L(R). 4. Conclusion The studies of fuzzy algebraic substructures were at their prime from the year 1981 to 1995. During the early part of nineties, a systematic investigation of modularity of the lattice of fuzzy normal subgroups was carried out by Ajmal and Thomas and the modularity was established in a stepwise manner. Finally in the year 1994, it was proved that the lattice of fuzzy normal subgroups is modular. The proof of this result displays a very beautiful and strong application of the notion of strong level subsets. This suggested further applications of this notion in the studies of fuzzy algebraic substructures. In the year 1995, Head in order to take care of the common procedure adopted in the proofs of the various results of fuzzy algebraic substructures, used the notion of strong level subset to formulate his metatheorem and subdirect product theorem. Using this subdirect product theorem, Head very easily extended the modularity of the lattice of normal subgroups to the fuzzy setting. At this stage of development of studies of L-algebraic substructures, a need is felt to establish a similar version of both metatheorem and subdirect product theorem in L-setting. The necessary machinery for such a development is provided in some papers by Seselja and Tepavcevic [12,14,15]. To this end the concept of strong level subset has to be replaced by level subset. Moreover, an evaluation lattice has to be chosen in order to establish a metatheorem or a subdirect product theorem. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
N. Ajmal, The lattice of fuzzy normal subgroups is modular, Inf. Sci. 83 (1995) 199–209. N. Ajmal, Fuzzy groups with sup property, Inf. Sci. 93 (1996) 247–264. N. Ajmal, K.V. Thomas, The lattice of fuzzy subgroups and fuzzy normal subgroups, Inf. Sci. 76 (1994) 1–11. N. Ajmal, K.V. Thomas, The lattice of fuzzy ideals of a ring R, Fuzzy Sets Syst. 74 (1995) 371–379. N. Ajmal, K.V. Thomas, A complete study of the lattices of fuzzy congruences and fuzzy normal subgroups, Inf. Sci. 82 (1995) 198–218. T. Head, A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets Syst. 73 (1995) 349–358. T. Head, Erratum to “A metatheorem for deriving fuzzy theorems from crisp versions”, Fuzzy Sets Syst. 79 (1996) 227–228. A. Jain, Tom Head’s join structure of fuzzy subgroups, Fuzzy Sets Syst. 125 (2002) 191–200. I. Jahan, Modularity of Ajmal for the lattices of fuzzy ideals of a ring, Iran. J. Fuzzy Syst. 5 (2008) 71–78. W. Liu, Operations on fuzzy ideals, Fuzzy Sets Syst. 11 (1983) 31–41. J.N. Mordeson, D.S. Malik, Fuzzy Commutative Algebra, World Scientific, 1998. M.G. Ranitovic, A. Tepavcevic, General form of lattice-valued fuzzy sets under the cutworthy approach, Fuzzy Sets Syst. 158 (2007) 1213–1216.
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A. Rosenfeld, Fuzzy subgroups, J. Math. Anal. Appl. 35 (1971) 512–517. B. Seselja, A. Tepavcevic, Completion of ordered structures by cuts of fuzzy sets: an overview, Fuzzy Sets Syst. 136 (2003) 1–19. B. Seselja, D. Stojic, A. Tepavcevic, On existence of P-valued fuzzy sets with a given collection of cuts, Fuzzy Sets Syst. 161 (2010) 763–768. A. Weinberger, Embedding lattices of fuzzy subalgebras into lattices of crisp sub algebras, Inf. Sci. 108 (1998) 51–70. Q. Zhang, Fuzzy Sets Syst. 125 (2002) 209–214. Q. Zhang, G. Meng, On the lattice of fuzzy ideals of a ring, Fuzzy Sets Syst. 112 (2000) 349–353.
Fuzzy Sets and Systems 199 (2012) 121 – 129 www.elsevier.com/locate/fss
The lattice of L-ideals of a ring is modular Iffat Jahan Department of Mathematics, Ramjas College, University of Delhi, Delhi 110007, India Received 30 January 2009; received in revised form 15 December 2011; accepted 20 December 2011 Available online 29 December 2011
Abstract In this paper, we extend the notion of a tip-extended pair of fuzzy subgroups to L-ideals of a ring. We prove that the sum of two tip-extended L-ideals of an arbitrary pair of L-ideals of a ring is the least L-ideal containing the union of the given L-ideals. Using this construction of join of L-ideals, we prove that the lattice of all L-ideals of a given ring is modular. © 2011 Elsevier B.V. All rights reserved. Keywords: Algebra; L-ideals; Ring; Sum of L-ideals; Lattice; Modularity
1. Introduction Rosenfeld initiated the studies of fuzzy algebraic substructures by introducing the notions of a fuzzy subgroupoid and a fuzzy subgroup of a group in his pioneering paper [13]. In the same paper, the idea of a least fuzzy subgroupoid containing a given fuzzy set was also introduced. Consequently, Rosenfeld constructed the lattice of all fuzzy subgroupoids of a given group. In a recent paper [9], this author has established that the lattice of all fuzzy ideals of a ring is modular. The technique of the proof in this paper is different from those of earlier authors. In fact, the proof of modularity is heavily based on the property of the unit interval that it is a dense chain. Ajmal and Thomas in 1994 initiated the construction of various types of lattices and sublattices of fuzzy subgroups [2,3]. Modularity of the lattice of all fuzzy normal subgroups was established in a systematic and stepwise manner in the papers [1–3,5]. However, not much attention was paid to the studies of the lattices of fuzzy ideals of a ring. The only notable attempts can be found in [4,9,16–18]. Head also discussed modularity of the lattice of all fuzzy normal subgroups in his outstanding work [6], wherein he formulated the well known metatheorem and the subdirect product theorem. In his erratum [7], the concept of a tip-extended pair of fuzzy subgroups was introduced in order to construct the join of two fuzzy normal subgroups of a group. In fact, modularity of the lattice of all fuzzy normal subgroups follows as a consequence of the subdirect product theorem in view of the fact that the Rep function commutes with the set product of two tip-extended fuzzy normal subgroups which is the join of those subgroups. Jain [8] employed this technique to provide the direct proof of modularity of the lattice of all fuzzy normal subgroups. Modularity of the lattices of L-normal subgroups of a group and L-ideals of a ring still remains an open question. The main feature of the techniques, so far employed in the proofs of modularity in the fuzzy setting, either involves E-mail address: [email protected]. 0165-0114/$ - see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.12.012
122
I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
the applications of strong level subsets or subdirect product theorem introduced by Head. In the L-fuzzy setting, both of these above techniques fail in view of the fact that the properties of L-subgroups, L-normal subgroups and L-ideals cannot be characterized in terms of strong level subsets and the metatheorem or the subdirect product theorem formulated by Head are only applicable to fuzzy algebraic substructures. In this paper, 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. 2. Preliminaries Throughout this paper we denote by R a commutative ring and by L = L , ≤, ∨, ∧ a completely distributive lattice where ‘≤’ denotes the partial ordering of L, the join (sup) and meet (inf) of the elements of L are denoted by ‘∨’ and ‘∧’ respectively. We also write 1 and 0 for maximal and minimal elements of L respectively. In this section, we first introduce some basic definitions and results which are used in the sequel. An L-subset of X is a function from X into L. The set of all L-subsets of X is called the L-power set of X and is denoted by L X . We say that an L-subset of X is contained in an L-subset of X if (x) ≤ (x) for every x ∈ X and is denoted by ⊆ . In particular when L is [0,1], the L-subsets of X are called fuzzy subsets of X and the set [0, 1] X is referred as the fuzzy power set of X . Definition 2.1. Let , ∈ L R . Define + ∈ L R as follows: + (x) = ∨{(y) ∧ (z) : x = y + z; y, z ∈ R}. + is called sum of and . By the definition of sum it follows that + = + . Definition 2.2. Let ∈ L R . Then is said to be an L-subring of R if (i) (x − y) ≥ (x) ∧ (y), (ii) (x y) ≥ (x) ∧ (y); for all x, y ∈ R. It can be easily verified that if is an L-subring of R then (x) ≤ (0) and (x) = (−x) for all x ∈ R. We denote the set of L-subrings of a ring R by L(R). Definition 2.3 ([10]). Let ∈ L(R). Then is called an L-(left, right) ideal of R if ((x y) ≥ (y), (x y) ≥ (x)), (x y) ≥ (x) ∨ (y) for all x, y ∈ R. We denote the set of L-ideals of a ring R by L(R). Proposition 2.4. The intersection
i
i of any family {i } of L-ideals of a ring R is an L-ideal of R.
Definition 2.5. Let be an L-subset of a ring R. 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 ∈ L(R)}. = ⊆i
Proposition 2.6. The set of all L-ideals L(R) of a ring R is a complete lattice under the ordering of L-set inclusion, where the infima and the suprema of a family of L-ideals of R are defined as the intersection of the family and the ideal generated by their union respectively.
I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
123
3. The tip-extended pair of L-ideals and modularity Head in his erratum [7] of the paper [6] 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. Jain [8] demonstrated the utility of this join structure to establish a direct proof of modularity of the lattice of fuzzy normal subgroups. So far this approach of formation of join structure is not investigated in the studies of L-algebraic substructures. Here we introduce the idea of a tip-extended pair of L-ideals. We first prove the following proposition: Proposition 3.1. Let ∈ L(R) and t ∈ L. Then the L-subset t of R defined by (x) if x 0, t (x) = (x) ∨ t if x = 0 is also an L-ideal of R. Proof. Let x, y ∈ R. Case 1: x − y 0. Then either x 0 and y 0 or one of x or y is 0. In the first case we have t (x − y) = (x − y) ≥ (x) ∧ (y) = t (x) ∧ t (y). And in the later case either x − y = x or x − y = y, so that t (x − y) = t (x) or t (y). Hence t (x − y) ≥ t (x) ∧ t (y). Case 2: x − y = 0. Then t (0) = (0) ∨ t ≥ (0) ≥ (x) = t (x) for all 0 x ∈ R. Thus t (0) ≥ t (x) for all x ∈ R. Now, t (x − y) = t (0) ≥ t (x) and t (x − y) = t (0) ≥ t (y). This implies t (x − y) ≥ t (x) ∧ t (y). Next, we consider Case 1: x y 0 then x 0 and y 0. Thus t (x y) = (x y) ≥ (x) ∨ (y) = t (x) ∨ t (y). Case 2: x y = 0. Then as before t (x y) = t (0) ≥ t (x) and t (x y) = t (0) ≥ t (y). Therefore t (x y) ≥ t (x) ∨ t (y). Consequently t ∈ L(R). Definition 3.2. Let and ∈ L(R). Define L-subsets and of R as follows: (x) = (x) and (x) = (x) for all x 0, and (0) = (0) = (0) ∨ (0).
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I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
From the above Proposition 3.1 it follows that if , ∈ L(R), then and ∈ L(R). We call the pair , the tip-extended pair of L-ideals. In L-ideal theory, it is well known (see [11]) that for two L-ideals and , the sum + is the least L-ideal containing and provided their tips are identical. That is (0) = (0). However, the above assertion fails if and have different tips. In order to rectify the above situation, we use the concept of a tip-extended pair of L-ideals. We also notice that + (0) = (0) ∨ (0). For the sake of completeness, we prove that + is an L-ideal of R and also establish the following: Proposition 3.3. If , are L-ideals of R then + is the least L-ideal of R such that + = ∪ , where ∪ is the ideal generated by the union ∪ . Proof. First, we show that + is an L-ideal of R. Let x, y ∈ R. Define the following subsets of R × R: D(x) = {( p, q) ∈ R × R : x = p + q}, D(y) = {(r, s) ∈ R × R : y = r + s}, D(x + y) = {(u, v) ∈ R × R : x + y = u + v}, and D(x y) = {(u, v) ∈ R × R : x y = u + v}. Now we first take ( p, q) ∈ D(x) and (r, s) ∈ D(y), then x + y = ( p + q) + (r + s) = ( p + r ) + (q + s). This implies ( p + r, q + s) ∈ D(x + y). Therefore, we have + (x + y) = ( (u) ∧ (v)) (u,v)∈D(x+y)
≥
( ( p + r ) ∧ (q + s))
( p+r,q+s)∈D(x+y) ( p,q)∈D(x),(r,s)∈D(y)
≥
( ( p) ∧ (r ) ∧ (q) ∧ (s))
( p,q)∈D(x) (r,s)∈D(y)
=
( ( p) ∧ (q) ∧ (r ) ∧ (s)).
( p,q)∈D(x) (r,s)∈D(y)
Thus + (x + y) ≥
( ( p) ∧ (q) ∧ (r ) ∧ (s)).
( p,q)∈D(x) (r,s)∈D(y)
Now, we prove that the R.H.S. of the above inequality is + (x) ∨ + (y). As L is a completely distributive lattice, for a fixed ( p, q) ∈ D(x) we have ( ( p) ∧ (q) ∧ (r ) ∧ (s)) (r,s)∈D(y)
⎛
= ( ( p) ∧ (q)) ∧ ⎝
(r,s)∈D(y)
⎞ ( (r ) ∧ (s))⎠ .
I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
Consequently ( p,q)∈D(x)
⎧ ⎨ ⎩
(r,s)∈D(y)
⎧ ⎨
=
( p,q)∈D(x)
=
⎧ ⎨ ⎩
⎩
125
⎫ ⎬ { ( p) ∧ (q) ∧ (r ) ∧ (s)} ⎭ ⎛
( ( p) ∧ (q)) ∧ ⎝
( ( p) ∧ (q))
( p,q)∈D(x)
⎫ ⎬ ⎭
(r,s)∈D(y)
∧
⎧ ⎨
⎞⎫ ⎬ ( (r ) ∧ (s))⎠ ⎭
⎩
( (r ) ∧ (s))
(r,s)∈D(y)
⎫ ⎬ ⎭
= + (x) ∧ + (y). Thus + (x + y) ≥ + (x) ∧ + (y). Next, we observe that if (u, v) ∈ D(y) then x y = xu + xv. That is, (xu, xv) ∈ D(x y). Now + (y) = ( (u) ∧ (v)) (u,v)∈D(y)
≤
( (xu) ∧ (xv)) (as and ∈ L(R))
(u,v)∈D(y)
=
( (xu) ∧ (xv))
(xu,xv)∈D(x y)
≤
( (r ) ∧ (s))
(r,s)∈D(x y)
= + (x y). Similarly we can obtain + (x) ≤ + (x y). Consequently + is an L-ideal of R. Again, we consider + (x) = ( (y) ∧ (z)) ≥ (x) ∧ (0). (y,z)∈D(x)
Also, + (x) =
( (y) ∧ (z)) ≥ (0) ∧ (x).
(y,z)∈D(x)
Moreover, for all x ∈ R we have (0) = (0) = (0) ∨ (0) and this implies that (0) ≥ (x), (0) ≥ (x). Consequently it follows that (x) ∧ (0) = (x) and (0) ∧ (x) = (x). Thus + (x) ≥ (x) and + (x) ≥ (x). That is , ⊆ + . Also ⊆ and ⊆ . So ∪ ⊆ + . Thus + is an L-ideal containing ∪ . Now to prove that + is the smallest ideal containing ∪ , let ∈ L(R) be such that ∪ ⊆ and x ∈ R. If x = 0, then + (0) = (0) ∨ (0) ≤ (0).
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I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
So we let x 0 and consider ( ( p) ∧ (q)) + (x) = ( p,q)∈D(x)
=
⎧ ⎪ ⎨
⎪ ⎩ ( p,q)∈D(x)
( ( p) ∧ (q))
0 p,0 q
=
⎧ ⎪ ⎨
⎪ ⎩ ( p,q)∈D(x)
( ( p) ∧ (q))
0 p,0 q
=
⎧ ⎪ ⎨
⎪ ⎩ ( p,q)∈D(x) 0 p,0 q
≤
⎧ ⎪ ⎨
⎪ ⎩ ( p,q)∈D(x)
Hence
+ (x)
⎪ ⎭ ⎫ ⎪ ⎬ ⎪ ⎭
∨ {( (x) ∧ (0)) ∨ ( (0) ∧ (x))}
∨ { (x) ∨ (x)}
⎫ ⎪ ⎬
(( p) ∧ (q)) ∨{(x) ∨ (x)} ⎪ ⎭
(x)
0 p,0 q
⎫ ⎪ ⎬
⎫ ⎪ ⎬ ⎪ ⎭
∨ {(x) ∨ (x)} = (x). (As , ⊆ )
≤ (x) for all x ∈ R. That is + ⊆ . Consequently + = ∪ .
Proposition 3.4. The set of all L-ideals L(R) of a ring R is a lattice under the ordering of L-set inclusion, where the join ‘∨’ and the meet ‘∧’ in L(R) are defined as follows: ∨ = + and ∧ = ∩ . Theorem 3.5. The lattice of all L-ideals L(R) of a ring R is modular. Proof. Since modular inequality holds in every lattice, it is sufficient to establish that if , and ∈ L(R) and ⊇ then ∧ ( ∨ ) ⊆ ∨ ( ∧ ). In view of Proposition 3.3, we have ∨ = + and ∨ ( ∧ ) = ∧ + ( ∧ ) . Thus we shall prove ∧ ( + ) ⊆ ∧ + ( ∧ ) . Let x ∈ R. If x = 0, then ∧ ( + )(0) = (0) ∧ ((0) ∨ (0)) = (0) ∨ ((0) ∧ (0)) = (0) ∨ ( ∧ )(0) = (∧ + ( ∧ ) )(0). So we assume that x 0. Now as (0) = (0) = (0) ∨ (0), we have (x) ∧ (0) = (x) = (x)
(1)
(0) ∧ (x) = (x) = (x).
(2)
and
I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
127
Also as ⊇ , (x) ∧ (x) = (x).
(3)
Next, if x = y + z then (x) ∧ (y) ≤ (x − y) = (z).
(4)
Again as (x) ≤ (0) and ( ∧ )(x) ≤ ( ∧ )(0), we have (x) ∨ ( ∧ )(x) ≤ ( ∧ )(0) ∨ (0). By the definition of the tip-extended pair ∧ and ( ∧ ) , we have ∧ (0) = ( ∧ ) (0) = ( ∧ )(0) ∨ (0).
(5)
That is (x) ∨ ( ∧ )(x) ≤ ∧ (0) = ( ∧ ) (0), so that (x) ∨ ( ∧ )(x) = ∧ (0) ∧ ((x) ∨ ( ∧ )(x)). Define a subset of R × R as D(x) = {( p, q) ∈ R × R : x = p + q} and consider ( ∧ ( + ))(x) = (x) ∧ ( + )(x) ⎧ ⎫ ⎨ ⎬ = (x) ∧ ( (y) ∧ (z)) ⎩ ⎭ =
(y,z)∈D(x)
((x) ∧ (y) ∧ (z)) (as L is a completely distributive lattice)
(y,z)∈D(x)
=
⎧ ⎪ ⎨
⎪ ⎩ (y,z)∈D(x) 0 y,0 z
=
⎧ ⎪ ⎨
⎪ ⎩ (y,z)∈D(x)
⎫ ⎪ ⎬
((x) ∧ (y) ∧ (z)) ∨{{(x) ∧ (x) ∧ (0)} ∨ {(x) ∧ (0) ∧ (x)}} ⎪ ⎭ ⎫ ⎪ ⎬
((x) ∧ (y) ∧ (z))
⎪ ⎭
0 y,0 z
∨ {((x) ∧ (x)) ∨ ((x) ∧ (x))}
(using the definition of the tip-extended pair , , (1) and (2)) ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ ((x) ∧ (y) ∧ (y) ∧ (z)) ∨{(x) ∨ ((x) ∧ (x)} (using (3)) = ⎪ ⎪ ⎭ ⎩ (y,z)∈D(x) 0 y,0 z
≤
⎧ ⎪ ⎨
⎪ ⎩ (y,z)∈D(x) 0 y,0 z
((z) ∧ (y) ∧ (z))
⎫ ⎪ ⎬ ⎪ ⎭
∨ {(x) ∨ ( ∧ )(x)} (using (4))
(6)
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I. Jahan / Fuzzy Sets and Systems 199 (2012) 121 – 129
=
⎧ ⎪ ⎨ ⎪ ⎩ (y,z)∈D(x)
((y) ∧ ( ∧ )(z))
0 y,0 z
=
⎧ ⎪ ⎨
⎪ ⎭
∨ {∧ (0) ∧ ((x) ∨ (( ∧ )(x))} (using (6)) ⎫ ⎪ ⎬
⎪ ⎩ (y,z)∈D(x) 0 y,0 z
⎫ ⎪ ⎬
(∧ (y) ∧ ( ∧ ) (z) ∨{∧ (0) ∧ ∧ (x) ∨ (∧ (0) ∧ ( ∧ ) (x)} ⎪ ⎭
(as L is a completely distributive lattice) ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ ∧ ( (y) ∧ ( ∧ ) (z) ∨{(( ∧ ) (0) ∧ ∧ (x)) ∨ (∧ (0) ∧ ( ∧ ) (x))} (using (5)) = ⎪ ⎪ ⎭ ⎩ (y,z)∈D(x) 0 y,0 z
=
(∧ (y) ∧ ( ∧ ) (z)
(y,z)∈D(x) ∧
= (
+ ( ∧ ) )(x).
Hence ∧ ( + ) ⊆ ∧ + ( ∧ ) . This establishes of the modularity of L(R). 4. Conclusion The studies of fuzzy algebraic substructures were at their prime from the year 1981 to 1995. During the early part of nineties, a systematic investigation of modularity of the lattice of fuzzy normal subgroups was carried out by Ajmal and Thomas and the modularity was established in a stepwise manner. Finally in the year 1994, it was proved that the lattice of fuzzy normal subgroups is modular. The proof of this result displays a very beautiful and strong application of the notion of strong level subsets. This suggested further applications of this notion in the studies of fuzzy algebraic substructures. In the year 1995, Head in order to take care of the common procedure adopted in the proofs of the various results of fuzzy algebraic substructures, used the notion of strong level subset to formulate his metatheorem and subdirect product theorem. Using this subdirect product theorem, Head very easily extended the modularity of the lattice of normal subgroups to the fuzzy setting. At this stage of development of studies of L-algebraic substructures, a need is felt to establish a similar version of both metatheorem and subdirect product theorem in L-setting. The necessary machinery for such a development is provided in some papers by Seselja and Tepavcevic [12,14,15]. To this end the concept of strong level subset has to be replaced by level subset. Moreover, an evaluation lattice has to be chosen in order to establish a metatheorem or a subdirect product theorem. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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