Tom Head's join structure of fuzzy subgroups

Fuzzy Sets and Systems 125 (2002) 191–200www.elsevier.com/locate/fss

Tom Head’s join structure of fuzzy subgroupsAparna Jain ∗

Department of Mathematics, Deen Dayal Upadhyaya College, University of Delhi, Shivaji Marg, Karampura,New Delhi-110015, India

Received 2 June 1999; received in revised form 8 June 2000; accepted 7 July 2000

Abstract

Tom Head formulated a “Metatheorem” in 1995 [Fuzzy Sets and Systems 73 (1995) 349] with an idea of directlyproving the fuzzy analogs of results of classical Mathematics. In the process he de5ned a vital concept of tip extendedpair of fuzzy subgroups. In this paper we use this concept to de5ne the join of fuzzy subgroups. Using this de5nitionof join, we have provided a much simpler and direct proof of modularity of the lattice of fuzzy normal subgroups.Moreover, we have also demonstrated other applications regarding quasinormality and fuzzy submonoid generated by twofuzzy submonoids using the rede5ned sup-min product due to Kim [Inform. Sci. 91 (1996) 77]. c© 2002 Elsevier ScienceB.V. All rights reserved.

Keywords: Fuzzy algebra; Fuzzy subgroup; Join of fuzzy subgroups; Lattice; Strong level subset; Fuzzy normalsubgroup; Product of fuzzy sets; Modularity

1. Introduction

After the paper [11] on fuzzy groups by Rosenfeld in 1971, it was realised that numerous results whichwere true in classical algebraic set up, were also true after being converted into parallel fuzzy set up. Butlooking at the structural similarity between the proofs of all such theorems, the need was constantly felt fora universal theorem, which would automatically take care of the common procedure occurring in most of theresults. For this purpose, a “metatheorem” was formulated by Tom Head in 1995, which can be regarded asthe most signi5cant development in this subject till date.Tom Head’s paper [12] demonstrates a beautiful interplay of three classes P(X ), F(X ) and C(X ) which

are, respectively, the ordinary power set of X , the fuzzy power set of X (i.e. the collection of all fuzzysubsets of X ) and the crisp power set of X (i.e. the collection of characteristic functions of all subsets ofX ) for a nonempty set X . He de5ned the concept of a convolutional extension (see De5nition 2.5) of an

∗ Correspondence address: D-184, Phase-1, Ashok Vihar, Delhi-110052, India. Tel.: +91-11-7135-935; fax: +91-11-7235-150.E-mail address: [email protected] (A. Jain).

0165-0114/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(00)00128 -7

192 A. Jain / Fuzzy Sets and Systems 125 (2002) 191–200

n-ary operation “∗” on X to an n-ary operation “∗” on F(X ) and P(X ). Then a representation function“Rep” is de5ned from F(X ) to the cartesian product C(X )J where J = [0; 1). The function Rep is shownto commute with 5nite infs and arbitrary sups of fuzzy sets. Consequently, I(X ) which is the image ofF(X ) under the function Rep is a sublattice and also a complete upper subsemilattice of C(X )J and further,Rep becomes an order isomorphism of F(X ) onto I(X ). The function Rep also provides an algebra isomor-phism between F(X ) and I(X ) such that for each r in J , the projection of F(X ) into the rth coordinatespace of C(X )J is a surjection. All these facts lead to the formulation of a subdirect product theorem andmetatheorem.In the last section of his paper, Tom Head tries to establish with the help of subdirect product theorem,

the fact that for any group X with binary operation ‘∗’ and unary operation −1, the lattice L of all fuzzynormal subgroups of X is modular. In the process, he de5ned the join of two fuzzy normal subgroups f andg as f ∗ g (where ∗ is the convolutional extension of the binary operation ∗ in X ), which was infact not thecase. Rather we have, sup(f; g)=f ∗ g if and only if f and g have the same tip. With this idea Tom Headbeautifully de5ned a tip extended pair of fuzzy subgroups f and g (see De5nition 3.1) and called them f′

and g′. Consequently, the least upper bound of f and g in L turns out to be f′ ∗ g′.The purpose of this paper is to look more deeply into this vital concept of tip extended pair of fuzzy

subgroups and explore a few areas of its applications.

2. Preliminaries

Throughout this paper, G will denote a group, unless otherwise mentioned.

De�nition 2.1 (Rosenfeld [11]). A fuzzy subset � of G is said to be a fuzzy subgroup of G if the followingaxioms are satis5ed:(i) �(xy)¿min{�(x); �(y)},(ii) �(x−1) = �(x); ∀x; y∈G.It follows that �(x)6�(e) for each x in G, where e is the identity of G.

De�nition 2.2 (Ajmal and Thomas [3]). Let � be a fuzzy subset of G. Then the fuzzy subgroup generatedby � in G is de5ned as the least fuzzy subgroup of G containing � and is denoted by 〈�〉.

Let L(G) denote the collection of all fuzzy subgroups of G. Then it is known from [3] that L(G) is acomplete lattice under the ordering of fuzzy set inclusion ‘⊆’. Here, meet and join are de5ned by

(∧�i)(x) = inf �i(x) ∀x ∈ G and ∨�i = 〈∪�i〉;

where 〈∪�i〉 is the fuzzy subgroup generated by ∪�i.

De�nition 2.3 (Liu [9]). Let �∈L(G). Then � is said to be a fuzzy invariant (or fuzzy normal) if �(xy)=�(yx) for each x; y in G.

Let Ln(G) denote the collection of all fuzzy normal subgroups of G. Then it is also known from [4] thatLn(G) forms a sublattice of L(G).

Theorem 2.4 (Mukherjee and Bhattacharya [10]). Let �∈L(G). Then �∈Ln(G) if and only if � is constanton each conjugate class of G. i.e. �(xyx−1) = �(y) ∀x; y∈G.

A. Jain / Fuzzy Sets and Systems 125 (2002) 191–200 193

De�nition 2.5 (Head [12]). Let X have an n-ary operation ∗ :X n→X , where n is a positive integer. Then theconvolutional extension of ∗ to the fuzzy power set F(X ) is de5ned as an n-ary operation ∗ :F(X )n→F(X )given by

∗ (f1; : : : ; fn)(x) = supminx=∗(x1 ;:::; xn)

{f1(x1); : : : ; fn(xn)}:

The operation ∗ on X also gives an n-ary operation ∗ :P(X )n→P(X ) de5ned for subsets X1; : : : ; Xn of X by

∗ (X1; : : : ; Xn) = {∗(x1; : : : ; xn) | x1 ∈ X1; : : : ; xn ∈ Xn}:

The following de5nition by Liu [9] is a particular case of the Tom Head’s convolutional extension.

De�nition 2.6 (Liu [9]). The set product of fuzzy subsets � and � of G denoted by � ◦ � is a fuzzy subsetof G, de5ned as follows:

� ◦ �(x) = supminx=yz

{�(y); �(z)} ∀x ∈ G:

Theorem 2.7 (Ajmal [5]). Let �; �∈L(G). Then the set product � ◦ �∈L(G) if and only if � ◦ �= � ◦ �.

Theorem 2.8 (Liu [9]). Let �∈Ln(G). Then � ◦ �=� ◦ � ∀�∈L(G).

In classical group theory, a subgroup H of a group G is said to be quasinormal if it commutes with everysubgroup of G. Furthermore, a group G is said to be quasi Hamiltonian if every subgroup of G is quasinormal.

De�nition 2.9 (Ajmal and Thomas [6]). Let �∈L(G). Then � is said to be fuzzy quasinormal if � ◦ �=� ◦ � ∀�∈L(G).

Let Lq(G) denote the collection of all fuzzy quasinormal subgroups of G. It is known from [6] that if theset of all quasinormal subgroups of G forms a lattice, then the set Lq(G) is a modular sublattice of L(G).

De�nition 2.10 (Ajmal [5]). Let � be a fuzzy subset of G and let t ∈ [0; 1]. Then the strong level subset �¿tis de5ned as follows:

�¿t = {x ∈ G | �(x)¿ t}:

Theorem 2.11 (Ajmal and Thomas [6]). Let � be a fuzzy subset of G. Then � is a fuzzy (normal) subgroupof G if and only if each nonempty strong level subset �¿t is a (normal) subgroup of G.

Theorem 2.12 (Ajmal and Thomas [6]). Let �∈L(G). Then � is fuzzy quasinormal if and only if eachnonempty strong level subset �¿t is a quasinormal subgroup of G.

As a consequence of the above theorems, if G is a quasi-Hamiltonian group, then not only the set of allquasinormal subgroups coincides with the lattice of all subgroups of G, but the set Lq(G) also coincides withthe fuzzy subgroup lattice L(G).

Lemma 2.13 (Ajmal and Thomas [3]). Let �∈L(G). If �(x)¡�(y) then �(xy)= �(x)= �(yx).


3. The tip extended pair of fuzzy subgroups and modularity

The following notion is taken from [13] and is going to play a crucial role in rest of the section.

De�nition 3.1. For �; �∈L(G), de5ne fuzzy subsets �′ and �′ of G as follows:

�′(x) = �(x) and �′(x) = �(x) ∀x �= e

and

�′(e) = �′(e) = sup {�(e); �(e)}:

Clearly, �⊆ �′ and �⊆ �′.

If � and � are fuzzy subgroups of G, then the pair �′, �′ is called the tip extended pair of fuzzy subgroupscorresponding to � and �. Remark 3.4 provides an explanation for this. It can also be observed here that eachof �′ and �′ depends on both � and �. In order to rectify this situation, we prove the following:

Proposition 3.2. If �∈L(G) and t ∈ [0; 1]; then the fuzzy subset �t of G de4ned by

�t(x) ={�(x) if x �= e;max{�(e); t} if x = e:

is also a fuzzy subgroup of G.

Proof. If t6 �(e), there is nothing to prove. So, let us assume that t¿�(e). It is clear that �t(x−1)= �t(x) foreach x in G. Let x; y∈G. If x= e and y �= e, then �t(xy)=�t(y)=�(y)=min{t; �(y)}=min{�t(x); �t(y)}.Similarly, if x �= e and y= e, we have �t(xy)= �t(x)=�(x)=min{t; �(x)}=min{�t(y); �t(x)}. Further, ifx �= e �=y and xy= e, then �t(xy)= t¿�(e)¿min{�(x); �(y)}=min{�t(x); �t(y)}. If x �= e �=y and xy �= e,then �t(xy)= �(xy)¿min{�(x); �(y)}=min{�t(x); �t(y)}. Finally, if x=y= e, then �t(xy)= t=min{�t(x),�t(y)}. Hence, �t ∈L(G).

Using the above result, we see that �′= �t for t= �(e) and �′= �t for t= �(e).The proof of the following proposition is straightforward.

Proposition 3.3. If �; �∈L(G); then � ◦ �⊆ �′ ◦ �′.

In view of Proposition 3.2, the following remark is clear.

Remark 3.4. Let t ∈ [0; 1] and � be a fuzzy subset of G. Then(1) if �∈L(G), then �t ∈L(G),(2) if �∈Ln(G), then �t ∈Ln(G),(3) if �∈Lq(G), then �t ∈Lq(G).

Theorem 3.5 (Ajmal and Thomas [2]). Let �; �∈Lq(G). Then � ◦ �∈Lq(G).

Proposition 3.6. Let �; �∈Lq(G). Then �′ ◦ �′ ∈Lq(G).

Proof. This is a direct consequence of Remark 3.4(3) and Theorem 3.5.


We recall here that the fuzzy subgroup of G generated by �∪ � is the least fuzzy subgroup of G containing�∪ � (see De5nition 2.2).

Theorem 3.7. Let �; �∈L(G) such that �◦�∈L(G). Then �′ ◦ �′ is the fuzzy subgroup generated by �∪ �.

Proof. We 5rst prove that �′ ◦ �′ ∈L(G). In view of Remark 3.4 and Theorem 2.7, it is enough to provethat �′ ◦ �′= �′ ◦ �′. Clearly, (�′ ◦�′)(e)=min{�′(e); �′(e)}=min{�′(e); �′(e)}=(�′ ◦�′)(e).Suppose if possible �′ ◦ �′(z)¿�′ ◦ �′(z) for some e �= z ∈G. Then there exist x◦, y◦ such that z= x◦ y◦

and

min{�′(x◦); �′(y◦)}¿ supminz=xy

{�′(x); �′(y)}: (1)

Case 1: x◦ �= e, y◦ �= e. Then (1) implies that

min{�(x◦); �(y◦)}¿max

supminz=xy

x �=e �=y

{�(x); �(y)};min{�′(z); �′(e)};min{�′(e); �′(z)}

¿max

{supminz=xy; x �=e �=y

{�(x); �(y)};min{�(z); �(e)};min{�(e); �(z)}}

= supminz=xy

{�(x); �(y)} = � ◦ �(z) = � ◦ �(z) (as � ◦ � ∈ L(G))

= max

supminz=xy

x �=e �=y

{�(x); �(y)};min{�(z); �(e)};min{�(e); �(z)}

:

Therefore, min{�(x◦); �(y◦)}¿min{�(x); �(y)} ∀x; y such that z= xy and x �= e �=y. In particular,min{�(x◦); �(y◦)}¿min{�(x◦); �(y◦)}, which is a contradiction.Case 2: x◦= e; y◦ �= e. Then z= ez. By (1), we get

min{�′(e); �(z)}¿min{�′(x); �′(y)} ∀x; y such that z = xy:

In particular, min{�′(e); �(z)}¿min{�′(z); �′(e)}. This implies �(z)¿�(z), which is a contradiction.Case 3: x◦ �= e; y◦= e. This is similar to Case 2.Hence �′ ◦ �′(z)= �′ ◦ �′(z) for each z in G and consequently, �′ ◦ �′ ∈L(G). Further, since �⊆ �′ and

�′ ◦ �′(z)¿min{�′(z); �′(e)}= �′(z) for each z in G, we have �⊆ �′ ◦ �′. Similarly, �⊆ �′ ◦ �′. Now, let �be any fuzzy subgroup of G such that �∪ �⊆ �. Then

�′ ◦ �′(e) = max{�(e); �(e)} = � ∪ �(e)6�(e):

Note that, if z= xy is in G, then

�(z)= �(xy)¿min{�(x); �(y)}¿min{�(x); �(y)}:


Therefore, if z �= e is in G, then

�′ ◦ �′(z) = supminz=xy

{�′(x); �′(y)}

=max

supminz=xy

x �=e �=y

{�(x); �(y)};min{�(z); �′(e)};min{�′(e); �(z)}

=max

supminz=xy

x �=e �=y

{�(x); �(y)}; �(z); �(z)

6max{�(z); �(z); �(z)} = �(z):

Hence, �′ ◦ �′ ⊆ �. This proves that �′ ◦ �′ is the least fuzzy subgroup of G containing � and �.

Next we oMer a new proof of a well-known result.

Theorem 3.8. Let �; �∈Ln(G). Then � ◦ �∈Ln(G).

Proof. Clearly, � ◦ �∈L(G). Suppose if possible � ◦ � =∈Ln(G). Then for some x; y∈G, � ◦ �(yxy−1)¿� ◦ �(x). This implies

supminyxy−1 = ab

{�(a); �(b)}¿� ◦ �(x):

Therefore, there exist a◦; b◦ such that yxy−1 = a◦b◦, and

min{�(a◦); �(b◦)}¿� ◦ �(x);i.e.,

min{�(a◦); �(b◦)}¿min{�(a′); �(b′)} ∀a′; b′ ∈ G such that x= a′b′:

But x=y−1a◦yy−1b◦y. Therefore,

min{�(a◦); �(b◦)}¿min{�(y−1a◦y); �(y−1b◦y)} = min{�(a◦); �(b◦)}:This contradiction gives the desired result.

Corollary 3.9. Let �; �∈Ln(G). Then �′ ◦ �′ is the fuzzy normal subgroup generated by � and �.

Proof. Let �; �∈Ln(G). Then by Remark 3.4 and Theorem 3.8, �′ ◦ �′ ∈Ln(G). Hence by Theorem 3.7,�′ ◦ �′ is the least fuzzy normal subgroup containing � and �.

As a consequence of the above theorem, Ln(G) is a lattice, with meet and join de5ned as follows:

� ∧ � = � ∩ � and � ∨ � = �′ ◦ �′:Let us recall here that the proof of modularity of Ln(G) in [4] depends heavily on the fact that the set of allnormal subgroups of G forms a modular lattice. Also, a rather indirect proof of modularity of Ln(G) occursin [8] where the join of � and � is de5ned by the fuzzy set �∪ �∪ (� ◦ �). Here in Theorem 3.10, using


the fact that the join of two fuzzy normal subgroups � and � of G is �′ ◦ �′, we present a direct proof ofmodularity of Ln(G) which is independent of modularity of the lattice of all normal subgroups of G.

Theorem 3.10. The lattice Ln(G) is modular.

Proof. Let �; �; �∈Ln(G) such that �¿�. We have � ∧ (� ∨ �)¿� ∨ (� ∧ �).By using the modularity of closed unit interval [0, 1] under the usual ordering of reals, we get

� ∧ (� ∨ �)(e) = min{�(e);max{�(e); �(e)}} = max{�(e);min{�(e); �(e)}} = � ∨ (� ∧ �)(e):Suppose now for some z �= e, � ∧ (� ∨ �)(z)¿� ∨ (� ∧ �)(z).This implies �(z)¿�′ ◦ (� ∧ �)′(z), and �′ ◦ �′(z)¿�′ ◦ (� ∧ �)′(z), i.e.,

�(z)¿min{�′(x); (� ∧ �)′(y)}and

�′ ◦ �′(z)¿min{�′(x); (� ∧ �)′(y)} ∀x; y: z= xy:Therefore, there exists a decomposition z= x◦y◦ of z such that

�(z)¿min{�′(x); (� ∧ �)′(y)}and

min{�′(x◦); �′(y◦)}¿min{�′(x); (� ∧ �)′(y)} ∀x; y: z = xy: (1)

Case 1: x◦ �= e and y◦ �= e.Considering the particular case of z= x◦y◦ in (1), we get

�(z); �(x◦); �(y◦)¿min{�(x◦); (� ∧ �)(y◦)};i.e.,

�(z); �(x◦); �(y◦)¿min{�(x◦); �(y◦); �(y◦)}:As �(x◦)��(x◦) and �(y◦)��(y◦), therefore, from the above inequality we get min{�(x◦), �(y◦); �(y◦)}=�(y◦). Thus, �(z); �(x◦)¿�(y◦). Consequently, we have �(y−1◦ )= �(y◦)¡�(z)= �(x◦y◦). Hence byLemma 2.13, �(y◦)= �(x◦y◦y−1◦ )= �(x◦). This implies �(x◦)¿�(x◦), which contradicts �¿ �.Case 2: x◦= e and y◦= z.Then by (1), �(z);min{�′(e); �′(z)}¿min{�′(x); (� ∧ �)′(y)}∀x; y: z= xy.In particular, for z= ez, we have

�(z); min{�′(e); �′(z)}¿ min{�′(e); (� ∧ �)′(z)}:Now, min{�′(e); �′(z)}= �(z), (as �(z)= �′(z)6 �′(e)= �′(e)). Similarly, min{�′(e); (�∧�)′(z)}=(�∧�)(z).Therefore, we get �(z); �(z)¿(� ∧ �)(z). This is a contradiction.Case 3: x◦= z; y◦= e.Again by (1), �(z);min{�′(z); �′(e)}¿min{�′(x); (�∧�)′(y)}∀x; y: z= xy. In particular, for z= ze, we have

�(z);min{�′(z); �′(e)}¿min{�′(z); (� ∧ �)′(e)}. Now as in Case 2, min{�′(z); �′(e)}= �(z) and min{�′(z);(� ∧ �)′(e)}= �(z). Therefore we get �(z)¿�(z), which is a contradiction.Hence �∧ (�∨ �)(z)= �∨ (�∧ �)(z) for each z in G. This establishes that Ln(G) is a modular lattice.


4. Other applications

De�nition 4.1 (Kim [7]). A fuzzy subgroupoid � of a monoid (i.e. a semigroup with identity e) G is said tobe a fuzzy submonoid of G if, for each x in G

�(e)¿�(x):

Let FD(G) denote the collection of all fuzzy submonoids of a monoid G.

Proposition 4.2. If G is a monoid and �; �∈ FD(G); then �′; �′ ∈ FD(G).

Proof. The proof is straightforward.

In [7], Kim has rede5ned the sup min product of two fuzzy subsets of a semigroup G as follows:

De�nition 4.3 (Kim [7]). Let � and � be fuzzy subsets of a semigroup G. Then the product �˝ � of � and� is the fuzzy subset of G de5ned by

�˝ �(x) = Sup minx=a1b1 :::anbn

n∈N

{�(a1); �(b1); : : : ; �(an); �(bn)}:

Clearly, �◦�⊆ �˝ �⊆ �′˝ �′. Therefore, �◦�⊆ �′˝�′. Also note that least upper bound of �˝� is �(e)∧�(e). Let us call it t◦. In the same paper [7], Kim characterised the strong level subsets of �˝� asfollows:

(�˝ �)¿t = {a1b1 : : : anbn | ai ∈ �¿t ; bi ∈ �¿t ; n ∈ N} for each t ∈ [0; t◦]:

This particular characterisation yields the following interesting theorem.

Theorem 4.4 (Kim [7]). If � and � are fuzzy subsets of a semigroup G; then �˝� is a fuzzy subgroupoidof G.

The following result is established for fuzzy subgroups in [1]. However, it is found to be valid in a moregeneral setting.

Lemma 4.5 (Ajmal [1]). Let � and � be fuzzy submonoids of a monoid G. Then

�⊆ � ◦ � and �⊆ � ◦ � if and only if �(e) = �(e):

Lemma 4.6 (Kim [7]). Let � be a fuzzy subset of a semigroup G. Then � is a fuzzy subgroupoid of G ifand only if �˝�⊆ �.

The following lemma is easy to verify.

Lemma 4.7. If G is a monoid and �; �; �; �∈FD(G); such that �⊆ � and �⊆ �; then �˝�⊆ �˝�.

Theorem 4.8. Let � and � be fuzzy submonoids of a monoid G. Then �′˝�′ is the fuzzy submonoidgenerated by � and �.


Proof. Let G be a monoid and �; �∈FD(G). Then by Proposition 4.2 and Theorem 4.4, �′˝�′ is a fuzzy sub-groupoid of G. We shall prove that for each x in G, �′˝�′(e)¿�′˝�′(x). Suppose (�′˝�′)(e)¡(�′˝�′)(x)for some e �= x∈G. Then there exists a decomposition x= a1b1 : : : anbn of x in G, such that

�′˝ �′(e)¡min{�′(a1); �′(b1); : : : ; �′(an); �′(bn)}:This implies

min{�′(e); �′(e)}¡min{�′(a1); �′(b1); : : : ; �′(an); �′(bn)}:This contradicts that �′; �′ ∈FD(G). Hence, for every element x of G; (�′˝�′)(e)¿(�′˝�′)(x), and con-sequently �′˝�′ ∈FD(G). Now since �′, �′ ∈FD(G); by Lemma 4.5, �⊆ �′ ⊆ �′ ◦�′ ⊆ �′˝�′. Similarly,�⊆ �′˝�′. Further, let � be any fuzzy submonoid of G such that �; �⊆ �. Then clearly �′; �′ ⊆ � and thereforeby Lemmas 4.6 and 4.7, �′˝�′ ⊆ �˝�⊆ �. Hence, (�′˝�′) is the least fuzzy submonoid of G containing �and �. In other words, �′˝�′ is the fuzzy submonoid generated by � and �.

Thus FD (G) for a monoid G becomes a lattice with meet and join de5ned as

� ∧ � = � ∩ � and � ∨ � = �′˝ �′ for �; � ∈ FD(G):In [7], Kim established that the join of two fuzzy submonoids � and � of G is �∪ �∪ (�˝�). Thus the abovediscussion implies � ∨ � = �∪ �∪ (�˝�)= �′˝�′ in the lattice FD(G).The following theorem is easy to verify.

Theorem 4.9. If � and � are fuzzy submonoids of a monoid G; then �∪ �∪ (�◦�)= �′ ◦�′.

In [8], L(G) (if G is a quasi Hamiltonian group) is proved to be a modular lattice where the join of �and � is given by the fuzzy set �∪ �∪ (�◦�). This result can be modi5ed as follows:

Theorem 4.10. If G is a quasi-Hamiltonian group; then L(G) is a modular lattice under meet and joinde4ned by

� ∧ � = � ∩ � and � ∨ � = �′ ◦ �′:

The lattice structure of L(G) in the above theorem can be derived from Theorem 3.7 (using the fact that Gis a quasi-Hamiltonian group) and modularity can be proved using the same technique as in Theorem 3.10.Further, if G is an arbitrary group, then Ln(G) is proved to be a modular lattice in [8], with join of � and �given by �∪ �∪ (�◦�). This is a direct consequence of Theorems 3.10 and 4.9. We can conclude from thisdiscussion that if G is a quasi-Hamiltonian group, then Ln(G) is a modular sublattice of L(G). Whereas, ifG is an arbitrary group, then Ln(G) is a modular sublattice of Lq(G).

Acknowledgements

I am highly thankful to the learned referees for their constructive criticism and healthy suggestions whichmade the presentation of this paper better.

References

[1] N. Ajmal, Set product and fuzzy subgroups, in: R. Lowen, M. Roubens (Eds.), Proceedings of IFSA World Congress, Belgium,1991, pp. 3–7.

[2] N. Ajmal, K.V. Thomas, Quasinormality and fuzzy subgroups, Fuzzy Sets and Systems 58 (1993) 217–225.


[3] N. Ajmal, K.V. Thomas, The lattice of fuzzy subgroups and fuzzy normal subgroups, Inform. Sci. 76 (1994) 1–11.[4] N. Ajmal, The lattice of fuzzy normal subgroups is modular, Inform. Sci. 83 (1995) 199–209.[5] N. Ajmal, Fuzzy groups with sup property, Inform. Sci. 93 (1996) 247–264.[6] N. Ajmal, K.V. Thomas, The join of fuzzy algebraic substructures of a group and their lattices, Fuzzy Sets and Systems 99 (1998)

213–224.[7] J.G. Kim, Lattices of fuzzy subgroupoids, fuzzy submonoids and fuzzy subgroups, Inform. Sci. 91 (1996) 77–93.[8] K.C. Gupta, S. Ray, Modularity of the Quasi-Hamiltonian fuzzy subgroups, Inform. Sci. 79 (1994) 233–250.[9] W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133–139.[10] N.P. Mukherjee, P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225–239.[11] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512–517.[12] T. Head, A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and Systems 73 (1995) 349–358.[13] T. Head, Erratum to “A metatheorem for deriving fuzzy theorems from crisp versions”, Fuzzy Sets and Systems 79 (1996) 277–278.

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Tom Head's join structure of fuzzy subgroups