© Hindawi Publishing Corp. ON AN EQUIVALENCE OF FUZZY SUBGROUPS III

This paper is the third in a series of papers studying equivalence classes of fuzzy subgroups of a given group under a suitable equivalence relation. We introduce the notion of a pinned flag in order to study the operations sum, intersection and union, and their behavior with respect to the equivalence. Further, we investigate the extent to which a homomorphism preserves the equivalence. Whenever the equivalences are not preserved, we have provided suitable counterexamples.


Introduction.
For the benefit of the reader and also to fix notations, we recall the following from [3].
We use I = [0, 1], the real unit interval, as a chain with the usual ordering in which ∧ stands for infimum (or intersection) and ∨ stands for supremum (or union).
A fuzzy subset of a set X is a mapping µ : X → I.The union, intersection of two fuzzy sets, and complementation of a fuzzy set are defined using sup pointwise, inf pointwise, and 1−µ operator pointwise, respectively.We denote the set of all fuzzy subsets of X by I X .Further, we denote fuzzy sets by the Greek letters µ, ν, η, and so forth.If X is a finite group (which is what we will be interested in this paper), a fuzzy set µ is said to be a fuzzy subgroup if µ(x + y) ≥ µ(x) ∧ µ(y) for all x, y ∈ X and µ(x) = µ(−x).Without loss of generality, we assume that µ(0) = 1.From this assumption, we notice that the only admissible fuzzy subgroup of the trivial group is µ(0) = 1.
We define an equivalence relation ∼ on I X as follows: µ ∼ ν if and only if (i) for all x, y ∈ X, µ(x) > µ(y) if and only if ν(x) > ν(y), (ii) µ(x) = 0 if and only if ν(x) = 0.It is easily checked that this relation is indeed an equivalence relation on I X and when restricted to 2 X , this relation coincides with an equality of sets.We denote this equivalence relation by µ ∼ ν.Originally, this idea of equivalence was prompted by [3,4,5].
Note 1.1.The condition, µ(x) = 0 if and only if ν(x) = 0, simply says that the supports of µ and ν are equal, where by support we mean supp µ = {x ∈ X : µ(x) > 0}.The above condition cannot be made redundant since it is an essential part of the equivalence relation.Throughout this paper, by a group G we mean a finite group.By a flag we mean a chain of subgroups of the form 0 For some examples of flags used in this paper, we refer to [1].In [1], flags were referred to as series.
We recall from [4] that a keychain l means a set of real numbers in where the λ i 's are not all necessarily distinct.The λ i 's are called pins.By a pinned flag, we mean a pair (Ꮿ,l), of a flag Ꮿ and a keychain l, written as follows: We associate the following fuzzy subgroup with such a pinned flag (Ꮿ,l): where the component G n is the whole group G.We denote this simply by G n λn = G λn .That µ is indeed a fuzzy subgroup on G may be quickly verified using the definition.In this case we say that µ is represented by the pinned flag Conversely, every fuzzy subgroup µ may be decomposed into a pinned flag as above by considering suitable α-cuts.For further details, see [3,4].Similar techniques have been used in [2].

Homomorphisms and equivalences.
In this section, given a homomorphism between two groups, we look at the equivalence classes of homomorphic images and preimages of fuzzy subgroups.Firstly, we recall that if f : G → H is a homomorphism, by f (µ) we mean the image of a fuzzy subset µ of G and it is a fuzzy subset of H defined by, for The subgroup property is transferred to images and preimages by a homomorphism between groups.For further properties of images and preimages of fuzzy sets under a mapping, see [6].Throughout this section, we suppose that f : Also, one could consider the behavior of inequivalent fuzzy subgroups under a homomorphism.In general, one would expect inequivalent fuzzy subgroups to have inequivalent images, but the following example illustrates that two inequivalent fuzzy subgroups may have equivalent images under a homomorphism.
Similarly, we may have inequivalent fuzzy subgroups giving rise to equivalent preimages under a homomorphism. (2.3) which are equivalent.
It is clear that if f : G → H is an epimorphism, then f (f −1 (µ)) = µ; therefore, if µ and ν are inequivalent fuzzy subgroups of H, then f −1 (µ) and f −1 (ν) are inequivalent fuzzy subgroups of G. Similarly, it is clear that if f : G → H is a monomorphism, then f −1 (f (µ)) = µ; therefore, if µ and ν are inequivalent fuzzy subgroups of G, then f (µ) and f (ν) are inequivalent fuzzy subgroups of H. Therefore, equivalences and unequivalences of fuzzy subgroups are preserved under an isomorphism, but the notion of a fuzzy isomorphism is very different (see [3]).

Equivalences under operation.
In general, the operations of infimum (the intersection), supremum (the union), and sum of fuzzy subgroups need not preserve the equivalence classes of fuzzy subgroups.We have the following example.
Let G be the group of integers Z under addition.Let , otherwise, , otherwise. ( Then Firstly, we notice that Secondly, it is easily seen that while Similarly, we can show by an example that in general µ ∼ µ and ν ∼ ν do not imply µ ∨ ν ∼ µ ∨ ν .
The next example deals with the operation of sum.
Example 3.2.Suppose that µ ∼ ν and µ ∼ ν .Then it is not necessary that Let G be the group of integers Z under addition.Let Then , otherwise, (3.8) Firstly, we notice that Secondly, it is easily seen that while (3.12)Note 3.3.Although in this example µ + ν ∼ ν, this needs not be true in general.For example, if 1/6 is replaced by 0 in both ν and ν above, then µ + ν ∼ ν and µ + ν ∼ µ.
In contrast to the above examples, if we take two fuzzy subgroups µ and ν from the same equivalence class determined by µ and ν, then the inf, sup, and sum of µ and ν determine the same equivalence class .
which in turn implies µ(x) > µ(y).Other cases are dealt with similarly.
For the following proposition, we require both µ and ν to be fuzzy subgroups of a finite group.
But, as for Zadeh's complement [6], we have the following proposition.

Intersection and sums of fuzzy subgroups.
In this section, we determine the equivalence class of fuzzy subgroups corresponding to intersection and sum of two fuzzy subgroups in terms of pinned flags associated with given fuzzy subgroups.Throughout this section, we require the number of components in a pinned flag to be at least 3, otherwise the discussions become trivial.Consequently, we assume that n ≥ 2.
Firstly, in the next proposition, we look at a special case, namely the characterization of intersection and sum of two equivalence classes of fuzzy subgroups whose pinned flags (Ꮿ,l) have the same underlying flag of subgroups (Ꮿ).
Suppose that µ and ν are two fuzzy subgroups whose pinned flags have the same underlying flag Ꮿ but have different keychains of the forms Note 4.2.We emphasize in this note that µ and ν are not necessarily equivalent.
Proof.Let x ∈ G. Then there is an index i such that 1 ≤ i ≤ n with x ∈ G i , but x ∈ G i−1 .Then µ(x) = λ i and ν(x) = β i .
(i) To prove this part, it suffices to check that (µ ∧ ν)(x) = λ i ∧ β i , which is clearly true.
Suppose that we have two flags differing in only one component, such as Proposition 4.3.Suppose that µ and ν are fuzzy subgroups of G whose representative keychains are of the form 1λ 1 λ 2 •••λ n and whose underlying flags are Ꮿ µ and Ꮿ ν , respectively, as described above.Then (i) µ ∧ ν is represented by the keychain (ii) µ + ν is represented by the keychain Before we give the proof, we note that Ꮿ µ can equivalently be replaced by Ꮿ ν without any loss of generality.

It suffices to prove the case for
(ii) Firstly, we observe that G k+1 = G k + H k by the maximality of chains.Secondly, similar to the proof of case (i), for the sum, it is enough to consider and in the former case, x = x 1 +x 2 ∈ G s + G s = G s .By the chain property, we conclude that x ∈ G k−1 , which is a contradiction to the choice of x.Thus, (µ This completes the proof.
Suppose that we have two flags differing in two or more components but not consecutively, such as where Then, using the same argument as in Proposition 4.3 inductively, we have the following corollary.
Corollary 4.4.Let µ and ν be two fuzzy subgroups whose underlying flags Ꮿ µ and Ꮿ ν , respectively, differ in two or more components but not consecutively as shown in (4.4).Then (i) µ ∧ ν is represented by the keychain (ii) µ + ν is represented by the keychain Now, we would like to consider two flags differing in two or more components consecutively, such as Then the flags Ꮿ µ∧ν and Ꮿ µ+ν are given by where F can be either G i+k or H i+k , and where E can be either G i or H i , respectively.In the above, we have only indicated the corresponding distinct components in Ꮿ µ , Ꮿ ν , Ꮿ µ∧ν , and Ꮿ µ+ν and as the suppressed corresponding components are assumed to be identical in the two flags.Proposition 4.5.Suppose that µ and ν are fuzzy subgroups of G whose representative keychains are of the form 1λ 1 λ 2 •••λ n and whose underlying flags are Ꮿ µ and Ꮿ ν , respectively, as described above.Then (i) µ ∧ ν is represented by the keychain (ii) µ + ν is represented by the keychain Then there exists a pin λ s representing the value of (µ ∧ν)(x) and it is such that λ i+j−1 > λ s > λ i+j , which is a contradiction as λ i+j−1 and λ i+j are two consecutive pins.Thus, (µ ∧ ν)(x) = λ i+j .
The determination of the pinned flags of intersection and sum of two fuzzy subgroups µ and ν, where the pins as well as the flags of the pinned flags Ꮿ µ and Ꮿ ν representing µ and ν are distinct, in general, does not seem to follow any particular pattern as we have derived above.This is illustrated by the following example.
respectively.A simple calculation reveals the pinned flags for µ ∧ ν and µ + ν to be respectively.In the above calculation, notice that the roles played by the pins and the components of the flags are equally important in a way in which they are tied to each other.Suppose that we retain the flags but the pins for µ and ν are changed to respectively.Then µ ∧ ν and µ + ν have the pinned flags given by 0 respectively.Similarly, we could retain the pins but change the flags of µ and ν; for example, in (4.13) above we retain the same pins but swop the underlying flags.Then a simple calculation shows that we arrive at different (from (4.14)) pinned flags for µ ∧ ν and µ + ν.The complete determination of the pinned flags for µ and ν in the most general case will be dealt with in another paper.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

Example 4 . 6 .
Let G = Z 72 .Let Ꮿ µ and Ꮿ ν be the pinned flags of µ and ν on G given by

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation