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Fuzzy greedoids were recently introduced as a fuzzy set generalization of (crisp) greedoids. We characterize fuzzy languages which define fuzzy greedoids, give necessary properties and sufficient properties of the fuzzy rank function of a fuzzy greedoid, give a characterization of the rank function for a weighted greedoid, and discuss the rank closure of a fuzzy greedoid.

In 1965 Zadeh [

In fact, starting in 1988, Goetschel Jr. and Voxman introduced the idea of a fuzzy matroid through a series of papers studying the many different cryptomorphic definitions of a crisp matroid [

Only recently, in 2011, Al-Hawary [

Similar to matroids and other set systems, greedoids can be defined in a variety of ways, including as a set system, as an ordered language, from a rank function, and from a “closure operator”. In this article, we consider some of the other ways that fuzzy greedoids may be defined. We obtain a characterization of fuzzy greedoids in terms of fuzzy languages, and obtain necessary conditions and also sufficient conditions for a function to be a rank function of a fuzzy greedoids. Finally, we discuss the existence of a closure operator for a fuzzy greedoid.

In Section 2, we state the basic definitions and give a large class of examples of fuzzy greedoids. In Section 3, we introduce the ordered language interpretation of fuzzy greedoids. In the fourth section we discuss the rank function, introducing both sufficiently and separately necessary conditions for the rank function of general fuzzy greedoids. Additionally we obtain a characterization for the rank function of weighted greedoids (a large class of fuzzy greedoids). In Section 5 we discuss the existence of a rank closure operator in fuzzy greedoids. We finish with some open areas of research relating to fuzzy greedoids.

Although it is possible to define fuzzy sets on any set, we are going to assume that the underlying set E is finite. A fuzzy set on E is a function

Definition 1. Suppose that

1)

2)

3)

4)

5)

6)

7)

8)

Finally, we let

There is one particular type of fuzzy set which we will use extensively throughout the article―a spike― which we define here:

Definition 2. Suppose

Every fuzzy set can be thought of as a fuzzy union of spikes, as the following lemma, whose proof is straightforward, shows.

Lemma 3. Suppose

We now want to generalize the definition of a (crisp) greedoid to a fuzzy set version. The traditional definition of a greedoid in term of its feasible sets is given below. A thorough introduction to crisp greedoid theory can be found in [

Definition 4. A finite set E and a collection

1)

2) For any

3) If

By considering this definition in terms of functions from E to

Definition 5. Given a finite set E and a family

(F1) For every

(F2) For any

1)

2)

Then the fuzzy set system

Because of their close relationship with crisp greedoids, a large number of examples of fuzzy greedoids can be obtained by adding a weight function to a greedoid, as the following proposition shows.

Proposition 6. Suppose

then

Proof. All that is required is to show that

Suppose

With this proposition, it becomes easy to generate a large number of classes of examples of fuzzy greedoids. We give two such classes.

Example 1. (Weighted Branching Fuzzy Greedoids)

Suppose that

In particular, if

and

Example 2. (Weighted Poset Greedoids)

Suppose P is a finite poset. The collection of ideals of P form the feasible sets for the poset greedoid

In particular, if P has the following Hasse diagram.

If

Although weighted greedoids give a large number of examples of fuzzy greedoids, not every fuzzy greedoid can be defined as a weighted greedoid, as the following example shows:

Example 3. Let

However, as weighted greedoids, by definition, must have a finite collection of feasible fuzzy sets, this fuzzy greedoid cannot be a weighted greedoid.

In addition to a set system of feasible sets, a crisp greedoid can be defined as an ordered language with given properties. By defining a fuzzy language on a finite set, a fuzzy greedoid can be defined in the same manner. In this section, we determine the conditions necessary for a fuzzy language to define a fuzzy greedoid.

We begin by defining a fuzzy language on a set E.

Definition 7. Suppose

where

In a standard abuse of notation, we will use

In order to show when a fuzzy language defines a fuzzy greedoid, we need to be able to go from a collection of fuzzy words to a collection of fuzzy sets, and vice versa. Suppose

For the other direction, Suppose

Given this ability to switch from a fuzzy language to a fuzzy set system, we now consider the properties that the languages for crisp greedoids have. Mirroring the properties possessed by crisp greedoids, we obtain the following theorem.

Theorem 1. Suppose E is a finite set and

(L1)

(L2) If

(L3) If

Proof. We need to show that

Let

Therefore

All that is left to show is that

Similarly to crisp greedoids, Theorem 1 gives another cryptomorphic definition of a fuzzy greedoid. Next, we consider some of the functions that also give cryptomorphic definitions of crisp greedoids and consider their fuzzy versions. The two most common such functions are the rank function and the rank closure operator. We begin with the rank function for a fuzzy greedoid.

As with crisp greedoids, there is a natural definition of a rank function for fuzzy sets originally given in [

Definition 8. If

Notice that for any fuzzy set

Although we cannot guarantee that for any fuzzy set

Lemma 9. If

We first consider the properties possessed by r. In crisp greedoids, the rank function r is characterized by the following four properties (Theorem V.1.1 from [

1)

2)

3) If

4) If

The fuzzy equivalents of these four properties are:

(R1)

(R2) If

(R3) If

(R4) If

The natural question is whether these properties still hold for fuzzy greedoids, and if they do, whether they characterize the rank function of a fuzzy greedoid.

Proposition 10. If

Proof. (R1) is trivial. Since

Suppose

Suppose

By choice of

Given any rank function

Example 4. Let

However, with the following additional property, we do obtain a fuzzy greedoid.

(R5) If

Before showing that a function satisfying (R1)-(R5) is the rank function of a fuzzy greedoid, we shall need the following lemma:

Lemma 11. Suppose

If

Proof. Suppose

If

This is clearly a contradiction, so

Thus

With this, we can now show:

Theorem 2. If a function

Proof. As above, let

The previous lemma shows that if

To show

Let

Contradicting the assumptions. Otherwise we obtain a pair

Continuing this process, we obtain a pair of feasible fuzzy sets

Thus, using (R4), it is straightforward to show that there exists

Regretfully, not every rank function satisfies (R5), as the following example shows:

Example 5. For the following weighted rooted graph (rooted at

Notice that

This leads to the natural question:

Question 1. Determine necessary and sufficient conditions for a function to be a rank function for a fuzzy greedoid.

To finish this section, we will obtain a partial answer to the above question. We will now characterize the properties of a rank function for a weighted greedoid. From the work above, the rank function of a weighted greedoid clearly satisfies properties (R1)-(R4). It is easy to see that such a rank function also satisfies

(R5') For all

All that remains is to show that a function satisfying properties (R1)-(R4) and (R5') defines a weighted greedoid. We first show that it must define a fuzzy greedoid.

Lemma 12. If a function

Proof. Once again, define

Suppose that

To show that

If

The proof to show that

Theorem 3. Suppose that

Proof. As above, let

is a fuzzy greedoid. The collection of supports of feasible fuzzy sets forms the collection of feasible sets in a greedoid. For each element

We want to end with an example of the difference between crisp and fuzzy greedoids. In addition to feasible sets, ordered languages, and rank functions, crisp greedoids can be characterized by a closure operator. We will now show how the natural extension of the rank closure operator to fuzzy greedoids loses properties which would give a characterization of fuzzy greedoids. The natural extension is stated below:

Definition 13. Let

For crisp greedoids, the rank of a subset and the rank of the closure of the subset are equal. This property leads to a collection of properties which can be used to characterize the closure operator of a crisp greedoid. Regretfully, this property no longer holds for fuzzy greedoids, as the following example shows.

Example 6. Consider the weighted branching greedoid on the following weighted graph:

Notice that

As fuzzy greedoids are a new area of interest, there are a lot of directions research can go to understand these objects. Most of these areas come directly from the crisp greedoid definitions. We now mention four such areas of investigation for fuzzy greedoids.

1) The poset greedoid mentioned in Section 2 is actually an example of a combinatorial structure called an antimatroid (see [

Definition 14. Given a finite set E and a collection

a) For every

b) If

A lot of work has been done in characterizing antimatroids in terms of a closure operator, shelling processes, etc. Determine the fuzzy versions of these properties for antimatroids.

2) In 1989 Gordon and McMahon [

3) In addition to the rank closure introduced here, greedoids possess another closure operator, the kernel closure. If

4) In addition to matroids and antimatroids, there are many classes of greedoids which have been studied extensively. In [

This research was supported by Summer Research Grants offered through Misericordia University.

Steven J.Tedford, (2015) Rank Functions of Fuzzy Greedoids. Open Journal of Discrete Mathematics,05,65-73. doi: 10.4236/ojdm.2015.54006