M -Fuzzifying span mappings in M -Fuzzifying matroids 1

Abstract

In this paper, we give a new characterization of M-fuzzifying span mappings defined by M-fuzzifying rank functions. Then we research the relations between M-fuzzifying span mappings and other M-fuzzifying mappings on M-fuzzifying matroid, and characterize the M-fuzzifying coindependent, nonspan and hyperplane mappings. Based on these characterizations, we put forward one class of special fuzzifying matroids.

Keywords

fuzzifying dual matroids

1 Introduction

Matroids were introduced by Whitney in 1935 as a generalization of vector linear correlation and graphs [14]. It is well known that matroids play an important role in mathematics, especially in applied mathematics. One example is that the greedy algorithm is one simple and efficient algorithm in matroid structures [1]. A characteristic of matroids is that they can be defined in many different but equivalent concepts such as bases, independent sets, spanning sets, rank function, closure operator, etc. Among these concepts, the notion of independent sets, which is a generalization of the notion of vector linear correlation, defined a matroid as a pair $(E, I)$ where E is a finite set and $I$ is a subfamily of 2^E satisfying three axioms [2, 13].

Now a fuzzifying method of the (crisp) matroid is using a mapping from 2^E to M instead of a family of sets, where M a completely distributive lattice. From this point of view, Shi introduced the notion of M-fuzzifying matroids [6] and further he introduced the notion of (L, M)-fuzzy matroids (L is a completely distributive lattice) in [7]. From then on, the axioms of M-fuzzifying matriod were studied and discussed by many scholars, for example, the axioms of bases and circuits of ([0, 1]-)fuzzifying matroids, M-fuzzifying bases, M-fuzzifying rank function, M-fuzzifying closure operator, M-fuzzifying nullity, etc. [3–5 , 15–21]. In [9, 11], the M-fuzzifying spanning and nonspanning axioms were introduced by Sun and Li. They further discussed the relation with dual M-fuzzifying independent sets [10]. But the relations among the known axioms of M-fuzzifying matroid have not been given. Li and Shi proposed the definition of dual M-fuzzifying rank function and characterize the dual, minors, and nullity for an M-fuzzifying matroid when M is the unit interval [0, 1] or a finite Boolean algebra [3]. And as the applications, the ([0, 1]-)fuzzifying matroid has been introduced to obtain knapsack problem in [21] and the fuzzifying greedy algorithm in [20].

The aim of this paper is to research the axioms and properties of M-fuzzifying matroid. In Section 2, we recall some basic properties and notions of M-fuzzy natural number, crisp matroids, lattices, and previous M-fuzzifying axioms. We begin Section 3 by using M-fuzzifying rank function to give a new characterization of M-fuzzifying span mapping, and then characterize the M-fuzzifying nonspan, hyperplane and coindependent mappings by researching the relations between M-fuzzifying span mapping and other M-fuzzifying mappings. In Section 4, we study the properties of fuzzifying dual matroids induced by fuzzifying span mappings. In Section 5, we propose a class of special fuzzifying matroid (i.e., RC matroid) and give its some equivalent conditions and properties.

2 Preliminaries

Throughout this paper (M, ⋁ , ⋀ , ′) denotes a completely distributive lattice with an order-reversing involution (.) ′, unless specifically stated otherwise. The smallest element and the largest element in M are denoted by ⊥ and ⊤, respectively. An element a in M is called a (meet-)prime element if a ≥ b ∧ c implies a ≥ b or a ≥ c. a in M is called co-prime or join-prime if a ≤ b ∨ c implies a ≤ b or a ≤ c . The set of non-unit prime elements in M is denoted by P (M) . The set of non-zero join-prime elements in M is denoted by J (M). Denote the set of all M-fuzzy sets on X by M^X . For A ∈ M^X and a ∈ M, define A_[a] = {x ∈ X : A (x) ≥ a}, A^(a) = {x ∈ X : A (x) ≰a}.

In [6], there is one class of fuzzy natural number to define the cardinality of a fuzzy set. Let N denote the set of all natural numbers. An M-fuzzy natural number is an antitone map λ : N → M satisfying λ (0) =⊤ , ⋀ _n∈Nλ (n) = ⊥. The set of all M-fuzzy natural numbers is denoted by N (M). For any m ∈ N, we define $\underline{m} \in N (M)$ such that m (t) =⊤ when t ≤ m and m (t) =⊥ when t ≥ m + 1. In the sequel, we shall not distinguish m from $\underline{m}$ .

Recall that the pseudocomplement [3] of a relative to b (a, b ∈ M) is an element a → b of M satisfying a ∧ x ≤ b if and only if x ≤ a → b, that is, $a \to b = ⋁ {x \in M : a \land x \leq b} .$

It follows easily from definition that ⊤ → a = a and a→ ⊤ = ⊥ → b = ⊤ for any a, b ∈ M. And it is easy to show that b ≤ c ⇒ a → b ≤ a → c.

Given a non-empty finite set E, we denote the set of all subsets of E by 2^E, and for any X ∈ E, the cardinality of X by |X|. Let $A \subseteq 2^{E}$ . The following symbols are useful in the sequel.

$Min (A)$ = { $A \in A$ : $A \subseteq B \in A$ implies A = B};

= { $A \in A$ : $A \supseteq B \in A$ implies A = B};

= {X ∈ 2^E: there exists an $A \in A$ such that A ⊆ X};

= {X ∈ 2^E: there exists an $A \in A$ such that X ⊆ A};

= {X ∈ 2^E: $X \notin A$ };

= {X ∈ 2^E: $E - X \in A$ }.

The families of independent sets $I$ , bases $B$ , dependent sets $D$ , minimal circuits $C$ , spanning sets $S$ , nonspanning sets $N$ , and rank function $R_{I}$ of crisp matroids have been extended to fuzzy environment. Now we just list the parts of the knowledge researched in this paper.

Definition 2.1. ([2, 13]). $H \subseteq 2^{E}$ is the family of hyperplanes of a matroid $(E, I)$ if and only if $H$ satisfies the following conditions.

$E \notin H$ .

If $H_{1}, H_{2} \in H$ , then H₁ ⊆ H₂ implies H₁ = H₂.

If $H_{1}, H_{2} \in H$ with H₁ ≠ H₂, then for any e ∈ E - (H₁ ∪ H₂), there exists an $H \in H$ such that H ⊇ (H₁ ∩ H₂) ∪ {e}.

Theorem 2.2. ([2, 13]). The above subfamilies of crisp matroids could be exchanged by the operations as follows:

C \binom{Upp}{\overset{⇌}{Min}} D \binom{Opp}{\overset{⇌}{Opp}} I \binom{Min}{\overset{⇌}{Low}} B \binom{Upp}{\overset{⇌}{Min}} S \binom{Opp}{\overset{⇌}{Opp}} N \binom{Max}{\overset{⇌}{Low}} H

Theorem 2.3. ([2, 13]). Let $(E, I)$ be a matroid, $B$ be the family of bases of $(E, I)$ , then $Com (B)$ satisfies the base axioms.

Definition 2.4. ([2, 13]). For a matroid $M = (E, I)$ , there exists a matroid $M^{*} = (E, I^{*})$ such that its family of bases is $Com (B)$ , then M^∗ is called the dual matroid for M. And a base of M^∗ is called a cobase of M, an independent set of M^∗ is called a coindependent set of M, a circuit of M^∗ is called a cocircuit of M and so on. The families of subsets of M^∗ are denoted by $I^{*}$ , $B^{*}$ , $C^{*}$ and so on.

Theorem 2.5. ([2, 13]). Let $(E, I)$ be a matroid and $(E, I^{*})$ be its dual, then the following statements are true.

$(I^{*})^{*} = I$ , $B = Com (B^{*})$ ;

$I = Com (S^{*})$ , $S = Com (I^{*})$ ;

$D = Com (N^{*})$ , $N = Com (D^{*})$ ;

$C = Com (H^{*})$ , $H = Com (C^{*})$ ;

$R_{I^{*}} (X) + R_{I} (E) = | X | + R_{I} (E - X)$ for any X ⊆ E, and $R_{I^{*}} (E) + R_{I} (E) = | E |$ .

On the basis of the above theory, the definition of M-fuzzifying matroids was first given by Shi in [16].

Definition 2.6. ([6, 7]). Let $I$ : 2^E → M be a mapping. The pair $(E, I)$ is called an M-fuzzifying matroid if $I$ satisfies the following conditions.

$I (\emptyset) = ⊤$ .

For any A, B ∈ 2^E and A ⊇ B implies $I (A) \leq I (B)$ .

For any A, B ∈ 2^E with |A| < |B|, $⋁_{e \in B - A} I (A \cup {e}) \geq I (A) \land I (B)$ .

A [0, 1]-fuzzifying matroid is also called a fuzzifying matroid for short.

Definition 2.7. ([6, 7]). Let $(E, I)$ be an M-fuzzifying matroid. The mapping $R_{I} : 2^{E} \to ℕ (M)$ defined by $R_{I} (A) (n) = ⋁ {I (B) : B \subseteq A, | B | \geq n},$ is called the M-fuzzifying rank function for $(E, I) .$

Theorem 2.8. ([6, 7]). Let $(E, I)$ be an M-fuzzifying matroid, then the M-fuzzifying rank function $R_{I}$ for $(E, I)$ satisfies the following conditions.

For any $A \in 2^{E}, \underline{0} \leq R_{I} (A) \leq \underline{| A |} .$

If A, B ∈ 2^E and A ⊆ B, then $R_{I} (A) \leq R_{I} (B) .$

For any $A, B \in 2^{E}, R_{I} (A) + R_{I} (B) \geq R_{I} (A \cap B) + R_{I} (A \cup B) .$

Moreover, it was proved that whenever a mapping $R : 2^{E} \to ℕ (M)$ satisfies (MFR1), (MFR2) and (MFR3), there is an M-fuzzifying matroid $(E, I_{R}),$ in which $I_{R} : 2^{E} \to M$ is determined by $I_{R} (X) = R (X) (| X |), \forall X \subseteq E .$

And as expected, R is the M-fuzzifying rank function for $(E, I_{R}) .$ In other words, the above gives a one-to-one corresponding between M-fuzzifying matroids and the mappings from 2^E to $ℕ (M)$ satisfying conditions (MFR1), (MFR2) and (MFR3), and hence the three conditions give a complete characterization of M-fuzzifying rank functions.

Theorem 2.9. ([6, 7]) Let E be a finite set and $I : 2^{E} \to M$ be a mapping. Then the following statements are equivalent.

$(E, I)$ is an M-fuzzifying matroid.

For each a ∈ J (M) , $(E, I_{[a]})$ is a matroid.

For each a ∈ P (M) , $(E, I^{(a)})$ is a matroid.

Theorem 2.10. ([8]). Let $(E, I)$ be an M-fuzzifying matroid. Then for any A ⊆ E,

for each a ∈ J (M) , $R_{I_{[a]}} (A) = R_{I} (A)_{[a]},$ where $R_{I_{[a]}}$ is the rank function of $(E, I_{[a]});$

for each a ∈ P (M) , $R_{I^{(a)}} (A) = R_{I} (A)^{(a)},$ where $R_{I^{(a)}}$ is the rank function of $(E, I^{(a)})$ ;

$\begin{matrix} R_{I} (A) = ⋀_{a \in P (M)} (a \lor R_{I^{(a)}} (A)) = ⋁_{a \in J (M)} (a \land R_{I_{[a]}} (A)) . \end{matrix}$

Then Shi and Wang gave the definition of M-fuzzifying dependent axioms in [8].

Definition 2.11. ([8]). A mapping $D$ : 2^E → M is called an M-fuzzy family of dependent sets if $D$ satisfies the following conditions.

$D (E) = ⊤$ .

For any A, B ∈ 2^E and A ⊆ B implies $D (A) \leq D (B)$ .

For any A, B ∈ 2^E, $D (A \cap B) \lor ⋀_{e \in E} D (A \cup B - {e}) \geq D (A) \land D (B)$ .

Especially, the relation between the M-fuzzy family of dependent sets and the M-fuzzy family of independent sets is a one-to-one corresponding described in the following theorems.

Theorem 2.12. ([8]) Let $(E, I)$ be an M-fuzzifying matroid. Define a mapping $D$ : 2^E → M by $D_{I} (A) = (I (A))^{'}$ for any A ∈ 2^E. Then $D_{I}$ is an M-fuzzy family of dependent sets on E.

Theorem 2.13. ([8]). Let E be a finite set and $D$ : 2^E → M be an M-fuzzy family of dependent sets on E. For any A ∈ 2^E, define $I_{D} (A) = (D (A))^{'}$ , then $I_{D}$ is an M-fuzzy family of independent sets and $D_{I_{D}} = D$ .

The fuzzifying base-mappings of fuzzifying matroids were introduced in [19 –21] as follows.

Definition 2.14. ([20, 21]). Let $(E, I)$ be a fuzzifying matroid. A mapping $B : 2^{E} \to [0, 1]$ is called a fuzzifying quasi-base-mapping of $(E, I)$ if for any A ⊆ E, $I (A) = ⋁_{A \subseteq B} B (B)$ . The minimal fuzzifying quasi-base-mapping is called the fuzzifying base-mapping.

Theorem 2.15. ([19 –21]). Let $(E, I)$ be a fuzzifying matroid. Define a mapping $B_{I} : 2^{E} \to [0, 1]$ by $B_{Ι} (B) = {\begin{matrix} I (B), \lor_{B \subset A} I (A) ≱ I (B), \\ 0, o t h e r w i s e . \end{matrix}$

Then $B_{I}$ is the fuzzifying base-mapping of $(E, I)$ .

Theorem 2.16. ([20]) Let $(E, I)$ be a fuzzifying matroid, and $B_{I}$ be the fuzzifying base-mapping. Then $B_{I}$ satisfies the following conditions.

There exists a B ∈ 2^E such that $B_{I} (B) = 1$ .

If $B_{I} (B_{1}) \neq 0$ and |B₁| < |B₂|, then $B_{I} (B_{1}) > B_{I} (B_{2})$ and there exists a B₃ ∈ 2^E such that B₁ ⊂ B₃ and $B_{I} (B_{3}) > B_{I} (B_{2})$ .

If B₁, B₂ ∈ 2^E, |B₁| = |B₂| and B₁ ≠ B₂, then $\begin{matrix} ⋀_{x \in B_{1} - B_{2}} (⋁_{y \in B_{2} - B_{1}} (B_{I} ((B_{1} - {x}) \cup {y}))) \\ \geq B_{I} (B_{1}) \land B_{I} (B_{2}) . \end{matrix}$

These three conditions are called the bases axioms of fuzzifying matroids.

Theorem 2.17. ([20]). Let E be a finite set and $B : 2^{E} \to [0, 1]$ be a mapping which satisfies (FB1), (FB2) and (FB3). Define a mapping $I_{C} : 2^{E} \to [0, 1]$ by

I_{B} (A) = \lor {B (B), A \subseteq B} .

Then $(E, I)$ is a fuzzifying matroid.

Theorem 2.18. ([20]). Let E be a finite set.

If $(E, I)$ is a fuzzifying matroid, then I_{B_I} = I.

If $B : 2^{E} \to [0, 1]$ be a mapping which satisfies (FB1), (FB2) and (FB3), then B_{I_B} = B.

The above give a one-to-one corresponding relation between fuzzifying matroids and the mappings which satisfy the fuzzifying basis axioms.

The fuzzifying circuit-mappings of fuzzifying matroids were introduced in [19 –21] as follows.

Definition 2.19. ([20, 21]). Let $(E, I)$ be a fuzzifying matroid. A mapping $C : 2^{E} \to [0, 1]$ is called a fuzzifying quasi-circuit-mapping of $(E, I)$ if for all A ⊆ E, $I (A) = (⋁_{C \subseteq A} C (C))^{'}$ . The minimal fuzzifying quasi-circuit-mapping is called the fuzzifying circuit-mapping.

Theorem 2.20. ([19 –21]). Let $(E, I)$ be a fuzzifying matroid. Define a mapping $C_{I} : 2^{E} \to [0, 1]$ by $C_{Ι} (C) = {\begin{matrix} (I (C))', \lor_{B \subset C} (I (V))' ≱ (I (C))', \\ 0, o t h e r w i s e . \end{matrix}$

Then $C_{I}$ is the fuzzifying circuit-mapping of $(E, I)$ .

Theorem 2.21. ([20]). Let $(E, I)$ be a fuzzifying matroid, and $C_{I}$ be the fuzzifying circuit-mapping. Then $C_{I}$ satisfies the following conditions.

$C_{I} (\emptyset) = 0$ .

Let C₁, C₂ ∈ 2^E such that $C_{I} (C_{i}) \neq 0$ for i = 1, 2. If C₁ ⊂ C₂ then $C_{I} (C_{1}) < C_{I} (C_{2})$ .

Let B₁, B₂ ∈ 2^E such that C₁∩ C₂ ≠ ∅ and C₁ ∩ C₂ ≠ C_i for i = 1, 2. Then $⋀_{e \in C_{1} \cap C_{2}} (⋁_{C \subseteq C_{1} \cup C_{2} - e} C_{I} (C)) \geq C_{I} (C_{1}) \land C_{I} (C_{2}) .$

These three conditions are called the circuit axioms of fuzzifying matroids.

Theorem 2.22. ([20]). Let E be a finite set and $C : 2^{E} \to [0, 1]$ be a mapping which satisfies (FC1), (FC2) and (FC3). Define a mapping $I_{C} : 2^{E} \to [0, 1]$ by

I_{C} (A) = (\lor {C (C), C \subseteq A})' .

Then $(E, I)$ is a fuzzifying matroid.

Theorem 2.23. ([20]). Let E be a finite set.

If $(E, I)$ is a fuzzifying matroid, then I_{B_I} = I.

If $C : 2^{E} \to [0, 1]$ be a mapping which satisfies (FC1), (FC2) and (FC3), then C_{I_C} = C.

The above give a one-to-one corresponding relation between fuzzifying matroids and the mappings which satisfy the fuzzifying circuit axioms.

The definition of M-fuzzifying span axioms was firstly given by Sun, Li and Hu as follows [9].

Definition 2.24. ([9]) Let E be a finite set and $S$ : 2^E → M be a mapping. The $S$ is called an M-fuzzy family of spanning sets if $S$ satisfies the following conditions.

$S (E) = ⊤$ .

For any A, B ∈ 2^E, and A ⊇ B implies $S (A) \geq S (B)$ .

For any A, B ∈ 2^E with |A| > |B|, $⋁_{e \in A - B} S (A - {e}) \geq S (A) \land S (B)$ .

In the sequel, an M-fuzzy family of spanning sets is also called an M-fuzzifying span mapping.

Theorem 2.25. ([9]). Let E be a finite set and $S : 2^{E} \to M$ be a mapping. Then the following statements are equivalent.

$S$ is an M-fuzzy family of spanning sets.

For each a ∈ J (M) , $S_{[a]}$ is a family of spanning sets on E.

For each a ∈ P (M) , $S^{(a)}$ is a family of spanning sets on E.

The definition of M-fuzzifying nonspan axioms was firstly given by Sun, Li, Meng and Xu as follow [11].

Definition 2.26 ([11]). Let E be a finite set and $N$ : 2^E → M be a mapping. The $N$ is called an M-fuzzy family of nonspanning sets if $N$ satisfies the following conditions.

$N (E) = ⊥$ .

For any A, B ∈ 2^E and A ⊆ B implies $N (A) \geq N (B)$ .

For any A, B ∈ 2^E,

N (A \cup B) \lor ⋀_{e \in E} N ((A \cap B) \cup {e}) \geq N (A) \land N (B) .

Then Sun, Xiu and Li researched the dual matroid and fuzzifying span mapping on fuzzifying matroids [10]. They gave some useful definitions and conditions as follows.

Definition 2.27. ([10]). For any a : 2^E → [0, 1] and X ∈ 2^E, we define the following functions:

COM (a) :2^E → [0, 1], COM (a) (X) = a (E - X);

UPP (a) :2^E → [0, 1], UPP (a) (X) = ⋁ _X⊆Aa (A);

LOW (a) :2^E → [0, 1], LOW (a) (X) = ⋁ _X⊇Aa (A)

Lemma 2.28. ([10]). For any $A : 2^{E} \to [0, 1]$ , we have that:

COM (COM (a)) = a;

COM (UPP (a)) = LOW (COM (a));

COM (LOW (a)) = UPP (COM (a)) .

In fact, the definition of COM (a) can be also applied to M-fuzzifying matroids, just by replacing [0, 1] with a completely distributive lattice.

Theorem 2.29. ([10]). Let (E, ı) be a fuzzifying matroid. Then (E, ı) has fuzzifying dual matroid if it satisfies the follow condition.

, ∀a ∈ (0, 1] .

3 Characterizations of M-fuzzy Spanning Sets

In a crisp matroid, a spanning set S contains one base B of the matroid, that means S has the same rank as the whole set E [2, 13]. $A \in S \Leftrightarrow R (A) = R (E) .$

So we give a characterization of M-fuzzifying span mappings defined by the rank functions of M-fuzzifying matroids.

Theorem 3.1. Let $(E, I)$ be an M-fuzzifying matroid and $R_{I}$ be the M-fuzzifying rank function. The mapping $S_{I} : 2^{E} \to M$ is defined by

\begin{matrix} S_{I} (A) & = R_{I} (E) \to R_{I} (A) \\ = ⋀_{n \in N} (R_{I} (E) (n) \to R_{I} (A) (n)), \end{matrix}

(1)

for any A ⊆ E. Then $S_{I}$ is an M-fuzzifying span mapping for $(E, I)$ .

Proof. (MFS1) $S_{I} (E) = R_{I} (E) \to R_{I} (E) = ⊤$ .

(MFS2) If A ⊆ B, then $R_{I} (A) \leq R_{I} (B)$ . So $\begin{matrix} S_{I} (A) & = R_{I} (E) \to R_{I} (A) \\ \leq R_{I} (E) \to R_{I} (B) = S_{I} (B) . \end{matrix}$ (MFS3) ∀A, B ∈ E with |A| > |B|, if $S_{I} (A) \land S_{I} (B) = ⊥,$ then (MFS3) is clearly set up.

Now we suppose that $S_{I} (A) \land S_{I} (B) = a > ⊥$ .

By $S_{I} (A) = ⋀_{n \in N} (R_{I} (E) (n) \to R_{I} (A) (n)) \geq a$ , for any n ∈ N, $R_{I} (E) (n) \to R_{I} (A) (n) \geq a$ ,

if $R_{I} (E) (n) \geq a$ , then $R_{I} (A) (n) \geq a$ , similarly, $R_{I} (B) (n) \geq a$ . Hence in the crisp matroid $(E, I_{[a]})$ , A and B have the same rank as E, both are spanning sets. Therefore, there exists an e₀ ∈ A - B such that A - {e₀} is a spanning set. In another word, there exists a base B_{e
₀} ⊆ A - {e₀}, $I (B_{e_{0}}) \geq a$ , so $\begin{matrix} ⋁_{e \in A - B} S_{I} (A - {e}) & \geq S_{I} (A - {e_{0}}) \\ = R_{I} (E) \to R_{I} (A - {e_{0}}) \\ \geq R_{I} (E) \to R_{I} (B_{e_{0}}) \\ \geq a = S_{I} (A) \land S_{I} (B) . \end{matrix}$

The proposition has been proved. □

By Theorem 2, in a crisp matroid $(E, I)$ and its dual $(E, I^{*})$ , $S = Com (I^{*}), I = Com (S^{*})$ .

Hence let $I^{*} = COM (S_{I})$ , and we get the following definition.

Definition 3.2. Let $(E, I)$ be an M-fuzzifying matroid and $R_{I}$ be the M-fuzzifying rank function. The mapping $I^{*} : 2^{E} \to M$ defined by $\begin{matrix} I^{*} (A) & = S_{I} (E - A) = R_{I} (E) \to R_{I} (E - A) \\ = ⋀_{n \in N} (R_{I} (E) (n) \to R_{I} (E - A) (n)) \end{matrix}$ (2) for any A ⊆ E. Then $I^{*}$ is called the M-fuzzifying coindependent mapping for $(E, I)$ , and $(E, I^{*})$ is called the M-fuzzifying dual matroid for $(E, I) .$

It’s easy to verify that $I^{*}$ satisfies (MFI1), (MFI2) and (MFI3).

Li and Shi had gave a characterization of M-fuzzifying dual matroids for M = [0, 1] or a finite Boolean algebra [3].

We can easily see that two characterizations of $I^{*}$ are equivalent for M = [0, 1] or a finite Boolean algebra when n = |X|.

Then through the M-fuzzifying rank function $R_{I^{*}}$ of the M-fuzzifying dual matroid $(E, I^{*})$ , we can get the M-fuzzifying span mapping $S^{*}$ . Further, the M-fuzzifying coindependent mapping $(I^{*})^{*}$ for $(E, I^{*})$ is defined by $(I^{*})^{*} (A) = S^{*} (E - A)$ for any A ⊆ E.

Similarly, by Theorem 2.5, in a crisp matroid and its dual, $N = Com (D^{*})$ . And we know that the relation between $I^{*}$ and $D^{*}$ is $I^{*} (A) = (D^{*} (A))^{'}$ for any A ∈ 2^E. So we can get the following theorems.

Theorem 3.3. Let E be a finite set and $S$ be an M-fuzzifying span mapping on E. Define a mapping $N_{S} : 2^{E} \to M$ by $N_{S} (A) = (S (A))^{'}$ for any A ∈ 2^E. Then $N_{S}$ is an M-fuzzifying nonspan mapping on E.

Theorem 3.4. Let E be a finite set and $N$ : 2^E → M be an M-fuzzy family of nonspanning sets on E. For any A ∈ 2^E, define $S_{N} (A) = (N (A))^{'}$ . Then $S_{N}$ is an an M-fuzzifying span mapping and $N_{S_{N}} = N$ .

Example 3.5. Let E = {a, b, c}, M = [0, 1]. For any x ∈ [0, 1], define x′ = 1 - x. Define the mapping ı:2^E → [0, 1] as follows:

$I (\emptyset) = 1,$

$I ({a}) = 0.4,$ $I ({b}) = 0.6, I ({c}) = 1,$

$I ({a, b}) = 0.4,$ $I ({a, c}) = 0.3, I ({b, c}) = 0.5,$

$I (E) = 0$ ,

such that

(E, I)

is a fuzzifying matroid.

By the properties of M-fuzzy natural number, for any A ⊆ E, the fuzzifying rank function R_ı (A) is written as $R_{ı} (A) = \frac{a_{1}}{1} + \dots + \frac{a_{n}}{n},$ in the formula, n = R_ı (A) ⁽⁰⁾, and for each {j : 1 ≤ j ≤ n, j ∈ N}, a_j = R_ı (A) (j). Especially, $R_{ı} (\emptyset) = \underline{0}$ , and if A≠ ∅, R_ı (A) is crisp, then $R_{ı} (A) = \underline{R_{ı} (A)^{(0)}}$ .(In the last example of this paper, we will also use this form.)

So in this example,

$R_{ı} (\emptyset) = \underline{0},$

$R_{ı} ({a}) = \frac{0.4}{1},$ $R_{ı} ({b}) = \frac{0.6}{1}, R_{ı} ({c}) = \underline{1},$

$R_{ı} ({a, b}) = \frac{0.6}{1} + \frac{0.4}{2},$ $R_{ı} ({a, c}) = \frac{1}{1} + \frac{0.3}{2},$

$R_{ı} ({b, c}) = \frac{1}{1} + \frac{0.5}{2},$ $R_{ı} (E) = \frac{1}{1} + \frac{0.5}{2} .$

By the formulas (1) and (2), let us calculate $S_{ı}$ and $I^{*}$ . For instance, we choose a set {a, b}, then $\begin{matrix} S_{ı} ({a, b}) & = R_{I} (E) \to R_{I} ({a, b}) \\ = (1 \to 0.6) \land (0.5 \to 0.4) = 0.4, \end{matrix}$ and $I^{*} ({c}) = S (E - {c}) = S ({a, b}) = 0.4$ .

Similar calculations are performed on all sets, we get the mappings as follows.

$S_{ı} (\emptyset) = 0,$

$S_{ı} ({a}) = 0,$ $S_{ı} ({b}) = 0, S_{ı} ({c}) = 0,$

$S_{ı} ({a, b}) = 0.4,$ $S_{ı} ({a, c}) = 0.3, S_{ı} ({b, c}) = 1,$

$S_{ı} (E) = 1;$

$I^{*} (\emptyset) = 1,$

$I^{*} ({a}) = 0.3,$ $I^{*} ({b}) = 1, I^{*} ({c}) = 0.4,$

$I^{*} ({a, b}) = 0,$ $I^{*} ({a, c}) = 0, I^{*} ({b, c}) = 0,$

$I^{*} (E) = 0 .$

Then by Theorem 3.3,

$N (\emptyset) = 1,$

$N ({a}) = 1,$ ] $N ({b}) = 1, N ({c}) = 1,$

$N ({a, b}) = 0.6,$ $N ({a, c}) = 0.7, N ({b, c}) = 0,$

$N (E) = 0 .$

In fact, by Theorem 2.2, there is $S = Upp (B) = Upp (Max (I))$ in crisp matroids. And by Definition 2, for M = [0, 1], we can easily get the extension characterization as follow: $\begin{array}{l} S_{B_{I}} (S) = UPP (B_{I}) = \underset{B \subseteq S}{\lor} B_{I} (B) \\ = {\begin{matrix} \lor_{B \subseteq S} I (B), & \lor_{B \subseteq S} I (A) I (B), \\ 0, & otherwise . \end{matrix} \end{array}$ (3)

But we don’t use this characterization because it isn’t necessarily right. The counterexample is as follow.

Example 3.6. Let E = {a, b, c}, M = [0, 1], and the fuzzy family of independent sets are as follows:

$I (\emptyset) = 1,$

$I ({a}) = 1,$ $I ({b}) = 0.8, I ({c}) = 0.8,$

$I ({a, b}) = 0.8,$ $I ({a, c}) = 0.8, I ({b, c}) = 0.8,$

$I (E) = 0 .$

such that

(E, I)

is a fuzzifying matroid.

By Theorem 2.15, the fuzzifying base-mapping is as follows:

B_I(∅) = 0,

B_I({a}) = 1, B_I({b}) = 0, B_I({c}) = 0,

B_I({a, b}) = 0.8, B_I({a, c}) = 0.8, B_I({b, c}) = 0.8,

B_I(E) = 0,

By (3), the M-fuzzifying span mapping is as follows:

S_{B_I}(∅) = 0,

S_{B_I}({a}) = 1, S_{B_I}({b}) = 0, S_{B_I}({c}) = 0,

S_{B_I}({a, b}) = 0.8, S_{B_I}({a, c}) = 0.8, S_{B_I}({b, c}) = 0.8,

S_{B_I}(E) = 0,

If A = {b, c} , B = {a}, the axiom (MFS3) is clearly not true.

In crisp matroids, the family of independent sets and the family of spanning sets both could be used to deduce a family of bases, and the operations of deduction are dual. In fact, we know that $I^{*} = COM (S)$ and it satisfies the M-fuzzfying independent axioms. For M = [0, 1], $B^{*}$ is the fuzzifying base-mapping of $(E, I^{*})$ and $LOW (B^{*}) = I^{*}$ . Let the fuzzifying mapping $B_{S} = COM (B^{*})$ , by Lemma 2.22, $UPP (B_{S}) = S$ . Then we can get the fuzzifying span-base-mapping, just like Definition 2.14 as follow:

Definition 3.7. Let $S$ be a fuzzifying spanning mapping. A mapping $B_{S} : 2^{E} \to [0, 1]$ is called a fuzzifying quasi-span-base-mapping induced by s if ∀S ⊆ E, $S (S) = ⋁_{B \subseteq S} B_{S} (B)$ . The minimal fuzzifying quasi-span-base-mapping is called the fuzzifying span-base-mapping induced by s.

The proofs of following theorems about B_S are analogous to the method in [20], and we just give the proof of Theorem 3.9.

Theorem 3.8. Let $S$ be a fuzzifying span mapping. Define a mapping $B_{S} : 2^{E} \to [0, 1]$ by

B_{S} (B) = {\begin{matrix} S (B), ⋁_{S \subset B} S (S) S (B), \\ 0, otherwise . \end{matrix}

(4)

Then $B_{S}$ is the span-base-mapping induced by s.

Theorem 3.9. Let $S$ be a fuzzifying span mapping, and $B_{S}$ be the fuzzifying span-base-map. Then $B_{S}$ satisfies the following conditions.

There exists a B ∈ 2^E such that $B_{S} (B) = 1$ .

If $B_{S} (B_{1}) \neq 0$ and |B₁| > |B₂|, then $B_{S} (B_{1}) > B_{S} (B_{2})$ and there exists a B₃ ∈ 2^E such that B₁ ⊃ B₃ and $B_{S} (B_{3}) \geq B_{S} (B_{2})$ .

If B₁, B₂ ∈ 2^E, |B₁| = |B₂| and B₁ ≠ B₂, then

\begin{matrix} ⋀_{x \in B_{1} - B_{2}} (⋁_{y \in B_{2} - B_{1}} (B_{S} ((B_{1} - {x}) \cup {y}))) \\ \geq B_{S} (B_{1}) \land B_{S} (B_{2}) . \end{matrix}

Proof. (FSB1) By (MFS1), we have $E \in {S \in 2^{E} : s(S) = 1} \neq \emptyset .$

Let B ∈ {S ∈ 2^E : s (S) =1} such that |B| is minimum. It is easy to verify that B_S(B) = 1.

(FSB2) Let $B_{S} (B_{1}) \neq 0$ and |B₁| > |B₂|.

If B₁ ⊃ B₂, by the definition of B_S, we have B_S(B₁) = S(B₁). Then $B_{S} (B_{1}) > B_{S} (B_{2})$ follows from $⋁_{S \subset B_{1}} S (S) S (B_{1})$ . Let B₃ = B₂, (FSB2) is true.

So we suppose that B₁nsupseteqB₂. By (MFI3), we know that there exists an e ∈ B₁ - B₂ such that $S (B_{1} - {e}) \geq S (B_{1}) \land S (B_{2}) .$

If $S (B_{1}) \leq S (B_{2})$ , then $S (B_{1} - {e}) = S (B_{2})$ . This implies $⋁_{S \subset B_{1}} S (S) \geq S (B_{1})$ . Hence $B_{S} (B_{1}) = 0$ , which is a contradiction. Therefore $B_{S} (B_{1}) > B_{S} (B_{2})$ .

And for any B ∈ 2^E, it is clear that ${A : A \supseteq B, s (A) = s (B)} \neq \emptyset .$ Let B₀ ∈ {A : A ⊆ B, s (A) = s (B)} such that |B₀| is minimum. By the construction of B₀, we get $S (A) > S (B) = S (B_{0})$ for any A ⊂ B₀, which means $⋁_{A \subset B_{0}} S (A) \leq S (B_{0})$ . Thus $B_{s} (B_{0}) = S (B_{0}) = S (B)$ .

So there exists a B₃ ∈ 2^E such that $B_{3} \subset (B_{1} - {e}) \subseteq B_{1}, B_{s} (B_{3}) = S (B_{1} - {e}),$ which implies $B_{S} (B_{3}) \geq B_{S} (B_{2})$ .

(FSB3) If $B_{S} (B_{1}) = 0$ or $B_{S} (B_{2}) = 0$ , then (FSB3) is clearly set up.

Now we suppose that $B_{S} (B_{i}) \neq 0$ for i = 1, 2, and let x ∈ B₁ - B₂. Denote $a = B_{S} (B_{1}) \land B_{S} (B_{2}) = S (B_{1}) \land S (B_{2}) .$

$S_{[a]}$ is a family of spanning sets on E, and let S_[a] be the family of bases deduced by $S_{[a]}$ . Then it is easy to verify that $B_{1}, B_{2} \in B_{S_{[a]}}$ since |B₁| = |B₂|. Hence there exists a y ∈ B₂ - B₁ such that $(B_{1} - {x}) \cup {y} \in B_{S_{[a]}}$ , which implies s ((B₁ - {x}) ∪ {y}) ≥ a and s (A) < a for any A ⊂ (B₁ - {x}) ∪ {y}. Therefore $\begin{matrix} B_{S} ((B_{1} - {x}) \cup {y}) \\ = & s((B_{1} - {x}) \cup {y}) \\ \geq & B_{S} (B_{1}) \land B_{S} (B_{2}) . \end{matrix}$ Hence $\begin{matrix} ⋀_{x \in B_{1} - B_{2}} (⋁_{y \in B_{2} - B_{1}} (B_{S} ((B_{1} - {x}) \cup {y}))) \\ \geq B_{S} (B_{1}) \land B_{S} (B_{2}) \end{matrix}$

The proposition has been proved. □

We call these three conditions the span-bases axioms of fuzzifying matroids.

Theorem 3.10. Let E be a finite set and $B : 2^{E} \to [0, 1]$ be a mapping which satisfies (FSB1), (FSB2) and (FSB3). Define a mapping $S_{B} : 2^{E} \to [0, 1]$ by

S_{B} (S) = \lor {B (B), S \supseteq B} .

Then $S_{B}$ is a fuzzifying span mapping on E.

Theorem 3.11. Let E be a finite set.

If $S$ is a fuzzifying span mapping on E, then S_{B_S} = S;

If $B : 2^{E} \to [0, 1]$ is a mapping which satisfies (FSB1), (FSB2) and (FSB3), then B_{S_B} = _B.

The above give a one-to-one corresponding relation between the fuzzifying span mapping and the mappings which satisfy the fuzzifying span-bases axioms.

Now let us revisit the formula (3).

Theorem 3.12. Let $(E, I)$ be a fuzzifying matroid, the mapping $S_{B_{I}} : 2^{E} \to [0, 1]$ defined by

S_{B_{I}} (S) = ⋁ {B_{I} (B), B \subseteq S}

is a fuzzifying span mapping, if and only if B₁, B₂ ∈ 2^E, B₁ ≠ B₂ and $B_{I} (B_{1}) \land B_{I} (B_{2}) \neq 0$

, then |B₁| = |B₂|. (In other words, the non-zero-image set of this fuzzifying base mapping is an antichain.)

Proof. By the definition, the fuzzy family of bases needs to satisfy two fuzzifying axioms of bases and span-bases. Then by (FB2) and (FSB2), the condition is natural. □

By Theorem 2.2 and Theorem 2.5, in a crisp matroid and its dual, the family of hyperplanes has the relations $H = Max (N) = Max (Com (D^{*})) = Com (C^{*})$ . □

So like the definition $S (A) = I^{*} (E - A)$ , let $N (A) = D^{*} (E - A)$ and $H (A) = C^{*} (E - A)$ for any A ⊆ E. Then we get the following definition and theorems (these proofs are also analogous to the method in [20]).

Definition 3.13. Let $S$ be a fuzzifying span mapping. A mapping $H : 2^{E} \to [0, 1]$ is called a fuzzifying quasi-hyperplane mapping induced by s if for all S ⊆ E, $S (S) = (⋁_{H \supseteq S} H (H))^{'}$ . The minimal quasi-hyperplane-mapping is called the fuzzifying hyperplane mapping induced by s.

Theorem 3.14. Let $S$ be a fuzzifying span mapping on E. Define a mapping $H_{S} : 2^{E} \to [0, 1]$ by

H_{S} (H) = {\begin{matrix} (S (H))^{'}, ⋁_{H \subset S} (S (S))^{'} (S (H))^{'}, \\ 0, otherwise . \end{matrix}

(5)

Then $H_{S}$ is the fuzzifying hyperplane mapping induced by s.

Theorem 3.15. Let $S$ be a fuzzifying span mapping on E, and $H_{S}$ be the fuzzifying hyperplane-mapping. Then $H_{S}$ satisfies the following conditions.

$H_{S} (E) = 0$ .

Let H₁, H₂ ∈ 2^E such that $H_{S} (H_{i}) \neq 0$ for i = 1, 2. If H₁ ⊂ H₂ then $H_{S} (H_{1}) > H_{S} (H_{2})$ .

Let H₁, H₂ ∈ 2^E such that H₁∩ H₂ ≠ ∅ and H₁ ∩ H₂ ≠ H_i for i = 1, 2. Then

$⋀_{e \notin H_{1} \cup H_{2}} (⋁_{H \supseteq (H_{1} \cap H_{2}) \cup e} H_{S} (H)) \geq H_{S} (H_{1}) \land H_{S} (H_{2}) .$

We call these three conditions the hyperplane axioms of fuzzifying matroids.

Theorem 3.16. Let E be a finite set and $H : 2^{E} \to [0, 1]$ be a mapping which satisfies (FH1), (FH2) and (FH3). Define a mapping $s_{H} : 2^{E} \to [0, 1]$ by

S_{H} (S) = (⋁ {H (H), H \subseteq S})^{'} .

Then $S_{H}$ is a fuzzifying span mapping on E.

Theorem 3.17. Let E be a finite set.

If $S$ be a fuzzifying span mapping on E, then $s_{H_{s}} = s$ .

If $H : 2^{E} \to [0, 1]$ is a mapping which satisfies (FH1), (FH2) and (FH3), then $H_{s_{H}} = H$ .

The above give a one-to-one corresponding relation between the fuzzifying span mapping and the mappings which satisfy the fuzzifying hyperplane axioms.

Example 3.18. Let E = {a, b, c}, M = [0, 1]. Define the mapping s : 2^E → [0, 1] as follows:

$S (\emptyset) = 0,$

$S ({a}) = 0.5,$ $S ({b}) = 0.3, S ({c}) = 0.4,$

$S ({a, b}) = 1,$ $S ({a, c}) = 0.6, S ({b, c}) = 0.4,$

$S (E) = 1,$

such that

S

is an M-fuzzifying span mapping on E.

By (4) and (5),

$B_{S} (\emptyset) = 0,$

$B_{S} ({a}) = 0.5,$ $B_{S} ({b}) = 0.3, B_{S} ({c}) = 0.4,$

$B_{S} ({a, b}) = 1,$ $B_{S} ({a, c}) = 0.6, B_{S} ({b, c}) = 0,$

$B_{S} (E) = 0;$

$H_{S} (\emptyset) = 0,$

$H_{S} ({a}) = 0.5,$ $H_{S} ({b}) = 0.7, H_{S} ({c}) = 0,$

$H_{S} ({a, b}) = 0,$ $H_{S} ({a, c}) = 0.4, H_{S} ({b, c}) = 0.6,$

$H_{S} (E) = 0 .$

4 The Dual of fuzzifying matroids

In this section, we will discuss the properties of dual fuzzifying matroids by the fuzzifying rank function (the formula (2) for M = [0, 1]).

Lemma 4.1. Let $(E, I)$ be a fuzzifying matroid and $R_{I}$ be the fuzzifying rank function, there exists a subset A of E such that $| A | = R_{I} (E)^{(0)}$ and $R_{I} (A) = R_{I} (E)$ .

Proof. Suppose that there exists a subset A₀ of E, such that $| A_{0} | = R_{I} (E)^{(0)}$ and $\begin{matrix} I (A_{0}) = & ⋁ {I (X) : | X | = R_{I} (E)^{(0)}} \\ = & R_{I} (E) (R_{I} (E)^{(0)}) . \end{matrix}$ If $R (A_{0}) (R_{I} (E)^{(0)} - 1) < R_{I} (E) (R_{I} (E)^{(0)} - 1),$ then there exists a subset $A_{0}^{'}$ of A₀ such that $| A_{0}^{'} | = R_{I} (E)^{(0)} - 1$ , and $I (A_{0}^{'}) = R (A_{0}) (R_{I} (E)^{(0)} - 1)$ .

We choose a set $A_{1}^{'}$ such that $| A_{1}^{'} | = R_{I} (E)^{(0)} - 1$ and $I (A_{1}^{'}) = R_{I} (E) (R_{I} (E)^{(0)} - 1)$ .

By (MFI2), $I (A_{1}^{'}) > I (A_{0}^{'}) \geq I (A_{0})$ .

By (MFI3), there exists an $e \in A_{0} - A_{1}^{'}$ such that $I (A_{1}^{'} \cup e) \geq I (A_{1}^{'}) \land I (A_{0}) = I (A_{0})$ .

Denote $A_{1} = A_{1}^{'} \cup {e}$ , then $| A_{1} | = R_{I} (E)^{(0)}$ , $I (A_{1}) \leq I (A_{0})$ .

So $I (A_{1}) = I (A_{0})$ .

Therefore, $R_{I} (A_{1})$ has the properties that

$R_{I} (A_{1}) (R_{I} (E)^{(0)})$ $= I (A_{0}) = R_{I} (E) (R_{I} (E)^{(0)}),$

$R (A_{1}) (R_{I} (E)^{(0)} - 1)$ $= R_{I} (E) (R_{I} (E)^{(0)} - 1) .$

Similarly, if $R (A_{1}) (R_{I} (E)^{(0)} - 2) < R_{I} (E) (R_{I} (E)^{(0)} - 2),$ we can find a set A₂ such that $| A_{2} | = R_{I} (E)^{(0)}$ , and

$R (A_{2}) (R_{I} (E)^{(0)})$ $= R_{I} (E) (R_{I} (E)^{(0)}),$

$R (A_{2}) (R_{I} (E)^{(0)} - 1)$ $= R_{I} (E) (R_{I} (E)^{(0)} - 1),$

$R (A_{2}) (R_{I} (E)^{(0)} - 2)$ $= R_{I} (E) (R_{I} (E)^{(0)} - 2) .$

And so on, finally we get a subset A of E such that $| A | = R_{I} (E)^{(0)}$ and $R_{I} (A) = R_{I} (E)$ . □

Lemma 4.2. Let $(E, I)$ be a fuzzifying matroid, $R_{I} (E) = \underline{R_{I} (E)^{(0)}}$ if and only if there exists a subset A of E such that $| A | = R_{I} (E)^{(0)}$ and $I (A) = 1$ .

Proof. (⇐) $| A | = R_{I} (E)^{(0)}$ and $I (A) = 1$ . By (FI2), for all X ⊆ A, $I (X) \geq I (A) = 1$ which implies $I (X) = 1$ . Then for each ${l \in N : 0 \leq l \leq R_{I} (E)^{(0)}}$ , let A_l ⊆ A and |A_l| = l, $\begin{matrix} R_{I} (E) (l) & = ⋁ {I (B) : B \subseteq E, | B | \geq l} \\ \geq I (A_{l}) = 1, \end{matrix}$ so $R_{I} (E) = \underline{R_{I} (E)^{(0)}}$ .

(⇒) This is natural. □

Lemma 4.3. Let $(E, I)$ be a fuzzifying matroid, there exists a subset A of E such that $| A | = R_{I} (E)^{(0)}$ and $I (A) = 1$ , if and only if for any X ⊆ E, if ı (X) ≠0, then there exists a subset B of E such that X ⊆ B, $| B | = R_{I} (E)^{(0)}$ , ı (B) = ı (A).

Proof. (⇐) Let X =∅, the condition is clearly true.

(⇒) By (MFI3), if $ı (X) \neq 0, | X | < | A | = R_{I} (E)^{(0)},$ then there exists an e ∈ A - X, such that $ı (X \cup {e}) \geq ı (X) \land ı (A) = ı (X) .$ And by (MFI2), X ⊆ X ∪ {e}, then ı (X) ≥ ı (X ∪ {e}), so ı (X) = ı (X ∪ {e}).

Denote X₁ = X ∪ {e}, if $| X_{1} | = | X | + 1 < | A | = R_{I} (E)^{(0)},$ we can repeat the above analysis to get a set X₂ ⊇ X₁ ⊇ X such that $| X_{2} | = | X_{1} | + 1, ı (X_{2}) = ı (X_{1}) = ı (X) .$

And so on, finally we get a set B satisfying the condition. □

Lemma 4.4. Let $(E, I)$ be a fuzzifying matroid, B_I be the fuzzifying base-mapping and $R_{I}$ be the fuzzifying rank function, $R_{I} (E) = \underline{R_{I} (E)^{(0)}}$ if and only if for any B₁, B₂ ∈ 2^E, B₁ ≠ B₂ and $\underline{(} B_{1}) \land \underline{(} B_{2}) \neq 0$ , then |B₁| = |B₂|. Therefore, for any A ⊆ E, $I (A) = ⋁ {I (B) : B \supseteq A, | B | = R_{I} (E)^{(0)}}$ .

Proof. (⇐) By (FB1) and Lemma 4, the condition is true.

(⇒) By Lemma 4 and Lemma 4, for any X ⊆ E, if ı (X) ≠0, there exists a subset B of E, such that X ⊆ B, $| B | = R_{I} (E)^{(0)}$ , ı (B) = ı (A). Then by the definition of B_I, $B_{I} (X) = {\begin{matrix} ı (X), | X | = R_{I} (E)^{(0)}, \\ 0, | X | < R_{I} (E)^{(0)} . \end{matrix}$ The proposition has been proved. □

Theorem 4.5. Let $(E, I)$ be a fuzzifying matroid and $(E, I^{*})$ be its dual fuzzifying matroid, the fuzzifying mapping $I^{*}$ is defined by (2), then $R_{I^{*}} (E) = \underline{R_{I^{*}} (E)^{(0)}}$ and $R_{I} (E)^{(0)} + R_{I^{*}} (E)^{(0)} = | E |$ .

Proof. By Lemma 4.1, there exists a subset A of E, $| A | = R_{I} (E)^{(0)}$ , such that $R_{I} (A) = R_{I} (E)$ . Then $I^{*} (E - A) = R_{I} (E) \to R_{I} (A) = 1$ .

And for any B ⊆ E, if $| B | < | A | = R_{I} (E)^{(0)}$ , then $| E - B | > | E - A | = | E | - R_{I} (E)^{(0)}$ . Therefore, $I^{*} (E - B) = R_{I} (E) \to R_{I} (B) = 0$ .

So $R_{I^{*}} (E)^{(0)} = | E - A | = | E | - R_{I} (E)^{(0)}$ , and $R_{I} (E)^{(0)} + R_{I^{*}} (E)^{(0)} = | E |$ . □

Theorem 4.6. Let $(E, I)$ be a fuzzifying matroid, then

$(I^{*})^{*} \geq I$ .

$(I^{*})^{*} = I$ if and only if $R_{I} (E) = \underline{R_{I} (E)^{(0)}}$ .

Proof. (i) By Theorem 4, $R_{I^{*}} (E) = \underline{R_{I^{*}} (E)^{(0)}}$ and $R_{(I^{*})^{*}} (E) = \underline{R_{I} (E)^{(0)}}$ . Then for any A ⊆ E, $\begin{matrix} (I^{*})^{*} (A) \\ = & R_{I^{*}} (E) \to R_{I^{*}} (E - A) \\ = & R_{I^{*}} (E - A) (R_{I^{*}} (E)^{(0)}) \\ = & ⋁ {I^{*} (E - B) : A \subseteq B, | E - B | = R_{I^{*}} (E)^{(0)}} \\ = & ⋁ {R_{I} (E) \to R_{I} (B) : B \supseteq A, | B | = R_{I} (E)^{(0)}} . \end{matrix}$

If $| A | > R_{I} (E)^{(0)}$ , then $(I^{*})^{*} (A) = 0 = ı (A)$ .

If $| A | = R_{I} (E)^{(0)}$ , then $(I^{*})^{*} (A) = R_{I} (E) \to R_{I} (A) \geq I (A) .$

If $| A | = k < R_{I} (E)^{(0)}$ , then by Lemma 4, there exists a subset B_n of E, such that $| B_{n} | = R_{I} (E)^{(0)}$ , $R_{I} (B_{n}) = R_{I} (E)$ .

Then if A ⊂ B_n, $(I^{*})^{*} (A) = 1 \geq ı (A)$ . Else, select the subsets {B_j : k < j < n, j ∈ N} of B_n such that for each j, |B_j| = j and ı (B_j) = R_ı (E) (j).

Firstly, by (MFI3), there exists an e₁ ∈ B_k+1 - A, such that ı (A ∪ {e₁}) ≥ ı (A) ∧ ı (B_k+1). Therefore, $\begin{matrix} R_{I} (E) (k + 1) \to ı (A \cup {e_{1}}) \\ \geq & (R_{I} (E) (k + 1) \to ı (A)) \land \\ (R_{I} (E) (k + 1) \to ı (B_{k + 1})) \\ \geq & ı (A) . \end{matrix}$

Then denote A₁ = A ∪ {e₁}, similarly, we can find an e₂ ∈ B_k+2 - A₁, such that $\begin{matrix} ı (A \cup {e_{2}}) \geq & ı (A_{1}) \land ı (B_{k + 2}) \\ \geq & ı (A) \land ı (B_{k + 1}) \land ı (B_{k + 2}) \\ = & ı (A) \land ı (B_{k + 2}) . \end{matrix}$

And then $\begin{matrix} R_{I} (E) (k + 2) \to ı (A_{1} \cup {e_{2}}) \\ \geq & (R_{I} (E) (k + 2) \to ı (A)) \land \\ (R_{I} (E) (k + 2) \to ı (B_{k + 2})) \\ \geq & ı (A) . \end{matrix}$

And so on, finally we get a set A_n-k, such that |A_n-k| = R_ı (E) ⁽⁰⁾, and for each j, $R_{I} (E) (j) \to R_{I} (A_{n - k}) (j) \geq ı (A) .$ Therefore, $(I^{*})^{*} (A) \geq R_{I} (E) \to R_{I} (A_{n - k}) \geq ı (A) .$ In summary, $(I^{*})^{*} \geq I$ .

(ii) Now, we prove the necessary and sufficient condition for $(I^{*})^{*} = I$ .

If $(I^{*})^{*} = I$ , $R_{I} (E) = R_{(I^{*})^{*}} (E) = \underline{R_{I} (E)^{(0)}}$ .

If $R_{I} (E) = \underline{R_{I} (E)^{(0)}}$ , by Lemma 4, for any A ⊆ E, $\begin{matrix} (I^{*})^{*} (A) \\ = & ⋁ {R_{I} (E) \to R_{I} (B) : B \supseteq A, | B | = R_{I} (E)^{(0)}} \\ = & ⋁ {I (B) : B \supseteq A, | B | = R_{I} (E)^{(0)}} = I (A) . \end{matrix}$ The proposition has been proved. □

5 One class of special fuzzifying matroids

On the above discussion, we can see the equivalence condition between a fuzzifying matroid and its dual (or fuzzifying span mapping), $R_{I} (E) = \underline{R_{I} (E)^{(0)}}$ . In other words, for any a ∈ [0, 1), the crisp matroid $(E, I^{(a)})$ has the certain rank. So we define a class of special fuzzifying matroids as follows.

Definition 5.1. Let $(E, I)$ be a fuzzifying matroid, if the fuzzifying rank function $R_{I} (E) = \underline{R_{I} (E)^{(0)}}$ , this matroid is called a rank-crisp (RC for short) matroid. In the previous researches, the special fuzzifying matroid has been shown to have some equivalence conditions and good properties.

Theorem 5.2. Let $(E, I)$ be a fuzzifying matroid and $(E, I^{*})$ be its dual fuzzifying matroid, then the following statements are equivalent.

$R_{ı} (E) = \underline{R_{I} (E)^{(0)}}$ , i.e., $(E, I)$ is an RC matroid.

(ı ^∗) ^∗ = ı.

$R_{I} (E) + R_{I^{*}} (E) = \underline{| E |}$ .

$R_{I^{(a)}} (E) = R_{I} (E)_{[1]}$ for any a ∈ [0, 1).

$R_{I_{[a]}} (E) = R_{I} (E)_{[1]}$ for any a ∈ (0, 1].

$(I^{*})^{(a)} = (I^{(a)})^{*}$ for any a ∈ [0, 1).

$(I^{*})_{[a]} = (I_{[a]})^{*}$ for any a ∈ (0, 1].

If B₁, B₂ ∈ 2^E, B₁ ≠ B₂ and $\underline{(} B_{1}) \land \underline{(} B_{2}) \neq 0$ , then |B₁| = |B₂|.

for any B ⊆ E.

for any a ∈ [0, 1).

(DFBS) B_[a] = Max{B_[a]} for any a ∈ (0, 1].

If a < b, then $Max {B^{(a)}} \supseteq Max {B^{(b)}}}$ for any a, b ∈ [0, 1).

If a < b, then $Max {B^{(a)}} \supseteq Max {B^{(b)}}$ for any a, b ∈ (0, 1].

There exists a subset A of E such that $| A | = R_{I} (E)^{(0)}$ and $I (A) = 1$ .

For any A ⊆ E, if ı (A) ≠0, then there exists a subset B of E such that A ⊆ B, $| B | = R_{I} (E)^{(0)}$ , ı (B) = ı (A).

The mapping S_{B_I}: 2^E → [0, 1] defined by $S_{B_{I}} (S) = \underset{B \subseteq S}{\lor} B_{I} (B)$ is a fuzzifying span mapping.

(The statements (ii)-(vii), (iv) and (x) have been mentioned in [3, 10])

Proof. (i)⇔(ii) By Theorem 4.

(i)⇔(iii) If $R_{ı} (E) = \underline{R_{I} (E)^{(0)}}$ , then by Theorem 4, we get that $R_{I} (E) + R_{I^{*}} (E) = \underline{R_{I} (E)^{(0)}} + \underline{R_{I^{*}} (E)^{(0)}} = \underline{| E |} .$ If (3) is true, then for any a ∈ [0, 1), $R_{I} (E)^{(a)} + R_{I^{*}} (E)^{(a)} = | E | .$ And by Theorem 4, $R_{I^{*}} (E) = \underline{R_{I^{*}} (E)^{(0)}}$ , hence $R_{I^{(a)}} (E) + R_{I^{*}} (E)^{(0)} = | E |,$ which implies $R_{I^{(a)}} (E) = R_{I} (E)^{(0)}$ .

So $R_{I} (E) = \underline{R_{I} (E)^{(0)}}$ .

(i)⇔(iv)⇔(v) These conditions are natural.

(i)⇔(viii)⇔(xiv)⇔(xv) By Lemma 4, Lemma 4 and Lemma 4.

(viii)⇔(ix) By Lemma 4 and Theorem 4, $R_{ı^{*}} (E) = \underline{R_{I^{*}} (E)^{(0)}}$ , and B^* satisfies (viii). Then (ix) is true if and only if B also satisfies (viii).

(viii)⇔(x)⇔(xi)⇔(xii)⇔(xiii) For the non-zero-image sets of fuzzifying base mapping is an antichain, these conditions are natural.

(viii)⇔(xvi) By Theorem 3.

(xii)⇔(vi) By Theorem 4 and the pervious equivalence conditions, in (E, ı ^∗), (B*)^(a) = Max{B*)^(a)} is the family of bases of (E, (ı ^∗) ^(a)) for any a, b ∈ [0, 1), and satisfies (xii). Therefore, by Definition 2.4, (vi) is true if and only if $Max {B^{(a)}}$ also satisfies (xii) for any a, b ∈ [0, 1).

Similarly, (xiii)⇔(vii). □

Theorem 5.3. Let $(E, I)$ be an RC matroid, then for any A ⊆ E,

$(I^{*})^{*} (A) = I (A)$ , $B (A) = B^{*} (E - A)$ ;

$I (A) = S^{*} (E - A)$ , $S (A) = I^{*} (E - A)$ ;

$D (A) = N^{*} (E - A)$ , $N (A) = D^{*} (E - A)$ ;

$C (A) = H^{*} (E - A)$ , $H (A) = C^{*} (E - A)$ ;

$R_{I^{*}} (A) + R_{I} (E) = \underline{| A |} + R_{I} (E - A)$ and $R_{I^{*}} (E) + R_{I} (E) = \underline{| E |}$ .

Proof. (i), (ii), (iii), (iv) and $R_{I^{*}} (E) + R_{I} (E) = \underline{| E |}$ have been included in the previous definitions and theorems.

Now we just prove $R_{I^{*}} (A) + R_{I} (E) = \underline{| A |} + R_{I} (E - A)$ . By Theorem 5.2, for any a ∈ [0, 1), $R_{I} (E)^{(a)} = R_{I} (E)^{(0)}$ , $(I^{*})^{(a)} = (I^{(a)})^{*}$ . Then by Theorem 2.5 and Theorem 2.10, for any A ∈ 2^E, $\begin{matrix} R_{(I^{*})} (A)^{(a)} + R_{I} (E)^{(a)} = R_{(I^{*})^{(a)}} (A) + R_{I} (E)^{(0)} \\ = | A | + R_{I^{(a)}} (E - A) = | A | + R_{I} (E - A)^{(a)} \end{matrix}$ So the property is true. □

Example 5.4. Let E = {a, b, c}. For any x ∈ [0, 1], define x′ = 1 - x. Define the mapping ı:2^E → [0, 1] as follows:

$I (\emptyset) = 1,$

$I ({a}) = 1,$ $I ({b}) = 1, I ({c}) = 0.8,$

$I ({a, b}) = 1,$ $I ({a, c}) = 0.8, I ({b, c}) = 0.7,$

$I (E) = 0,$

such that (E, ı) is an RC matroid. All fuzzifying mappings on fuzzifying matroids mentioned in this paper is shown in Table 1. The symbol F represents the fuzzifying mappings on (E, ı) and its dual (E, ı ^∗), such as the fuzzifying independent mapping ı, the fuzzifying base mapping B, the fuzzifying rank function R_ı and so on; T represents the subsets of E; F(T) represents the mapping result of F on T.

Table 1
The Fuzzifying Mappings of An RC Matroid

6 Conclusion

In this paper, firstly we introduce a new characterization of M-fuzzifying span mapping defined by M-fuzzifying rank function. Then we research that the relations between M-fuzzifying span mapping and the other M-fuzzifying mappings on M-fuzzifying matroid, and characterize the M-fuzzifying nonspan, hyperplane and coindependent mappings. Based on these characterizations, we put forward a class of special fuzzifying matroid (RC matroid), which has some equivalence conditions and good properties.

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M -Fuzzifying span mappings in M -Fuzzifying matroids 1

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Characterizations of M-fuzzy Spanning Sets

5 One class of special fuzzifying matroids

Table 1 The Fuzzifying Mappings of An RC Matroid

References

Table 1
The Fuzzifying Mappings of An RC Matroid