Open AccessArticle

New Applications of m-Polar Fuzzy Matroids

Musavarah Sarwar

and

Muhammad Akram

Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Pakistan

Author to whom correspondence should be addressed.

Symmetry 2017, 9(12), 319; https://doi.org/10.3390/sym9120319

Submission received: 1 November 2017 / Revised: 7 December 2017 / Accepted: 11 December 2017 / Published: 18 December 2017

(This article belongs to the Special Issue Fuzzy Techniques for Decision Making)

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Abstract

Mathematical modelling is an important aspect in apprehending discrete and continuous physical systems. Multipolar uncertainty in data and information incorporates a significant role in various abstract and applied mathematical modelling and decision analysis. Graphical and algebraic models can be studied more precisely when multiple linguistic properties are dealt with, emphasizing the need for a multi-index, multi-object, multi-agent, multi-attribute and multi-polar mathematical approach. An m-polar fuzzy set is introduced to overcome the limitations entailed in single-valued and two-valued uncertainty. Our aim in this research study is to apply the powerful methodology of m-polar fuzzy sets to generalize the theory of matroids. We introduce the notion of m-polar fuzzy matroids and investigate certain properties of various types of m-polar fuzzy matroids. Moreover, we apply the notion of the m-polar fuzzy matroid to graph theory and linear algebra. We present m-polar fuzzy circuits, closures of m-polar fuzzy matroids and put special emphasis on m-polar fuzzy rank functions. Finally, we also describe certain applications of m-polar fuzzy matroids in decision support systems, ordering of machines and network analysis.

Keywords:

m-polar fuzzy matroid; m-polar fuzzy uniform matroid; m-polar fuzzy linear matroid; m-polar fuzzy partition matroid; m-polar fuzzy cycle matroid; m-polar fuzzy rank function; closure; m-polar fuzzy circuit

MSC:

05C65; 05C85; 05C90; 03E72

1. Introduction

Matroid theory had its foundations laid in 1935 after the work of Whitney [1]. This theory constitutes a useful approach for linking major ideas of linear algebra, graph theory, combinatorics and many other areas of Mathematics. Matroid theory has been a focus of active research during the last few decades.

Zadeh’s fuzzy set theory [2,3] handles real life data having non-statistical uncertainty and vagueness. Petković et al. [4] investigated the accuracy of an adaptive neuro-fuzzy computing technique in precipitation estimation. Various applications of fuzzy sets in the field of automotive and railway level crossings for safety improvements are studied in [5,6]. The fuzzy set plays a vital role to solve various multi-criteria decision making problems. Some applications of fuzzy theory in multi-criteria models are discussed in [7,8]. Zhang [9] extended fuzzy set theory to bipolar fuzzy sets and discusses the bipolar behaviour of objects. The idea which lies behind such a description is connected with the existence of “bipolar information”. For illustration, profit and loss, hostility and friendship, competition and cooperation, conflicted interests and common interests , unlikelihood and likelihood, feedback and feedforward, and so on, are generally two sides in coordination and decision making. Just like that, bipolar fuzzy set theory indeed has considerable impacts on many fields, including computer science, artificial intelligence, information science, decision science, cognitive science, economics, management science, neural science, medical science and social science. Recently, bipolar fuzzy set theory has been applied and studied speedily and increasingly. Thus, bipolar fuzzy sets not only have applications in mathematical theories but also in real-world problems [10,11,12].

In a number of real world problems, data come from m sources or agents

(m \geq 2)

, that is, multi-indexed information arises which cannot be mathematically expressed by means of the existing approaches of classical set theory, the crisp theory of graphs, fuzzy systems and bipolar fuzzy systems. The research presented in this paper is mainly developed to handle the lack of a mathematical approach towards multi-index, multipolar and multi-attribute data. Nowadays, analysts believe that the natural world is approaching the ideas of multipolarity. Multipolarity in data and information plays an important role in various domains of science and technology. In information technology, multipolar technology can be oppressed to operate large scale systems. In neurobiology, multipolar neurons in brain assemble a lot of information from other neurons. For instance, over a noisy channel, a communication channel may have a different network range, radio frequency, bandwidth and latency. In a food web, species may be of different types including strong, weak, vegetarian and non-vegetarian, and preys may be energetic, harmful and digestive. In a social network, the influence rate of different people may be different with respect to socialism, proactiveness, and trading relationship. A company may have different market power from others according to its product quality, annum profit, price control of its product, etc. These are multipolar information which are fuzzy in nature. To discuss such network models, we need mathematical and theoretical approaches which deal with multipolar information.

In view of this motivation, Chen et al. [13] extended bipolar fuzzy set theory and introduced the powerful idea of m-polar fuzzy sets. The membership value of an object, in an m-polar fuzzy set, belongs to

{[0, 1]}^{m}

, which represents m different attributes of the object. Considering the idea of graphic structures,

m

-polar fuzzy sets can be used to describe the relationship among several individuals. In particular, m-polar fuzzy sets have found applications in the adaptation of accurate problems if it is necessary to make decisions and judgements with a number of agreements. For instance, the exact value of telecommunication safety of human beings is a point which lies in

{[0, 1]}^{m} (m \approx 7 \times 10^{9})

, since different people are monitored in different times. Some other applications include ordering and evaluation of alternatives and m-valued logic. m-polar fuzzy sets are shown to be useful to explore weighted games, cooperative games and multi-valued relations. In decision making issues, m-polar fuzzy sets are helpful for multi-criteria selection of objects in view of multipolar data. For example, m-polar fuzzy sets can be implemented when a country elects its political leaders, a company decides to manufacture an item or product, a group of friends wants to visit a country with multiple alternatives. In wireless communication, it can be used to discuss the conflicts and confusions of communication signals. Thus, m-polar fuzzy sets not only have applications in mathematical theories but also in real-world problems.

Akram and Younas [14] implemented the concept of m-polar fuzzy set into graph theory and discussed irregularity in m-polar fuzzy graphs. Several researchers have been applying this technique to explore various applications of m-polar fuzzy theory including grouping of objects [15], detecting human trafficking suspects [16], finding minimum number of locations [17] and decision support systems [18]. In 1988, Goetschel [19] studied the approach to the fuzzification of matroids and discussed the uncertain behaviour of matroids. The same authors [20] introduced the concept of bases of fuzzy matroids, fuzzy matroid structures and greedy algorithm in fuzzy matroids. Akram and Sarwar [15,21] have also discussed m-polar fuzzy hypergraphs, product formulae of distance for various types of m-polar fuzzy graphs and applications of m-polar fuzzy competition graphs in different domains. Akram and Waseem [22] constructed antipodal and self-median m-polar fuzzy graphs. Li et al. [23] considered different algebraic operations on m-polar fuzzy graphs. Hsueh [24] discussed independent axioms of matroids which preserve basic operational properties. Fuzzy matroids can be used to study the uncertain behaviour of objects but if the data have multipolar information to be dealt with, fuzzy matroids cannot give appropriate results. For this reason, we need the theory of m-polar fuzzy matroids to handle data and information with multiple uncertainties. In this research paper, we present the notion of m-polar fuzzy matroids and study various types of m-polar fuzzy matroids. We apply the concept of m-polar fuzzy matroids to graph theory, linear algebra and discuss their fundamental properties. We present the notion of closure of an m-polar fuzzy matroid and give special focus to the m-polar fuzzy rank function. We also describe certain applications of m-polar fuzzy matroids. We have used basic concepts and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [22,25,26,27,28,29,30,31,32,33,34].

Throughout this research paper, we will use the notation “mF set” for an m-polar fuzzy set, denote the elements of an m-polar fuzzy set A as

(y, A (y))

and use

A^{*}

as a crisp set and A as an m-polar fuzzy set.

2. Preliminaries

The term crisp matroid has various equivalent definitions. We use here the simplest definition of matroid.

Definition 1.

If Y is a non-empty universe and I is a subset of

P (Y)

, power set of Y, satisfying the following conditions,

1.: $If D_{1} \in I and D_{2} \subset D_{1} then, D_{2} \in I$ ,
2.: $If D_{1}, D_{2} \in I and | D_{1} | < | D_{2} | then there exists D_{3} \in I such that D_{1} \subset D_{3} \subseteq D_{1} \cup D_{2}$ .

The pair

M = (Y, I)

is a matroid and I is known as the family of independent sets of M.

Definition 2.

([19]) If

M = (Y, I)

is a matroid then the mapping

R : P (Y) \to {0, 1, 2, \dots, | Y |}

defined by

R (D) = max {| F | : F \subseteq D, F \in I}

is a rank function for M. If

D \in P (Y)

, R is known as rank of D.

Definition 3.

([19]) For any non-empty universe Y, a mapping

μ : P (Y) \to [0, \infty)

is called submodular if for each,

D, F \in P (Y)

μ (D) + μ (F) \geq μ (D \cup F) + μ (D \cap F) .

Definition 4.

([2,3]) A fuzzy set

τ

in a non-empty universe Y is a mapping

τ : Y \to [0, 1]

. A fuzzy relation on Y is a fuzzy subset δ in

Y \times Y

. If

τ

is a fuzzy set in Y and δ is a fuzzy relation on Y then we can say that δ is a fuzzy relation on

τ

δ (y, z) \leq

min {τ (y), τ (z)} for all y, z \in Y

Definition 5.

([19]) If

F (Y)

is a power set of fuzzy subsets on Y and

I \subseteq F (X)

which satisfy the following conditions,

1.

τ_{1} \in I

and

τ_{2} \subset τ_{1}

then,

τ_{2} \in I

where,

τ_{2} \subset τ_{1} \Rightarrow τ_{2} (y) < τ_{1} (y)

, for every

y \in X

2.

τ_{1}, τ_{2} \in I

and

| s u p p (τ_{1}) | < | s u p p (τ_{2}) |

then there exists

τ_{3} \in I

such that

a.: $τ_{1} \subset τ_{3} \subseteq τ_{1} \cup τ_{2}$ , for any $y \in X$ , $τ_{1} \cup τ_{2} (y) = max {τ_{1} (y), τ_{2} (y)}$ ,
b.: $m (τ_{3}) \geq min {m (τ_{1}), m (τ_{2})}$ where, $m (ν) = min {ν (y) : y \in s u p p (ν)}$ .

The pair

M = (X, I)

is called a fuzzy matroid.

I

is known as the collection of independent fuzzy sets of

M

Definition 6.

([13]) An mF set C on a non-empty set Y is a mapping

C = (P_{1} \circ C (z), P_{2} \circ C (z), \dots,

P_{m} {\circ C (z)) : Y \to [0, 1]}^{m}

where, the jth projection mapping is defined as

P_{j} \circ C : {[0, 1]}^{m} \to [0, 1]

Definition 7.

([22]) An mF relation

D = (P_{1} \circ D, P_{2} \circ D, \dots, P_{m} \circ D)

on C is a function

D : C \to C

such that,

D (y z) \leq inf {C (y), C (z)}

, for all

y, z \in Y

. That is, for all

y, z \in Y

P_{j} \circ D (y z) \leq inf {P_{j} \circ C (y), P_{j} \circ C (z)}

, for each

1 \leq j \leq m

, where

P_{j} \circ C (z)

and

P_{j} \circ D (y z)

represent the jth membership values of the element z and the relation

y z .

Definition 8.

([13,22]) An mF graph

G = (C, D)

in a universe Y consists of two mappings

C : Y \to {[0, 1]}^{m}

and

D : Y \times Y \to {[0, 1]}^{m}

such that,

D (y z) \leq inf {C (y), C (z)}

, for all

y, z \in Y

. That is,

P_{j} \circ D (y z) \leq inf {P_{j} \circ C (y), P_{j} \circ C (z)}

, for each

1 \leq j \leq m

. Note that

P_{j} \circ D (y z) = 0

for all

y z \in Y \times Y - E, 1 \leq j \leq m

where, E is the set of edges having non-zero degree of membership. mF relation, D, is called symmetric if

P_{j} \circ D (y z) = P_{j} \circ D (z y)

for all

y, z \in Y .

3. Matroids Based on $m$ F Sets

In this section, we define mF vector spaces, mF matroids and study their properties.

Definition 9.

An mF vector space over a field K is defined as a pair

\tilde{Y} = (Y, C_{v})

where,

C_{v} : Y \to {[0, 1]}^{m}

is a mapping and Y is a vector space over K such that for all

c, d \in F

and

y, z \in Y

C_{v} (c y + d z) \geq inf {C_{v} (y), C_{v} (z)}

, i.e., for all

1 \leq i \leq m

P_{i} \circ C_{v} (c y + d z) \geq inf {P_{i} \circ C_{v} (y), P_{i} \circ C_{v} (z)} .

Example 1.

Let Y be a vector space of

2 \times 1

column vectors over

R

. Define a mapping

C_{v} : Y \to {[0, 1]}^{3}

such that for each

z = {[\begin{matrix} x & y \end{matrix}]}^{t}

C_{v} (z) = \{\begin{matrix} (1, 1, 1), & z = {[\begin{matrix} 0 & 0 \end{matrix}]}^{t}, \\ (1, \frac{1}{3}, \frac{2}{3}), & z = {[\begin{matrix} x & 0 \end{matrix}]}^{t} or r z = {[\begin{matrix} 0 & y \end{matrix}]}^{t}, \\ (1, 1, 1), & x \neq 0 a n d y \neq 0 . \end{matrix}

It remains only to show that

\tilde{Y} = (Y, C_{v})

is a

3

-polar fuzzy vector space. For

z = {[\begin{matrix} 0 & 0 \end{matrix}]}^{t}

, the case is trivial. So the following cases are to be discussed.

Case 1.

Consider two column vectors

z = {[\begin{matrix} x & y \end{matrix}]}^{t}

and

u = {[\begin{matrix} u & v \end{matrix}]}^{t}

then, for any scalars c and d,

C_{v} (c z + d u) = C_{v} ([\begin{matrix} c x + d u \\ c y + d v \end{matrix}]) .

If either exactly one of c or d is zero or both are non-zero then,

c x + d u \neq 0

and

c y + d v \neq 0

and so

C_{v} (c z + d u) = (1, 1, 1) = inf {C_{v} (z), C_{v} (u)} .

Also if

c = 0

and

d = 0

then ,

C_{v} (c z + d u) = (1, 1, 1) .

Case 2.

z = {[\begin{matrix} x & 0 \end{matrix}]}^{t}

and

u = {[\begin{matrix} 0 & v \end{matrix}]}^{t}

then,

c z + d u = {[\begin{matrix} c x & d v \end{matrix}]}^{t}

. If either both c and d are zero or both are non-zero then,

C_{v} (c z + d u) = (1, 1, 1) > inf {C_{v} (z), C_{v} (u)} .

If exactly one of c or d is zero then,

C_{v} (c z + d u) = (1, \frac{1}{3}, \frac{2}{3}) = inf {C_{v} (z), C_{v} (u)} .

Hence

\tilde{Y}

is a 3-polar fuzzy vector space.

Definition 10.

Let

\tilde{Y} = (Y, C_{v})

be an mF vector space over K. A set of vectors

{x_{k}}_{k = 1}^{n}

is known as mF linearly independent in

\tilde{Y}

1.: ${x_{k}}_{k = 1}^{n}$ is linearly independent,
2.: $C_{v} (\sum_{k = 1}^{n} c_{k} x_{k}) = ⋀_{k = 1}^{n} C_{v} (c_{k} x_{k})$ for all ${c_{k}}_{k = 1}^{n} \subset K$ .

Definition 11.

A set of vectors

B = {x_{k}}_{k = 1}^{n}

is known to be an mF basis in

\tilde{Y}

B

is a basis in Y and condition 2 of Definition 10 is satisfied.

Proposition 1.

\tilde{Y} = (Y, C_{v})

is an mF vector space then any set of vectors with distinct jth, for each

1 \leq j \leq m

, degree of membership is linearly independent and mF linearly independent.

Proposition 2.

Let

\tilde{Y} = (Y, C_{v})

be an mF vector space then,

$C_{v} (0) = {sup}_{y \in Y} C_{v} (y)$ ,
$C_{v} (a y) = C_{v} (y)$ for all $a \in K ∖ {0}$ and $y \in Y$ ,
If $C_{v} (y) \neq C_{v} (z)$ for some $y, z \in Y$ then $C_{v} (y + z) = C_{v} (y) \land C_{v} (z)$ .

Remark 1.

B

is an mF basis of

\tilde{Y}

then the membership value of every element of Y can be calculated from the membership values of basis elements, i.e., if

u = \sum_{k = 1}^{n} c_{k} u_{k}

then,

C_{v} (u) = C_{v} (\sum_{k = 1}^{n} c_{k} u_{k}) = ⋀_{k = 1}^{n} C_{v} (c_{k} u_{k}) = ⋀_{k = 1}^{n} C_{v} (u_{k}) .

We now come to the main idea of this research paper called mF matroids.

Definition 12.

Let Y be a non-empty finite set of elements and

C \subseteq P (Y)

be a family of mF subsets,

P (Y)

is an mF power set of Y, satisfying the following the conditions,

1.: If $η_{1} \in C$ , $η_{2} \in P (Y)$ and $η_{2} \subset η_{1}$ then, $η_{2} \in C$ ,
where, $η_{2} \subset η_{1} \Rightarrow η_{2} (y) < η_{1} (y)$ for every $y \in Y$ .
2.: If $η_{1}, η_{2} \in C$ and $| s u p p (η_{1}) | < | s u p p (η_{2}) |$ then there exists $η_{3} \in C$ such that
a. $η_{1} \subset η_{3} \subseteq η_{1} \cup η_{2}$ ,
where for any $y \in Y, (η_{1} \cup η_{2}) (y) = sup {η_{1} (y), η_{2} (y)}$ ,
b. $m (η_{3}) \geq inf {m (η_{1}), m (η_{2})}$ ,
$m (η_{i}) = inf {η_{i} (x) | x \in s u p p (η_{i})}$ , $i = 1, 2, 3$ .

Then the pair

M (Y) = (Y, C)

is called an mF matroid on Y, and

C

is a family of independent mF subsets of

M (Y)

{δ : δ \in P (Y), δ \notin C}

is the family of dependent mF subsets in

M (Y)

. A minimal mF dependent set is called an

m

-polar fuzzy circuit. The family of all mF circuits is denoted by

C_{r} (M)

. An mF circuit having n number of elements is called an mF

n

-circuit. An mF matroid can be uniquely determined from

C_{r} (M)

because the elements of

C

are those members of

P (Y)

that contain no member of

C_{r} (M)

. Therefore, the members of

C_{r} (M)

can be characterized with the following properties:

$Ø \notin C_{r} (M)$ ,
If $δ_{1}$ and $δ_{2}$ are distinct and $δ_{1} \subseteq δ_{2}$ then, $s u p p (δ_{1}) = s u p p (δ_{2})$ ,
If $δ_{1}, δ_{2} \in C_{r} (G)$ and for $A \in P (Y)$ , $A (e) = inf {δ_{1} (e), δ_{2} (e)}$ , $e \in s u p p (δ_{1} \cap δ_{2})$ then there exists $δ_{3}$ such that $δ_{3} \subseteq δ_{1} \cup δ_{2} - {(e, A (e)}$ .

Proposition 3.

\tilde{Y} = (Y, C_{v})

is an mF vector space of

p \times q

column vectors over

R

, and

C

is the family of linearly independent mF subsets

η_{i}

\tilde{Y}

then

(Y, C)

is an mF matroid on Y.

Proposition 4.

M = (Y, C)

is an mF matroid and y is an element of Y such that

C \cup {(y, A (y))}

A \in P (Y)

is dependent. Then

M (Y)

has a unique mF circuit contained in

C \cup {(y, A (y))}

and this mF circuit contains

{(y, A (y))}

Definition 13.

Let Y be a non-empty universe. For any mF matroid, the mF rank function

μ_{r} : P (Y) \to {[0, \infty)}^{m}

is defined as,

μ_{r} (ξ) = sup {| η | : η \subseteq ξ and η \in C}

where,

| η | = \sum_{y \in Y} η (y)

. Clearly the mF rank function of an mF matroid possesses the following properties:

1.: If $η_{1}, η_{2} \in P (Y)$ and $η_{1} \subseteq η_{2}$ then $μ_{r} (η_{1}) \leq μ_{r} (η_{2})$ ,
2.: If $η \in P (Y)$ then, $μ_{r} (η) \leq | η |$ ,
3.: If $η \in C$ then, $μ_{r} (η) = | η |$ .

We now describe the concept of mF matroids by various examples.

A trivial example of an mF matroid is known as an mF uniform matroid which is defined as,

$C = {η \in P (Y) : | s u p p (η) | \leq l} .$

It is denoted by $U_{l, n} = (Y, C)$ where, l is any positive integer and $| Y | = n$ . The mF circuit of $U_{l, n}$ contains those mF subsets $δ$ such that $| s u p p (δ) | = l + 1$ .
Consider the example of a $2$ -polar fuzzy uniform matroid $M = (Y, C)$ where, $Y = {e_{1}, e_{2}, e_{3}}$ and $C = {η \in P (Y) : | s u p p (η) | \leq 2}$ such that for any $η \in P (Y)$ , $η (y) = τ (y)$ , for all $y \in Y$ where,

$τ (y) = \{\begin{matrix} (0.2, 0.3), & y = e_{1} \\ (0.4, 0.5), & y = e_{2} \\ (0.1, 0.3), & y = e_{3} \end{matrix} .$

$\begin{matrix} C = & {Ø, {(e_{1}, 0.2, 0.3)}, {(e_{2}, 0.4, 0.5)}, {(e_{3}, 0.1, 0.3)}, {(e_{1}, 0.2, 0.3), (e_{2}, 0.4, 0.5)}, {(e_{2}, 0.4, 0.5), (e_{3}, 0.1, 0.3)}, {(e_{1}, 0.2, 0.3), (e_{3}, 0.1, 0.3)}} . \end{matrix}$

The $2$ -polar fuzzy circuit of $M$ is $C_{r} (M) = {(e_{1}, 0.2, 0.3), (e_{2}, 0.4, 0.5), (e_{3}, 0.1, 0.3)} .$ For $η = {(e_{2}, 0.4, 0.5), (e_{1}, 0.2, 0.3)}$ , $μ_{r} (η) = (0.6, 0.8)$ .
mF linear matroid is derived from an mF matrix. Assume that Y represents the column labels of an mF matrix and $η_{x}$ denotes an mF submatrix having those columns labelled by Y. It is defined as,

$C = {η_{x} \in P (Y) : columns of η_{x} are m -polar fuzzy linearly independent} .$

For any $η_{x} \in P (Y)$ , $| η_{x} | = \sum_{k = 1}^{r} sup {η_{x} (a_{k 1}), η_{x} (a_{k 2}), \dots, η_{x} (a_{k c})}$ , $η_{x}^{*} = {[a_{i j}]}_{r \times c}$ .
Let $A = {1, 2, 3, 4}$ be a set of $3$ -polar fuzzy $2 \times 1$ vectors over $R$ such that for any $η_{x} \in P (Y)$ , $η_{x} (y) = A (y)$ where,

$A = [\begin{matrix} 1 & 2 & 3 & 4 \\ (0.1, 0.2, 0.3) & (0.3, 0.4, 0.5) & (0.5, 0.6, 0.7) & (0.7, 0.8, 0.9) \\ (0.2, 0.3, 0.4) & (0.4, 0.5, 0.6) & (0.6, 0.7, 0.8) & (0.8, 0.9, 1.0) \end{matrix}]$

.
Take $C = {Ø, {1}, {2}, {4}, {1, 2}, {2, 4}}$ then, $M (A) = (A, C)$ is a $3$ -polar fuzzy matroid on A. The family of dependent $3$ -polar fuzzy subsets of matroid $M (A)$ is ${{3}$ , ${1, 3}$ , ${1, 4}$ , ${2, 3}$ , ${3, 4}} \cup {η : η \subseteq A, | s u p p (η) | \geq 3}$ . For $η = {2, 4}$ , $μ_{r} (η) = (1.5, 1.7, 1.9) .$
An mF partition matroid in which the universe Y is partitioned into mF sets $α_{1}, α_{2}, \dots, α_{r}$ such that

$C = {η \in P (Y) : | s u p p (η) \cap s u p p (α_{i}) | \leq l_{i}, for all 1 \leq i \leq r}$

for given positive integers $l_{1}, l_{2}, \dots, l_{r}$ . The circuit of an mF partition matroid is the family of those mF subsets $δ$ such that $| s u p p (δ) \cap s u p p (α_{i}) | = l_{i} + 1 .$
The very important class of mF matroids are derived from mF graphs. The detail is discussed in Proposition 5. The mF matroid derived using this method is known as m-polar fuzzy cycle matroid, denoted by $M (G)$ . Clearly $C$ is an independent set in G if and only if for each $η \in C$ , $s u p p (η)$ is not edge set of any cycle. Equivalently, the members of $M (G)$ are mF graphs $η$ such that $s u p p (η)$ is a forest.
Consider the example of an mF fuzzy cycle matroid $(Y, C)$ where, $Y = {y_{1}, y_{2}, y_{3}, y_{4}, y_{5}}$ and for any, $η \in C$ , $β (y) = D (y)$ , $(C, D)$ is an mF multigraph on Y as shown in Figure 1.
By Proposition 5, $C_{r} (G) =$ ${{(y_{5}, 0.2, 0.3, 0, 4)}$ , ${(y_{2}, 0.1, 0.2, 0.3)$ , $(y_{3}, 0.1, 0.2, 0.3)}$ , ${(y_{1},$ $0.1,$ $0.2, 0.3)$ , $(y_{2}, 0.1, 0.2, 0.3)$ , $(y_{4}, 0.5, 0.6, 0.7)}$ , ${(y_{1}, 0.1, 0.2, 0.3)$ , $(y_{3}, 0.1, 0.2, 0.3)$ , $(y_{4}, 0.5, 0.6, 0.7)}} .$

$\begin{matrix} C = & {Ø, {(y_{1}, 0.1, 0.2, 0.3)}, {(y_{2}, 0.1, 0.2, 0.3)}, {(y_{3}, 0.1, 0.2, 0.3)}, {(y_{1}, 0.1, 0.2, 0.3), \\ (y_{2}, 0.1, 0.2, 0.3)}, {(y_{1}, 0.1, 0.2, 0.3), (y_{4}, 0.5, 0.6, 0.7)}, {(y_{4}, 0.5, 0.6, 0.7)}, {(y_{2}, 0.1, 0.2, 0.3), \\ (y_{4}, 0.5, 0.6, 0.7)}, {(y_{1}, 0.1, 0.2, 0.3), (y_{3}, 0.1, 0.2, 0.3)}, {(y_{3}, 0.1, 0.2, 0.3), (y_{4}, 0.5, 0.6, 0.7)}} \end{matrix}$

For $η = {(y_{2}, 0.1, 0.2, 0.3), (y_{4}, 0.5, 0.6, 0.7)}$ , $μ_{r} (η) = (0.6, 0.8, 1.0)$ .

Proposition 5.

For any mF graph

G = (C, D)

on Y, if

C_{r}

is the family of mF edge sets δ such that

s u p p (δ)

is the edge set of a cycle in

G^{*}

. Then

C_{r}

is the family of mF circuits of an mF matroid on Y.

Proof.

Clearly conditions 1 and 2 of Definition 12 hold. To prove condition 3, let

δ_{1}

and

δ_{2}

be mF edge sets of distinct cycles that have

y z

as a common edge. Clearly,

δ_{3} = δ_{1} \cup δ_{2} - {(y z, D (y z))}

is an mF edge set of a cycle and so condition 3 is satisfied. ☐

Example 2.

For any mF graph

G = (C, D)

and

0 \leq t \leq 1

define,

$E_{t} = {y z \in s u p p (D) | D (y z) \geq t}$ ,
$F_{t} = {H | H is a forest in the crisp graph (Y, E_{t})}$ ,
$C_{t} = {E (F) | F \in F_{t}}$ , $E (F)$ is the edge set of F.
Clearly $(E_{t}, C_{t})$ is a matroid for each $0 \leq t \leq 1$ . Define $D = {η \in P (Y) | η_{t} \in C_{t}, 0 \leq t \leq 1}$ then, $(Y, D)$ is an mF cycle matroid.

Theorem 1.

Let

M = (Y, C)

be an mF matroid and, for each

0 \leq t \leq 1

, define

C_{t} = {η_{t} | η \in C}

. Then

(Y, C_{t})

is a matroid on Y.

Proof.

We prove conditions 1 and 2 of Definition 12. Assume that

η_{1 t} \in C_{t}

and

α \subseteq η_{1 t}

. Define an mF set

η_{2} \in P (Y)

η_{2} (y) = \{\begin{matrix} t & y \in α, \\ 0 & otherwise . \end{matrix}

Clearly

η_{2} \subseteq η_{1}

η_{2} \in C

and

η_{2 t} = α

therefore,

α \in C_{t}

. To prove condition 2, let

α_{1}, α_{2} \in C_{t}

and

| α_{1} | < | α_{2} |

. Then there exist

η_{1}

and

η_{2}

such that

η_{1 t} = α_{1}

and

η_{2 t} = α_{2}

. Define

{\hat{η}}_{1}

and

{\hat{η}}_{2}

{\hat{η}}_{1} (y) = \{\begin{matrix} t & y \in η_{1}, \\ 0 & otherwise . \end{matrix} {\hat{η}}_{2} (y) = \{\begin{matrix} t & y \in η_{2}, \\ 0 & otherwise . \end{matrix}

It is clear that

s u p p ({\hat{η}}_{1}) < s u p p ({\hat{η}}_{2})

. Since

M

is an mF matroid, there exists

η_{3}

such that

{\hat{η}}_{1} \subseteq η_{3} \subseteq {\hat{η}}_{1} \cup {\hat{η}}_{2}

. Since

{\hat{η}}_{1} \cup {\hat{η}}_{2} (y) = \{\begin{matrix} t & y \in α_{1} \cup α_{2}, \\ 0 & otherwise . \end{matrix}

Therefore, there exists a set

α_{3}

such that

η_{3} (y) = \{\begin{matrix} t & y \in α_{3}, \\ 0 & otherwise . \end{matrix}

Also,

α_{1} \subseteq α_{3} \subseteq α_{1} \cup α_{2}

α_{3} \in C_{t}

. Hence

M_{t}

is a matroid on Y. ☐

Remark 2.

Let

M = (Y, C)

be an mF matroid and, for each

0 \leq t \leq 1

M_{t} = (Y, C_{t})

be the matroid on a finite set Y as given in Theorem 1. As Y is finite therefore, there is a finite sequence

0 < t_{1} < t_{2} < \dots < t_{n}

such that

M_{t_{i}} = (Y, C_{t_{i}})

is a crisp matroid, for each

1 \leq i \leq n

, and

1.: $t_{0} = 0, t_{n} \leq 1$ ,
2.: $C_{w} \neq Ø$ if $0 < w \leq t_{n}$ and $C_{w} = Ø$ if $w > t_{n}$ ,
3.: If $t_{i} < w, s < t_{i + 1}$ then, $C_{w} = C_{s}$ , $0 \leq i \leq n - 1$ ,
4.: If $t_{i} < w < t_{i + 1} < s < t_{i + 2}$ then, $C_{w} \supset C_{s}$ , $0 \leq i \leq n - 2$ .

The sequence

0, t_{1}, t_{2}, \dots, t_{n}

is known as fundamental sequence of

M

. Let

{\bar{t}}_{i} = \frac{1}{2} (t_{i - 1} + t_{i})

for

1 \leq i \leq n

. The decreasing sequence of crisp matroids

M_{t_{1}} \supset M_{t_{2}} \supset \dots \supset M_{t_{n}}

is known as

M

-indeced matroid sequence.

Theorem 2.

If Y is a finite set and

0 = t_{0} < t_{1} < t_{2} < \dots < t_{n} \leq 1

is a finite sequence such that

(Y, C_{t_{1}})

(Y, C_{t_{2}})

, …,

(Y, C_{t_{n}})

is a sequence of crisp matroids. For each

m

-tuple

t

, where,

t_{i - 1} < t \leq t_{i} (1 \leq i \leq n)

, assume that

C_{t} = C_{t_{i}}

and

C_{t} = Ø

t_{n} < t \leq 1

Define

C^{*} = {η \in P (Y) | η_{t} \in C_{t}, 0 < t \leq 1}

then

M = (Y, C^{*})

is an mF matroid.

Proof.

Let

η_{1} \in C^{*}

η_{2} \in P (Y)

, and

η_{2} \subseteq η_{1}

. Clearly

η_{1 t} \in C_{t}

η_{2 t} \subseteq η_{1 t}

, and since

(Y, C_{t})

is a crisp matroid therefore,

η_{2 t} \in C_{t}

, so

η_{2} \in C^{*}

Assume that

η_{1}, η_{2} \in C^{*}

and

| s u p p (η_{2}) | < | s u p p (η_{1}) |

. Define

β = inf {inf_{y \in s u p p (η_{1})} C^{*} (y), inf_{y \in s u p p (η_{2})} C^{*} (y)} .

It is easy to see that

s u p p (η_{1}), s u p p (η_{2}) \in C_{β}

. Since

C_{β}

is the family of independent sets of a crisp matroid therefore, there exists an independent set

A \in C_{β}

such that

s u p p (η_{2}) \subset A \subseteq s u p p (η_{1}) \cup s u p p (η_{2}) .

Let

η_{3} (y) = \{\begin{matrix} η_{2} (y) & y \in s u p p (η_{2}), \\ β & y \in A ∖ s u p p (η_{2}), \\ 0 & otherwise . \end{matrix}

The mF set

η_{3}

satisfies condition 2 of Definition 12 and hence

(Y, C^{*})

is an mF matroid. ☐

Theorem 3.

Let

M = (Y, C)

be an mF matroid and for each

0 < t \leq 1

M_{t} = (Y, C_{t})

is a crisp matroid by Theorem 2. Let

C^{*} = {η \in P (Y) : η_{t} \in C_{t}, 0 < t \leq 1}

. Then

C = C^{*}

Proof.

It is clear from the definition of

C^{*}

that

C \subseteq C^{*}

. To prove the converse part, we proceed on the following steps.

Suppose that

{α_{1}, α_{2}, \dots, α_{p}}

is the non-zero range of

η \in C

such that

α_{1} > α_{2} > \dots > α_{p} > 0

. For each

1 \leq i \leq p

η_{α_{i}} \in C_{α_{i}}

and

η_{α_{i - 1}} \subset η_{α_{i}}

. Define

f_{i} \in P (Y)

f_{i} (y) = \{\begin{matrix} α_{i} & if y \in η_{α_{i}}, \\ 0 & otherwise . \end{matrix}

Since

η_{α_{i}} \in C_{α_{i}}

therefore,

f_{i} \in C

and

⋃_{i = 1}^{q} f_{i} = η

. Assume that

s u p p (η) = {y_{1}, y_{2}, \dots, y_{n_{p}}}

. We use the induction method to show that

η \in C

. Since

f_{1} \in C

therefore, it remains to show that if

⋃_{i = 1}^{l - 1} f_{i} \in C

then,

⋃_{i = 1}^{l} f_{i} \in C

, for each

l < p

. Define

g_{1} (y) = \{\begin{matrix} α_{l} & if y \in {y_{1}, y_{2}, \dots, y_{n_{l - 1}}, y_{n_{l - 1} + 1}}, \\ 0 & otherwise . \end{matrix}

Since for each

1 \leq i \leq l - 1

α_{i} > α_{l}

therefore,

g_{1} \subseteq f_{l}

which implies that

g_{1} \in C

. Define

h_{1} \in P (Y)

h_{1} (y) = \{\begin{matrix} η (y_{n_{l - 1} + 1}) = α_{l} & if y = y_{n_{l - 1} + 1}, \\ 0 & otherwise . \end{matrix}

Since by induction method

⋃_{i = 1}^{l - 1} f_{i} \in C

and

s u p p (⋃_{i = 1}^{l - 1} f_{i}) = {y_{1}, y_{2}, \dots, y_{n_{l - 1}}}

m (⋃_{i = 1}^{l - 1} f_{i}) > α_{l}

therefore, condition 2(b) of Definition 12 implies that

⋃_{i = 1}^{l - 1} f_{i} \cup h_{1} \in C

. If

n_{l - 1} + 1 = n_{l}

then,

⋃_{i = 1}^{l} f_{i} \in C

and we are done. But if on the other hand,

n_{l - 1} + 1 < n_{l}

then define,

g_{2} (y) = \{\begin{matrix} α_{l} & if y \in {y_{1}, y_{2}, \dots, y_{n_{l - 1}}, y_{n_{l - 1} + 1}, y_{n_{l - 1} + 2}}, \\ 0 & otherwise . \end{matrix}

Since for each

1 \leq i \leq l - 1

α_{i} > α_{l}

therefore,

g_{2} \subseteq f_{l}

which implies that

g_{2} \in C

. Define

h_{2} \in P (Y)

h_{2} (y) = \{\begin{matrix} η (y_{n_{l - 1} + 2}) = α_{l} & if y = y_{n_{l - 1} + 2}, \\ 0 & otherwise . \end{matrix}

Since

s u p p (⋃_{i = 1}^{l - 1} f_{i} \cup h_{1}) = {y_{1}, y_{2}, \dots, y_{n_{l - 1}}, y_{n_{l - 1} + 1}}

m (⋃_{i = 1}^{l - 1} f_{i} \cup h_{1}) > α_{l}

therefore, condition 2(b) of Definition 12 implies that

⋃_{i = 1}^{l - 1} f_{i} \cup h_{1} \cup h_{2} \in C

. If

n_{l - 1} + 1 = n_{l}

then,

⋃_{i = 1}^{l} f_{i} \in C

and we are done. If o

n_{l - 1} + 2 < n_{l}

then we continue the process and obtain an mF set

β_{n} = ⋃_{i = 1}^{l - 1} f_{i} \cup h_{1} \cup h_{2} \cup \dots \cup h_{n}

such that

β_{n} = ⋃_{i = 1}^{l} f_{i}

which completes the induction procedure and the proof. ☐

The submodularity of an mF rank function

μ_{r}

is quiet difficult and it depends on Theorem 3 and the following definition.

Definition 14.

Let

t_{0}, t_{1}, \dots, t_{n}

be the fundamental sequence of an mF matroid. For any

m

-tuple

t, 0 < t \leq 1

, define

{\bar{C}}_{t} = C_{{\bar{t}}_{i}}

where,

t_{i - 1} < t \leq t_{i}

and

{\bar{t}}_{i} = \frac{1}{2} (t_{i - 1} + t_{i})

. If

t > t_{n} t a k e {\bar{C}}_{t} = C_{t}

. Define

\bar{C} = {η \in P (Y) : η_{t} \in {\bar{C}}_{t}, for each t, 0 < t \leq 1} .

Then

\bar{M} = (Y, \bar{C})

is known as closure of

M = (Y, C)

Example 3.

We now explain the concept of closure by an example of a

3

-polar fuzzy uniform matroid

M = (Y, C)

where,

Y = {y_{1}, y_{2}, y_{3}}

and

C = {η \in P (Y) : | s u p p (η) | \leq 1}

such that for any

η \in P (Y)

η (y) = τ (y)

, for all

y \in Y

where,

τ (y) = \{\begin{matrix} (0.1, 0.2, 0.3), & y = y_{1} \\ (0.2, 0.3, 0.4), & y = y_{2} \\ (0.3, 0.4, 0.5), & y = y_{3} \end{matrix} .

C = \{Ø, {(y_{1}, 0.1, 0.2, 0.3)}, {(y_{2}, 0.2, 0.3, 0.4)}, {(y_{3}, 0.3, 0.4, 0.5)}\} .

The fundamental sequence of

M

{t_{0} = 0, t_{1} = (0.1, 0.2, 0.3), t_{2} = (0.2, 0.3, 0.4), t_{3} = (0.3, 0.4, 0.5)}

. From routine calculations,

{\bar{t}}_{1} = (0.05, 0.1, 0.15)

{\bar{t}}_{2} = (0.15, 0.25, 0.35)

{\bar{t}}_{3} = (0.25, 0.35, 0.45)

. Since for any

0 < t \leq 1

{\bar{C}}_{t} = C_{{\bar{t}}_{i}}

1 \leq i \leq 3

, therefore,

C_{{\bar{t}}_{1}} = C_{t_{1}}

C_{{\bar{t}}_{2}} = {{y_{2}}, {y_{3}}}

C_{{\bar{t}}_{2}} = {{y_{3}}}

. Hence the closure of

C

can be defined as,

\begin{matrix} \bar{C} = & {Ø, {(y_{1}, 0.1, 0.2, 0.3)}, {(y_{2}, 0.2, 0.3, 0.4)}, {(y_{3}, 0.3, 0.4, 0.5)}, {(y_{1}, 0.1, 0.2, 0.3), (y_{2}, 0.2, 0.3, 0.4)}, \\ {(y_{1}, 0.1, 0.2, 0.3), (y_{3}, 0.3, 0.4, 0.5)}, {(y_{2}, 0.2, 0.3, 0.4), (y_{3}, 0.3, 0.4, 0.5)}} . \end{matrix}

Theorem 4.

The closure

\bar{M} = (Y, \bar{C})

of an mF matroid

M = (Y, C)

is also an mF matroid.

The proof of this theorem is a clear consequence of Theorem 1.

Definition 15.

An mF matroid with fundamental sequence

t_{0}, t_{1}, \dots, t_{n}

is known as a closed mF matroid if for each

t_{i - 1} < t \leq t_{i}

C_{t} = C_{t_{i}}

Remark 3.

Note that the closure of an mF matroid is closed and that it is the smallest closed mF matroid containing

M

. Also the fundamental sequence of

M

and

\bar{M}

is same.

Lemma 1.

{\bar{μ}}_{r}

and

μ_{r}

are mF rank functions of

\bar{M} = (Y, \bar{C})

and

M = (Y, C)

, respectively then

{\bar{μ}}_{r} = μ_{r}

Assume that

M = (Y, C)

is an mF matroid with fundamental sequence

t_{0}, t_{1}, \dots, t_{n}

and rank function

μ_{r}

. To prove that

μ_{r}

is submodular, we now define a function

{\hat{μ}}_{r} : P (Y) \to {[0, \infty)}^{m}

which is also submodular.

For any

η \in P (Y)

, let

0 < α_{1} < α_{2} < \dots < α_{p}

be the non-zero range of

η

and

β_{1} < β_{2} < \dots < β_{q}

be the common refinement of

t_{i}^{'} s

and

α_{j}^{'} s

defined as,

{β_{1}, β_{2}, \dots, β_{q}} = {α_{1}, α_{2}, \dots, α_{p}} \cup {t_{1}, t_{2}, \dots, t_{n}} .

R_{i}

is the rank function of crisp matroid

M_{t_{i}} = (Y, C_{t_{i}})

, for all

1 \leq i \leq n

. For each integer j, there is an integer i,

1 \leq i \leq n

, such that

t_{i - 1} \leq β_{j - 1} < β_{j} \leq t_{i}

. Then

(i, j)

is known as a correspondence pair. For each correspondence pair

(i, j)

, define

γ_{j} (η) = \{\begin{matrix} (β_{j} - β_{j - 1}) R_{i} (η_{β_{j}}) & if β_{j} \leq t_{n}, \\ 0 & if β_{j} > t_{n} . \end{matrix}

Since for each

β_{j - 1} < β < β_{j}

η_{β} = η_{β_{j}}

. Define a new function

{\hat{μ}}_{r} : P (Y) \to {[0, \infty)}^{m}

{\hat{μ}}_{r} = \sum_{j = 1}^{q} γ_{j} (η) .

(1)

Lemma 2.

Assume that

0 < ρ_{1} < ρ_{2} < \dots < ρ_{p}

and

{β_{1}, β_{2}, \dots, β_{q}} \subseteq {ρ_{1}, ρ_{2}, \dots, ρ_{p}}

. For each i,

1 \leq i \leq n

, let

(i, j)

be the correspondence pair if

t_{i - 1} \leq ρ_{j - 1} < ρ_{j} \leq t_{i}

. For each correspondence pair

(i, j)

, define

γ_{j}^{*} : P (Y) \to R^{m}

γ_{j}^{*} (η) = \{\begin{matrix} (ρ_{j} - ρ_{j - 1}) R_{i} (η_{ρ_{j}}) & i f ρ_{j} \leq t_{n}, \\ 0 & i f ρ_{j} > t_{n} . \end{matrix}

Then

\sum_{j = 1}^{q} γ_{j} (η) = \sum_{j = 1}^{q} γ_{j}^{*} (η)

Theorem 5.

t_{0}, t_{1}, \dots, t_{n}

is the fundamental sequence of an mF matroid

M = (Y, C)

and

{\hat{μ}}_{r}

is defined by (1) then,

{\hat{μ}}_{r}

is submodular.

Proof.

Let

η_{1}, η_{2} \in P (Y)

and

{α_{1}, α_{2}, \dots, α_{s}}

{β_{1}, β_{2}, \dots, β_{r}}

be the non-zero ranges of

η_{1}

and

η_{2}

, respectively. Define

{ρ_{1}, ρ_{2}, \dots, ρ_{p}} = {α_{1}, α_{2}, \dots, α_{s}} \cup {β_{1}, β_{2}, \dots, β_{r}} \cup {t_{0}, t_{1}, \dots, t_{n}} .

Lemma 2 implies that

{\hat{μ}}_{r} = \sum_{j = 1}^{q} γ_{j}^{*} (η)

. Since

ρ_{j} - ρ_{j - 1} > 0

, for each j therefore, by the submodularity of the crisp rank function

R_{i}

\begin{matrix} \sum_{j = 1}^{p} (ρ_{j} - ρ_{j - 1}) R_{i} (η_{1 t_{j}}) - \sum_{j = 1}^{p} (ρ_{j} - ρ_{j - 1}) R_{i} (η_{2 t_{j}}) & \geq \sum_{j = 1}^{p} (ρ_{j} - ρ_{j - 1}) R_{i} (η_{1 t_{j}} \cup η_{2 t_{j}}) \\ + \sum_{j = 1}^{p} (ρ_{j} - ρ_{j - 1}) R_{i} (η_{1 t_{j}} \cap η_{2 t_{j}}) . \\ \Rightarrow {\hat{μ}}_{r} (η_{1}) + {\hat{μ}}_{r} (η_{1}) \geq {\hat{μ}}_{r} (η_{1} \cup η_{2}) + {\hat{μ}}_{r} (η_{1} \cap η_{2}) . \end{matrix}

☐

Example 4.

Consider a

3

-polar fuzzy matroid given in Example 3. For

η = {(y_{2}, 0.2, 0.3, 0.4)}

, the non-zero range of η is

{α_{1} = (0.2, 0.3, 0.4)}

. Define

{β_{1}, β_{2}, β_{3}} = {t_{0}, t_{1}, t_{2}, t_{3}} \cup {α_{1}} = {β_{1} = (0.1, 0.2, 0.3), β_{2} = (0.2, 0.3, 0.4), β_{3} = (0.3, 0.4, 0.5)} .

Since

t_{1} = β_{1} < β_{2} = t_{2}

therefore,

(2, 2)

is correspondence pair. Similarly

(3, 3)

is also a correspondence pair. Now

γ_{1} (η) = 0

γ_{2} (η) = (β_{2} - β_{1}) R_{2} (η_{β_{2}}) = (0.1, 0.1, 0.1), γ_{3} (η) = (β_{3} - β_{2}) R_{3} (η_{β_{3}}) = (0, 0, 0) .

Thus

{\hat{μ}}_{r} (η) = (0.1, 0.1, 0.1)

Theorem 6.

For any mF matroid,

μ_{r} \geq {\hat{μ}}_{r}

Proof.

Since

μ_{r} = {\bar{μ}}_{r}

therefore, assume that

M

is a closed mF matroid and

μ_{r} (η_{1}) \neq 0

for some

η_{1} \in P (Y)

. Suppose that there exists

η_{2} \in C

η_{2} \subseteq η_{1}

such that

μ_{r} (η_{1}) = | η_{2} | .

We will show that

{\hat{μ}}_{r} (η_{1}) \leq | η_{2} | .

Take

t_{0} < t_{1} < \dots < t_{n}

as the fundamental sequence of

M

and

α_{1} < α_{2} < \dots < α_{p}

as the non-zero range of

η_{1}

. Let

β_{1} < β_{2} < \dots < β_{q}

be defined by

{β_{1}, β_{2}, \dots, β_{q}} = {α_{1}, α_{2}, \dots, α_{p}} \cup {t_{0}, t_{1}, \dots, t_{n}} .

For each

0 < β \leq 1

, define

C_{β}^{η_{1}} = {C \in C_{β} : C \subseteq η_{1 β}}, β^{*} = sup {β : C_{β}^{η_{1}} \neq Ø} .

Remark 2 implies that

β^{*} = β_{i^{*}}

, for some

β_{i^{*}} \in {β_{j}}_{j = 1}^{q}

. The following properties of

β_{i^{*}}

always hold:

(i): $β_{i^{*}} \leq t_{n}$ , ${\hat{μ}}_{r} (η_{1}) = \sum_{i = 1}^{i^{*}} γ_{i} (η_{1})$ .
(ii): For $η_{2} \in C, η_{2} \subseteq η_{1}$ we have, $0 < η_{2} (y) \leq β_{i^{*}}$ for each $y \in s u p p (η_{2})$ .

For each integer

i \leq i^{*}

, let

| C_{β_{i}} | = R_{j} (η_{β_{i}})

where,

A_{β_{i}} \in C_{β_{i}}^{η_{1}}

t_{i - 1} \leq β_{j - 1} < β_{j} \leq t_{i}

and

R_{i}

is rank function of

M_{t_{i}}

. Clearly,

| C_{β_{i^{*}}} | < | C_{β_{i^{*} - 1}} | < \dots < | C_{β_{1}} |

and define a new sequence

D_{β_{i^{*}}} \subseteq D_{β_{i^{*} - 1}} \subseteq \dots \subseteq D_{β_{1}}

such that

D_{β_{i^{*}}} = C_{β_{i^{*}}}

and

D_{β_{i^{*} - 1}} = \{\begin{matrix} D_{β_{i^{*}}} & if | D_{β_{i^{*}}} | = | C_{β_{i^{*} - 1}} |, \\ C_{β_{i^{*} - 1}}^{'} & if | D_{β_{i^{*}}} | < | C_{β_{i^{*} - 1}} |, \end{matrix}

where,

| C_{β_{i^{*} - 1}}^{'} | = | C_{β_{i^{*} - 1}} |

and

D_{β_{i^{*}}} \subseteq C_{β_{i^{*} - 1}}^{'}

which is by condition 2 of Definition 12. Proceeding in this way, we can find a sequence

{D_{β_{i^{*}}}}_{i = 1}^{i^{*}}

such that

(i): $D_{β_{i}}$ is maximal in $(Y, C_{β_{i}}^{η_{1}})$
(ii): $| D_{β_{i}} | = R_{j} (η_{β_{i}})$ where, i and j are such that $t_{i - 1} \leq β_{j - 1} < β_{j} \leq t_{i}$ .

For each positive integer i,

1 \leq i \leq i^{*}

, define

η_{2 i}

as mF set such that

s u p p (η_{2 i}) = D_{β_{i}}

with non-zero range

{β_{i}}

. Let

η_{2} = ⋃_{i = 1}^{i^{*}} η_{2 i}

. Since

η_{2} \subseteq η_{1}

and

η_{2} \in C^{*}

therefore, by Theorem 3,

μ_{r} (η_{1}) = | η_{2} | \geq \sum_{i = 1}^{i^{*}} (β_{i} - β_{i - 1}) | D_{β_{i}} | = {\hat{μ}}_{r} (η_{1}) .

☐

4. Applications

mF matroids have interesting applications in graph theory, combinatorics and algebra. mF matroids are used to discuss the uncertain behaviour of objects if the data have multipolar information and have many applications in addition to Mathematics.

4.1. Decision Support Systems

mF matroids can be used in decision support systems to find the ordering of n tasks if each task constitutes m linguistic values. All tasks are available at 0 time and each task has a profit p associated with its m properties and a deadline d. The profit

p_{j}

can be gained if each mF task j is completed at the deadline

d_{j}

. The problem is to find the mF ordering of tasks to maximize the total profit. mF matroids can also be used in the secret sharing problem to share parts of secret information among different participants such that we have multipolar information about each participant.

It doesn’t look like an mF matroid problem because the mF matroid problem asks to find an optimal mF subset, but this problem requires one to find an optimal schedule. However, this is an mF matroid problem. The profit, penalty and expense of any ordering can be determined by an mF subset of tasks that are on or before time. For an mF subset S of deadlines

{d_{1}, d_{2}, \dots, d_{n}}

corresponding to tasks

t = {t_{1}, t_{2}, \dots, t_{n}}

, if there is a ordering such that every task in S is on or before time, and all tasks out of S are late. The procedure for the selection of tasks has net time complexity is O

(n 2^{n})

4.2. Ordering of Machines/Workers for Certain Tasks

An important application is to divide a set of workers into different groups to perform a specific task for which they are eligible. Consider the example of allocating a collection of tasks to a set of workers

W_{1}, W_{2}, \dots, W_{7}

who are eligible to perform that task. The problem is to assign a task to a group of workers to be fulfilled in required time, accuracy and cost. The 3-polar fuzzy set of workers is,

\begin{matrix} W^{'} = & {(W_{1}, 0.8, 0.9, 0.9), (W_{2}, 0.7, 0.9, 0.7), (W_{3}, 0.7, 0.7, 0.6), (W_{4}, 0.7, 0.9, 0.8), (W_{5}, 0.6, 0.9, 0.8), \\ (W_{6}, 0.6, 0.8, 0.75), (W_{7}, 0.7, 0.7, 0.6)} . \end{matrix}

The degree of membership of each worker shows the time taken by them, the accuracy of the output if they work on the task and cost of the worker for service. The problem is to determine a collection of workers for tasks

T_{1}

and

T_{2}

such that,

\begin{matrix} T_{1} & = {(W_{i}, W^{'} (W_{i})) ∣ P_{1} \circ W_{i} \leq 0.7, P_{2} \circ W_{i} \geq 0.7, P_{3} \circ W_{i} \leq 0.7}, \\ T_{2} & = {(W_{i}, W^{'} (W_{i})) ∣ P_{1} \circ W_{i} \leq 0.8, P_{2} \circ W_{i} \geq 0.9, P_{3} \circ W_{i} \leq 0.9} . \end{matrix}

The 3-polar fuzzy set of workers for both the tasks are,

\begin{matrix} T_{1} & = {(W_{2}, 0.7, 0.9, 0.7), (W_{3}, 0.7, 0.7, 0.6), (W_{6}, 0.6, 0.8, 0.75), (W_{7}, 0.7, 0.7, 0.6)}, \\ T_{2} & = {(W_{1}, 0.8, 0.9, 0.9), (W_{3}, 0.7, 0.7, 0.6), (W_{4}, 0.7, 0.9, 0.8)} . \end{matrix}

The workers

W_{2}, W_{3}, W_{6}, W_{7}

are preferable for task

t_{1}

and

W_{1}, W_{3}, W_{4}

are preferable for task

t_{2}

4.3. Network Analysis

mF models can be used in network analysis problems to determine the minimum number of connections for wireless communication. The procedure for the selection of minimum number of locations from a wireless connection is explained in the following steps.

Input the n number of locations $L_{1}, L_{2}, \dots, L_{n}$ of wireless communication network.
Input the adjacency matrix $ξ = {[L_{i j}]}_{n^{2}}$ of membership values of edges among locations.
From this adjacency matrix, arrange the membership values in increasing order.
Select an edge having minimum membership value.
Repeat Step 4 so that the selected edge does not create any circuit with previous selected edges.
Stop the procedure if the connection between every pair of locations is set up.

Here we explain the use of mF matroids in network analysis. The 2-polar fuzzy graph in Figure 2 represents the wireless communication between five locations

L_{1}, L_{2}, L_{3}, L_{4}, L_{5}

The degree of membership of each edge shows the time taken and cost for sending a message from one location to the other. Each pair of vertices is connected by an edge. However, in general we do not need connections among all the vertices because the vertices linked indirectly will also have a message service between them, i.e., if there is a connection from

L_{2}

L_{3}

and

L_{3}

L_{4}

, then we can send a message from

L_{2}

L_{4}

, even if there is no edge between

L_{2}

and

L_{4}

. The problem is to find a set of edges such that we are able to send message between every two vertices under the condition that time and cost is minimum. The procedure is as follows. Arrange the membership values of edges in increasing order as,

{(0.5, 0.28)

(0.6, 0.33)

(0.6, 0.37)

(0.7, 0.41)

(0.7, 0.44)

(0.7, 0.46)

(0.7, 0.48)

(0.8, 0.5)

(0.8, 0.51)

(0.8, 0.53)}

. At each step, select an edge having minimum membership value so that it does not create any circuit with previous selected edges. The

2

-polar fuzzy set of selected edges is,

{(L_{3} L_{4}, 0.5, 0.28), (L_{3} L_{5}, 0.6, 0.33), (L_{1} L_{5}, 0.6, 0.37), (L_{2} L_{4}, 0.7, 0.41), (L_{1} L_{4}, 0.7, 0.46)} .

The communication network with minimum number of locations and cost is shown in Figure 3.

Figure 3 shows that only five connections are needed to communicate among given locations in order to minimize the cost and improve the network communication.

5. Conclusions

In this research paper, we have applied the powerful technique of mF sets to extend the theory of vector spaces and matroids. The mF models give more accuracy, precision and compatibility to the system when more than one agreements are to be dealt with. We have mainly introduced the idea of the mF matroid, implemented this concept to graph theory, linear algebra and have studied various examples including the mF uniform matroid, mF linear matroid, mF partition matroid and mF cycle matroid. We have also presented the idea of mF circuit, closure of mF matroid and put special emphasis on mF rank function. The paper is concluded with some real life applications of mF matroids in decision support system, ordering of machines to perform specific tasks and detection of minimum number of locations in wireless network in order to motivate the idea presented in this research paper. We are extending our work to (1) decision support systems based on intuitionistic fuzzy soft circuits, (2) fuzzy rough soft circuits, (3) and neutrosophic soft circuits.

Author Contributions

Musavarah Sarwar and Muhammad Akram conceived of the presented idea. Musavarah Sarwar developed the theory and performed the computations. Muhammad Akram verified the analytical methods.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. 3-polar fuzzy multigraph.

Figure 2. Wireless communication.

Figure 3. Communication network with minimum connections.

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Sarwar, M.; Akram, M. New Applications of m-Polar Fuzzy Matroids. Symmetry 2017, 9, 319. https://doi.org/10.3390/sym9120319

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Sarwar, M., & Akram, M. (2017). New Applications of m-Polar Fuzzy Matroids. Symmetry, 9(12), 319. https://doi.org/10.3390/sym9120319

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New Applications of m-Polar Fuzzy Matroids

Abstract

1. Introduction

2. Preliminaries

3. Matroids Based on $m$ F Sets

4. Applications

4.1. Decision Support Systems

4.2. Ordering of Machines/Workers for Certain Tasks

4.3. Network Analysis

5. Conclusions

Author Contributions

Conflicts of Interest

References

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New Applications of m-Polar Fuzzy Matroids

Abstract

1. Introduction

2. Preliminaries

3. Matroids Based on m F Sets

4. Applications

4.1. Decision Support Systems

4.2. Ordering of Machines/Workers for Certain Tasks

4.3. Network Analysis

5. Conclusions

Author Contributions

Conflicts of Interest

References

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3. Matroids Based on $m$ F Sets