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On modified soft rough sets on a complete atomic Boolean lattice

Abstract

Uncertainty has been used in different areas in our life such as medical diagnosis, social studies, pharmacology and others. The soft set theory and rough set theory are treated as mathematical approaches to deal with uncertainty. In 2011, Feng et al. introduced the notion of soft rough sets. In 2013, we introduced the notion of soft rough sets on a complete atomic Boolean lattice as a generalization of soft rough sets. In this paper, we strengthen the concept of soft rough set on a complete atomic Boolean lattice by defining the concept of modified soft rough set (MRS-set) on a complete atomic Boolean lattice. In this model, some properties which were not satisfied in soft rough sets on a complete atomic Boolean lattice can be proved. Finally, we will use this concept to introduce the concept of modified soft rough topology and we apply this concept in diabetes mellitus.

Introduction

In recent years, scientists, engineers, and mathematicians have shown great interest in uncertainty as it found many fields like decision-making, engineering, environmental science, social sciences, and medical science. Probability theory, fuzzy set theory [1], rough set theory [2, 3], and other mathematical tools have been used successfully to describe uncertainty. Each of these ideas has its inherent difficulties as pointed out in [4, 5]. Consequently, Molodtsov proposed a novel concept for modeling vagueness and uncertainty called soft set theory.

Theory of soft sets has enough parameters, so that it is free from abovementioned difficulties. It deals with uncertainty and vagueness on the one hand while on the other it has enough parametrization tools. These qualities of soft set theory make it popular among researchers and experts working in diverse areas. Some applications of soft set theory can be seen in [68]. Research on soft set theory is growing rapidly [913]. In [14], we presented soft sets on a complete atomic Boolean lattice as a generalization of soft sets and obtained the lattice structure of these soft sets.

Rough set theory, introduced by Pawlak [3] in the 1980s, is a powerful machine learning tool that has applications in many areas [15, 16]. Rough set theory was proposed as a formal tool for modeling and processing incomplete information in information system. Pawlak rough set is mainly based on equivalence relation. But in practical, it is very difficult to find an equivalence relation among the element of a set. So, some other general relations such as tolerance one and dominance ones are considered to define rough set models. Equivalence relations can be replaced by tolerance relations [17], similarity relations [18], and binary relations [19]. The properties of the rough approximations in more general setting of complete atomic Boolean lattice were studied by Järvinen in [20]. All these proposals share the common feature that they deal with approximations of concepts in terms of granules.

The major criticism on rough set theory is that it lacks parametrization tools. In order to make parametrization tools available in rough sets, a major step is taken by Feng et al. in [21]. They introduce the concept of soft rough sets, where instead of equivalence classes parameterized subsets of a set serve the purpose of finding lower and upper approximations of a subset. In [22], Xueling and Jianming apply rough soft sets to BL-algebras. In [23, 24], the concept of bipolar soft rough set and bipolar soft rough relation are proposed In [25], Shabir introduced a new approach to soft rough sets called modified soft rough set (MSR-set) and studied some of their basic properties. In [14], we defined two pairs of soft rough approximation operators on a complete atomic Boolean lattice Band gave their properties. These operators suffer from unexpected properties such as soft upper approximation of non zero element might be equal zero and soft upper approximation of any element might not greater than this element. To resolve this problem, we introduce the notion of modified soft rough set on a complete atomic Boolean lattice and its application in decision making was analyzed.

This paper is arranged as follows, in “Preliminaries” section, some basic concepts of soft sets and rough sets on a complete atomic Boolean lattice are discussed. Also, we discuss the notion of soft rough set on a complete atomic Boolean lattice. The purpose of “Modified soft rough sets (MSR-sets) on a complete atomic Boolean lattice” section is to generalize soft rough set theory by introducing the notion of modified soft rough approximation operators on a complete atomic Boolean lattice B. Moreover, we study their properties and introduce a new rough set model, which is an improvement of Järvinen’s model [20]. We study the relations between soft rough approximations on B [14] and current approximations. Also, we introduce the deviation of some properties of the previous soft model [14] and our new model supported by counter examples. In “Relation between MSR sets and rough sets on a complete atomic Boolean lattice” section, we introduce the notion of Boolean lattice information system. Also, the relationship between the Boolean lattice information system and soft set on a complete atomic Boolean lattice is discussed. Also, the concept of approximations of Boolean lattice information system with respect to another Boolean lattice information system is studied and we introduce an applicable example to illustrate this notion. In the last section we introduce the concept of modified soft rough topology and we apply it and its base in diabetes mellitus(DM).

Preliminaries

We assume that the reader is familiar with the usual lattice-theoretical notation and conventions, which can be found in [26, 27].

Lemma 1

[26] Let B=(B,≤) be a complete lattice, S, TB and {Xi:iI}(B)

  1. (i)

    If ST, then \(\bigvee S\subseteq \bigvee T\).

  2. (ii)

    \(\bigvee {(S\cup {T})}={(\bigvee {S})}\bigvee {(\bigvee {T})}\).

  3. (iii)

    \(\bigvee {(\bigcup \{X_{i}:i\in I \})}=\bigvee {\{\bigvee X_{i}\in I\}}\).

Lemma 2

[26] Let B=(B,≤) be a Boolean lattice, then for all x, yB

  1. (i)

    0=1 and 1=0,

  2. (ii)

    x′′=x,

  3. (iii)

    (xy)=xy, and (xy)=xy,

  4. (iv)

    xy iff xy=0.

Lemma 3

[20] Let B=(B,≤) be a complete Boolean lattice. Then for all {xi:iI}B and yB

$$y\wedge{({\bigvee_{i\in{I}}}{x_{i}})}=\bigvee_{i\in I}{(y\wedge{x_{i}})}$$

and

$$y\vee{({\bigwedge_{i\in{I}}}{x_{i}})}=\bigwedge_{i\in I}{(y\vee{x_{i}})}$$

Definition 1

[20] Let B=(B,≤) be an ordered set and x, yB, we say that x is covered by y (or that y covers x), and write, xy if x<y and there is no element z in B with x<z<y.

Definition 2

[20] Let B=(B,≤) be a lattice with a least element 0. Then aB is called an atom if 0a. The set of atoms of B is denoted by A(B). The lattice B is called atomic if every element of B is the supremum of the atoms below it, that is \(x=\bigvee {\{a\in {A(B)}: a\prec {x}\}}\).

Definition 3

[14] Let B=(B,≤) be a complete atomic Boolean lattice and E be a set of parameters. Let A be a non empty subset of E. A soft set over B, with support A, denoted by fA on B is defined by the set of ordered pairs

$$f_{A}=\{(e,f_{A}(e)):e\in A, f_{A}(e)\in B\},$$

or is a function fA:AB s.t

$$f_{A}(e)\neq{0} \hspace{0.3cm} \forall \hspace{0.3cm} e\in A \subseteq E \hspace{0.1cm}and\hspace{0.1cm} f_{A}(e)=0 \hspace{0.1cm}if\hspace{0.1cm} e\not\in A.$$

In other words, a soft set over B is a parameterized family of elements of B. For each eA, f(e) is considered as e-approximate element of fA

Definition 4

[14] Let B=(B,≤) be a complete atomic Boolean lattice. Let AE and let fA be a soft set over B. The complement of fA, denoted by (fA)c is defined by (fA)c=(fc,A), where fc:AB is a mapping given by fc(e)=f(e) for every eA.

Definition 5

[14] Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B.

  1. (i)

    fA is called full, if \({\bigvee _{e\in A}}f(e)=1\);

  2. (ii)

    fA is keeping infimum, if for any e1,e2A, there exists e3A such that f(e1)f(e2)=f(e3);

  3. (iii)

    fA is keeping supremum, if for any e1,e2A, there exists e3A such that f(e1)f(e2)=f(e3);

  4. (iv)

    fA is called partition of B if

    1. (1)

      \({\bigvee _{e\in A}}f(e)=1\),

    2. (2)

      For every eA, f(e)≠0,

    3. (3)

      For every e1,e2A either f(e1)=f(e2) or f(e1)f(e2)=0.

Definition 6

[14] Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. For any element xB, we define a pair of operators x,x:BB as follows:

  • \(x^{\vee }=\bigvee {\{b\in {A(B)}:\exists {e\in {A}} \hspace {0.1cm}s.t \hspace {0.1cm}b\leq {f(e)}\hspace {0.1cm}and\hspace {0.1cm} f(e)\leq {x}\}}\),

  • \(x^{\wedge }=\bigvee {\{b\in {A(B)}:\exists {e\in {A}} \hspace {0.1cm}s.t \hspace {0.1cm}b\leq {f(e)}\hspace {0.1cm}and\hspace {0.1cm} {f(e)\wedge {x}}\neq {0}\}}.\)

The elements x and x are called the soft lower and the soft upper approximations of x over B. Two elements x and y are called soft equivalent if they have the same soft upper and soft lower approximations over B. The resulting equivalence classes are called soft rough sets over B.

Example 1

Let B={0,a,b,c,d,e,f,1} and let the order ≤ be defined as in Fig. 1.

Fig. 1
figure1

The complete atomic Boolean lattice B

The set of atoms of a complete atomic Boolean lattice B=(B,≤) is {a,b,c}. Let A={e1,e2,e3,} and fA be a soft set over B defined as follows: f(e1)=a, f(e2)=b, and f(e3)=d. So, fA is not full. Many odd situation occurs. For example,

If x=c≠0, then x=x=0. Also if x=e, then x=ab=d̸e Moreover c̸x or c̸x for any xB

In order to avoid these situations, we introduced in [14] the notion of full soft sets on B. Moreover the concept of soft postive, soft negative and soft boundary are meaningful in the case of full soft sets.

In the following, we show that negative element of any xB cannot be avoided.

Proposition 1

Let fA be a soft set over B which is not full. Then there exists at least bA(B) such that bneg(x)=(x) for all xB.

Proof

Since fA is not full, i.e., \({\bigvee _{_{_{_{a\in {A}}}}}}{f(a)}\not \leq 1\). So bA(B) s.t b̸f(a) aA. Let xB s.t bx. If bx, then aA s.t bf(a) and f(a)x≠0 which is a contradiction. Hence b̸x. By a similar argument when b̸x it can be shown that b̸x. Therefore b≤1−x=(x). □

Modified soft rough sets (MSR-sets) on a complete atomic Boolean lattice

Some properties of generalized soft rough model

In this section, we will generalize the soft rough model on a complete atomic Boolean lattice B by defining modified soft rough sets on B. Also, we will present some properties of modified soft rough approximation operators on B, and introduce a new soft rough model on B, which is an improvement of previous models [14, 20].

Definition 7

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let φ:A(B)→(A) be another map defined as φ(b)={aA:bf(a)}. Then the pair (A(B),φ) is called MSR-approximation space on B and for any element xB, lower MSR-approximation on B is defined as

  • \(x^{\vee }_{\varphi }=\bigvee {\{a\in {A(B)}: a\leq {x} \hspace {0.1cm},\varphi (a)\neq \varphi (b) \hspace {0.1cm}\forall \hspace {0.1cm}b\in A(B)\hspace {0.1cm}s.t\hspace {0.1cm}b\not \leq x\}}\),

  • and its upper MSR-approximation over B is defined as

  • \(x^{\wedge }_{\varphi }=\bigvee {\{a\in {A(B)}:\varphi (a)= \varphi (b) \hspace {0.1cm} for \hspace {0.1cm}some \hspace {0.1cm} b\in A(b)\hspace {0.1cm}s.t\hspace {0.1cm} b\leq x\}}.\)

  • If \(x^{\vee }_{\varphi }\neq x^{\wedge }_{\varphi }\), Then x is said to be MSR-element on B.

Remark 1

Lower MSR-approximation of x over B can be defined as \(x^{\vee }_{\varphi }=\bigvee {\{a\in {A(B)}}: {\varphi (a)\neq \varphi (b) \hspace {0.1cm}\forall \hspace {0.1cm}b\in A(B)\hspace {0.1cm}s.t\hspace {0.1cm}b\not \leq x\}}\) because if we let φ(a)≠φ(b)bA(B) s.t b̸x and a̸x, then by hypothesis φ(a)≠φ(a) which is impossible.

Lemma 4

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. Then for all cA(B) and xB

  1. (i)

    \({c\leq {x^{\vee }_{\varphi }}}\Longleftrightarrow {c\leq x\hspace {0.1cm}}\) and φ(c)≠φ(b) bA(B) s.t b̸x;

  2. ii)

    \({c\leq {x^{\wedge }_{\varphi }}}\Longleftrightarrow { \varphi (c)=\varphi (b)\hspace {0.1cm}}\) for some bA(B) s.t bx.

Proof

(i) () Suppose that \(c\leq {x^{\vee }_{\varphi }}=\bigvee \left \{a\in {A(B)}: a\leq {x} \hspace {0.1cm},\varphi (a)\neq \varphi (b) \hspace {0.1cm}\forall \hspace {0.1cm}b\in A(B)\hspace {0.1cm} s.t\hspace {0.1cm}b\not \leq x\right \}\). So cx. If φ(c)≠φ(b) bA(B) s.t b̸x, then \(c\wedge {{x^{\vee }_{\varphi }}}=c\wedge {\bigvee {\{a\in {A(B)}: a\leq {x} \hspace {0.1cm},\varphi (a)\neq \varphi (b)}} {{ \hspace {0.1cm}\forall \hspace {0.1cm}b\in A(B)\hspace {0.1cm} s.t\hspace {0.1cm}b\not \leq x\}}}\)= \(\bigvee {\{a\wedge c: a\in {A(B)}, c\leq {x} \hspace {0.1cm},\varphi (a)\neq \varphi (b) \hspace {0.1cm}\forall \hspace {0.1cm}b\in A(B)\hspace {0.1cm} s.t\hspace {0.1cm}b\not \leq x\}}\). Since φ(c)≠φ(b), then ca, i.e., ca=0. Hence \(c\leq ({x^{\vee }_{\varphi }})^{\prime }\), a contradiction.

() Suppose that cx and φ(c)≠φ(b) bA(B) s.t b̸x, then \(c\leq {\bigvee {\{a\in {A(B)}: a\leq {x} \hspace {0.1cm}}}\), \(\varphi (a)\neq \varphi (b) \hspace {0.1cm}\forall \hspace {0.1cm}b\not \leq x\}=x^{\vee }_{\varphi }\). Condition (ii) can be proved similarly. □

Proposition 2

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. If φ(a)=φ(b) for some a,bA(B), then for any xB either \(a,b\leq x^{\wedge }_{\varphi }\) or \(a,b\not \leq x^{\wedge }_{\varphi }\).

Proof

If \(a\leq x^{\wedge }_{\varphi }\), then φ(a)=φ(c) for some cA(B) s.t cx. Since φ(a)=φ(b), then φ(b)=φ(c), cx. This implies that \(b\leq x^{\wedge }_{\varphi }\). □

Proposition 3

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. Then for all xB

  1. (i)

    \(x^{\vee }_{\varphi }\leq x\leq x^{\wedge }_{\varphi }\)

  2. (ii)

    \(0^{\vee }_{\varphi }=0=0^{\wedge }_{\varphi }\)

  3. (iii)

    \(1^{\vee }_{\varphi }=1=1^{\wedge }_{\varphi }\)

Proof

(i) Assume that bA(B), s.t \(b\leq x^{\vee }_{\varphi }\), then bx. For the other inclusion, let bA(B) s.t bx. Then φ(b)=φ(b) for some bx. Thus \(b\leq x^{\wedge }_{\varphi }\).

(ii) \(0^{\vee }_{\varphi }=\bigvee {\{a\in {A(B)}: a\leq {0} \hspace {0.1cm},\varphi (a)\neq \varphi (b) \hspace {0.1cm}\forall b\in A(B)\hspace {0.1cm}s.t\hspace {0.1cm}b\not \leq 0\}}=0\). Also, \(0^{\wedge }_{\varphi }=\bigvee {\{a\in {A(B)}:} {\varphi (a)= \varphi (b)} \hspace {0.1cm} for \hspace {0.1cm}some \hspace {0.1cm} b\in A(B)\hspace {0.1cm}s.t\hspace {0.1cm} b\leq 0\}=0.\) Claim (iii) can be proved similarly. □

Proposition 4

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. Then for all x,yB

  1. (i)

    The mappings \(^{\vee }_{\varphi }:B\longrightarrow {B}\) and \(^{\wedge }_{\varphi }:B\longrightarrow {B}\) are order preserving.

  2. (ii)

    The mappings \(^{\vee }_{\varphi }:B\longrightarrow {B}\) and \(^{\wedge }_{\varphi }:B\longrightarrow {B}\) are mutually dual.

Proof

(i) Assume that xy and \(a\leq x^{\vee }_{\varphi }\). Let bA(B) s.t b̸y. Since xy, then b̸x. Since \(a\leq x^{\vee }_{\varphi }\), then φ(a)≠φ(b). So \(a\leq y^{\vee }_{\varphi }\) and we get \(x^{\vee }_{\varphi }\leq y^{\vee }_{\varphi }\). Now let \(b\leq x^{\wedge }_{\varphi }\), that is φ(b)=φ(c) for some cA(B) s.t cx. Since xy, then φ(b)=φ(c) for some cA(B) s.t cy. Thus \(b\leq y^{\wedge }_{\varphi }\). Consequently, \(x^{\wedge }_{\varphi }\leq y^{\wedge }_{\varphi }\).

(ii) We must show that \(x^{\wedge }_{\varphi }=((x^{{\prime }})^{\vee }_{\varphi })^{{\prime }}\) and \(((x^{{\prime }})^{\wedge }_{\varphi })^{{\prime }}=x^{\vee }_{\varphi }\). Let aA(B) s.t \(a\leq ((x^{{\prime }})^{\vee }_{\varphi })^{{\prime }}\), then \(a\not \leq (x^{{\prime }})^{\vee }_{\varphi }\). So, either a̸x or φ(a)=φ(b) for some bA(B) s.t b≤(x)=x, that is \(a\leq x^{\wedge }_{\varphi }\). Conversely, Let \(a\leq x^{\wedge }_{\varphi }\), then φ(a)=φ(b) for some bA(B) s.t bx. Thus \(a\not \leq (x^{{\prime }})^{\vee }_{\varphi }\), that is \(a\leq ((x^{{\prime }})^{\vee }_{\varphi })^{{\prime }}\). So \(x^{\wedge }_{\varphi }=((x^{{\prime }})^{\vee }_{\varphi })^{{\prime }}\).

Let aA(B) s.t \(a\leq ((x^{{\prime }})^{\wedge }_{\varphi })^{{\prime }}\), then \(a\not \leq (x^{{\prime }})^{\wedge }_{\varphi }\). Hence, φ(a)≠φ(b) for all bA(B) s.t bx. That is ax because otherwise, if ax, then φ(a)≠φ(a), a contradiction. Thus, ax, φ(a)≠φ(b) for all bA(B) s.t bx and consequently, \(a\leq x^{\vee }_{\varphi }\). Conversely, let aA(B) s.t \(a\leq x^{\vee }_{\varphi }\), then ax and φ(a)≠φ(b) for all bA(B) s.t bx. Thus \(a\not \leq (x^{{\prime }})^{\wedge }_{\varphi }\), i.e \(a\leq ((x^{{\prime }})^{\wedge }_{\varphi })^{{\prime }}\). So \(((x^{{\prime }})^{\wedge }_{\varphi })^{{\prime }}=x^{\vee }_{\varphi }\).

For all SB, we denote \(S^{\vee }_{\varphi }={\{x^{\vee }_{\varphi }: x\in {S}\}}\) and \({S^{\wedge }_{\varphi }}={\{x^{\wedge }_{\varphi }: x\in {S}\}}\). □

Proposition 5

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B, then

  1. i)

    For all SB, \(\vee {S^{\wedge }_{\varphi }}=(\vee {S})^{\wedge }_{\varphi }\) and \(\wedge {S^{\wedge }_{\varphi }}\geq (\wedge {S})^{\wedge }_{\varphi }\).

  2. ii)

    For all SB, \(\wedge {S^{\vee }_{\varphi }}=(\wedge {S})^{\vee }_{\varphi }\) and \(\vee {S^{\vee }_{\varphi }}\leq (\vee {S})^{\vee }_{\varphi }\).

  3. (iii)

    \((B^{\wedge }_{\varphi },\leq)\) is a complete lattice; 0 is the least element and 1 is the greatest element of \((B^{\wedge }_{\varphi },\leq)\).

  4. (iv)

    \((B^{\vee }_{\varphi },\leq)\) is a complete lattice; 0 is the least element and 1 is the greatest element of\((B^{\vee }_{\varphi },\leq)\).

  5. (v)

    The kernal \(\Theta ^{\vee }_{\varphi }=\{(x,y):x^{\vee }_{\varphi }=y^{\vee }_{\varphi }\}\) of the map \(^{\vee }_{\varphi }:B\longrightarrow {B}\) is a congruence on the semi lattice (B,) such that the \(\Theta ^{\vee }_{\varphi }\)-class of any x has a least element.

  6. (vi)

    The kernal \(\Theta ^{\wedge }_{\varphi }=\{(x,y):x^{\wedge }_{\varphi }=y^{\wedge }_{\varphi }\}\) of the map \(^{\wedge }_{\varphi }:B\longrightarrow {B}\) is a congruence on the semi lattice (B,) such that the \(\Theta ^{\wedge }_{\varphi }\)-class of any x has a greatest element.

Proof

(i) Let SB. The mapping \(^{\wedge }_{\varphi }\): BB is order preserving, which implies that \(\vee {S^{\wedge }_{\varphi }}\leq {(\vee {S})^{\wedge }_{\varphi }}\) and \(\wedge {S^{\wedge }_{\varphi }}\geq (\wedge {S})^{\wedge }_{\varphi }\). Let bA(B) and assume that \(a\leq {(\vee {S})^{\wedge }_{\varphi }}\). So, φ(a)=φ(b) for some bA(B) s.t bS. So, φ(a)=φ(b) for some bA(B) and xS s.t bx. So {aA(B):φ(a)=φ(b) forsome bA(B) s.t bS}

\(\subseteq {\cup _{_{x\in {S}}}\{a\in {A(B)}:\varphi (a)= \varphi (b)\hspace {0.1cm} \\for some \hspace {0.1cm}b\in A(B)\hspace {0.1cm} s.t \hspace {0.1cm}b\leq x\}}\). Then

\((\vee {S})^{\wedge }_{\varphi }=\bigvee {\{a\in {A(B)}:\varphi (a)= \varphi (b) \hspace {0.1cm} for \hspace {0.1cm}some \hspace {0.1cm} b\in A(b)\hspace {0.1cm}s.t\hspace {0.1cm} b\leq \vee {S}\}}\)

\(\hspace {1.3 cm}\leq {\bigvee (\cup _{_{x\in {S}}}\{a\in {A(B)}:\varphi (a)= \varphi (b) \hspace {0.1cm} for \hspace {0.1cm}some \hspace {0.1cm} b\in A(B)\hspace {0.1cm}s.t\hspace {0.1cm} b\leq x\}}\))

\(\hspace {1.3 cm}={\bigvee _{_{x\in {S}}}(\bigvee \{a\in {A(B)}:\varphi (a)= \varphi (b) \hspace {0.1cm} for \hspace {0.1cm}some \hspace {0.1cm} b\in A(b)\hspace {0.1cm}s.t\hspace {0.1cm} b\leq x\}}\)) (by Lemma 1)

\(=\bigvee \{x^{\wedge }_{\varphi }:x\in {S}\}=\bigvee {S^{\wedge }_{\varphi }}\)

(ii) Let SB. The mapping \(^{\vee }_{\varphi }\): BB is order preserving, which implies that \((\wedge {S})^{\vee }_{\varphi }\leq {\wedge {S^{\vee }_{\varphi }}}\) and \(\vee {S^{\vee }_{\varphi }}\leq (\vee {S})^{\vee }_{\varphi }\). Let aA(B) s.t \(a\leq {\wedge {S^{\vee }_{\varphi }}}=\wedge {\{x^{\vee }_{\varphi }: x\in {S}\}}\). So, ax and φ(a)≠φ(b) for all bA(B) s.t b̸x for every xS. Hence φ(a)≠φ(b) for all bA(B) s.t b̸S. In fact if b̸S, then xS s.t b̸x. So, φ(a)≠φ(b). Therefore \(b\leq {(\wedge {S})^{\vee }_{\varphi }}\). Consequently, \(\wedge {S^{\vee }_{\varphi }}\leq {(\wedge {S})^{\vee }_{\varphi }}\). Assertions (iii) and (iv) follow easily from (i), (ii) and Proposition 3(i). The proof of (v) and (vi) follow by (i) and (ii). □

The inequalities in Proposition 5 may be proper. This can be seen in the following example.

Example 2

Let B={0,a,b,c,d,e,f,1} and let the order ≤ be defined as in Fig. 1. Let A={e1,e2,e3,} and fA be a soft set over B defined as follows:

f(e1)=0, f(e2)=c, and f(e3)=d. Then the map φ of MSR-approximation space (A(B),φ) on B will be φ(a)={e3}, φ(b)={e3}, and φ(c)={e2}.

If we take x=e and y=f. Then xy=1 and \((x\vee y)^{\vee }_{\varphi }=1\). Also, \(x^{\vee }_{\varphi }=c\), \(y^{\vee }_{\varphi }=c\) and \(x^{\vee }_{\varphi }\vee y^{\vee }_{\varphi }=c\). Thus \((x\vee y)^{\vee }_{\varphi }{>}_{_{_{_{\neq }}}}{x^{\vee }_{\varphi }}\vee y^{\vee }_{\varphi }\).

Now \(x^{\wedge }_{\varphi }=a\vee b\vee c=1\), \(y^{\wedge }_{\varphi }=a\vee b\vee c=1\). So \(x^{\wedge }_{\varphi }\wedge y^{\wedge }_{\varphi }=1\). On the other hand xy=ef=c and \((x\wedge y)^{\wedge }_{\varphi }=0\). Thus \(x^{\wedge }_{\varphi }\wedge y^{\wedge }_{\varphi }{>}_{_{_{_{\neq }}}}(x\wedge y)^{\wedge }_{\varphi }\).

Proposition 6

Let B=(B,≤) be a complete atomic Boolean lattice and let fn be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. Then, \((B^{\vee }_{\varphi },\geq)\cong {(B^{\wedge }_{\varphi },\leq)}\)

Proof

We show that \(x^{\wedge }_{\varphi }{\longrightarrow {(x^{{\prime }})^{\vee }_{\varphi }}}\) is the required dual order isomorphism. It is obvious that \(x^{\wedge }_{\varphi }{\longrightarrow {(x^{{\prime }})^{\vee }_{\varphi }}}\) is onto \((B^{\vee }_{\varphi },\geq)\). We show that \(x^{\wedge }_{\varphi }{\longrightarrow {(x^{{\prime }})^{\vee }_{\varphi }}}\) is order embedding. Suppose that \(x^{\wedge }_{\varphi }\leq {y^{\wedge }_{\varphi }}\). Then for all aA(B), \(a\leq {x^{\wedge }_{\varphi }}\) implies \(a\leq {y^{\wedge }_{\varphi }}\). So, for all aA(B) such that φ(a)=φ(c) for some cA(B) s.t cx implies φ(a)=φ(b) for some bA(B) s.t by. Suppose that \({(y^{{\prime }})^{\vee }_{\varphi }}{\not \leq }{(x^{{\prime }})^{\vee }_{\varphi }}\). So there exists aA(B) such that \(a\leq {{(y^{{\prime }})^{\vee }_{\varphi }}}\) and \(a\not \leq {(x^{{\prime }})^{\vee }_{\varphi }}\). Hence ay′ and φ(a)≠φ(b) for all bA(B) s.t b≤(y)=y. Also \(a\not \leq (x^{{\prime }})^{\vee }_{\varphi }\) implies either a̸x′ or φ(a)=φ(c) for some cA(B) s.t c≤(x)=x, a contradiction. Hence \((y^{{\prime }})^{\vee }_{\varphi }\leq {(x^{{\prime }})^{\vee }_{\varphi }}\). On the other hand, assume that \((y^{{\prime }})^{\vee }_{\varphi }\leq {(x^{{\prime }})^{\vee }_{\varphi }}\) and \(x^{\wedge }_{\varphi }{\not \leq {y^{\wedge }_{\varphi }}}\). So there exists aA(B) such that \(a\leq {x^{\wedge }_{\varphi }}\) and \(a\not \leq {y^{\wedge }_{\varphi }}\). So φ(a)=φ(b) for some bA(B) s.t bx. But φ(a)≠φ(c) for all cA(B) s.t cy. So \(a\leq (y^{{\prime }})^{\vee }_{\varphi }\) and \(a\not \leq {(x^{{\prime }})^{\vee }_{\varphi }}\), a contradiction. □

Proposition 7

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. Then for all xB

  1. i)

    \((x^{\vee }_{\varphi })^{\vee }_{\varphi }=x^{\vee }_{\varphi }\);

  2. ii)

    \((x^{\wedge }_{\varphi })^{\wedge }_{\varphi }=x^{\wedge }_{\varphi }\).

  3. iii)

    \((x^{\vee }_{\varphi })^{\wedge }_{\varphi }=x^{\vee }_{\varphi }\)

  4. iv)

    \((x^{\wedge }_{\varphi })^{\vee }_{\varphi }=x^{\wedge }_{\varphi }\)

Proof

i) \((x^{\vee }_{\varphi })^{\vee }_{\varphi }=\bigvee {\{a\in {A(B)}: a\leq {x^{\vee }_{\varphi }} \hspace {0.1cm},\varphi (a)\neq \varphi (b) \hspace {0.1cm}\forall \hspace {0.1cm} b\in A(B)\hspace {0.1cm} s.t\hspace {0.1cm}b\not \leq x^{\vee }_{\varphi }\}}\). By Proposition 3\(x^{\vee }_{\varphi }\leq x\) which gives \({x}^{{\prime }}\leq ({x^{\vee }_{\varphi }})^{{\prime }}\). So if bx, then \(b\leq ({x^{\vee }_{\varphi }})^{{\prime }}\). Therefore \((x^{\vee }_{\varphi })^{\vee }_{\varphi }=\bigvee {\{a\in {A(B)}: a\leq {x^{\vee }_{\varphi }} \hspace {0.1cm},\varphi (a)\neq \varphi (b) \hspace {0.1cm}\forall \hspace {0.1cm} b\in A(B)\hspace {0.1cm} s.t\hspace {0.1cm}b\not \leq x\}}=x^{\vee }_{\varphi }\).

ii)By Proposition 3\(x^{\wedge }_{\varphi }\leq (x^{\wedge }_{\varphi })^{\wedge }_{\varphi }\). For the reverse inclusion, let \(a\leq (x^{\wedge }_{\varphi })^{\wedge }_{\varphi }\), then φ(a)=φ(b) for some bA(B) s.t \(b\leq x^{\wedge }_{\varphi }\), that is φ(b)=φ(c) for some cA(B) s.t cx. This implies that φ(a)=φ(c) for some cA(B) s.t cx. Hence \(a\leq x^{\wedge }_{\varphi }\) and therefore \((x^{\wedge }_{\varphi })^{\wedge }_{\varphi }\leq x^{\wedge }_{\varphi }\). Hence \((x^{\wedge }_{\varphi })^{\wedge }_{\varphi }=x^{\wedge }_{\varphi }\).

iii)\(x^{\vee }_{\varphi }\leq (x^{\vee }_{\varphi })^{\wedge }_{\varphi }\) by Proposition 3. For the converse assume that \(a\leq (x^{\vee }_{\varphi })^{\wedge }_{\varphi }\). So φ(a)=φ(b) for some bA(B) s.t \(b\leq x^{\vee }_{\varphi }\). Hence φ(b)≠φ(c) for all cA(B) s.t cx. Since φ(a)=φ(b), then φ(a)≠φ(c) for all cA(B) s.t cx. Since φ(a)=φ(a), then a̸x, i.e ax. So \(a\leq x^{\vee }_{\varphi }\). Hence \((x^{\vee }_{\varphi })^{\wedge }_{\varphi }\leq x^{\vee }_{\varphi }\) and we conclude that \((x^{\vee }_{\varphi })^{\wedge }_{\varphi }=x^{\vee }_{\varphi }\).

iv)By Proposition 3\((x^{\wedge }_{\varphi })^{\vee }_{\varphi }\leq x^{\wedge }_{\varphi }\). Conversely, if \(a\not \leq (x^{\wedge }_{\varphi })^{\vee }_{\varphi }\), then either \(a\not \leq x^{\wedge }_{\varphi }\) or φ(a)=φ(b) for some bA(B) s.t \(b\leq (x^{\wedge }_{\varphi })^{{\prime }}\). If \(a\not \leq x^{\wedge }_{\varphi }\), then we get our required result. In later case, φ(a)=φ(b) for some bA(B) s.t \(b\leq (x^{\wedge }_{\varphi })^{{\prime }}\). So φ(b)≠φ(z) for all zA(B) s.t zx. Therefore φ(a)≠φ(z) for all zA(B) s.t zx. Also, ax because if a̸x, then ax. So φ(a)≠φ(a), a contradiction. Hence \(a\leq (x^{{\prime }})^{\vee }_{\varphi }\). That is \(a\not \leq ((x^{{\prime }})^{\vee }_{\varphi })^{{\prime }}=x^{\wedge }_{\varphi }\) by Proposition 4(ii). □

Relations between soft rough approximations on B and current approximations

The following proposition shows a relation between soft lower approximation operators over a complete atomic Boolean lattice B and lower MSR-approximation over B.

Proposition 8

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. Then for any xB, \(x^{\vee }\leq x^{\vee }_{\varphi }\).

Proof

Let bA(B) s.t bx. Then aA s.t bf(a)≤x. So aφ(b) and bx. If \(b\not \leq x^{\vee }_{\varphi }\), then φ(b)=φ(c) for cA(B) s.t c̸x. Since aφ(b) and φ(b)=φ(c), then aφ(c) which implies that cf(a)≤x. Hence cx which is a contradiction. Therefore \(x^{\vee }\leq x^{\vee }_{\varphi }\). □

The following example shows that the relation ≤ between \(x^{\vee }_{\varphi }\) and x may be proper.

Example 3

Let B={0,a,b,c,d,e,f,1} and let the order ≤ be defined as in Fig. 1. Let A={e1,e2,e3,} and fA be a soft set over B defined as follows:

f(e1)=a, f(e2)=0, and f(e3)=e. Then the map φ of MSR-approximation space (A(B),φ) on B will be φ(a)={e1,e3}, φ(b)=ϕ, and φ(c)={e3}.

Let x=d. Then \(x^{\vee }_{\varphi }=a\vee b=d\) and x=a. Thus \(x^{\vee }{<}_{_{_{\neq }}} x^{\vee }_{\varphi }\).

In the following proposition we study a necessary and sufficient condition for \(x^{\vee }_{\varphi }\leq x^{\vee }\) to be hold for any xB.

Proposition 9

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. Then for any xB, \(x^{\vee }_{\varphi }\leq x^{\vee }\) iff for every bA(B)eA s.t \(f(e)=\bigvee \{a\in A(B):\varphi (a)=\varphi (b)\}\).

Proof

() Assume that \(x^{\vee }_{\varphi }\leq x^{\vee }\) for any xB. Let bA(B) and \(x=\bigvee \{a\in A(B):\varphi (a)=\varphi (b)\}\). So \(x^{\vee }_{\varphi }=\bigvee \{a\in A(B):\varphi (a)=\varphi (c) \hspace {0.1cm}for\hspace {0.1cm} some\hspace {0.1cm}c\in A(B)\hspace {0.1cm}c\leq x \} =\bigvee \{a\in A(B):\varphi (a)=\varphi (b)\}=x\). Since \(b\leq x=x^{\vee }_{\varphi }\) and \(x^{\vee }_{\varphi }\leq x^{\vee }\), then bx. Therefore eA s.t bf(e) and f(e)≤x. Also, for all aA(B) s.t ax, we have φ(a)=φ(b), thus eφ(b)=φ(a). So af(e) and therefore xf(e). Consequently, \(f(e)=x=\bigvee \{a\in A(B):\varphi (a)=\varphi (b)\}\).

() Let xB and bA(B) s.t \(b\leq {x^{\vee }_{\varphi }}\). So for every aA(B), if φ(a)=φ(b), then ax. Hence \(\bigvee \{a\in A(B):\varphi (a)=\varphi (b)\}\leq x\). By assumption, eA s.t \(f(e)=\bigvee \{a\in A(B):\varphi (a)=\varphi (b)\}\). Hence bf(e) and f(e)≤x. Therefore bx and consequently, \(x^{\vee }_{\varphi }\leq x^{\vee }\). □

Remark 2

In general, there is no relation between \(x^{\wedge }_{\varphi }\) and x. If x=e in Example 1, then x=d and \(x^{\wedge }_{\varphi }=e\not \leq d\).

In Propositions 10 and 11 and Corollary 2 we show that there is a relation between \(x^{\wedge }_{\varphi }\) and x if some specific conditions hold.

Proposition 10

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. Then fA is full iff \(x^{\wedge }_{\varphi }\leq x^{\wedge }\) for every xB.

Proof

() Assume that fA is full and xB. Let aA(B) s.t \(a\leq {x^{\wedge }_{\varphi }}\), then bA(B) s.t bx and φ(a)=φ(b). Since \(b\leq {1}={\bigvee _{_{_{_{e\in {A}}}}}}f(e)\), then eA s.t bf(e). Hence bf(e)x and thus f(e)x≠0. By, bf(e), we have eφ(b)=φ(a) and therefore af(e). Consequently, \(x^{\wedge }_{\varphi }\leq x^{\wedge }\).

() Suppose that \(x^{\wedge }_{\varphi }\leq x^{\wedge }\) for every xB, we show that \(1\leq {\bigvee _{_{_{_{e\in {A}}}}}}f(e)\). Let aA(B), then \(a\leq {a^{\wedge }_{\varphi }}\leq a^{\wedge }\). Therefore eA s.t f(e)a≠0 and thus af(e) because aA(B). Consequently, fA is a full soft set over B. □

Corollary 1

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. Then fA is full iff xx for every xB.

Proof

() Assume that fA is full and xB. Then \(x\leq {x^{\wedge }_{\varphi }}\leq {x^{\wedge }}\) (by Propositions 3 and 10).

() Assume that xx for every xB. Let bA(B), then \(b\leq {b^{\wedge }}\leq {\bigvee \{f(e): e\in {A}\hspace {0.1cm}} and {f(e)\wedge {b}}\neq {0}\}={\bigvee \{f(e): e\in {A} \hspace {0.1cm}and\hspace {0.1cm} b\leq {f(e)}\}}\). Therefore eA s.t bf(e) and consequently, fA is full. □

Proposition 11

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. Then \(x^{\wedge }\leq x^{\wedge }_{\varphi }\) for every xB iff for every e1,e2A, f(e1)f(e2)=0 whenever f(e1)≠f(e2).

Proof

() Assume that \(x^{\wedge }\leq x^{\wedge }_{\varphi }\) for every xB. Let e1,e2A, if f(e1)f(e2)≠0, then bA(B) s.t bf(e1)f(e2). Since bf(e1), then \(f(e_{1})\leq {\bigvee \{f(e): b\leq {f(e)}\}}={\bigvee \{f(e): b\wedge {f(e)}\neq {0}\}}=b^{\wedge }\leq b^{\wedge }_{\varphi }=\bigvee \{a\in A(B): \varphi (a)=\varphi (b)\}\). On the other hand we show that \({\bigvee \{a\in A(B): \varphi (a)=\varphi (b)\}}\leq {f(e_{1})}\). Let cA(B) s.t \(c\leq {\bigvee \{a\in A(B): \varphi (a)} {=\varphi (b)\}}\), then φ(c)=φ(b) and thus e1φ(b)=φ(c). Therefore cf(e1) and consequently, \(f(e_{1})=\bigvee \{a\in A(B): \varphi (a)=\varphi (b)\}\). Similarly, by bf(e2), \(f(e_{2})=\bigvee \{a\in A(B): \varphi (a)=\varphi (b)\}\) and hence f(e1)=f(e2).

() Assume that for every e1,e2A, f(e1)f(e2)=0 whenever f(e1)≠f(e2). Let xB and aA(B) s.t ax, then e1A s.t af(e1) and f(e1)x≠0. So, bA(B) s.t bf(e1)x. We show that φ(a)={e2A:f(e2)=f(e1)}. If f(e2)≠f(e1), then f(e1)f(e2)=0 by assumption. Thus a̸f(e2) because af(e1) and therefore e2φ(a). So φ(a){e2A:f(e2)=f(e1)}. On the other hand, if f(e2)=f(e1), then af(e2) and hence e2φ(a). Consequently, φ(a)={e2A:f(e2)=f(e1)}. Similarly we can show that φ(b)={e2A:f(e2)=f(e1)} and thus φ(a)=φ(b). Since bx, then \(a\leq x^{\wedge }_{\varphi }\). □

Corollary 2

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B. If f(e)≠0 for every eA, then fA is a partition soft set iff \(x^{\wedge }=x^{\wedge }_{\varphi }\) for every xB.

Proof

Obvious. □

Deviation of some properties of the previous soft rough approximations and the current approximations

In this section we will introduce the deviations of some properties of soft rough approximations on a complete atomic Boolean lattice B [14] and the current approximations.

On accounting of Proposition 3, 5, 6, and 7[parts (ii) and (iv)], there are deviations between some of the properties of lower and upper approximations on B and MSR lower and upper approximations on B, as follows

Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Then for all xB

  1. (i)

    x̸x

  2. (ii)

    \(1^{\vee }_{\varphi }\neq 1\neq 1^{\wedge }\)

  3. (iii)

    For all SB, S≠(S).

  4. (iv)

    (x)̸x.

  5. (v)

    (x)̸x

  6. (vi)

    In general (B,≥)(B,≤)

The following counter example support our claims about the above deviations.

Example 4

Let B={0,a,b,c,d,e,f,1} and let the order ≤ be defined as in Fig. 1. Let A={e1,e2,e3,}.

(1) Let fA be a soft set over B defined as follows:

f(e1)=a, f(e2)=b, f(e3)=d and f(e4)=0. So fA is not full. Let x=f, then x=d̸f. So, x̸x. Also, \(1^{\vee }_{\varphi }=d=1^{\wedge }\neq 1\).

(2) Let fA be a soft set over B defined as follows: f(e1)=a, f(e2)=f, f(e3)=e and f(e4)=0. So fA is not keeping infumum. Let S={f,e}, then S=fe=c and (S)=c=0. Also, S={f,e} and hence S=fe=c̸0=(S). Let x=a, then x=e and (x)=e=1̸e=x. Also x=a and (x)=e̸a=x.

Let x=a and y=f; then x=e and y=1. Therefore xy. On the other hand y=a̸f=x. Hence (B,≥)(B,≤).

Remark 3

(1)It is mentioned that in order to prove that xx and 1=1=1 in [14] we employed a strong condition on soft set fA over a complete atomic Boolean lattice to be full. However in proving \(x\leq x^{\wedge }_{\varphi }\) and \(1^{\vee }_{\varphi }=1= 1^{\wedge }_{\varphi }\) in Proposition 3 no such condition is required.

(2) Also to prove that for all SB, Sn=(S) in [14] we employed a strong condition on soft set fA over a complete atomic Boolean lattice to be keeping infimum. However in proving \(\wedge {S^{\vee }_{\varphi }}=(\wedge {S})^{\vee }_{\varphi }\) in Proposition 5 no such condition is required.

(3) Finally to prove that (x)=x, (x)=x and (B,≥)(B,≤) we employed a strong condition on soft set fA over a complete atomic Boolean lattice to be a partition. However in proving \((x^{\vee }_{\varphi })^{\wedge }_{\varphi }=x^{\vee }_{\varphi }\), \((x^{\wedge }_{\varphi })^{\wedge }_{\varphi }=x^{\wedge }_{\varphi }\) and \((B^{\vee }_{\varphi },\geq)\cong {(B^{\wedge }_{\varphi },\leq)}\) in Propositions 6 and 7 no such condition is required.

(4) It is clear that MSR-element over a complete atomic Boolean lattice satisfies all the basic properties Järvinen’s approximations [20]. Thus, MSR-element over a complete atomic Boolean lattice provides a good combination of roughness and parametrization.

Relation between MSR sets and rough sets on a complete atomic Boolean lattice

In the following, we introduce the notion of Boolean lattice information system and we show that every soft sets on a complete atomic Boolean lattice induces a Boolean lattice information system and vice versa.

Definition 8

Let B=(B,≤) be a complete atomic Boolean lattice and A be a finite set of attributes. The pair (A(B),A,V,g) is called lattice information system, if g is an information function from A(BA to \(V={\bigcup _{e\in {A}}} {V_{e}}\) where Ve={g(b,e):bA(B),eA} is the values of the attribute set e.

Definition 9

A lattice information system (A(B),A,V,g) is called Boolean lattice information system if V={0,1}.

Definition 10

Let B=(B,≤) be a complete atomic Boolean lattice and let S=fA be a soft set over B. Then fA induces a Boolean lattice information system Is=(A(B),A,V,gs), where gs:A(BAV={0,1}, For any bA(B) and eA,

$$g_{s}(b,e)=\left\{ \begin{array}{cc} 1 & \text{if}\,\, b\leq f(e),\\ 0 & \text{if}\,\, b\not\leq f(e) \end{array}\right. $$

Definition 11

Let B=(B,≤) be a complete atomic Boolean lattice and I=(A(B),A,V,g) be a Boolean lattice information system. Then \(S^{I}=f^{I}_{A}\) is called a soft set over B induced by I, where \(f^{I}_{A}:A\longrightarrow {B}\) and for eA, fI(e)={bA(B):g(b,e)=1}.

Proposition 12

Let B=(B,≤) be a complete atomic Boolean lattice and S=fA be a soft set over B. Let Is=(A(B),A,V,gs) be a Boolean lattice information system induced by S and \(\phantom {\dot {i}\!}S^{I_{s}}=f^{I_{s}}_{A}\) be a soft set over B induced by Is. Then \(f^{I_{s}}_{A}=f_{A}\).

Proof

By Definition 11, for any eA, \(f^{I_{s}}_{A}(e)=\vee \{b\in A(B):g_{s}(b,e)=1\}\).

By Definition 10, for any bA(B) and eA,

$$g_{s}(b,e)=\left\{ \begin{array}{cc} 1 & \text{if}\,\, b\leq f(e),\\ 0 & \text{if}\,\, b\not\leq f(e) \end{array}\right. $$

This implies that gs(b,e)=1bf(e). So, for any bA(B)eA, \(\phantom {\dot {i}\!}f(e)=f^{I_{s}}(e)\).

Proposition 13

Let B=(B,≤) be a complete atomic Boolean lattice and I=(A(B),A,V,g) be a Boolean lattice information system. Let \(S^{I}=f^{I}_{A}\) be a soft set over B induced by I and \(\phantom {\dot {i}\!}{I_{s^{I}}}=(A(B), A, V, g_{s^{I}})\) be a Boolean lattice information system induced by SI. Then \(\phantom {\dot {i}\!}I={I_{s^{I}}}\).

Proof

By Definition 10, for any bA(B) and eA,

$$g_{s^{I}}(b,e)=\left\{ \begin{array}{cc} 1 & \text{if}\,\, b\leq f^{I}(e),\\ 0 & \text{if}\,\, b\not\leq f^{I}(e) \end{array}\right. $$

By Definition 11, for any eA, fI(e)={bA(B):g(b,e)=1}. Since I=(A(B),A,V,g) be a Boolean lattice information system, then g(b,e)=0 if b̸fI(e). This implies that

$$g(b,e)=\left\{ \begin{array}{cc} 1 & \text{if}\,\, b\leq f^{I}(e),\\ 0 & \text{if}\,\, b\not\leq f^{I}(e) \end{array}\right. $$

So for any bA(B) and eA, \(\phantom {\dot {i}\!}g_{s^{I}}(b,e)=g(b,e)\). Hence \(\phantom {\dot {i}\!}g_{s^{I}}=g\) and Consequently, \(\phantom {\dot {i}\!}I=I_{s^{I}}\). □

Definition 12

Let B=(B,≤)be a complete atomic Boolean lattice and let S=fA be a soft set over B. Let Is=(A(B),A,V,gs) be a Boolean lattice information system induced by S. Then Is induces a mapping ψs:A(B)→B as follows; for every a,bA(B)

$$a\leq\psi_{s}(b) \Leftrightarrow{\hspace{0.1cm}\forall\hspace{0.1cm} e\in A\hspace{0.1cm} g_{s}(a,e)=g_{s}(b,e)} $$

In [14], we define a mapping induced by a soft set fA on a complete atomic Boolean lattice B as follows

Definition 13

[14] Let B=(B,≤)be a complete atomic Boolean lattice and let fA be a soft set over B. Define a mapping ψf:A(B)→B by

$$a\leq{\psi_{f}(b)}\Leftrightarrow{\exists {e\in{A}}, \hspace{0.1cm}s.t\hspace{0.1cm}a\leq{f(e)} \hspace{0.1cm}and \hspace{0.1cm}b\leq{f(e)}}$$

for every a,bA(B). Then ψf is called the mapping induced by fA on B.

Definition 14

[14] Let B=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let ψf:A(B)→B be the mapping induced by fA on B. We define a pair of soft approximation operators \(^{\nabla _{f}},^{\triangle _{f}}:B\rightarrow {B}\) as follows s

  • \(x^{\nabla _{f} }=\bigvee {\{b\in A(B):\psi _{f}(b)\leq {x}\}}\), and

  • \(x^{\triangle _{f} }=\bigvee {\{b\in A(B):{\psi _{f}(b)\wedge {x}}\neq {0}\}}\).

The elements \(\phantom {\dot {i}\!}x^{\nabla _{f}}\) and \(\phantom {\dot {i}\!}x^{\triangle _{f}}\) are called the soft lower and the soft upper approximations of x with respect to the mapping ψf induced by fA respectively. Two elements x and y are called equivalent if they have the same soft upper and lower approximations with respect to the mapping ψf induced by fA on B. The resulting equivalence classes are called soft rough sets with respect to the mapping ψf induced by fA on B.

Proposition 14

Let B=(B,≤)be a complete atomic Boolean lattice and S=fA be a partition soft set over B. Let Is=(A(B),A,V,gs) be a Boolean lattice information system induced by fA. Then

$$a\leq\psi_{s}(b) \hspace{0.1cm}\Leftrightarrow{a\leq{\psi_{f}(b)}} \hspace{0.1cm}\forall\hspace{0.1cm}a,b\in A(B) $$

Proof

() Let a,bA(B) s.t aψs(b). Then eA, gs(a,e)=gs(b,e). Since a≤1 and fA be a partition soft set, then eA s.t af(e). So gs(b,e)=gs(a,e)=1 and therefore bf(e). Consequently, aψf(b).

()Let a,bA(B) s.t aψf(b). Then e1A s.t af(e1) and bf(e1). So gs(a,e1)=gs(b,e1). For every e2A−{e1}, if f(e1)=f(e2), then af(e2) and bf(e2) and thus gs(a,e2)=1=gs(b,e2). If f(e1)≠f(e2), then f(e1)f(e2)=0 because fA is a partition. Since af(e1) and bf(e1), then a̸f(e2) and b̸f(e2) and therefore gs(a,e2)=0=gs(b,e2). So, gs(a,e)=gs(b,e)eA and consequently aψs(b). □

Proposition 15

Let B=(B,≤) be a complete atomic Boolean lattice and let S=fA be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B and Is=(A(B),A,V,gs) be a Boolean lattice information system induced by S. Then

  1. (i)

    a,bA(B)aψs(b)φ(a)=φ(b).

  2. (ii)

    xB, \({x^{\vee }_{\varphi }}={x^{\nabla }_{s}}\), where \({x^{\nabla }_{s}}=\bigvee {\{b\in A(B):\psi _{s}(b)\leq {x}\}}\).

  3. (iii)

    xB, \({x^{\wedge }_{\varphi }}={x^{\triangle }_{s}}\), where \({x^{\triangle }_{s}}=\bigvee {\{b\in A(B):\psi _{s}(b)\wedge {x}\neq 0\}}\)

Proof

i) () Let a,bA(B) s.t aψs(b). So gs(a,e)=gs(b,e)eA. Let eφ(a), so af(e) and hence gs(a,e)=1. Therefore gs(b,e)=1 and thus bf(e). Consequently, eφ(b). So φ(a)φ(b), similarly we can show that φ(b)φ(a).

() Let a,bA(B) s.t φ(a)=φ(b). Let eA, if gs(a,e)=1, then af(e) and hence eφ(a)=φ(b). Also, bf(e) and so gs(b,e)=1. If gs(a,e)=0, then a̸f(e) and hence eφ(a)=φ(b). Therefore b̸f(e) and thus gs(b,e)=0. So gs(a,e)=gs(b,e)eA and consequently, aψs(b).

ii) Let xB and bA(B) s.t \(b\leq {{x^{\vee }_{\varphi }}}\). We show that ψs(b)≤x. Let aA(B) s.t aψs(b), then φ(a)=φ(b) by i). But \(b\leq {{x^{\vee }_{\varphi }}}\) implies that φ(b)≠φ(c)cA(B) s.t c̸x. Since φ(a)=φ(b), then ax and therefore ψs(b)≤x. This implies that \({x^{\vee }_{\varphi }}\leq {x^{\nabla }_{s}}\).

Conversely, Let bA(B) s.t \(b\leq {x^{\nabla }_{s}}\). So ψs(b)≤x. Let aA(B) s.t a̸x, thus a̸ψs(b). Therefore φ(a)≠φ(b) by i) and so \(b\leq {{x^{\vee }_{\varphi }}}\). Consequently, \({x^{\nabla }_{s}}\leq {x^{\vee }_{\varphi }}\).

iii) Let xB and bA(B) s.t \(b\leq {{x^{\wedge }_{\varphi }}}\), then aA(B) s.t ax and φ(a)=φ(b). So aψs(b) by i) and therefore ψs(b)x≠0. Consequently, \(b\leq {x^{\triangle }_{s}}\).

Conversely, let bA(B) s.t \(b\leq {x^{\triangle }_{s}}\), then ψs(b)x≠0. Thus aA(B) s.t ax and aψs(b). Thus φ(a)=φ(b) by i) and therefore \(b\leq {{x^{\wedge }_{\varphi }}}\). □

In the following, we introduce the concept of Boolean lattice information system with respect to another Boolean lattice information system. We study upper and lower MSR-approximations of soft set on a complete atomic Boolean lattice with respect to another soft set.

Definition 15

Let B=(B,≤) be a complete atomic Boolean lattice and let \(f_{A_{1}} \) be a soft set over B. Let (A(B),φ) be a MSR-approximation space on B where φ:A(B)→P(A1) is defined as φ(b)={aA:bf(a)}. Let \(g_{A_{2}}\) be another soft set over B. For any eA2, lower and upper MSR approximations of \(g_{A_{2}}\) over B are denoted by \((g_{A_{2}})^{\vee }_{\varphi }\) and \((g_{A_{2}})^{\wedge }_{\varphi }\) defined as

  • \(g(e)^{\vee }_{\varphi }=\bigvee {\{a\in {A(B)}: a\leq {g(e)} \hspace {0.1cm},\varphi (a)\neq \varphi (b) \hspace {0.1cm}\forall \hspace {0.1cm}b\in A(B)\hspace {0.1cm}s.t\hspace {0.1cm}b\not \leq g(e)\}}\) eA2,

  • \(g(e)^{\wedge }_{\varphi }=\bigvee {\{a\in {A(B)}:\varphi (a)= \varphi (b) \hspace {0.1cm} for \hspace {0.1cm}some \hspace {0.1cm} b\in A(b)\hspace {0.1cm}s.t\hspace {0.1cm} b\leq g(e)\}} \forall e\in A_{2}\).

In order to understand this concept consider the following example:

Example 5

Let B={0,a,b,c,d,e,f,1} representing 8 patients, where 0 denotes patient who drink mineral water only, a denotes patient who drink coffee, b denotes patient who drink cola, c denotes patient who drink tea, d denotes patient who drink caffeine liquids, e denotes patient who drink antioxidant liquids, f denotes patient who drink cold liquids and 1 denotes patient who drink all liquids. So, the order ≤ can be defined as in Fig. 1.

Let A1={e1,e2,e3,}, where e1 denotes temperature, e2 denotes headache and e3 denotes stomach problem and let \(f_{A_{1}}\) be a soft set over B representing the diagnosis of doctor M, defined as follows:

f(e1)=b, f(e2)=e and f(e3)=b. Then the Boolean lattice information system of \(f_{A_{1}}\) can be given by Table 1, where 1 and 0 denote yes and no respectively.

Table 1 Lattice information system of the soft set fA

Then the map φ of MSR-approximation space (A(B),φ) will be φ(a)={e2}, φ(b)={e1,e3}, and φ(c)={e2}.

Let A2={e1,e2,e3,e4} and \(g_{A_{2}}\) be another soft set over B representing the diagnosis of doctor N, where A2={e1,e2,e3,e4} and e4 represents cough, defined as follows:

g(e1)=d, g(e2)=b, g(e3)=c and g(e4)=e. So lower MSR-approximations of \(g_{A_{2}}\) over B are \(g(e_{1})^{\vee }_{\varphi }=b\), \(g(e_{2})^{\vee }_{\varphi }=b\), \(g(e_{3})^{\vee }_{\varphi }=0\) and \(g(e_{4})^{\vee }_{\varphi }=a\). Also, upper MSR-approximations of \(g_{A_{2}}\) over B are \(g(e_{1})^{\wedge }_{\varphi }=a\vee b\vee c=1\), \(g(e_{2})^{\wedge }_{\varphi }=b\), \(g(e_{3})^{\wedge }_{\varphi }=a\vee c=e\) and \(g(e_{4})^{\wedge }_{\varphi }=e\).

So, for example the patient a will be diagnosed cough by doctor N.

Application

In this section, we introduce the concepts of modified soft rough topology based on the notion of modified soft lower and upper approximations

Proposition 16

Let U be the universe, XUB=(U) and FA be a soft set over U. Then, the collection \(\tau ^{\varphi }_{SR}(X)=\{U,\phi,X^{\vee }_{\varphi }, X^{\wedge }_{\varphi }, yy{\varphi }(X)=X^{\wedge }_{\varphi }-{X^{\vee }_{\varphi }}\}\), forms a topology on U called the modified soft rough topology on U w.r.t X.

Proposition 17

Let \(\tau ^{\varphi }_{SR}(X)\) be a modified soft rough topology on U w.r.t X. Then the collection \(\beta ^{\varphi }_{SR}(X)=\{U, X^{\vee }_{\varphi }, Bnd_{\varphi }(X) \}\) forms a base for \(\tau ^{\varphi }_{SR}(X)\).

Proof

Obvious □

Now, we will apply the concept of soft rough topology in Diabetes mellitus (DM), commonly referred to as diabetes, is a group of metabolic diseases in which there are high blood sugar levels over a prolonged period. Symptoms of high blood sugar include frequent urination, increased thirst and increased hunger. If left untreated, diabetes can cause many complications. Acute complications can include diabetic ketoacidosis, nonketotic hyperosmolar coma or death. Serious long-term complications include heart disease, stroke, chronic kidney failure, foot ulcers, and damage to the eyes.

Consider the following information table (Table 2) giving data about 6 patients as a random representative. The rows of the table represent the attributes (the symptoms for Diabetes) and the columns represent the objects (the patients). Let U={p1,p2,p3,p4,p5,p6} and A={e1(Frequent Urination),e2(Increased Hunger),e3(Increased Thirst)}. Let FA be a soft over U given by Table 2 and (U,φ) be a MSR-approximation space.

Table 2 Tabular representation of the soft set FA

Let X={p1,p4,p5} be the set of patients having diabetes. Then, we have \(X^{\vee }_{\varphi }=\{p_{1}, p_{4}, p_{5}\}\), \(X^{\wedge }_{\varphi }=\{p_{1}, p_{4}, p_{5}\}\) and Bndφ(X)=ϕ. Therefore \(\tau ^{\varphi }_{SR}=\{U,\phi,\{p_{1}, p_{4}, p_{5}\}\}\) is a modified soft rough topology on U and its soft basis \(\beta ^{\varphi }_{SR}(X)= \{U,\varphi,\{p_{1}, p_{4}, p_{5}\}\}\).

If the attribute frequent urination is removed, we have \(X^{\vee }_{\varphi }=\{ p_{4}, p_{5}\}\), \(X^{\wedge }_{\varphi }=\{p_{1}, p_{3}, p_{4}, p_{5}\}\) and Bndφ(X)={p1,p3}. Therefore, \(\tau ^{\varphi }_{SR}(X)=\{U,\phi,\{ p_{4}, p_{5}\},\{p_{1}, p_{3}, p_{4}, p_{5}\}, \{p_{1}, p_{3}\}\}\) is a modified soft rough topology on U and its soft basis \(\beta ^{\varphi }_{SR}(X)-{e_{1}}= \{U,\{ p_{4}, p_{5}\},\{p_{1}, p_{3}\}\}\neq {\beta ^{\varphi }_{SR}(X)}\).

Again, if the attribute increased hunger is removed, we have \(X^{\vee }_{\varphi }=\{p_{1}, p_{4}, p_{5}\}\), \(X^{\wedge }_{\varphi }=\{p_{1}, p_{4}, p_{5}\}\) and Bndφ(X)=ϕ. Therefore, \(\tau ^{\varphi }_{SR}(X)=\{U,\phi,\{p_{1}, p_{4}, p_{5}\}\}\) is a modified soft rough topology on U and its soft basis \({\beta ^{\varphi }_{SR}(X)}-{e_{2}}= \{U,\phi,\{p_{1}, p_{4}, p_{5}\}\}=\beta ^{\varphi }_{SR}(X)\).

Finally, if the attribute Increased Thirst is removed, we have \(X^{\vee }_{\varphi }=\{p_{1}, p_{4}, p_{5}\}\), \(X^{\wedge }_{\varphi }=\{p_{1}, p_{4}, p_{5}\}\) and Bndφ(X)=ϕ. Therefore \(\tau ^{\varphi }_{SR}(X)=\{U,\phi,\{p_{1}, p_{4}, p_{5}\}\}\) is a modified soft rough topology on U and its soft basis \({\beta ^{\varphi }_{SR}(X)}-{e_{3}}= \{U,\phi,\{p_{1}, p_{4}, p_{5}\}\}=\beta ^{\varphi }_{SR}(X)\).

Therefore, Core(SRφ)(X)={e1}, i.e., frequent urination is the key attribute that has close connection to disease diabetes.

Algorithm:

Step 1: Given a finite universe U, a finite set A of attributes represent the data as an information table, rows of which are labeled by attributes (C),columns by objects and entries of the table are attribute values.

Step 2: Find the lower MSR approximation, upper MSR approximation and the soft MSR-boundary region of XU.

Step 3: Generate the soft rough topology \(\tau ^{\varphi }_{SR}(X)\) on U and its soft basis \(\beta ^{\varphi }_{SR}(X)\).

Step 4: Remove an attribute x from conditions of attributes (C) and find the lower and upper MSR approximations and the MSR-boundary region of X on C−(x).

Step 5: Generate the soft rough topology \(\tau ^{\varphi }_{SR}\) on U and its soft basis \({\beta ^{\varphi }_{SR}(X)} - {x}\).

Step 6: Repeat steps 4 and 5 for all attributes in C.

Step 7: Those attributes in C for which \({\beta ^{\varphi }_{SR}(X)} - {x} \neq \beta ^{\varphi }_{SR}(X)\) forms the Core(SRφ)(X).

Conclusion

Lattice is a very important structure in mathematics. In [14], we introduced the concept of soft sets on a complete atomic Boolean lattice B. We combine soft set and rough set by introducing the concept of soft rough set on B. Some shortcoming became the part of soft rough sets on B. In this paper, we introduced the concept of Modified soft rough sets(MSR) on a complete atomic Boolean lattice. Some important properties of MSR on B have been discussed. Similar results which require some strong conditions for their proof in soft rough sets on B can be proved in MSR sets without these conditions. Furthermore, we used the modified soft rough approximation operators to introduce the concept of modified soft rough topology and apply this concept in diabetes mellitus.

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Mustafa, H.I. On modified soft rough sets on a complete atomic Boolean lattice. J Egypt Math Soc 27, 15 (2019). https://doi.org/10.1186/s42787-019-0019-8

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Keywords

  • Complete atomic Boolean lattice
  • Soft rough approximation operators on a complete atomic Boolean lattice
  • MSR sets on a complete atomic Boolean lattice
  • Boolean lattice information system
  • Modified soft rough topology

2010 MSC

  • 06B23
  • 54A05
  • 30E99