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On modified soft rough sets on a complete atomic Boolean lattice
Journal of the Egyptian Mathematical Society volume 27, Article number: 15 (2019)
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 (MRSset) 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 decisionmaking, 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 [6–8]. Research on soft set theory is growing rapidly [9–13]. 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 BLalgebras. 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 (MSRset) 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 (MSRsets) 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 latticetheoretical notation and conventions, which can be found in [26, 27].
Lemma 1
[26] Let B=(B,≤) be a complete lattice, S, T⊆B and {X_{i}:i∈I}⊆℘(B)

(i)
If S⊆T, then \(\bigvee S\subseteq \bigvee T\).

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

(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, y∈B

(i)
0^{′}=1 and 1^{′}=0,

(ii)
x^{′′}=x,

(iii)
(x∨y)^{′}=x^{′}∧y^{′}, and (x∧y)^{′}=x^{′}∨y^{′},

(iv)
x≤y iff x∧y^{′}=0.
Lemma 3
[20] Let B=(B,≤) be a complete Boolean lattice. Then for all {x_{i}:i∈I}⊆B and y∈B
and
Definition 1
[20] Let B=(B,≤) be an ordered set and x, y∈B, we say that x is covered by y (or that y covers x), and write, x≺y 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 a∈B is called an atom if 0≺a. 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 f_{A} on B is defined by the set of ordered pairs
or is a function f_{A}:A→B s.t
In other words, a soft set over B is a parameterized family of elements of B. For each e∈A, f(e) is considered as eapproximate element of f_{A}
Definition 4
[14] Let B=(B,≤) be a complete atomic Boolean lattice. Let A⊆E and let f_{A} be a soft set over B. The complement of f_{A}, denoted by (f_{A})^{c} is defined by (f_{A})^{c}=(f^{c},A), where f^{c}:A→B is a mapping given by f^{c}(e)=f(e)^{′} for every e∈A.
Definition 5
[14] Let B=(B,≤) be a complete atomic Boolean lattice and let f_{A} be a soft set over B.

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

(ii)
f_{A} is keeping infimum, if for any e_{1},e_{2}∈A, there exists e_{3}∈A such that f(e_{1})∧f(e_{2})=f(e_{3});

(iii)
f_{A} is keeping supremum, if for any e_{1},e_{2}∈A, there exists e_{3}∈A such that f(e_{1})∨f(e_{2})=f(e_{3});

(iv)
f_{A} is called partition of B if

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

(2)
For every e∈A, f(e)≠0,

(3)
For every e_{1},e_{2}∈A either f(e_{1})=f(e_{2}) or f(e_{1})∧f(e_{2})=0.

(1)
Definition 6
[14] Let B=(B,≤) be a complete atomic Boolean lattice and let f_{A} be a soft set over B. For any element x∈B, we define a pair of operators x^{∨},x^{∧}:B→B 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.
The set of atoms of a complete atomic Boolean lattice B=(B,≤) is {a,b,c}. Let A={e_{1},e_{2},e_{3},} and f_{A} be a soft set over B defined as follows: f(e_{1})=a, f(e_{2})=b, and f(e_{3})=d. So, f_{A} is not full. Many odd situation occurs. For example,
If x=c≠0, then x^{∨}=x^{∧}=0. Also if x=e, then x^{∧}=a∨b=d≧̸e Moreover c≦̸x^{∨} or c≦̸x^{∧} for any x∈B
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 x∈B cannot be avoided.
Proposition 1
Let f_{A} be a soft set over B which is not full. Then there exists at least b∈A(B) such that b≤neg(x)=(x^{∧})^{′} for all x∈B.
Proof
Since f_{A} is not full, i.e., \({\bigvee _{_{_{_{a\in {A}}}}}}{f(a)}\not \leq 1\). So ∃b∈A(B) s.t b≦̸f(a) ∀ a∈A. Let x∈B s.t b≤x. If b≤x^{∧}, then ∃a∈A s.t b≤f(a) and f(a)∧x≠0 which is a contradiction. Hence b≦̸x^{∧}. By a similar argument when b≦̸x it can be shown that b≦̸x^{∧}. Therefore b≤1−x^{∧}=(x^{∧})^{′}. □
Modified soft rough sets (MSRsets) 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 f_{A} be a soft set over B. Let φ:A(B)→℘(A) be another map defined as φ(b)={a∈A:b≤f(a)}. Then the pair (A(B),φ) is called MSRapproximation space on B and for any element x∈B, lower MSRapproximation 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 MSRapproximation 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 MSRelement on B.
Remark 1
Lower MSRapproximation 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)∀b∈A(B) s.t b≦̸x and a≦̸x, then by hypothesis φ(a)≠φ(a) which is impossible.
Lemma 4
Let B=(B,≤) be a complete atomic Boolean lattice and let f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. Then for all c∈A(B) and x∈B

(i)
\({c\leq {x^{\vee }_{\varphi }}}\Longleftrightarrow {c\leq x\hspace {0.1cm}}\) and φ(c)≠φ(b) ∀ b∈A(B) s.t b≦̸x;

ii)
\({c\leq {x^{\wedge }_{\varphi }}}\Longleftrightarrow { \varphi (c)=\varphi (b)\hspace {0.1cm}}\) for some b∈A(B) s.t b≤x.
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 c≤x. If φ(c)≠φ(b)∀ b∈A(B) s.t b≦̸x, 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 c≠a, i.e., c∧a=0. Hence \(c\leq ({x^{\vee }_{\varphi }})^{\prime }\), a contradiction.
(⇐) Suppose that c≤x and φ(c)≠φ(b) ∀ b∈A(B) s.t b≦̸x, 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 f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. If φ(a)=φ(b) for some a,b∈A(B), then for any x∈B 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 c∈A(B) s.t c≤x. Since φ(a)=φ(b), then φ(b)=φ(c), c≤x. This implies that \(b\leq x^{\wedge }_{\varphi }\). □
Proposition 3
Let B=(B,≤) be a complete atomic Boolean lattice and let f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. Then for all x∈B

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

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

(iii)
\(1^{\vee }_{\varphi }=1=1^{\wedge }_{\varphi }\)
Proof
(i) Assume that b∈A(B), s.t \(b\leq x^{\vee }_{\varphi }\), then b≤x. For the other inclusion, let b∈A(B) s.t b≤x. Then φ(b)=φ(b) for some b≤x. 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 f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. Then for all x,y∈B

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

(ii)
The mappings \(^{\vee }_{\varphi }:B\longrightarrow {B}\) and \(^{\wedge }_{\varphi }:B\longrightarrow {B}\) are mutually dual.
Proof
(i) Assume that x≤y and \(a\leq x^{\vee }_{\varphi }\). Let b∈A(B) s.t b≦̸y. Since x≤y, then b≦̸x. 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 c∈A(B) s.t c≤x. Since x≤y, then φ(b)=φ(c) for some c∈A(B) s.t c≤y. 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 a∈A(B) s.t \(a\leq ((x^{{\prime }})^{\vee }_{\varphi })^{{\prime }}\), then \(a\not \leq (x^{{\prime }})^{\vee }_{\varphi }\). So, either a≦̸x or φ(a)=φ(b) for some b∈A(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 b∈A(B) s.t b≤x. 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 a∈A(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 b∈A(B) s.t b≤x^{′}. That is a≤x because otherwise, if a≤x^{′}, then φ(a)≠φ(a), a contradiction. Thus, a≤x, φ(a)≠φ(b) for all b∈A(B) s.t b≤x^{′} and consequently, \(a\leq x^{\vee }_{\varphi }\). Conversely, let a∈A(B) s.t \(a\leq x^{\vee }_{\varphi }\), then a≤x and φ(a)≠φ(b) for all b∈A(B) s.t b≤x^{′}. 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 S⊆B, 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 f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B, then

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

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

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

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

(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.

(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 S⊆B. The mapping \(^{\wedge }_{\varphi }\): B→B 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 b∈A(B) and assume that \(a\leq {(\vee {S})^{\wedge }_{\varphi }}\). So, φ(a)=φ(b) for some b∈A(B) s.t b≤∨S. So, φ(a)=φ(b) for some b∈A(B) and x∈S s.t b≤x. So {a∈A(B):φ(a)=φ(b) forsome b∈A(B) s.t b≤∨S}
\(\subseteq {\cup _{_{x\in {S}}}\{a\in {A(B)}:\varphi (a)= \varphi (b)\hspace {0.1cm} \\for some \hspace {0.1cm}b\in A(B)\hspace {0.1cm} s.t \hspace {0.1cm}b\leq x\}}\). 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 S⊆B. The mapping \(^{\vee }_{\varphi }\): B→B 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 a∈A(B) s.t \(a\leq {\wedge {S^{\vee }_{\varphi }}}=\wedge {\{x^{\vee }_{\varphi }: x\in {S}\}}\). So, a≤x and φ(a)≠φ(b) for all b∈A(B) s.t b≦̸x for every x∈S. Hence φ(a)≠φ(b) for all b∈A(B) s.t b≦̸∧S. In fact if b≦̸∧S, then ∃x∈S s.t b≦̸x. 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={e_{1},e_{2},e_{3},} and f_{A} be a soft set over B defined as follows:
f(e_{1})=0, f(e_{2})=c, and f(e_{3})=d. Then the map φ of MSRapproximation space (A(B),φ) on B will be φ(a)={e_{3}}, φ(b)={e_{3}}, and φ(c)={e_{2}}.
If we take x=e and y=f. Then x∨y=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 x∧y=e∧f=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 MSRapproximation 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 a∈A(B), \(a\leq {x^{\wedge }_{\varphi }}\) implies \(a\leq {y^{\wedge }_{\varphi }}\). So, for all a∈A(B) such that φ(a)=φ(c) for some c∈A(B) s.t c≤x implies φ(a)=φ(b) for some b∈A(B) s.t b≤y. Suppose that \({(y^{{\prime }})^{\vee }_{\varphi }}{\not \leq }{(x^{{\prime }})^{\vee }_{\varphi }}\). So there exists a∈A(B) such that \(a\leq {{(y^{{\prime }})^{\vee }_{\varphi }}}\) and \(a\not \leq {(x^{{\prime }})^{\vee }_{\varphi }}\). Hence a≤y′ and φ(a)≠φ(b) for all b∈A(B) s.t b≤(y^{′})^{′}=y. Also \(a\not \leq (x^{{\prime }})^{\vee }_{\varphi }\) implies either a≦̸x′ or φ(a)=φ(c) for some c∈A(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 a∈A(B) such that \(a\leq {x^{\wedge }_{\varphi }}\) and \(a\not \leq {y^{\wedge }_{\varphi }}\). So φ(a)=φ(b) for some b∈A(B) s.t b≤x. But φ(a)≠φ(c) for all c∈A(B) s.t c≤y. 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 f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. Then for all x∈B

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

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

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

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 b≤x^{′}, 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 b∈A(B) s.t \(b\leq x^{\wedge }_{\varphi }\), that is φ(b)=φ(c) for some c∈A(B) s.t c≤x. This implies that φ(a)=φ(c) for some c∈A(B) s.t c≤x. 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 b∈A(B) s.t \(b\leq x^{\vee }_{\varphi }\). Hence φ(b)≠φ(c) for all c∈A(B) s.t c≤x^{′}. Since φ(a)=φ(b), then φ(a)≠φ(c) for all c∈A(B) s.t c≤x^{′}. Since φ(a)=φ(a), then a≦̸x^{′}, i.e a≤x. 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 b∈A(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 b∈A(B) s.t \(b\leq (x^{\wedge }_{\varphi })^{{\prime }}\). So φ(b)≠φ(z) for all z∈A(B) s.t z≤x. Therefore φ(a)≠φ(z) for all z∈A(B) s.t z≤x. Also, a≤x^{′} because if a≦̸x^{′}, then a≤x. 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 MSRapproximation over B.
Proposition 8
Let B=(B,≤) be a complete atomic Boolean lattice and let f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. Then for any x∈B, \(x^{\vee }\leq x^{\vee }_{\varphi }\).
Proof
Let b∈A(B) s.t b≤x^{∨}. Then ∃ a∈A s.t b≤f(a)≤x. So a∈φ(b) and b≤x. If \(b\not \leq x^{\vee }_{\varphi }\), then φ(b)=φ(c) for c∈A(B) s.t c≦̸x. Since a∈φ(b) and φ(b)=φ(c), then a∈φ(c) which implies that c≤f(a)≤x. Hence c≤x 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={e_{1},e_{2},e_{3},} and f_{A} be a soft set over B defined as follows:
f(e_{1})=a, f(e_{2})=0, and f(e_{3})=e. Then the map φ of MSRapproximation space (A(B),φ) on B will be φ(a)={e_{1},e_{3}}, φ(b)=ϕ, and φ(c)={e_{3}}.
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 x∈B.
Proposition 9
Let B=(B,≤) be a complete atomic Boolean lattice and let f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. Then for any x∈B, \(x^{\vee }_{\varphi }\leq x^{\vee }\) iff for every b∈A(B)∃e∈A s.t \(f(e)=\bigvee \{a\in A(B):\varphi (a)=\varphi (b)\}\).
Proof
(⇒) Assume that \(x^{\vee }_{\varphi }\leq x^{\vee }\) for any x∈B. Let b∈A(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 b≤x^{∨}. Therefore ∃e∈A s.t b≤f(e) and f(e)≤x. Also, for all a∈A(B) s.t a≤x, we have φ(a)=φ(b), thus e∈φ(b)=φ(a). So a≤f(e) and therefore x≤f(e). Consequently, \(f(e)=x=\bigvee \{a\in A(B):\varphi (a)=\varphi (b)\}\).
(⇐) Let x∈B and b∈A(B) s.t \(b\leq {x^{\vee }_{\varphi }}\). So for every a∈A(B), if φ(a)=φ(b), then a≤x. Hence \(\bigvee \{a\in A(B):\varphi (a)=\varphi (b)\}\leq x\). By assumption, ∃e∈A s.t \(f(e)=\bigvee \{a\in A(B):\varphi (a)=\varphi (b)\}\). Hence b≤f(e) and f(e)≤x. Therefore b≤x^{∨} 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 f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. Then f_{A} is full iff \(x^{\wedge }_{\varphi }\leq x^{\wedge }\) for every x∈B.
Proof
(⇒) Assume that f_{A} is full and x∈B. Let a∈A(B) s.t \(a\leq {x^{\wedge }_{\varphi }}\), then ∃b∈A(B) s.t b≤x and φ(a)=φ(b). Since \(b\leq {1}={\bigvee _{_{_{_{e\in {A}}}}}}f(e)\), then ∃e∈A s.t b≤f(e). Hence b≤f(e)∧x and thus f(e)∧x≠0. By, b≤f(e), we have e∈φ(b)=φ(a) and therefore a≤f(e). Consequently, \(x^{\wedge }_{\varphi }\leq x^{\wedge }\).
(⇐) Suppose that \(x^{\wedge }_{\varphi }\leq x^{\wedge }\) for every x∈B, we show that \(1\leq {\bigvee _{_{_{_{e\in {A}}}}}}f(e)\). Let a∈A(B), then \(a\leq {a^{\wedge }_{\varphi }}\leq a^{\wedge }\). Therefore ∃e∈A s.t f(e)∧a≠0 and thus a≤f(e) because a∈A(B). Consequently, f_{A} is a full soft set over B. □
Corollary 1
Let B=(B,≤) be a complete atomic Boolean lattice and let f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. Then f_{A} is full iff x≤x^{∧} for every x∈B.
Proof
(⇒) Assume that f_{A} is full and x∈B. Then \(x\leq {x^{\wedge }_{\varphi }}\leq {x^{\wedge }}\) (by Propositions 3 and 10).
(⇐) Assume that x≤x^{∧} for every x∈B. Let b∈A(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 ∃e∈A s.t b≤f(e) and consequently, f_{A} is full. □
Proposition 11
Let B=(B,≤) be a complete atomic Boolean lattice and let f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. Then \(x^{\wedge }\leq x^{\wedge }_{\varphi }\) for every x∈B iff for every e_{1},e_{2}∈A, f(e_{1})∧f(e_{2})=0 whenever f(e_{1})≠f(e_{2}).
Proof
(⇒) Assume that \(x^{\wedge }\leq x^{\wedge }_{\varphi }\) for every x∈B. Let e_{1},e_{2}∈A, if f(e_{1})∧f(e_{2})≠0, then ∃b∈A(B) s.t b≤f(e_{1})∧f(e_{2}). Since b≤f(e_{1}), 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 c∈A(B) s.t \(c\leq {\bigvee \{a\in A(B): \varphi (a)} {=\varphi (b)\}}\), then φ(c)=φ(b) and thus e_{1}∈φ(b)=φ(c). Therefore c≤f(e_{1}) and consequently, \(f(e_{1})=\bigvee \{a\in A(B): \varphi (a)=\varphi (b)\}\). Similarly, by b≤f(e_{2}), \(f(e_{2})=\bigvee \{a\in A(B): \varphi (a)=\varphi (b)\}\) and hence f(e_{1})=f(e_{2}).
(⇐) Assume that for every e_{1},e_{2}∈A, f(e_{1})∧f(e_{2})=0 whenever f(e_{1})≠f(e_{2}). Let x∈B and a∈A(B) s.t a≤x^{∧}, then ∃e_{1}∈A s.t a≤f(e_{1}) and f(e_{1})∧x≠0. So, ∃b∈A(B) s.t b≤f(e_{1})∧x. We show that φ(a)={e_{2}∈A:f(e_{2})=f(e_{1})}. If f(e_{2})≠f(e_{1}), then f(e_{1})∧f(e_{2})=0 by assumption. Thus a≦̸f(e_{2}) because a≤f(e_{1}) and therefore e_{2}∉φ(a). So φ(a)⊆{e_{2}∈A:f(e_{2})=f(e_{1})}. On the other hand, if f(e_{2})=f(e_{1}), then a≤f(e_{2}) and hence e_{2}∈φ(a). Consequently, φ(a)={e_{2}∈A:f(e_{2})=f(e_{1})}. Similarly we can show that φ(b)={e_{2}∈A:f(e_{2})=f(e_{1})} and thus φ(a)=φ(b). Since b≤x, then \(a\leq x^{\wedge }_{\varphi }\). □
Corollary 2
Let B=(B,≤) be a complete atomic Boolean lattice and let f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B. If f(e)≠0 for every e∈A, then f_{A} is a partition soft set iff \(x^{\wedge }=x^{\wedge }_{\varphi }\) for every x∈B.
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 f_{A} be a soft set over B. Then for all x∈B

(i)
x≦̸x^{∧}

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

(iii)
For all S⊆B, ∧S^{∨}≠(∧S)^{∨}.

(iv)
(x^{∧})^{∧}≦̸x^{∧}.

(v)
(x^{∨})^{∧}≦̸x^{∨}

(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={e_{1},e_{2},e_{3},}.
(1) Let f_{A} be a soft set over B defined as follows:
f(e_{1})=a, f(e_{2})=b, f(e_{3})=d and f(e_{4})=0. So f_{A} is not full. Let x=f, then x^{∧}=d≧̸f. So, x≦̸x^{∧}. Also, \(1^{\vee }_{\varphi }=d=1^{\wedge }\neq 1\).
(2) Let f_{A} be a soft set over B defined as follows: f(e_{1})=a, f(e_{2})=f, f(e_{3})=e and f(e_{4})=0. So f_{A} is not keeping infumum. Let S={f,e}, then ∧S=f∧e=c and (∧S)^{∨}=c^{∨}=0. Also, S^{∨}={f,e} and hence ∧S^{∨}=f∧e=c≦̸0=(∧S)^{∨}. Let x=a, then x^{∧}=e and (x^{∧})^{∧}=e^{∧}=1≦̸e=x^{∧}. Also x^{∨}=a and (x^{∨})^{∧}=e≦̸a=x^{∨}.
Let x=a and y=f; then x^{∧}=e and y^{∧}=1. Therefore x^{∧}≤y^{∧}. On the other hand y^{′∨}=a≦̸f=x^{′∨}. Hence (B^{∨},≥)≇(B^{∧},≤).
Remark 3
(1)It is mentioned that in order to prove that x≤x^{∧} and 1^{∨}=1=1^{∧} in [14] we employed a strong condition on soft set f_{A} 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 S⊆B, ∧Sn=(∧S)^{∨} in [14] we employed a strong condition on soft set f_{A} 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 f_{A} 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 MSRelement over a complete atomic Boolean lattice satisfies all the basic properties Järvinen’s approximations [20]. Thus, MSRelement 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(B)×A to \(V={\bigcup _{e\in {A}}} {V_{e}}\) where V_{e}={g(b,e):b∈A(B),e∈A} 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=f_{A} be a soft set over B. Then f_{A} induces a Boolean lattice information system I_{s}=(A(B),A,V,g_{s}), where g_{s}:A(B)×A→V={0,1}, For any b∈A(B) and e∈A,
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 e∈A, f^{I}(e)=∨{b∈A(B):g(b,e)=1}.
Proposition 12
Let B=(B,≤) be a complete atomic Boolean lattice and S=f_{A} be a soft set over B. Let I_{s}=(A(B),A,V,g_{s}) 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 I_{s}. Then \(f^{I_{s}}_{A}=f_{A}\).
Proof
By Definition 11, for any e∈A, \(f^{I_{s}}_{A}(e)=\vee \{b\in A(B):g_{s}(b,e)=1\}\).
By Definition 10, for any b∈A(B) and e∈A,
□
This implies that g_{s}(b,e)=1⇔b≤f(e). So, for any b∈A(B)e∈A, \(\phantom {\dot {i}\!}f(e)=f^{I_{s}}(e)\).
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 S^{I}. Then \(\phantom {\dot {i}\!}I={I_{s^{I}}}\).
Proof
By Definition 10, for any b∈A(B) and e∈A,
By Definition 11, for any e∈A, f^{I}(e)=∨{b∈A(B):g(b,e)=1}. Since I=(A(B),A,V,g) be a Boolean lattice information system, then g(b,e)=0 if b≦̸f^{I}(e). This implies that
So for any b∈A(B) and e∈A, \(\phantom {\dot {i}\!}g_{s^{I}}(b,e)=g(b,e)\). 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=f_{A} be a soft set over B. Let I_{s}=(A(B),A,V,g_{s}) be a Boolean lattice information system induced by S. Then I_{s} induces a mapping ψ_{s}:A(B)→B as follows; for every a,b∈A(B)
In [14], we define a mapping induced by a soft set f_{A} on a complete atomic Boolean lattice B as follows
Definition 13
[14] Let B=(B,≤)be a complete atomic Boolean lattice and let f_{A} be a soft set over B. Define a mapping ψ_{f}:A(B)→B by
for every a,b∈A(B). Then ψ_{f} is called the mapping induced by f_{A} on B.
Definition 14
[14] Let B=(B,≤) be a complete atomic Boolean lattice and let f_{A} be a soft set over B. Let ψ_{f}:A(B)→B be the mapping induced by f_{A} 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 ψ_{f} induced by f_{A} respectively. Two elements x and y are called equivalent if they have the same soft upper and lower approximations with respect to the mapping ψ_{f} induced by f_{A} on B. The resulting equivalence classes are called soft rough sets with respect to the mapping ψ_{f} induced by f_{A} on B.
Proposition 14
Let B=(B,≤)be a complete atomic Boolean lattice and S=f_{A} be a partition soft set over B. Let I_{s}=(A(B),A,V,g_{s}) be a Boolean lattice information system induced by f_{A}. Then
Proof
(⇒) Let a,b∈A(B) s.t a≤ψ_{s}(b). Then ∀ e∈A, g_{s}(a,e)=g_{s}(b,e). Since a≤1 and f_{A} be a partition soft set, then ∃e∈A s.t a≤f(e). So g_{s}(b,e)=g_{s}(a,e)=1 and therefore b≤f(e). Consequently, a≤ψ_{f}(b).
(⇐)Let a,b∈A(B) s.t a≤ψ_{f}(b). Then ∃e_{1}∈A s.t a≤f(e_{1}) and b≤f(e_{1}). So g_{s}(a,e_{1})=g_{s}(b,e_{1}). For every e_{2}∈A−{e_{1}}, if f(e_{1})=f(e_{2}), then a≤f(e_{2}) and b≤f(e_{2}) and thus g_{s}(a,e_{2})=1=g_{s}(b,e_{2}). If f(e_{1})≠f(e_{2}), then f(e_{1})∧f(e_{2})=0 because f_{A} is a partition. Since a≤f(e_{1}) and b≤f(e_{1}), then a≦̸f(e_{2}) and b≦̸f(e_{2}) and therefore g_{s}(a,e_{2})=0=g_{s}(b,e_{2}). So, g_{s}(a,e)=g_{s}(b,e)∀e∈A and consequently a≤ψ_{s}(b). □
Proposition 15
Let B=(B,≤) be a complete atomic Boolean lattice and let S=f_{A} be a soft set over B. Let (A(B),φ) be a MSRapproximation space on B and I_{s}=(A(B),A,V,g_{s}) be a Boolean lattice information system induced by S. Then

(i)
∀a,b∈A(B)a≤ψ_{s}(b)⇔φ(a)=φ(b).

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

(iii)
∀x∈B, \({x^{\wedge }_{\varphi }}={x^{\triangle }_{s}}\), where \({x^{\triangle }_{s}}=\bigvee {\{b\in A(B):\psi _{s}(b)\wedge {x}\neq 0\}}\)
Proof
i) (⇒) Let a,b∈A(B) s.t a≤ψ_{s}(b). So g_{s}(a,e)=g_{s}(b,e)∀e∈A. Let e∈φ(a), so a≤f(e) and hence g_{s}(a,e)=1. Therefore g_{s}(b,e)=1 and thus b≤f(e). Consequently, e∈φ(b). So φ(a)⊆φ(b), similarly we can show that φ(b)⊆φ(a).
(⇐) Let a,b∈A(B) s.t φ(a)=φ(b). Let e∈A, if g_{s}(a,e)=1, then a≤f(e) and hence e∈φ(a)=φ(b). Also, b≤f(e) and so g_{s}(b,e)=1. If g_{s}(a,e)=0, then a≦̸f(e) and hence e∉φ(a)=φ(b). Therefore b≦̸f(e) and thus g_{s}(b,e)=0. So g_{s}(a,e)=g_{s}(b,e)∀e∈A and consequently, a≤ψ_{s}(b).
ii) Let x∈B and b∈A(B) s.t \(b\leq {{x^{\vee }_{\varphi }}}\). We show that ψ_{s}(b)≤x. Let a∈A(B) s.t a≤ψ_{s}(b), then φ(a)=φ(b) by i). But \(b\leq {{x^{\vee }_{\varphi }}}\) implies that φ(b)≠φ(c)∀c∈A(B) s.t c≦̸x. Since φ(a)=φ(b), then a≤x and therefore ψ_{s}(b)≤x. This implies that \({x^{\vee }_{\varphi }}\leq {x^{\nabla }_{s}}\).
Conversely, Let b∈A(B) s.t \(b\leq {x^{\nabla }_{s}}\). So ψ_{s}(b)≤x. Let a∈A(B) s.t a≦̸x, thus a≦̸ψ_{s}(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 x∈B and b∈A(B) s.t \(b\leq {{x^{\wedge }_{\varphi }}}\), then ∃a∈A(B) s.t a≤x and φ(a)=φ(b). So a≤ψ_{s}(b) by i) and therefore ψ_{s}(b)∧x≠0. Consequently, \(b\leq {x^{\triangle }_{s}}\).
Conversely, let b∈A(B) s.t \(b\leq {x^{\triangle }_{s}}\), then ψ_{s}(b)∧x≠0. Thus ∃a∈A(B) s.t a≤x and a≤ψ_{s}(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 MSRapproximations 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 MSRapproximation space on B where φ:A(B)→P(A_{1}) is defined as φ(b)={a∈A:b≤f(a)}. Let \(g_{A_{2}}\) be another soft set over B. For any e∈A_{2}, 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

\(g(e)^{\vee }_{\varphi }=\bigvee {\{a\in {A(B)}: a\leq {g(e)} \hspace {0.1cm},\varphi (a)\neq \varphi (b) \hspace {0.1cm}\forall \hspace {0.1cm}b\in A(B)\hspace {0.1cm}s.t\hspace {0.1cm}b\not \leq g(e)\}}\) ∀e∈A_{2},

\(g(e)^{\wedge }_{\varphi }=\bigvee {\{a\in {A(B)}:\varphi (a)= \varphi (b) \hspace {0.1cm} for \hspace {0.1cm}some \hspace {0.1cm} b\in A(b)\hspace {0.1cm}s.t\hspace {0.1cm} b\leq g(e)\}} \forall e\in A_{2}\).
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 A_{1}={e_{1},e_{2},e_{3},}, where e_{1} denotes temperature, e_{2} denotes headache and e_{3} denotes stomach problem and let \(f_{A_{1}}\) be a soft set over B representing the diagnosis of doctor M, defined as follows:
f(e_{1})=b, f(e_{2})=e and f(e_{3})=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.
Then the map φ of MSRapproximation space (A(B),φ) will be φ(a)={e_{2}}, φ(b)={e_{1},e_{3}}, and φ(c)={e_{2}}.
Let A_{2}={e_{1},e_{2},e_{3},e_{4}} and \(g_{A_{2}}\) be another soft set over B representing the diagnosis of doctor N, where A_{2}={e_{1},e_{2},e_{3},e_{4}} and e_{4} represents cough, defined as follows:
g(e_{1})=d, g(e_{2})=b, g(e_{3})=c and g(e_{4})=e. So lower MSRapproximations 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 MSRapproximations 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, X⊆UB=℘(U) and F_{A} 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 longterm 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={p_{1},p_{2},p_{3},p_{4},p_{5},p_{6}} and A={e_{1}(Frequent Urination),e_{2}(Increased Hunger),e_{3}(Increased Thirst)}. Let F_{A} be a soft over U given by Table 2 and (U,φ) be a MSRapproximation space.
Let X={p_{1},p_{4},p_{5}} 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 Bnd_{φ}(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 Bnd_{φ}(X)={p_{1},p_{3}}. 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 Bnd_{φ}(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 Bnd_{φ}(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(SR^{φ})(X)={e_{1}}, 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 MSRboundary region of X⊆U.
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 MSRboundary 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(SR^{φ})(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/s4278701900198
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DOI: https://doi.org/10.1186/s4278701900198
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