# Soft Rough Approximation Operators on a Complete Atomic Boolean Lattice

Authors: Heba I. Mustafa
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The concept of soft sets based on complete atomic Boolean lattice, which can be seen as a generalization of soft sets, is introduced. Some operations on these soft sets are discussed, and new types of soft sets such as full, keeping infimum, and keeping supremum are defined and supported by some illustrative examples. Two pairs of new soft rough approximation operators are proposed and the relationship among soft set is investigated, and their related properties are given. We show that Järvinen's approximations can be viewed as a special case of our approximation. IfB=℘(U), then our soft approximations coincide with crisp soft rough approximations (Feng et al. 2011).


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## Body

## 1. Introduction

Most of traditional methods for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. However, many practical problems within fields such as economics, engineering, environmental science, medical science, and social sciences involve data that contain uncertainties. We cannot use traditional methods because of various types of uncertainties present in these problems.There are several theories probability theory, fuzzy set theory, theory of interval mathematics, and rough set theory [1], which we can be considered as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties (see [2]). For example, theory of probabilities can deal only with stochastically stable phenomena. To overcome these kinds of difficulties, Molodtsov [2] proposed a completely new approach, which is called soft set theory, for modelling uncertainty.Presently, works on soft set theory are progressing rapidly. Maji et al. [3–5] further studied soft set theory, used this theory to solve some decision making problems, and devoted fuzzy soft sets combining soft sets with fuzzy sets. Roy and Maji [6] presented a fuzzy soft set theoretic approach towards decision making problems. Jiang et al. [7] extended soft sets with description logics. Aktas and Cagman [8] defined soft groups. Shabir and Naz [9] investigated soft topological spaces. Ge et al. [10] discussed relationships between soft sets and topological spaces.Rough set theory was initiated by Pawlak [1] for dealing with vagueness and granularity in information systems. This theory handles the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. It has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems, and many other fields (see [1, 11]). Since many classes of information granules are lattice ordered [12, 13], lattice theory [14–16] has found renewed interest and applications in diverse areas such as mathematical morphology [17], fuzzy set theory [18, 19], computational intelligence [20], automated decision making [21], and formal concept analysis [22]. In [23, 24] Järvinen studied properties of approximations in a more general setting of complete atomic Boolean lattices. He defined in a lattice theoretical setting two maps which mimic the rough approximation operators and noted that this setting is suitable also for other operators based on binary relations.It has been found that soft set and rough set are closely related concepts. Based on the equivalence relation on the universe of discourse, Feng et al. [25, 26] investigated the relationships among soft sets, rough sets, and fuzzy sets, obtaining three types of hybrid models: rough soft sets, soft rough sets, and soft rough fuzzy sets. They show that Pawlak’s rough set can be viewed as a special case of soft rough sets. Soft rough sets, which could provide a better approximation than rough sets do, can be seen as a generalized rough set model, and defining soft rough sets and some related concepts needs using soft rough approximation operators based on soft sets. Thus, soft rough approximation operators deserve further research.This paper is arranged as follows. In Section2 we recall and develop some notions and notations concerning lattice, ordered set, and properties of maps. Also we discuss the generalization of rough sets in a more general setting of complete atomic Boolean lattices which was studied by Järvinen [23, 24]. The purpose of Section 3 is to introduce the new concept of soft sets on a complete atomic Boolean lattice as a generalization of soft sets, discuss some operations and define new types of theses soft sets. At the end of this section, we obtain the algebraic structure (i.e., the lattice structure) of our new soft sets. In Section 4, we consider two pairs of soft rough approximations based on a complete atomic Boolean lattice as a generalization of soft rough approximations and give their properties. In Section 5 another pair of soft rough approximations is investigated, and the fact that Järvinen’s approximations can be viewed as a special case of our soft approximations is proved. The conclusion is in Section 6.


## 2. Preliminaries

We assume that the reader is familiar with the usual lattice-theoretical notation and conventions, which can be found in [27, 28].First we recall some definitions and properties of maps. LetB=(B,≤) be an ordered set. A mapping f:B→B is said to be extensive, if x≤f(x) for all x∈B. The map f is order preserving if x≤y implies f(x)≤f(y). Moreover, f is idempotent if f(f(x))=f(x) for all x∈B. A map c:B→B is said to be a closure operator on B, if c is extensive, order preserving, and idempotent. An element x∈B is c closed if c(x)=x. Furthermore, if i:B→B is a closure operator on Bϑ=(B,≥) then i is an interior operator on B. Let B=(B,≤) and Q=(Q,≤) be ordered sets. f:B→Q is an order embedding, if for any a,b∈B,  a≤b in B if and only if f(a)≤f(b) in Q; note that an order embedding is always an injection. An order-embedding f onto Q is called an order-isomorphism between B and Q; we say that B and Q are order-isomorphic and write B≅Q. If B=(B,≤) and Q=(Q,≤) are order-isomorphic, then B and Q are said to be dually order-isomorphic. A pair (∇,Δ) of maps ∇:B→B and Δ:B→B is called a dual Galois connection on B if ∇ and Δ are order preserving and x∇Δ≤x≤xΔ∇ for all x∈B.Before we consider the Boolean lattices, we present the following lemma, where℘(B) denotes the power set of B, that is, the set of all subsets of B.Lemma 1 (see [23]).
LetB=(B,≤) be a complete lattice, S,T⊆B, and {Xi:i∈I}⊆℘(B).(i)
IfS⊆T, then ⋁S⊆⋁T.(ii)
⋁(S∪T)=(⋁S)⋁(⋁T).(iii)
⋁(⋃{Xi:i∈I})=⋁{⋁Xi∈I}.Next we recall the concept of Boolean lattices. They are bounded distributive lattices with a complementation operation.Definition 2 (see [27]).
A latticeB=(B,≤) is called a Boolean lattice, if(i)
B is distributive;(ii)
B has a least element 0 and a greatest element 1, and;(iii)
eachx∈B has a complement x′∈B such that x∨x′=1 and x∧x′=0.Lemma 3 (see [27]).
LetB=(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.Let us recall some definitions and results that are useful in our consideration given in [23].Lemma 4 (see [23]).
LetB=(B,≤) be a complete Boolean lattice. Then for all {xi:i∈I}⊆B and y∈B(1)y∧(⋁i∈Ixi)=⋁i∈I(y∧xi),y∨(⋀i∈Ixi)=⋀i∈I(y∨xi).Definition 5 (see [23]).
LetB=(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 6 (see [23]).
LetB=(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=⋁{a∈A(B):a≤x}.It is obvious that in a latticeB=(B,≤) with a least element 0,
(2)a∧x≠0⟺a≤x
for all a∈A(B) and x∈B. This implies that a∧b=0 for all a,b∈A(B) s.t a≠b. Furthermore, if B is atomic, then for all x≠0 there exists an atom a∈A(B) s.t a≤x. Namely, if {a∈A(B):a≤x}=ϕ, then x=⋁{a∈A(B):a≤x}=⋁ϕ=0.Definition 7 (see [23]).
LetB=(B,≤) be a complete atomic Boolean lattice. We say that φ:A(B)→B is(i)
extensive, ifa≤φ(a) for all a∈A(B),(ii)
symmetric, ifa≤φ(b) implies b≤φ(a) for all a,b∈A(B),(iii)
closed, ifb≤φ(a) implies φ(b)≤φ(a) for all a,b∈A(B).Definition 8 (see [23]).
LetB=(B,≤) be a complete atomic Boolean lattice and let φ:A(B)→B be any mapping. For any element x∈B, let
(3)x∇=⋁{a∈A(B):φ(a)≤x},xΔ=⋁{a∈A(B):φ(a)∧x≠0}.
The elementsx∇ and xΔ are called the lower and the upper approximations of x with respect to φ, respectively. Two elements x and y are called equivalent if they have the same upper and lower approximations. The resulting equivalence classes are called rough sets.The following results are shown in [23, 24]. The ordered sets (BΔ,≤) and (BΔ,≤) are always complete lattices. They are distributive sublattices of (B,≤) if φ is extensive and closed. If the map φ is extensive, symmetric, and closed, then the ordered sets (BΔ,≤) and (BΔ,≤) are mutually equal complete atomic Boolean lattices.Proposition 9 (see [23]).
LetB=(B,≤) be a complete atomic Boolean lattice and let φ:A(B)→B be any mapping. Then for all a∈A(B) and x∈B,(i)
a≤x∇⇔φ(a)≤x;(ii)
a≤xΔ⇔φ(a)∧x≠0.Proposition 10 (see [23]).
LetB=(B,≤) be a complete atomic Boolean lattice and let φ:A(B)→B be an extensive mapping. Then for all x∈B,(i)
x∇≤x;(ii)
x≤xΔ.Proposition 11 (see [23]).
LetB=(B,≤) be a complete atomic Boolean lattice and let φ:A(B)→B be extensive and closed mapping. Then for all x∈B,(i)
x∇=x∇∇;(ii)
xΔΔ=xΔ.Proposition 12 (see [23]).
LetB=(B,≤) be a complete atomic Boolean lattice and let φ:A(B)→B be an extensive, symmetric and closed mapping. Then for all x∈B,(i)
x∇Δ=x∇;(ii)
xΔ∇=xΔ.Next, we recall the definitions of Pawlak rough sets, soft sets, and soft rough approximation operators.Definition 13 (see [29]).
An information system (or a knowledge representation system) is a pairγ=(U,A) of nonempty finite sets U and A, where U is a set of objects and A is a set of attributes; each attribute a∈A is a function a:U→Va, where Va is the set of values (called domain) of attribute a.LetU be a non-empty finite universe and let R be an equivalence relation on U. The pair (U, R) is called a Pawlak approximation space. The equivalence relation R is often called an indiscernibility relation and related to an information system. Specifically, if γ=(U,A) is an information system and B⊆A, then an indiscernibility relation R=I(B) can be defined by
(4)(x,y)∈I(B)⟺a(x)=a(y)∀a∈B,
where x,y∈U and a(x) denotes the value of attribute a for object x.Using the indiscernibility relationR, one can define the following two operations:
(5)R*X={x∈U:[x]R⊆X},R*X={x∈U:[x]R∩X≠∅}
assigning to every subset X⊆U two sets R*X and R*X called the R-lower and the R-upper approximation of X, respectively.IfR*X=R*X, then X is said to be R-definable; otherwise, X is said to be R-rough.Let us recall now the soft set notion, which is a newly emerging mathematical approach to vagueness.Definition 14 (see [2]).
LetU be a universal set and let E be a set of parameters. Let A be a nonempty subset of E. A soft set over A, with support A, denoted by fA on U is defined by the set of ordered pairs
(6)fA={(e,fA(e)):e∈E,fA(e)∈℘(U)}
or is a function fA:E→℘(U) s.t
(7)fA(e)≠ϕ,∀e∈A⊆E,fA(e)=ϕife∉A.Example 15.
Suppose thatU is the set of houses under consideration and A and B are both parameter sets. Let there be four houses in the universe U given by U={h1,h2,h3,h4}. And A={expensive,modern} and B={modern}. The soft sets fA and gB describe the ‘‘attractiveness of the houses.’’ For the sake of ease of designation, we use e, instead of expensive and m instead of modern. The soft set fA is defined as follows f(e) means expensive houses, and f(m) means modern houses. The soft set fA is the collection of approximations as below:
(8)fA={(e,{h1,h2}),(m,{h4})}.
The soft setgB is defined as g(m), which means the modern houses. The soft set gB is the collection of approximations as below:
(9)gB={(m,{h1,h4})}.Definition 16 (see [25, 26]).
LetU be a universal set and let fA be a soft set over U. Then the pair P=(U,fA) is called soft approximation space. We define a pair of operators apr_P,apr¯P:℘(U)→℘(U) as follows:
(10)apr_P(X)={u∈U:∃a∈A,s.tu∈f(a)⊆X},apr¯P(X)={u∈U:∃a∈A,s.tu∈f(a),f(a)∩X≠∅}.
The elementsapr_P(X) and apr¯P(X) are called the softP-lower and the softP-upperapproximations of X.
Ifapr_P(X)=apr¯P(X), X is said to be soft P-definable; otherwise X is called a soft P-rough set.Example 17.
Let us consider the following soft setS=fE which describes ‘‘life expectancy’’. Suppose that the universe U={u1,u2,u3,u4,u5,u6} consists of six persons and E={e1,e2,e3,e4} is a set of decision parameters. The ei (i=1,2,3,4) stands for “under stress,” “young,” “drug addict” and “healthy.” Set f(e1)={u5}, f(e2)={u1,u2}, f(e3)=∅; and f(e4)={u1,u2,u3,u6}. The soft set fE can be viewed as the following collection of approximations:
(11)fE={(understress,{u5});(young,{u1,u2});(drugaddict,∅);(healthy;{u1,u2,u3,u6})}.
On the other hand, “life expectancy” topic can also be described using rough sets as follows: the evaluation will be done in terms of attributes: “sex”, “age category”, “living area”, and “habits”, characterized by the value sets “{man, woman}”, “{baby, young, mature age, old}”, “{village, city}”, and “{smoke, drinking, smoke and drinking, no smoke and no drinking}”. We denote “smoke and drinking” by SD and “no smoke and no drinking” by NSND. The information will be given by Table1, where the rows are labeled by attributes and the table entries are the attribute values for each person. From here we obtain the following equivalence classes, induced by the above mentioned attributes:
(12)[u1]R=[u2]R={u1,u2},[u3]R=[u5]R={u3,u5},[u4]R={u4},[u6]R={u6}.
LetX be a target subset of U, that we wish to represent using the above equivalence classes. Hence we analyze the upper and lower approximations of X, in some particular cases.(1)
SetX={u1,u2,u3,u6}. It follows that
(13)R*X={u1,u2,u6},R*X={u1,u2,u3,u5,u6}.
Let us calculate now the soft P-lower and P-upper approximations of X, where P=(U,S). We obtain
(14)apr_P(X)=&{u1,u2,u3,u6}=X,apr¯P(X)={u1,u2,u3,u6}=X;
hence X is soft P-definable.(2)
SetX={u5}. It follows that R*X={u3,u5}. On the other hand, apr_P(X)=apr¯P(X)=X, hence X; is soft P-definable.Table 1
An information table.
u
1
u
2
u
3
u
4
u
5
u
6
Sex
Woman
Woman
Man
Man
Man
Man
Age category
Young
Young
Mature age
Old
Mature age
Baby
Living area
City
City
City
Village
City
Village
Habits
NSND
NSND
Smoke
SD
Smoke
NSNDThe above results show that soft rough set approximation is a worth considering alternative to the rough set approximation. Soft rough sets could provide a better approximation than rough sets do, depending on the structure of the equivalence classes and of the subsetsF(e), where e∈E.


## 3. Soft Sets on a Complete Atomic Boolean Lattice

Definition 18.
LetB=(B,≤) be a complete atomic Boolean lattice and let E be a set of parameters. Let A be a non empty subset of E. A soft set over A, with support A, denoted by fA on B is defined by the set of ordered pairs
(15)fA={(e,fA(e)):e∈E,fA(e)∈B},
or is a function FA:E→B s.t
(16)fA(e)≠0,∀e∈A⊆E,fA(e)=0ife∉A.
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 e-approximate element of fA.Definition 19.
LetB=(B,≤) be a complete atomic Boolean lattice. Let A1,A2⊆E and let fA1 and gA2 be two soft sets over B.(i)
fA1 is a soft subset of gA2, denoted by fA1⊑gA2 if A1⊆A2 and f(e)≤g(e) for every e∈A1.(ii)
fA1 and gA2 are called soft equal, denoted by fA1=gA2 if fA1⊑gA2 and gA2⊑fA1.Definition 20.
LetB=(B,≤) be a complete atomic Boolean lattice. Let A⊆E and let fA be a soft set over B.(i)
fA is called null, denoted by 0A if f(e)=0 for every e∈A.(ii)
fA is called absolute, denoted by 1A if f(e)=1 for every e∈A.
We stipulate that0ϕ is also a soft set over B with 0:ϕ→B.LetA⊆E and let fA be a soft set over B. Obviously,
(17)0A⊑fA⊑1A.Below, we introduce some operations on soft sets onB and investigate their properties.Definition 21.
LetB=(B,≤) be a complete atomic Boolean lattice. Let A1,A2⊆E and let fA1 and gA2 be two soft sets over B.(i)
hA3 is called the intersection of fA1 and gA2, denoted by fA1⊓gA2=hA3 if A3=A1∩A2 and h(e)=f(e)∧g(e) for every e∈A3.(ii)
hA3 is called the union of fA1 and gA2, denoted by fA1⊔gA2=hA3 if A3=A1∪A2 and h(e)=f(e) if e∈A-B, h(e)=g(e) if e∈B-A and h(e)=f(e)∨g(e) if e∈A∩B.Definition 22.
LetB=(B,≤) be a complete atomic Boolean lattice. Let A⊆E 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:A→B is a mapping given by fc(e)=f(e)′ for every e∈A.Proposition 23.
LetB=(B,≤) be a complete atomic Boolean lattice. Let A1,A2,A3⊆E and let fA1, gA2, and hA3 be three soft sets over B. Then(i)
fA1⊔fA1=fA1,(ii)
fA1⊔gA2=gA2⊔fA1,(iii)
(fA1⊔gA2)⊔hA3=fA1⊔(gA2⊔hA3).Proof.
(i) and (ii) are obvious. We only prove (iii). Put(18)(fA1⊔gA2)⊔hA3=kA1∪A2∪A3,fA1⊔(gA2⊔hA3)=lA1∪A2∪A3,fA1⊔gA2=sA1∪A2,gA2⊔hA3=tA2∪A3.
For anye∈A1∪A2∪A3 it follows that e∈A1, or e∈A2, or e∈A3.
Case  1  (e∈A3).(a)
Ife∉A1 and e∉A2, then k(e)=h(e)=t(e)=l(e).(b)
Ife∉A1 and e∈A2, then k(e)=s(e)∨h(e)=g(e)∨h(e)=t(e)=l(e).(c)
Ife∈A1 and e∉A2, then k(e)=s(e)∨h(e)=f(e)∨h(e)=f(e)∨t(e)=l(e).(d)
Ife∈A1 and e∈A2, then k(e)=s(e)∨h(e)=f(e)∨g(e)∨h(e)=f(e)∨t(e)=l(e).
Case  2  (e∉A3).(a)
Ife∉A1 and e∈A2, then k(e)=s(e)=g(e)=t(e)=l(e).(b)
Ife∈A1 and e∉A2, then k(e)=s(e)=f(e)=l(e).(c)
Ife∈A1 and e∈A2, then k(e)=s(e)=f(e)∨g(e)=f(e)∨t(e)=l(e).
Thus(fA1⊔gA2)⊔hA3=fA1⊔(gA2⊔hA3).Proposition 24.
LetB=(B,≤) be a complete atomic Boolean lattice. Let A1,A2,A3⊆E and let fA1, gA2; and hA3 be three soft sets over B. Then(i)
fA1⊓fA1=fA1,(ii)
fA1⊓gA2=gA2⊓fA1,(iii)
(fA1⊓gA2)⊓hA3=fA1⊓(gA2⊓hA3).Proof.
(i) and (ii) are obvious. We only prove (iii). Put(19)(fA1⊓gA2)⊓hA3=kA1∩A2∩A3,fA1⊓(gA2⊔hA3)=lA1∩A2∩A3.
For anye∈A1∩A2∩A3, it follows that e∈A1,  e∈A2, and e∈A3. Since k(e)=(f(e)∧g(e))∧h(e)=f(e)∧(g(e)∧h(e))=l(e), then (fA1⊓gA2)⊓hA3=fA1⊓(gA2⊓hA3).Proposition 25.
LetB=(B,≤) be a complete atomic Boolean lattice. Let A1,A2,A3⊆E and let fA1, gA2, and hA3 be three soft sets over B. Then(i)
(fA1⊔gA2)⊓hA3=(fA1⊓hA3)⊔(gA2⊓hA3),(ii)
(fA1⊓gA2)⊔hA3=(fA1⊔hA3)⊓(gA2⊔hA3).Proof.
(i) Put(fA1⊔gA2)⊓hA3=k(A1∪A2)∩A3, (fA1⊔hA3)⊓(gA2⊔hA3)=l(A1∩A3)∪(A2∩A3). Obviously, (A1∪A2)∩A3=(A1∩A3)∪(A2∩A3). For any e∈(A1∪A2)∩A3, it follows that e∈A1∩A3 or e∈A2∩A3.(a)
Ife∉A1∩A3 and e∈A2∩A3, then e∉A1, e∈A2, and e∈A3. So k(e)=g(e)∧h(e)=l(e).(b)
Ife∈A1∩A3 and e∉A2∩A3, then e∈A1, e∉A2, and e∈A3. So k(e)=f(e)∧h(e)=l(e).(c)
Ife∈A1∩A3 and e∈A2∩A3, then e∈A1, e∈A2, and e∈A3. So k(e)=(f(e)∨g(e))∧h(e)=(f(e)∧h(e))∨(g(e)∧h(e))=l(e).
Thus(fA1⊔gA2)⊓hA3=(fA1⊓hA3)⊔(gA2⊓hA3).
(ii) This is similar to the proof of (i).Proposition 26.
LetB=(B,≤) be a complete atomic Boolean lattice. Let A1,A2⊆E and let fA and gA be two soft sets over B.(i)
((fA)c)c=fA.(ii)
fA⊔(fA)c=1A.(iii)
fA⊓(fA)c=0A.(iv)
(fA⊔gA)c=(fA)c⊓(gA)c.(v)
(fA⊓gA)c=(fA)c⊔(gA)c.Proof.
(i) Put(fA)c=gA, (gA)c=hA.
For anye∈A, h(e)=gc(e)=g(e)′, g(e)=fc(e)=f(e)′. So, h(e)=g(e)′=(f(e)′)′=f(e) (by Lemma 3). This Shows that hA=fA; that is ((fA)c)c=fA.
(ii) PutfA⊔(fA)c=hA.
For anye∈A, h(e)=f(e)∨fc(e)=f(e)∨f(e)′=1. Hence fA⊔(fA)c=1A.
(iii) This is similar to the proof of (ii).
(iv) Put(fA⊔gA)c=hA, (fA)c⊓(gA)c=lA.
For anye∈A, h(e)=(f(e)∨g(e))′, l(e)=f(e)′∧g(e)′. Hence h(e)=l(e) by Lemma 3.
(v) This is similar to the proof of (iv).Definition 27.
LetB=(B,≤) be a complete atomic Boolean lattice and Let fE be a soft set over B.(i)
fE is called full, if ⋁e∈Ef(e)=1;(ii)
fE is keeping infimum, if for any e1,e2∈E, there exists e3∈E such that f(e1)∧f(e2)=f(e3);(iii)
fE is keeping supremum, if for any e1,e2∈E, there exists e3∈E such that f(e1)∨f(e2)=f(e3);(iv)
fE is called partition of B if
(1)
⋁e∈Ef(e)=1,(2)
for everye∈E, f(e)≠0,(3)
for everye1,e2∈E either f(e1)=f(e2) or f(e1)∧f(e2)=0.
Obviously, every partition soft set is full andfE is keeping infimum (resp., keeping supremum) if and only if for every E*⊆E, there exists e*∈E such that ⋀e∈E*f(e)=f(e*) (resp., ⋁e∈E*f(e)=f(e*)).Example 28.
LetB={0,a,b,c,d,e,f,1} and let the order ≤ be defined as in Figure 1.
The set of atoms of a complete atomic Boolean latticeB=(B,≤) is {a,b,c}. Let A={e1,e2,e3,e4} and let fA be a soft set over B defined as follows:
(20)f(e1)=e,f(e2)=b,f(e3)=c,f(e4)=0.
Obviously,fA is not a partition since f(e4)=0. Also, fA is full since ⋁e∈Af(e)=e∨b∨c=1. Also, fA is keeping infimum. In fact f(e1)∧f(e2)=f(e1)∧f(e4)=f(e3)∧f(e4)=f(e2)∧f(e4)=f(e4)=0.
f
(
e
1
)
∧
f
(
e
3
)
=
e
∧
c
=
c
=
f
(
e
3
) and f(e2)∧f(e3)=b∧c=0=f(e4). Consequently, fA is keeping infimum. On the other hand, fA is not keeping supremum since f(e1)∨f(e2)=e∨b=1≠f(e) for every e∈A.Figure 1LetgA be a soft set over B defined as follows:g
(
e
1
)
=
d,  g(e2)=a,  g(e3)=e, and g(e4)=1; then gA is a partition, keeping infimum, and keeping supremum.Next, we investigate the lattice structure of soft sets on a complete atomic Boolean LatticeB. We denote
(21)S(B,E)={fE:fEissoftsetoverB},S1(B,E)={fA:A⊆EandfAissoftsetoverB}.Obviously,(22)S1(B,E)⊆S(B,E).Theorem 29.
For anyfA,gB∈S(B,E), define
(23)fA≤gB⟺fA⊑gB,fA∨gB=fA⊔gB,fA∧gB=fA⊓gB.
ThenS(B,E) is a distributive lattice with smallest element 0Σ=0ϕ and greatest element 1Σ=1E.Proof.
DenoteΣ=S(B,E). It is easily proved that
(24)0Σ=0ϕ,1Σ=1E.
By Proposition25  S(X,E) is a distributive lattice with 1Σ and 0Σ.Theorem 30.
For anyfA,gB∈S1(B,E), define
(25)fA≤gB⟺fA⊑gB,fA∨gB=fA⊔gB,fA∧gB=fA⊓gB.
ThenS1(B,E) is a Boolean lattice.Proof.
DenoteΣ1=S1(B,E). It is easily proved that S1(B,E) is a distributive lattice with 0Σ1=0E and 1Σ1=1E.
LetfE∈Σ1. Put hE=fE∨fEc. Since hE=fE⊔fEc, then for any e∈E,
(26)h(e)=f(e)∨fc(e)=f(e)∨f(e)′=1.
So,hE=1E=1Σ1. This shows that fE∨fE′=1Σ1. Similarly, we can prove that fE∧fE′=0Σ1. Hence (fE)′=fEc and therefore S1(B,E) is a Boolean lattice.


## 4. Soft Rough Approximation Operators on a Complete Atomic Boolean Lattice

Definition 31.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. For any element x∈B, we define a pair of operators x∨,x∧:B→B as follows:
(27)x∨=⋁{b∈A(B):∃e∈As.tb≤f(e),f(e)≤x},x∧=⋁{b∈A(B):∃e∈As.tb≤f(e),f(e)∧x≠0}.
The elementsx∨ 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.Lemma 32.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Then for all c∈A(B) and x∈B(i)
c≤x∨⇔∃e∈As.tc≤f(e)andf(e)≤x;(ii)
c≤x∧⇔∃e∈As.tc≤f(e)andf(e)∧x≠0.Proof.
(i)(⇒) Suppose that c≤x∨=⋁{b∈A(B):∃e∈As.tb≤f(e)andf(e)≤x}. Assume that for all e∈A either c≰f(e) or f(e)≰x. If ∀e∈A, f(e)≰x, then c≰x∨, a contradiction. If ∀e∈A, c≰f(e), then c∧x∨=c∧⋁{b∈A(B):∃e∈As.tb≤f(e)andf(e)≤x} =  ⋁{c∧b:b∈A(B),∃e∈As.tb≤f(e)andf(e)≤x}. Since c≰f(e), then, c≠b. So c∧b=0 because c,b∈A(B). Hence c∧x∨=0. This implies that c≤(x∨)′, which is a contradiction.
(
⇐
) Suppose that ∃e∈As.tc≤f(e)andf(e)≤x; then c≤⋁{b∈A(B):∃e∈As.tb≤f(e)andf(e)≤x}=x∨.
Condition (ii) can be proved similarly.Proposition 33.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Then for all x∈B(i)
x∨=⋁{f(e):e∈Aandf(e)≤x}≤x;(ii)
x∧=⋁{f(e):e∈Aandf(e)∧x≠0}.Proof.
(i) Letc∈A(B), s.t c≤x∨; then ∃e∈As.tc≤f(e)andf(e)≤x. So, c≤⋁{f(e):e∈Aandf(e)≤x}≤x. On the other hand, let c∈A(B), s.t c≤⋁{f(e):e∈Aandf(e)≤x}. Hence, ∃e∈As.tc≤f(e)andf(e)≤x. In fact, if e∈A and f(e)≤x implies c≰f(e), then c∧f(e)′≠0. Therefore c≤f(e)′ because c∈A(B). Thus c≤⋁{f(e)′:e∈Aandf(e)≤x}. So, c≤⋁{f(e)∧f(e)′:e∈Aandf(e)≤x}=0, a contradiction. So, ∃e∈As.tc≤f(e)andf(e)≤x and consequently, c≤x∨.
Condition (ii) can be proved similarly.Proposition 34.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B.(i)
0∨=0∧=0 and 1∨=1∧=⋁e∈Af(e);(ii)
x≤y implies x∨≤y∨ and x∧≤y∧.Proof.
Obvious.
For allS⊆B, we denote S∨={x∨:x∈S} and S∧={x∧:x∈S}.Proposition 35.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B; then(i)
for allS⊆B, ∨S∧=(∨S)∧;(ii)
iffA is keeping infimum, then for all S⊆B, ∧S∨=(∧S)∨;(iii)
(B∧,≤) is a complete lattice; 0 is the least element and 1∧ is the greatest element of  (B∧,≤);(iv)
iffA is keeping infimum, then (B∨,≤) is a complete lattice; 0 is the least element and 1∨ is the greatest element of (B∨,≤);(v)
iffA is keeping infimum, the kernal Θ∨={(x,y):x∨=y∨} of the map :∨B→B is a congruence on the semi lattice (B,∧) such that the Θ∨-class of any x has a least element;(vi)
the kernalΘ∧={(x,y):x∧=y∧} of the map :∧B→B is a congruence on the semilattice (B,∨) such that the Θ∧-class of any x has a least element.Proof.
(i) LetS⊆B. The map ∧:B→B is order preserving, which implies that ∨S∧≤(∨S)∧. Let b∈A(B) and assume that b≤(∨S)∧. So, ∃e∈As.tb≤f(e)andf(e)∧∨S≠0. Then 0≠f(e)∧⋁S=⋁{f(e)∧x:x∈S}, which implies that f(e)∧x≠0 for some x∈S. Thus {b∈A(B):∃e∈As.tb≤f(e)andf(e)∧⋁S≠0}⊆∪x∈S{b∈A(B):∃e∈As.tb≤f(e)andf(e)∧x≠0}. Then
(28)(∨S)∧=⋁{b∈A(B):∃e∈As.tb≤f(e),f(e)∧⋁S≠0}≤⋁(∪x∈S{b∈A(B):∃e∈As.tb≤f(e),f(e)∧x≠0})=⋁x∈S(⋁{b∈A(B):∃e∈Es.tb≤f(e),f(e)∧x≠0})(byLemma1)=⋁{x∧:x∈S}=⋁S∧.
(ii) Let S⊆B. The map :∨B→B is order preserving, which implies that (∧S)∨≤∧S∨. Let b∈A(B) s.t b≤∧S∨=∧{x∨:x∈S}. So, ∃e∈As.tb≤f(e)andf(e)≤x for every x∈S. Hence ⋀{f(e):b≤f(e)andf(e)≤x}≤x for every x∈S. This implies that ⋀e∈A{f(e):b≤f(e)andf(e)≤x}≤∧{x:x∈S}=∧S. Since fA is keeping infimum, then ∧e∈Af(e)=f(e1) for e1∈A. So we show that ∃e1∈As.tb≤f(e1)andf(e1)≤∧S. Therefore b≤(∧S)∨. Consequently, ∧S∨≤(∧S)∨. Assertions (iii), and (iv) follow easily from (i), (ii) and Proposition 23(i). The proof of (v) and (vi) follows by (i) and (ii).In the following example, we show that in general(B∧,≤) and (B∨,≥) are not dually order-isomorphic.Example 36.
LetB={0,a,b,c,d,e,f,1} and let the order ≤ be defined as in Figure 1.
LetA={e1,e2,e3,e4} and let fA be a soft set over B defined as follows:
(29)f(e1)=e,f(e2)=b,f(e3)=c,f(e4)=0.
ThenfA is not a partition since f(e4)=0. Let x=c and y=d; then x∧=a∨c=e and y∧=b∨a∨c=1. Therefore x∧≤y∧. On the other hand y′∨=c∨=c≰x′∨=d∨=b.Next, we show that(B∧,≤) and (B∨,≥) are dually order-isomorphic if fA is a partition.Proposition 37.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. If fA is a partition, then (B∨,≥)≅(B∧,≤).Proof.
We show thatx∧→(x′)∨ is the required dual order isomorphism. It is obvious that x∧→(x′)∨ is onto (B∨,≥). We show that x∧→(x′)∨ is order embedding. Suppose that x∧≤y∧. Then for all b∈A(B), b≤x∧ implies b≤y∧. So, b∈A(B) such that ∃e1∈A,b≤f(e1) and f(e1)∧x≠0, implies ∃e2∈A, s.t b≤f(e2) and f(e2)∧y≠0. Since fA is a partition and b≤f(e1)∧f(e2), then f(e1)=f(e2). Hence if ∃e∈A,s.tb≤f(e) and f(e)∧x≠0, then f(e)∧y≠0. Suppose that (y′)∨≰(x′)∨. So there exists b∈A(B) such that b≤(y′)∨ and b≰(x′)∨. Since b≤(y′)∨, then ∃e∈A,s.tb≤f(e) and f(e)≤y′. Since b≤f(e) and b≰(x′)∨, then f(e)≰x′. Since f(e)≰x′ is equivalent to f(e)∧x≠0, then by hypothesis f(e)∧y≠0. But this means that f(e)≰y′, a contradiction. Hence (y′)∨≤(x′)∨. On the other hand, assume that (y′)∨≤(x′)∨. Since fA is a partition, then b∈A(B) s.t ∃e∈A,b≤f(e) and f(e)≤y′ implies f(e)≤x′. Suppose that x∧≰y∧. So there exists b∈A(B) such that b≤x∧ and b≰y∧. So ∃e∈A,b≤f(e),f(e)∧x≠0, and f(e)∧y=0. But this implies that f(e)≤y′. Since x∧≰y∧, then f(e)≤x′. This is equivalent to f(e)∧x=0, a contradiction.Next we study the properties of soft approximations more closely in cases when the soft setfA is full, keeping union, keeping intersection, and partition.Proposition 38.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Then the following properties hold.(i)
IffAis full, then
(a)
x∨≤x≤x∧;(b)
1∨=1∧=1.(ii)
IffAis keeping supremum, then
(a)
for allx∈B, ∃e∈A,s.tx∨=f(e);(b)
for allx∈B, ∃e∈A,s.tx∧=f(e).(iii)
IffAis full and keeping supremum, then
x∧=1
for everyx∈Bandx≠0.Proof.
(i)(a) By Proposition41  x∨≤x. Suppose that b∈A(B) and b≤x. Since fA is full, then ∃e∈A,s.tb≤f(e). So, b≤f(e)∧x and therefore f(e)∧x≠0. Consequently, b≤x∧. (b) Obvious.
(ii) It follows by Proposition33.
(iii) Letx∈B, then in general x≤1. Since fA is full and keeping supremum, then ∃e*∈A,s.t⋁e∈Af(e)=f(e*)=1. So, b≤f(e*) and f(e*)∧x≠0. Consequently, b≤x∧ and so x∧=1.Proposition 39.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. If fA is a partition, then,(i)
x∨∧=x∨;(ii)
x∧∧=x∧;(iii)
the map:∧B→Bis a closure operator.Proof.
(i) SincefA is full, then x∨≤x∨∧ by Proposition 38(1). Let b∈A(B) and b≤x∨∧; then ∃e1∈A,s.tb≤f(e1) and f(e1)∧x∨≠0. Hence, ∃c∈A(B),s.tc≤f(e1)andc≤x∨. But c≤x∨ implies ∃e2∈A,s.tc≤f(e2)andf(e2)≤x. Since fA is a partition and c≤f(e1)∧f(e2), then f(e1)=f(e2). So, ∃e1∈A,s.tb≤f(e1)andf(e1)≤x and therefore, b≤x∨. Consequently, x∨∧≤x∨. Claim (ii) can be proved similarly.
(iii) SincefA is full, then by Proposition 38 the map :∧B→B is extensive, and it is order preserving by Proposition 34. By (ii), x∧∧=x∧.


## 5. Another Soft Rough Approximation Operators on a Complete Atomic Boolean Lattice

Definition 40.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Define a mapping φf:A(B)→B by
(30)c≤φf(b)⟺∃e∈A,s.tc≤f(e),b≤f(e)
for every c,b∈A(B). Then φf is called the mapping induced by fA on B.Proposition 41.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let φf:A(B)→B be the mapping induced by fA on B. Then the following properties hold.(i)
φfis symmetric.(ii)
IffAis full, thenφfis extensive.(iii)
IffAis a partition, thenφfis extensive, symmetric, and closed.Proof.
(i) Obvious.
(ii) Letb∈A(B). Since fA is full, then ∃e∈A, s.t b≤f(e). Hence b≤φf(b).
(iii) IffA is a partition, then fA is full and hence φf is extensive. Since φf is symmetric, it remains to show that φf is closed. Let c,b∈A(B) s.t c≤φf(b). We show that φf(c)≤φf(b). Since c≤φf(b), then ∃e1∈A,s.tc≤f(e1)andb≤f(e1). Suppose that φf(c)≰φf(b). So, ∃d∈A(B), s.t d≤φf(c) and d≰φf(b). But d≰φf(b) implies that for every e∈A, either d≰f(e) or b≰f(e). Since d≤φf(c), then ∃e2∈A, s.t d≤f(e2) and c≤f(e2). Since fA is a partition and c≤f(e1)∧f(e2), then f(e1)=f(e2). Hence we show that ∃e1∈A, s.t d≤f(e1) and b≤f(e1), a contradiction. Consequently, φf(c)≤φf(b) and thus φf is closed.Proposition 42.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let φf:A(B)→B be the mapping induced by fA on B. Then the following properties hold.(i)
Ifb≤f(e)fore∈Aandb∈A(B), thenf(e)≤φf(b).(ii)
IffAis a partition andb≤f(e)fore∈Aandb∈A(B), thenf(e)=φf(b).(iii)
IffAis keeping supremum, then for allb∈A(B)∃e∈A,  s.tφf(b)=f(e).Proof.
(i) Letc∈A(B) s.t c≤f(e). Since b≤f(e), then c≤φf(b). Hence f(e)≤φf(b).
(ii) Suppose thatfA is a partition and assume that b≤f(e) for e∈A and b∈A(B). By (i) f(e)≤φf(b). On the other hand, let c∈A(B) s.t c≤φf(b). Then ∃e1∈A,s.tc≤f(e1) and b≤f(e1). So, b≤f(e)∧f(e1), and since fA is a partition, then f(e)=f(e2). Hence c≤f(e) and therefore φf(b)≤f(e). Consequently, φf(b)=f(e).
(iii) Suppose thatfA is keeping supremum and b∈A(B). Let c∈A(B) s.t c≤φf(b). Then ∃ec∈A, s.t c≤f(ec) and b≤f(ec). So f(ec)≤φf(b) by (i). Hence, φf(b)=⋁c∈A(B){f(ec):c≤φf(b)}. Since fA is keeping supremum, then ⋁c∈A(B){f(ec):c≤φf(b)}=f(e) for e∈A. Therefore φf(b)=f(e).Definition 43.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let φf:A(B)→B be the mapping induced by fA on B. We define a pair of soft approximation operators ∇f,Δf:B→B as follows:
(31)x∇f=⋁{b∈A(B):φf(b)≤x},xΔf=⋁{b∈A(B):φf(b)∧x≠0}.
The elementsx∇f and xΔf are called the soft lower and the soft upper approximations of x with respect to the mapping φf 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 φf induced by fA on B. The resulting equivalence classes are called soft rough sets with respect to the mapping φf induced by fA on B.Proposition 44.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let φf:A(B)→B be the mapping induced by fA.(i)
b≤x∇f⇔φf(b)≤x.(ii)
b≤xΔf⇔φf(b)∧x≠0.(iii)
IffAis full, thenx∇f≤x≤xΔf.(iv)
0Δf=0and1∇f=1. IffAis full, then0∇f=0Δf=0and1∇f=1Δf=1.(v)
x≤yimpliesx∇f≤y∇fandxΔf≤yΔf.(vi)
The mappings∇f:B→BandΔf:B→Bare mutually dual.(vii)
For allS⊆B,∨SΔf=(∨S)Δf.(viii)
For allS⊆B,∧S∇f=(∧S)∇f.(ix)
(BΔf,≤)is a complete lattice; 0 is the least element and  1Δfis the greatest element of(BΔf,≤).(x)
The pair(∇f,Δf)is a dual Galois connection onB.(xi)
(B∇f,≥)≅(BΔf,≤).Proof.
It follows by Propositions41, 9, 10, 11, and 12; see [23].In the following we study the relation between the above two pairs of soft rough approximation operators given in Definitions31 and 40Proposition 45.
LetB=(B,≤) be a complete atomic Boolean lattice and let fA be a soft set over B. Let φf:A(B)→B be the mapping induced by fA. Then the following properties hold.(i)
IffAis full, thenx∇f≤x∨.(ii)
IffAis full and keeping supremum, thenxΔf≤x∧.(iii)
IffAis a partition, then
(a)
x∇f=x∨,(b)
xΔf=x∧.Proof.
(i) Letb∈A(B) s.t b≤x∇f. Then φf(b)≤x. Since fA is full, then ∃e∈A,s.tb≤f(e). By Proposition 42(i) f(e)≤φf(b). Thus b≤f(e)≤x and hence b≤x∨. Consequently, x∇f≤x∨.
(ii) Ifx=0, then x∧=0∧=0=xΔf. If x≠0 and fA is keeping supremum, then by Proposition 38(3) x∧=1. Hence xΔf≤x∧.
(iii) (a) IffA is a partition, then it is full. So x∇f≤x∨ by (i). On the other hand, let b∈A(B) s.t b≤x∨. So ∃e∈A,s.tb≤f(e)≤x. Since fA is a partition and b≤f(e), then by Proposition 34(ii) f(e)=φf(b). This implies that b≤x∇f and therefore x∨≤x∇f. Consequently, x∇f=x∨.
(b) This is similar to the proof of (a).Example 46.
LetB={0,a,b,c,d,e,f,1} and let the order ≤ be defined as in Figure 1. Let A={e1,e2,e3,e4} and let fA be a soft set over B defined as follows:
(32)f(e1)=a,f(e2)=b,f(e3)=d,f(e4)=0.
Obviously,fA is not full. Also φf(a)=⋁{b∈A(B):b≤φf(a)}=a∨b=d, φf(b)=a∨b=d, and φf(c)=0.
Letx=b, y=a. So x∇f=⋁{d∈A(B):φf(d)≤b}=c, and xΔf=⋁{d∈A(B):φf(d)∧b≠0}=a∨b=d. On the other hand
y
∇
f
=
c and yΔf=d. Hence x∇f≤y∇f and xΔf≤yΔf, but x≰y.Example 47.
LetB={0,a,b,c,d,e,f,1} and let the order ≤ be defined as in Figure 1. Let A={e1,e2,e3,e4} and let gA be a soft set over B defined as follows:
(33)g(e1)=a,g(e2)=e,g(e3)=c,g(e4)=f.
Obviously.gA is full. Also φg(a)=a∨c=e,  φg(b)=b∨c=f, and φg(c)=a∨b∨c=1. Let x=f, then x∇g=b and xΔg=a∨b∨c=1. Hence x∇g≤x≤xΔg.In the following, we give a relation between soft rough approximation operators and Järvinen’s approximation operators on a complete atomic Boolean lattice.Definition 48.
LetB=(B,≤) be a complete atomic Boolean lattice. Let φ:A(B)→B be extensive, symmetric, and closed mapping. Define a mapping fφ:A→B by fφ(e)=φ(e) for every e∈A, where A=A(B). Then (fφ)A is called the soft set induced by φ on B.Theorem 49.
LetB=(B,≤) be a complete atomic Boolean lattice. Let (fφ)A be the soft set induced by φ on B. Then for every x∈B, x∨φ=x∇ and x∧φ=xΔ, where
(34)x∨φ=⋁{b∈A(B):∃e∈As.tb≤(fφ)(e),(fφ)(e)≤x},x∇=⋁{b∈A(B):φ(b)≤x},x∧φ=⋁{b∈A(B):∃e∈As.tb≤(fφ)(e),(fφ)(e)∧x≠0},xΔ=⋁{b∈A(B):φ(b)∧x≠0}.Proof.
Obvious.Theorem 50.
LetB=(B,≤) be a complete atomic Boolean lattice. Let fA be a partition soft set over B. Then for every x∈B, x∇f=x∨ and xΔf=x∧.Proof.
It follows immediately by Propositions41(iii) and 45(iii).Remark 51.
Theorems49 and 50 illustrate that Järvinen’s approximations can be viewed as a special case of our soft rough approximations on a complete atomic Boolean lattice.


## 6. Conclusion

In this paper, we introduced the concept of soft sets on a complete atomic Boolean lattice as a generalization of soft sets and obtained the lattice structure of these soft sets. Two pairs of soft rough approximation operators on a complete atomic Boolean lattice were considered, and their properties were given. We show that Järvinen’s approximations can be viewed as a special case of our soft rough approximations. We may mention that soft rough sets on a complete atomic Boolean lattice can be used in object evaluation and group decision making. It should be noted that the use of soft rough sets could, to some extent, automatically reduce the noise factor caused by the subjective nature of the expert’s evaluation. We will investigate these problems in future papers.



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