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. If , then our soft approximations coincide with crisp soft rough approximations (Feng et al. 2011).
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 Section 2 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. Let be an ordered set. A mapping is said to be extensive, if for all . The map is order preserving if implies . Moreover, is idempotent if for all . A map is said to be a closure operator on , if is extensive, order preserving, and idempotent. An element is closed if . Furthermore, if is a closure operator on then is an interior operator on . Let and be ordered sets. is an order embedding, if for any , in if and only if in ; note that an order embedding is always an injection. An order-embedding onto is called an order-isomorphism between and ; we say that and are order-isomorphic and write . If and are order-isomorphic, then and are said to be dually order-isomorphic. A pair of maps and is called a dual Galois connection on if and are order preserving and for all .
Before we consider the Boolean lattices, we present the following lemma, where denotes the power set of , that is, the set of all subsets of .
Lemma 1 (see [23]). Let be a complete lattice, , and .(i)If , then . (ii). (iii).
Next we recall the concept of Boolean lattices. They are bounded distributive lattices with a complementation operation.
Definition 2 (see [27]). A lattice is called a Boolean lattice, if(i) is distributive;(ii) has a least element 0 and a greatest element 1, and;(iii)each has a complement such that and .
Lemma 3 (see [27]). Let be a Boolean lattice; then for all (i) and ,(ii),(iii), and , (iv) iff .
Let us recall some definitions and results that are useful in our consideration given in [23].
Lemma 4 (see [23]). Let be a complete Boolean lattice. Then for all and
Definition 5 (see [23]). Let be an ordered set and ; we say that is covered by (or that covers ), and write if and there is no element in with .
Definition 6 (see [23]). Let be a lattice with a least element 0. Then is called an atom if . The set of atoms of is denoted by . The lattice is called atomic if every element of is the supremum of the atoms below it; that is, .
It is obvious that in a lattice with a least element 0, for all and . This implies that for all s.t . Furthermore, if is atomic, then for all there exists an atom s.t . Namely, if , then .
Definition 7 (see [23]). Let be a complete atomic Boolean lattice. We say that is(i)extensive, if for all ,(ii)symmetric, if implies for all ,(iii)closed, if implies for all .
Definition 8 (see [23]). Let be a complete atomic Boolean lattice and let be any mapping. For any element , let
The elements and are called the lower and the upper approximations of with respect to , respectively. Two elements and 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 and are always complete lattices. They are distributive sublattices of if is extensive and closed. If the map is extensive, symmetric, and closed, then the ordered sets and are mutually equal complete atomic Boolean lattices.
Proposition 9 (see [23]). Let be a complete atomic Boolean lattice and let be any mapping. Then for all and ,(i);(ii).
Proposition 10 (see [23]). Let be a complete atomic Boolean lattice and let be an extensive mapping. Then for all ,(i);(ii).
Proposition 11 (see [23]). Let be a complete atomic Boolean lattice and let be extensive and closed mapping. Then for all ,(i);(ii).
Proposition 12 (see [23]). Let be a complete atomic Boolean lattice and let be an extensive, symmetric and closed mapping. Then for all ,(i);(ii).
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 pair of nonempty finite sets and , where is a set of objects and is a set of attributes; each attribute is a function , where is the set of values (called domain) of attribute .
Let be a non-empty finite universe and let be an equivalence relation on . The pair (, ) is called a Pawlak approximation space. The equivalence relation is often called an indiscernibility relation and related to an information system. Specifically, if is an information system and , then an indiscernibility relation can be defined by where and denotes the value of attribute a for object .
Using the indiscernibility relation , one can define the following two operations: assigning to every subset two sets and called the -lower and the -upper approximation of , respectively.
If , then is said to be -definable; otherwise, is said to be -rough.
Let us recall now the soft set notion, which is a newly emerging mathematical approach to vagueness.
Definition 14 (see [2]). Let be a universal set and let be a set of parameters. Let be a nonempty subset of . A soft set over , with support , denoted by on is defined by the set of ordered pairs or is a function s.t
Example 15. Suppose that is the set of houses under consideration and and are both parameter sets. Let there be four houses in the universe given by . And and . The soft sets and describe the ‘‘attractiveness of the houses.’’ For the sake of ease of designation, we use , instead of expensive and instead of modern. The soft set is defined as follows means expensive houses, and means modern houses. The soft set is the collection of approximations as below:
The soft set is defined as , which means the modern houses. The soft set is the collection of approximations as below:
Definition 16 (see [25, 26]). Let be a universal set and let be a soft set over . Then the pair is called soft approximation space. We define a pair of operators as follows:
The elements and are called the soft -lower and the soft -upper approximations of .
If , is said to be soft -definable; otherwise is called a soft -rough set.
Example 17. Let us consider the following soft set which describes ‘‘life expectancy’’. Suppose that the universe consists of six persons and is a set of decision parameters. The () stands for “under stress,” “young,” “drug addict” and “healthy.” Set , , ; and . The soft set can be viewed as the following collection of approximations:
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 Table 1, 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:
Let be a target subset of , that we wish to represent using the above equivalence classes. Hence we analyze the upper and lower approximations of , in some particular cases.(1)Set . It follows that
Let us calculate now the soft -lower and -upper approximations of , where . We obtain
hence is soft -definable.(2)Set . It follows that . On the other hand, , hence ; is soft -definable.
The 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 subsets , where .
3. Soft Sets on a Complete Atomic Boolean Lattice
Definition 18. Let be a complete atomic Boolean lattice and let be a set of parameters. Let be a non empty subset of . A soft set over , with support , denoted by on is defined by the set of ordered pairs or is a function s.t In other words, a soft set over is a parameterized family of elements of . For each , is considered as -approximate element of .
Definition 19. Let be a complete atomic Boolean lattice. Let and let and be two soft sets over .(i) is a soft subset of , denoted by if and for every . (ii) and are called soft equal, denoted by if and .
Definition 20. Let be a complete atomic Boolean lattice. Let and let be a soft set over .(i) is called null, denoted by if for every . (ii) is called absolute, denoted by if for every .
We stipulate that is also a soft set over with .
Let and let be a soft set over . Obviously,
Below, we introduce some operations on soft sets on and investigate their properties.
Definition 21. Let be a complete atomic Boolean lattice. Let and let and be two soft sets over .(i) is called the intersection of and , denoted by if and for every .(ii) is called the union of and , denoted by if and if , if and if .
Definition 22. Let be a complete atomic Boolean lattice. Let and let be a soft set over . The complement of , denoted by is defined by , where is a mapping given by for every .
Proposition 23. Let be a complete atomic Boolean lattice. Let and let , , and be three soft sets over . Then(i),(ii),(iii).
Proof. (i) and (ii) are obvious. We only prove (iii). Put
For any it follows that , or , or .
Case 1 .(a)If and , then . (b)If and , then . (c)If and , then . (d)If and , then .
Case 2 .(a)If and , then . (b)If and , then . (c)If and , then .
Thus .
Proposition 24. Let be a complete atomic Boolean lattice. Let and let , ; and be three soft sets over . Then(i),(ii),(iii).
Proof. (i) and (ii) are obvious. We only prove (iii). Put
For any , it follows that , , and . Since , then .
Proposition 25. Let be a complete atomic Boolean lattice. Let and let , , and be three soft sets over . Then(i),(ii).
Proof. (i) Put , . Obviously, . For any , it follows that or .(a)If and , then , , and . So .(b)If and , then , , and . So . (c)If and , then , , and . So .
Thus .
(ii) This is similar to the proof of (i).
Proposition 26. Let be a complete atomic Boolean lattice. Let and let and be two soft sets over .(i).(ii).(iii).(iv).(v).
Proof. (i) Put , .
For any , , . So, (by Lemma 3). This Shows that ; that is .
(ii) Put .
For any , . Hence .
(iii) This is similar to the proof of (ii).
(iv) Put , .
For any , , . Hence by Lemma 3.
(v) This is similar to the proof of (iv).
Definition 27. Let be a complete atomic Boolean lattice and Let be a soft set over .(i) is called full, if ; (ii) is keeping infimum, if for any , there exists such that ; (iii) is keeping supremum, if for any , there exists such that ; (iv) is called partition of if(1), (2)for every , ,(3)for every either or .
Obviously, every partition soft set is full and is keeping infimum (resp., keeping supremum) if and only if for every , there exists such that (resp., ).
Example 28. Let and let the order be defined as in Figure 1.
The set of atoms of a complete atomic Boolean lattice is . Let and let be a soft set over defined as follows:
Obviously, is not a partition since . Also, is full since . Also, is keeping infimum. In fact .
and . Consequently, is keeping infimum. On the other hand, is not keeping supremum since for every .
Let be a soft set over defined as follows:
, , , and ; then is a partition, keeping infimum, and keeping supremum.
Next, we investigate the lattice structure of soft sets on a complete atomic Boolean Lattice . We denote
Obviously,
Theorem 29. For any , define
Then is a distributive lattice with smallest element and greatest element .
Proof. Denote . It is easily proved that
By Proposition 25 is a distributive lattice with and .
Theorem 30. For any , define
Then is a Boolean lattice.
Proof. Denote . It is easily proved that is a distributive lattice with and .
Let . Put . Since , then for any ,
So, . This shows that . Similarly, we can prove that . Hence and therefore is a Boolean lattice.
4. Soft Rough Approximation Operators on a Complete Atomic Boolean Lattice
Definition 31. Let be a complete atomic Boolean lattice and let be a soft set over . For any element , we define a pair of operators as follows:
The elements and are called the soft lower and the soft upper approximations of over . Two elements and are called soft equivalent if they have the same soft upper and soft lower approximations over . The resulting equivalence classes are called soft rough sets over .
Lemma 32. Let be a complete atomic Boolean lattice and let be a soft set over . Then for all and (i); (ii).
Proof. (i) Suppose that . Assume that for all either or . If , , then , a contradiction. If , , then = . Since , then, . So because . Hence . This implies that , which is a contradiction.
Suppose that ; then .
Condition (ii) can be proved similarly.
Proposition 33. Let be a complete atomic Boolean lattice and let be a soft set over . Then for all (i); (ii).
Proof. (i) Let , s.t ; then . So, . On the other hand, let , s.t . Hence, . In fact, if and implies , then . Therefore because . Thus . So, , a contradiction. So, and consequently, .
Condition (ii) can be proved similarly.
Proposition 34. Let be a complete atomic Boolean lattice and let be a soft set over .(i) and ; (ii) implies and .
Proof. Obvious.
For all , we denote and .
Proposition 35. Let be a complete atomic Boolean lattice and let be a soft set over ; then(i)for all , ;(ii)if is keeping infimum, then for all , ;(iii) is a complete lattice; 0 is the least element and is the greatest element of ;(iv)if is keeping infimum, then is a complete lattice; is the least element and is the greatest element of ;(v)if is keeping infimum, the kernal of the map is a congruence on the semi lattice such that the -class of any x has a least element;(vi)the kernal of the map is a congruence on the semilattice such that the -class of any x has a least element.
Proof. (i) Let . The map : is order preserving, which implies that . Let and assume that . So, . Then , which implies that for some . Thus . Then (ii) Let . The map is order preserving, which implies that . Let s.t . So, for every . Hence for every . This implies that . Since is keeping infimum, then for . So we show that . Therefore . Consequently, . 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 and are not dually order-isomorphic.
Example 36. Let and let the order be defined as in Figure 1.
Let and let be a soft set over defined as follows:
Then is not a partition since . Let and ; then and . Therefore . On the other hand .
Next, we show that and are dually order-isomorphic if is a partition.
Proposition 37. Let be a complete atomic Boolean lattice and let be a soft set over . If is a partition, then .
Proof. We show that is the required dual order isomorphism. It is obvious that is onto . We show that is order embedding. Suppose that . Then for all , implies . So, such that and , implies , s.t and . Since is a partition and , then . Hence if and , then . Suppose that . So there exists such that and . Since , then and . Since and , then . Since is equivalent to , then by hypothesis . But this means that , a contradiction. Hence . On the other hand, assume that . Since is a partition, then s.t and implies . Suppose that . So there exists such that and . So , and . But this implies that . Since , then . This is equivalent to , a contradiction.
Next we study the properties of soft approximations more closely in cases when the soft set is full, keeping union, keeping intersection, and partition.
Proposition 38. Let be a complete atomic Boolean lattice and let be a soft set over . Then the following properties hold.(i)If is full, then(a); (b). (ii)If is keeping supremum, then(a)for all , ;(b)for all , .(iii)If is full and keeping supremum, then for every and .
Proof. (i)(a) By Proposition 41 . Suppose that and . Since is full, then . So, and therefore . Consequently, . (b) Obvious.
(ii) It follows by Proposition 33.
(iii) Let , then in general . Since is full and keeping supremum, then . So, and . Consequently, and so .
Proposition 39. Let be a complete atomic Boolean lattice and let be a soft set over . If is a partition, then,(i); (ii); (iii)the map is a closure operator.
Proof. (i) Since is full, then by Proposition 38(1). Let and ; then and . Hence, . But implies . Since is a partition and , then . So, and therefore, . Consequently, . Claim (ii) can be proved similarly.
(iii) Since is full, then by Proposition 38 the map is extensive, and it is order preserving by Proposition 34. By (ii), .
5. Another Soft Rough Approximation Operators on a Complete Atomic Boolean Lattice
Definition 40. Let be a complete atomic Boolean lattice and let be a soft set over . Define a mapping by for every . Then is called the mapping induced by on .
Proposition 41. Let be a complete atomic Boolean lattice and let be a soft set over . Let be the mapping induced by on . Then the following properties hold.(i) is symmetric.(ii)If is full, then is extensive. (iii)If is a partition, then is extensive, symmetric, and closed.
Proof. (i) Obvious.
(ii) Let . Since is full, then , s.t . Hence .
(iii) If is a partition, then is full and hence is extensive. Since is symmetric, it remains to show that is closed. Let s.t . We show that . Since , then . Suppose that . So, , s.t and . But implies that for every , either or . Since , then , s.t and . Since is a partition and , then . Hence we show that , s.t and , a contradiction. Consequently, and thus is closed.
Proposition 42. Let be a complete atomic Boolean lattice and let be a soft set over . Let be the mapping induced by on . Then the following properties hold.(i)If for and , then .(ii)If is a partition and for and , then .(iii)If is keeping supremum, then for all , s.t .
Proof. (i) Let s.t . Since , then . Hence .
(ii) Suppose that is a partition and assume that for and . By (i) . On the other hand, let s.t . Then and . So, , and since is a partition, then . Hence and therefore . Consequently, .
(iii) Suppose that is keeping supremum and . Let s.t . Then , s.t and . So by (i). Hence, . Since is keeping supremum, then for . Therefore .
Definition 43. Let be a complete atomic Boolean lattice and let be a soft set over . Let be the mapping induced by on . We define a pair of soft approximation operators as follows:
The elements and are called the soft lower and the soft upper approximations of with respect to the mapping induced by , respectively. Two elements and are called equivalent if they have the same soft upper and lower approximations with respect to the mapping induced by on . The resulting equivalence classes are called soft rough sets with respect to the mapping induced by on .
Proposition 44. Let be a complete atomic Boolean lattice and let be a soft set over . Let be the mapping induced by .(i). (ii). (iii)If is full, then .(iv) and . If is full, then and . (v) implies and .(vi)The mappings and are mutually dual.(vii)For all , .(viii)For all , .(ix) is a complete lattice; 0 is the least element and 1 is the greatest element of .(x)The pair is a dual Galois connection on .(xi).
Proof. It follows by Propositions 41, 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 Definitions 31 and 40
Proposition 45. Let be a complete atomic Boolean lattice and let be a soft set over . Let be the mapping induced by . Then the following properties hold.(i)If is full, then .(ii)If is full and keeping supremum, then .(iii)If is a partition, then(a),(b).
Proof. (i) Let s.t . Then . Since is full, then . By Proposition 42(i) . Thus and hence . Consequently, .
(ii) If , then . If and is keeping supremum, then by Proposition 38(3) . Hence .
(iii) (a) If is a partition, then it is full. So by (i). On the other hand, let s.t . So . Since is a partition and , then by Proposition 34(ii) . This implies that and therefore . Consequently, .
(b) This is similar to the proof of (a).
Example 46. Let and let the order be defined as in Figure 1. Let and let be a soft set over defined as follows:
Obviously, is not full. Also , , and .
Let , . So , and . On the other hand
and . Hence and , but .
Example 47. Let and let the order be defined as in Figure 1. Let and let be a soft set over defined as follows:
Obviously. is full. Also , , and . Let , then and . Hence .
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. Let be a complete atomic Boolean lattice. Let be extensive, symmetric, and closed mapping. Define a mapping by for every , where . Then is called the soft set induced by on .
Theorem 49. Let be a complete atomic Boolean lattice. Let be the soft set induced by on . Then for every , and , where
Proof. Obvious.
Theorem 50. Let be a complete atomic Boolean lattice. Let be a partition soft set over . Then for every , and .
Proof. It follows immediately by Propositions 41(iii) and 45(iii).
Remark 51. Theorems 49 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.