Received 9 May 2016; accepted 22 August 2016; published 25 August 2016

1. Introduction

Rough set theory was first proposed by Pawlak for dealing with vagueness and granularity in information systems. Various generalizations of Pawlak s rough set have been made by replacing equivalence relations with kinds of binary relations and many results about generalized rough set with the universe being finite were obtained [1] - [7] . An interesting and natural research topic in rough set theory is studying it via topology [8] [9] . Neighborhood systems were first applied in generalizing rough sets in 1998 by T. Y. Lin as a generalization of topological connections with rough sets. Lin also introduced the concept of granular computing as a form of topological generalizations [10] - [13] . In this paper, we give the concept of g, b via topological ordered spaces and studied their properties which may be viewed as a generalization of previous studies in general approximation spaces, as if we take the partially ordered relation as an equal relation, we obtain the concepts in general approximation spaces [14] .

2. Preliminaries

In this section, we give an account of the basic definitions and preliminaries to be used in the paper.

Definition 2.1 [15] . A subset A of U, where is a partially ordered set is said to be increasing (resp. decreasing) if for all and such that (resp.) imply.

Definition 2.2 [15] . A triple is said to be a topological ordered space, where is a topological space and is a partial order relation on U.

Definition 2.3 [16] . Information system is a pair where U is a non-empty finite set of objects and is a non-empty finite set of attributes.

Definition 2.4 [17] . A non-empty set U equipped with a general relation which generates a topology on U and a partially order relation written as is said to be general ordered topological approximation space (for short, GOTAS).

Definition 2.5 [18] . Let be a GOTAS and. We define:

(1), is the greatest increasing open subset of A.

(2), is the greatest decreasing open subset of A.

(3), is the smallest increasing closed superset of A.

(4), is the smallest decreasing closed superset of A.

(5) (resp.) and )resp.) is R-increasing (resp. decreasing) accuracy.

Definition 2.6 [17] . Let be a GOTAS and. We define:

(1), is called R-increasing semi lower.

(2), is called R- increasing semi upper.

(3), is called R-decreasing semi lower.

(4), is called R-decreasing semi upper.

A is R- increasing (resp. decreasing) semi exact if (resp.), otherwise A is R- increasing (resp. decreasing) semi rough.

Proposition 2.7 [18] . Let be a GOTAS and. Then

(1).

(2).

3. New Approximations and Their Properties

In this section, we introduce some definitions and propositions about near approximations, near boundary regions via GOTAS which is essential for a present study.

Definition 3.1. Let be a GOTAS and. We define:

(1), is called R-increasing lower.

(2), is called R-increasing upper.

(3), is called R-decreasing lower.

(4), is called R-decreasing upper.

A is R-increasing (resp. R-decreasing) exact if (resp.) otherwise A is R-increasing (resp. R-decreasing) rough.

Proposition 3.2. Let be a GOTAS and. Then

(1) ().

(2) ().

(3) ().

Proof.

(1) Omitted.

One can prove the case between parentheses.

Proposition 3.3. Let be a GOTAS and. Then

(1) ().

(2) ().

(3) ().

Proof.

(1) Easy.

One can prove the case between parentheses.

Proposition 3.4. Let be a GOTAS and. If A is R-increasing (resp. decreasing) exact then A is R-increasing (resp. decreasing) exact.

Proof.

Let A be R-increasing exact. Then, thus and. Therefore.

One can prove the case between parentheses.

R-increasing (resp. decreasing) exact R-increasing (resp. decreasing) exact.

Proposition 3.5. Let be a GOTAS and. Then.

Proof.

Since and, then. There-

fore,. Thus.

One can prove the case between parentheses.

Proposition 3.6. Let be a GOTAS and. Then.

Proof. Since and, then. Thus

.

Therefore. Hence.

Proposition 3.7. Let be a GOTAS and. Then.

Proof. Let. Then and. Therefore and

.

Thus. Hence.

One can prove the case between parentheses.

Proposition 3.8. Let be a GOTAS and. Then.

Proof.

Let. Then and. Therefore and

or.

Thus. Hence.

One can prove the case between parentheses.

Proposition 3.9. Let be a GOTAS and. Then.

Proof.

Let. Then and. Therefore

.

Thus.

Proposition 3.10. Let be a GOTAS and. Then

.

Proof. Omitted.

Definition 3.11. Let be a GOTAS and. We define:

(1), is called R-increasing lower.

(2), is called R-increasing upper.

(3), is called R-decreasing lower.

(4), is called R-decreasing upper.

A is R-increasing (decreasing) exact if (resp.), otherwise A is R-increasing (decreasing) rough.

Proposition 3.12. Let be a GOTAS and. Then

(1) ().

(2) ().

(3) ().

Proof.

(1) Omitted.

One can prove the case between parentheses.

Proposition 3.13. Let be a GOTAS and. Then

(1) ().

(2) ().

(3) ().

Proof.

(1) Easy.

One can prove the case between parentheses.

Proposition 3.14. Let be a GOTAS and. If A is R-increasing (resp. decreasing) exact then A is b-increasing (resp. decreasing) exact.

Proof.

Let A be R-increasing exact. Then. Therefore,. Thus. Hence A is R-increasing exact.

One can prove the case between parentheses.

Proposition 3.15. Let be a GOTAS and. Then

.

Proof.

Since and. Then

.

Therefore. Thus.

One can prove the case between parentheses.

Proposition 3.16. Let be a GOTAS and. Then

.

Proof. Since and. Then. Thus

.

Therefore. Hence.

Definition 3.17. Let be a GOTAS and. Then

(1) (resp.), is increasing (resp. decreasing) j boundary region.

(2) (resp.), is increasing (resp. decreasing) j positive region.

(3) ( resp.), is increasing (resp. decreasing) j negative region. Where the near lower approximations s.t..

Proposition 3.18. Let be a GOTAS and. Then

(1) ().

(2) ().

Proof.

One can prove the case between parentheses.

Proposition 3.19. Let be a GOTAS and. Then

(1) ().

(2) ().

Proof.

One can prove the case between parentheses.

Proposition 3.20. Let be a GOTAS and. Then

.

Proof.

Let. Then. Therefore. Thus

and thus.

Hence

(1).

Since, then. Therefore.

Thus, and thus. Hence

(2)

From (1) and (2) we have,

.

One can prove the case between parentheses.

Proposition 3.21. Let be a GOTAS and. Then

.

Proof.

Let. Then. Therefore or. Thus

or. So, and so.

Thus. Hence

. (1)

Since, or, then. Therefore

(2)

From (1) and (2) we have,.

One can prove the case between parentheses.

Definition 3.22. Let be a GOTAS and A is a non-empty finite subset of U. Then the increasing (decreasing) j accuracy of a finite non-empty subset A of U is given by:

,.

Proposition 3.23. Let be a GOTAS and non-empty finite subset of. Then we have

, for all, where

Proof. Omitted.

In the following example we illustrate most of the properties that have been proved in the previous propositions.

Example 3.24. Let, , ,

and

For, we have:

, , ,.

.

.

, ,

Proposition 3.25. Let be a GOTAS and. Then we have

Proof. Omitted.

Remark 3.26..

Remark 3.27..

Proposition 3.28. Let be a GOTAS and be a non-empty finite subset of. Then ().

Proof. Omitted.

Proposition 3.28. Let be a GOTAS and. Then

Proof. Let. Then and

.

Therefore and [or]. Thus and and

thus and. Hence. Therefore.

One can prove the case between parentheses.

4. Conclusion

In this paper, we generalize rough set theory in the framework of topological spaces. Our results in this paper became the results about of, approximation in [2] in the case of is the equal relation. Also, the new approximation which we give became as Pawlak s approximation in the case of is the equal relation and R is the equivalence relation. This theory brings in all these techniques to information analysis and knowledge processing.