In this paper, we introduce the concepts of g and b approximations via general ordered topological approximation spaces. Also, increasing (decreasing) g, b boundary, positive and negative regions are given in general ordered topological approximation spaces (GOTAS, for short). Some important properties of them were investigated. From this study, we can say that studying any properties of rough set concepts via GOTAS is a generalization of Pawlak approximation spaces and general approximation spaces.
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.
Abo-Elhamayel, M. (2016) γ and β Approximations via General Ordered Topological Spaces. Applied Mathematics, 7, 1580-1588. doi: 10.4236/am.2016.714136.
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