A study of generalized roughness in -fuzzy filters of ordered semigroups

ABSTRACT Concept of generalized rough approximations for fuzzy filters in ordered semigroups is introduced. Then this idea is extended to rough approximations of fuzzy bi-filters and fuzzy quasi-filters in ordered semigroups. -fuzzy filters are a generalization of fuzzy filters and these can be further generalized by -fuzzy filters. Generalized roughness has been studied for both and -fuzzy filters of ordered semigroups.


Introduction
In daily life, there are many situations where certain order exist among the elements of a set. For example, prices of commodities can be descried by terms like very cheap, cheap, affordable, costly, very costly. It is clear that there is an order among these terms and commodities can be arranged with the help of order among their prices. Algebraically ordered semigroups are sets with associative binary operation having a certain partial order. Ordered semigroups have wide ranging applications in computer science, automata theory and coding theory.
Rough set theory and fuzzy set theory are two distinct concepts, but both of them are very handy to deal with uncertainty. These two can be hybridized in a very fruitful manner. Therefore, concepts of fuzzy rough sets and rough fuzzy sets are introduced in [1].
Rosenfeld introduced the idea of fuzzy algebraic structures by introducing fuzzy subgroups [2]. Study of fuzzy semigroup is initiated by Kuroki [3,4]. However fuzzy ordered groupoids and ordered semigroup were investigated by Kehayopulu and Tsingelis [5,6] for the first time. The idea of (a, b)-fuzzy subgroups is presented by Bhakat and Das [7][8][9]. Among different (a, b)-fuzzy subgroups ([, [ _q)-fuzzy subgroup are the most interesting. In study of ([, [ _q)-fuzzy subgroups quasi coincident fuzzy points play a basic role. This notion is introduced in [10]. Fuzzy filters for various algebraic structures have been studied by many authors. Generalized fuzzy filters of R 0 algebra have been studied by Ma et al. [11]. Kehayopulu and Tsingelis presented the notion of fuzzy filter in ordered semigroups [6].
Pawlak is the founder of rough set theory [12]. Many applications of this theory have been reported. Actually it is a nice tool to discuss uncertainty among the elements of a set. Equivalence relations play a fundamental role in it. Due to limited knowledge about the elements of a set, it is too complicated to determine the equivalence relation among the objects of a set. So authors studied different models with less restrictions. Generalized rough sets has been studied in [13]. Instead of equivalence relation, set valued maps are used to define approximations of a set in generalized rough set theory. In algebraic structures roughness has be discussed by many authors. Kuroki studied roughness in semigroups and fuzzy semigroups in [14]. Then this concept is studied for prime ideals in semigroups [15]. Study of roughness in ([, [ _q)-fuzzy ideals of hemirings initiated in [16]. In ordered semigroups rough approximations as proposed in [14] can not be a good idea. As in ordered semigroup there is a partial order associated with the semigroup, therefore non-trivial equivalence relations for such semigroups are difficult to find. Perhaps, this is the major reason that no study of roughness in case of ordered semigroups has been reported till now according to our knowledge. Therefore in this paper some weaker tool to study roughness for fuzzy filters in ordered semigroups have been introduced. Set valued maps give rise to binary relations in general. These maps with monotone or isotone order help us to study roughness in fuzzy filters of ordered semigroups.
Organization of this paper is as the following. In Section 2, some basic concepts about ordered semigroups, fuzzy sets and rough sets are given. These notions will be useful in later sections. Section 3, is devoted for discussion about approximations of fuzzy filters, fuzzy bi-filters and quasi-filters in ordered semigroup. It is seen that set valued monotone and isotone homomorphism play basic role for approximation of fuzzy filters. In Section 4, concept of approximations is extended to ([, [ _q)-fuzzy filters, fuzzy bi-filters and fuzzy quasi-filters. Approximations of ([, [ _qk)-fuzzy filters, fuzzy bi-filters and fuzzy quasi-filters in ordered semigroup have been studied in Section 5.

Preliminaries
In this section some fundamental concepts about ordered semigroups, fuzzy sets and rough sets are presented. These notions will be helpful in later sections.
If K is any non-empty subset of a po-semigroup S, then K is an ordered subsemigroup of S, if K 2 # K.
F is known as a filter if it is both a left filter and a right filter of S.
Next definitions of bi-filters and quasi filters are given. These are actually generalizations of filters in po-semigroups. In the following some basic concepts about fuzzy filters and their generalizations are given.
represent fuzzy point with value t and support by y. It is written as y t .
A fuzzy point y t of S is said to "belong to" fuzzy subset C denoted as y t [ C, if C(y) ≥ t, and is said to "quasicoincident" to C denoted by y t qC, if C(y) + t . 1. From here onward in discussion below S stands for ordered semigroup and C for fuzzy subset on ordered semigroup S unless stated otherwise. Definition 2.6: C is known as a fuzzy ordered subsemigroup of S if it satisfies C(y 1 y 2 ) ≥ min{C(y 1 ), C(y 2 )}, ∀y 1 , y 2 [ S.
In the following this inequality will be denoted by (FF 2 ).
A fuzzy subset C is known as a fuzzy filter of S, if C is a fuzzy left and a fuzzy right filter of S.  (FF 5 ) y 1 y 2 = y 3 y 1 implies C(y 1 y 2 ) ≤ C(y 1 ), ∀y 1 , y 2 , y 3 [ S.
In the following we recall some basic concepts of rough set theory introduced by Pawlak [12].
Consider an equivalence relation K on a universal set U. The pair (U, K) is known as an approximation space. Let ∅ = A # U, then A is definable if we can express it in the form of some equivalence classes of U, else it is not definable. If A is not definable, then it may be approximated in the form of definable subsets called lower approximation and upper approximation of A, defined as The pair (app(A), app(A)) is known as a rough set. If app(A) = app(A), then A is a definable set.
This notion of lower and upper approximations can be generalized for fuzzy sets as well.
Definition 2.10 ( [20]): Let us consider the approximation space (U, K) and C as a fuzzy subset of U. Define the lower and upper approximation of a fuzzy subset C as the following

For any y [ U,
A rough fuzzy set is the pair (appC, appC), if appC = appC.
Definition 2.11: Consider the ordered semigroups S 1 and S 2 . A mapping H : Where P * (S 2 ) denotes the collection of all non-empty subsets of S 2 .
Definition 2.12: Suppose that S 1 and S 2 are ordered semigroups. A mapping H : In the following concept of roughness in fuzzy sets is being generalized by SVMH and SVIH.
Definition 2.14: Let H : S − P * (S) be an SVMH or SVIH. Then for every y [ S, we define the generalized lower and upper approximation of C with respect to mapping H as,

Approximations of fuzzy filters in ordered semigroups
In this section study of roughness for fuzzy filters of ordered semigroups is being initiated. The following result is for the upper approximation of a fuzzy ordered subsemigroup of a fuzzy ordered semigroup.
Hence H(C) is a fuzzy ordered subsemigroup of S. □ In the following, study of roughness in fuzzy filters of ordered semigroups is being initiated. Certain restrictions are required on SVH for this study.
It is easy to see from Theorem 3.1, that Hence H(C) is a fuzzy left filter of S. Similarly, it can be shown that H(C) is a fuzzy right filter of S. □ The following example shows that if H is an SVMH, then for a fuzzy filter C, its lower approximation H(C) may not be a fuzzy filter.
is not a fuzzy filter of S.
In above example, it has been seen that in case of SV MH lower approximation of a fuzzy filter may not be a fuzzy filter. Now we turn over attention to SV IH and have the following: From Theorem 3.1, it is easy to see that H(C)(y 1 y 2 ) ≥ min{H(C)(y 1 ), H(C)(y 2 )}.
Next consider Hence H(C) is a fuzzy left filter of S. Similarly, it can be shown that H(C) is a fuzzy right filter of S. □ The following example shows that, if H is an SVIH, then for a fuzzy filter C, its upper approximation H(C) may not be a fuzzy filter. Also H(C)(y 1 y 2 ) ≥ min{ H(C)(y 1 ), H(C)(y 2 )}. Therefore consider the following for any since Cis fuzzy bi-filter therefore by Definition 2.8.
implies H C ( ) y 1 y 2 y 1 ≥ H C ( ) y 1 Hence H(C) on S is a fuzzy bi-filter of S. □ Theorem 3.6: Let H : S − P * (S) be an SVIH and C be a fuzzy bi-filter of S. Then H(C) is a fuzzy bi-filter of S. □ Table 1. Multiplication table for S.
Hence H(C) on S is a fuzzy quasi filter of S. □

Approximations of ([, [ _q)-fuzzy filters in ordered semigroups
In the following, study of roughness in ([, [ _q)-fuzzy filters of ordered semigroups is being initiated.

Hence, it is clear that
that is H C ( ) y 2 ≥ min H C ( ) y 1 , 1 − k 2 .