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The notions of (strong) intersection-soft filters in

To solve complicated problem in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. Uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [

To overcome these difficulties, Molodtsov [

In this paper, we apply the notion of intersection-soft sets to the filter theory in

Let

(R1)

(R2)

(R3)

(R4)

(R5)

(R6)

Let

We refer the reader to the book [

Let

A nonempty subset

Let

Soft set theory was introduced by Molodtsov [

In what follows, let

By analogy with fuzzy set theory, the notion of soft set is defined as follows.

A soft set of

Let

In what follows, we denote by

A soft set

If

Let

We provide characterizations of an int-soft filter.

Let

Assume that

Conversely, suppose that

Let

Let

The necessity follows from

Conversely, suppose that

Let

Suppose that there exists

Let

Assume that

Conversely, suppose that

Let

Suppose that

Conversely, assume that

Every int-soft filter

Let

The following example shows that the converse of Proposition

Let

For an int-soft filter

Assume that (

Conversely, suppose that (

We make a new int-soft filter from old one.

Let

Assume that

Any filter of

Let

An int-soft filter

The int-soft filter

Let

Suppose that

Conversely, assume that

Let

We provide a condition for an int-soft filter to be strong.

Let

Let

We consider an extension property of a strong int-soft filter.

Let

Assume that

Using the notion of int-soft sets, we have introduced the concept of (strong) int-soft filters in

Work is ongoing. Some important issues for future work are (1) to develop strategies for obtaining more valuable results, (2) to apply these notions and results for studying related notions in other (soft) algebraic structures; and (3) to study the notions of implicative int-soft filters and Boolean int-soft filters.

The authors wish to thank the anonymous reviewer(s) for their valuable suggestions. This work (RPP-2012-021) was supported by the Fund of Research Promotion Program, Gyeongsang National University, 2012.