Sobhy A. El-Sheikh

On fuzzy bitopological spaces

Abstract In this paper, we have used the fuzzy topologies τ1 and τ2 to generate a family τs which is a supra fuzzy topology on X. Using the family τs, we introduce and study the concepts of separation axioms, continuity (resp. openness, closedness) of a mapping and compactness for a fuzzy bitopological space (X, τ1, τ2). Our definitions preserve much of the correspondence between concepts of fuzzy bitopological spaces and the associated fuzzy topological spaces. We then investigate the relationship between these concepts and their correspondence with the fuzzy bitopological spaces (Kandil and El-Shafee, 1991).

Decompositions of Some Types of Supra Soft Sets and Soft Continuity

In this paper we introduce the notion of supra soft topological spaces. We extend the notion of  -operation, pre-open soft sets,  -open soft sets, semi-open soft sets and  -open soft sets to such spaces and study their properties and the relations between them. Also, we introduce the concepts of supra pre (resp.  - , semi-,  -) continuous soft functions on these spaces and study some of their properties. We show that a mapping between two soft topological spaces is supra  -continuous soft if and only if it is supra pre-continuous soft and supra semi -continuous soft. The importance of this approach is that, the class of supra soft topological spaces is wider and more general than the class of soft topological spaces.

Generalized rough multiset via multiset ideals

In this paper, ideals in the context of multisets on the lattice of all submultisets with the order relation as the multiset inclusion have been introduced. Moreover, generalization of rough multiset model by defining new multiset approximation operators in more general setting via multiset ideal has been presented. The concepts of lower and upper multiset approxima- tions via multiset ideals have been mentioned. These new definitions decrease the upper multiset approximation and increase the lower multiset approximation and hence decreasing the boundary region and increasing the accuracy measure. Properties of these multiset approximations are studied and various examples are mentioned. Also, the comparison between the rough multiset approximations defined by Girish and John (11, 12) and the current multiset approximations has been presented. Its therefore shown that the current definitions are more generally. Finally, the multiset topology induced by the present methods is finer than the multiset topology induced by the old method (11, 12).

Covering-based rough fuzzy sets and binary relation

Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. In this paper we study covering-based rough fuzzy sets in which a fuzzy set can be approximated by the intersection of some elements in a covering of the universe of discourse. Some properties of the covering-based fuzzy lower and upper approximation operators are examined. We present the conditions under which two coverings generate the same covering-based fuzzy lower and upper approximation. We approximate fuzzy sets based on a binary relation and its properties are introduced. Finally, we establish the equivalency between rough fuzzy sets generated by a covering and rough fuzzy sets generated by a binary relation.

A new approach to fuzzy bitopological spaces

Abstract Given a fuzzy bitopological space ( X , τ 1 , τ 2 ), we have used the operator C 12 : I X → I X defined by C 12 (μ)=τ 1 − cl (μ)∩τ 2 − cl (μ), μ∈I X , which is a supra fuzzy closure operator [Fuzzy Sets and Systems 74 (1995) 353], to generate a family τ s which is a supra fuzzy topology [loc. cit., 1995]. So the space ( X , τ s ) is a supra fuzzy topological space associated to the fuzzy bitopological space ( X , τ 1 , τ 2 ). We extend the notions of α . FT i , strong FT i and ultra FT i due to [Fuzzy Sets and System 43 (1991) 95] to the space ( X , τ s ). The properties of fuzzy bitopological spaces have been studied by using their associated supra fuzzy topological spaces. We introduce the notions of α . FP * T i , S. FP * T i and U. FP * T i by using the level supratopology ι α ( τ s ) of τ s . We investigate the relationship between these notions, the notions FP * T i [loc. cit., 1995], i =1,2,3,4 and the notions α . FPT i (resp. S. FPT i ,U. FPT i ) [Fuzzy Sets and Systems 105 (1999) 459]. We give a number of examples, which illustrate that these concepts are not equivalent. Also we extend the notions of α F-continuous (open), strong F-continuous (open) and ultra F-continuous (open) due to [Fuzzy Sets and Systems 38 (1990) 115] to the class of supra fuzzy topological spaces associated to the class of fuzzy bitopological spaces. We study some properties of α . FP * T i (resp. S. FP * T i ,U. FP * T i ) under these mappings.

Strong and ultra separation axioms of fuzzy bitopological spaces

Given a fuzzy bitopological space (X, τ1, τ2), we introduce a new notion of fuzzy pairwise separation axioms by using the family of its level bitopologies ια(τ1), ια(τ2), α ϵ[0,1). We prove that these concepts are good extension and we compare them with its corresponding FPT; (Kandil and EI-Shafee, 1991) and FPT1∗ (Abu Safiya et al., 1994) (i = 0, 1, 2, 3, 4), respectively. We show that these notions are not equivalent and we give a number of examples which illustrate this fact.

Comment on “Rough Multisets and Information Multisystems”

We show that some results introduced in Girish and John (2011) are incorrect. Moreover, a counterexample is given to confirm our claim. Furthermore, the correction form of the incorrect results in Girish and John (2011) is presented.

A note on “separation axioms in fuzzy bitopological spaces”

In this note we show that some results introduced in (3) are incorrect. These results were first introduced in fuzzy topological spaces (1, 2), and very recently generalized into fuzzy bitopological spaces (3). We give some counter examples to confirm our claim.

Compactness in fuzzy (fuzzy supra) topological spaces

The notion of Qα-compact fuzzy topological spaces was first introduced by Zhongfu (Kexue Tongbao 29(5) (1984) 582–585). In this paper we extend this notion to fuzzy supra topological spaces and study some of their properties. We also introduce the notion of fuzzy Qα-almost compact (Qα-almost supracompact). This notion is defined for any fuzzy subset and is a good extension. Finally, we try to extend such a notion to L-fuzzy (supra) topological spaces and discuss some of their properties.