الفهرس 
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Abstract
The notion of fuzzy sets was created by L.A.Zadeh [61] in 1965.
This notion played an important role in applications and in such
mathematical theories as Logic and Topology.
In 1968, Chang [11] used this concept to introduce the concept of
fuzzy topological spaces. Subsequently, Wong [57, 58], Lowen [26], and
others extend some basic concepts from general topology to fuzzy
subsets and developed a theory of fuzzy topological spaces. We denote
by FTS the category of Lowen’s fuzzy topological spaces and their
continuous functions. In 1982, Lowen [30], introduced the concept of
fuzzy neighbourhood spaces the category of which is denoted by FNS.
Many papers have appeared analyzing and making use of this concept.
Some papers showed the distinguished internal characterizations of FNS
(e.g. [42], [59]).
Given a lower semi-continuous triangular norm T, Morsi [19, 46,
47] introduced the two categories of fuzzy T-Iocality spaces and of fuzzy
T-neighbourhood spaces, denoted by T-FLS and T-FNS, respectively. In
the special case T = Min, the categories T-FLS and T-FNS coincide with
the categories of fuzzy T-neighbourhood -base spaces, denoted by FNBS
[44] and FNS, respectively.
On another side, ill 1982, Z. Pawlak introduced the concept of
rough sets and of approximation spaces [50] in accordance of an
indescernibility relation (e.g. equivalence relation). This concept played
an important role in applications especially in medicine and artificial
intelligence (e.g.[5I]). In 1985, Z. Pawlak introduced the relationship
between rough sets and fuzzy sets [52].
In 1986, L. Farinas and H. Prade introduced the concept of fuzzy
rough sets and of fuzzy approximation spaces [17].
We now give a brief overview of the contents of the thesis. It
consists of five chapters:
Chapter I is devoted to give an exposition of some needed
definitions and results to be used throughout the thesis.
In Chapter II, we introduce the axioms of fuzzy T-upper (lower)
approximation operators on IX for a lower semi-continuous triangular
norm T and study the relationships between them and that between each
of them and T-similarity relation on X. We generalize the definition of
fuzzy upper approximation operators introduced by L. Farinas and H.
Prarle in [17]. In the case T = Min, the reader may notice that our
definition of the upper approximation operator AR coincides on the
corresponding definition introduced in [17]. We prove many properties
of A more than that mentioned in [17];·whence we extend the theory of
fuzzy rough sets. We introduce a definition of fuzzy T-Iower
approximation operator of fuzzy rough set (X, .R) differs that introduced
in [17]; in the case T = Min, the reader may notice that our definition of
the lower approximation operator &is different from the corresponding
definition introduced in [17]. Their definition does not satisfy our
identities AA~ = Afl and AAfl = Afl for all ~EIx which satisfied in the
classical case in [50]. On the other hand, their definition is designed to
satisfy A(l-fl) = I-A~ for all flEIx. This identity is not valid for our
definition.