In this paper , the authors introduce and investigate the concept of strongly generalized g -closed sets(briefly strongly g -closed set) and investigate the relation between the associated topology.

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Strongly g -Closed Sets in Topological Spaces

T he aim of this paper is to introduce and study new class of sets called g*α-closed sets. This new class of sets lies between closed sets and αg-closed sets. Applying these sets, we introduce new spaces namely, -Space, -Space, -Space -Space, -Spaces are also introduced.

• g*α-CLOSED SETS IN TOPOLOGICAL SPACES

In this paper, we introduce several types of generalized closed sets in generalized topological spaces (GTSs). Their interrelationships are investigated and several characterizations of μ-T0, μ-T1, μ-T1/2, μ-regular, μ-normal GTSs and extremally μ-disconnected GTSs are obtained.

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On generalized closed sets in generalized topological spaces

I n this paper we introduce and study a new class of generalized closed sets called y * 𝛼 -closed sets in topological spaces.  We analyze the relations   between y * 𝛼 -closed sets with already existing closed sets. We discuss some basic properties of y * 𝛼 -closed sets The class of y * 𝛼 -closed sets is properly  placed  between the class  of 𝛼 -closed sets and the class of (resp. y )-closed  sets. We prove that the class of y * 𝛼 - closed sets form a topology.

• Between -closed sets and g ̃_α -closed sets

In this paper, we introduce a new class of sets namely, h – closed sets and their properties, Applying these sets, we introduce new spaces namely αTh , cTh and sTh spaces.

ON h - CLOSED SETS IN TOPOLOGICAL SPACES

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Between Closed Sets and g-Closed Sets

Circulant matrix is a square matrix whose rows are obtained by cyclically rotating by its first row. In this paper, we define some operations on circulant interval valued fuzzy matrices (CIVFMs). Some elementary ope rators on circulant interval valued fuzzy matrices (CIVFMs) are presented here. The idea of reflexive, symmetric, transitive, determinant and adjoint of circulant in terval valued fuzzy matrices (CIVFMs) are also defined. Keywords. Circulant matrices, circulant interval valued fuzz y matrices determinant, fuzzy matrices, interval valued fuzzy matrices. AMS Mathematics Subject Classification (2010): 03E72,15B15 1. Introduction The concept of fuzzy set was introduced by Zadeh [1 7] in 1965. Fuzzy Matrix (FM) is a very important topic in Fuzzy Algebra. In FM, the el ments belongs to the interval [0,1]. When the elements of FM are subintervals of the un it interval [0,1], then the FM is known as interval valued Fuzzy Matrix(IVFM). Thomason [16] defined fuzzy Matrices for the firs t time in 1977 and discussed about the convergence of the powers of fuzzy matri x. Several authors presented number of results on the convergence of power sequence of fuzzy matrices [3, 6]. Ragab and Emam [12] presented some properties on determinant and adjoint of square fuzzy matrix. Ragab and Emam [13] introduced some properties of the min-max composition of fuzzy matrix. Kim [5] investigated some important results on determinant of a square fuzzy matrices. Pal [9,19] introduced the concept of interval-valu ed fuzzy matrices with intervalvalued fuzzy rows and columns. Shyamal and Pal [14, 15,18] introduced two new operators and applications of fuzzy matrices. Bhowm ik and Pal [2,20] introduced the concept of circulant triangular fuzzy number matri ces (TFNMs) and some result on TNFMs. Hemasinha, Pal and Bezdek [4] studies the m axin iterates of fuzzy circulant matrices. In this paper, the concept of circulant interval -v alued fuzzy matrices (CIVFMs) are defined with some of its properties. The determinan t, some binary operations on circulant C.Loganathan and V.Pushpalatha 314 interval -valued fuzzy matrix are defined and som e i portant theorems are proved with examples. 2. Circulant interval valued fuzzy matrices Definition 2.1. [14] An m×n matrix = [ ] whose components are in the unit interval [0, 1] is called a fuzzy matrix. Definition 2.2. The determinant | | of an × fuzzy matrix is defined as follows; | | = ∑ ( ) ( ) ...... ( ) ∈ where denotes the symmetric group of all permutations of the indices 1, 2,... . . Definition 2.3. [9] An interval-valued fuzzy matrix(IVFM) of order m ×n is defined as = ( ) × where = !, "# is the $%&' element of represents the membership value. All the elements of an IVFM are i nt rvals and all the intervals are the subintervals of the interval [0, 1]. Definition 2.4. [12] The interval-valued fuzzy determinant (IVFD) of an IVFM of order n×n is denoted by | | or ()*( ) and we defined as | | = ∑ ( ) ( ) ............ ( ) ∈ = ∑ ∏ ( ) , ∈ . where ( ) = [ ( )!, ( )"] and denotes the symmetric group of all permutations of the indices 1, 2, . . . . ... . Definition 2.5. [2] An Interval Valued Fuzzy Matrix (IVFM) is said to be circulant interval valued fuzzy matrix if all the elements of can be determined completely by its first row. The first row of is [[ !, "] [ ! , "]. . . . . . . . . . . . . [ ! , "]] Then any element = [ !, "] of can be determined (throughout the element of the first row) as  = ( . . ) with ( ./) = / A circulant IVFM is of the form = 01 11 11 2[ !, "] [ ! , "]......... ...... . . [ !, "] [ ! , "] [ ! , "]......... ( )!, ( )"# ......................................................... ...................................................... . . ...................................................... . . [ 3!, 3"] [ 4!, 4"]............ . . . [ !, "] [ !, "] [ 3!, 3"]......... ....... [ !, "] 56 66 66 7 . Remark 2.1. An IVFM is circulant if and only if [ !, "] = [ (/⊕ )(/⊕ )! , (/⊕ )(/⊕ )"] , for every $, %, 9 ∈ 1, 2, 3, . . . . . . . , where ⊕ is sum modulo n. Here all the diagonal elements are equal. Remark 2.2. For a circulant IVFM we notice that [ ! , "] = [ ( ⊕ ) !, ( ⊕ ) "] and !, "# = [ ( ⊕ )!, ( ⊕ )"], for every $, % ∈ 1, 2, . . . . . . . . . . . . Circulant Interval Valued Fuzzy Matrices 315 Remark 2.3. For a circulant IVFM we notice that ; < ⊕( )= !, < ⊕( )= "> = ( ⊕ )!, ( ⊕ ) ( ⊕ )"# for every $, % ∈ 1, 2, . . . . . . . . . . . . Theorem 2.1. An × IVFM is circulant if and only if @ = @ , where @ is the permutation matrix of IVFM. @ = 01 11 11 2[0,0] [0,0]......... [0,0] [1,1] [1,1] [0,0]......... [0,0] [0,0] .......................................... . .............................. . .......... . .......................................... . [0,0] [0,0]......... [0,0] [0,0] [0,0] [0,0]......... [1,1] [0,0]56 66 66 7

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C. Loganathan

Papers

Between Closed Sets and g-Closed Sets

Circulant Interval Valued Fuzzy Matrices