ON NEUTROSOPHIC PARACONSISTENT TOPOLOGY

ON NEUTROSOPHIC PARACONSISTENT TOPOLOGY FRANCISCO GALLEGO LUPIÁÑEZ Dept. of Mathematics Univ. Complutense 28040 Madrid, SPAIN e-mail: FG_Lupianez@Mat.UCM.Es July 10, 2009 Abstract Purpose- Recently, F.Smarandache generalized the Atanassov’s intuitionistic fuzzy sets and other kinds of sets to neutrosophic sets. Also, this author introduced a general de…nition of neutrosophic topology.

On the other hand, there exist various kinds of paraconsistent Logics, where some contradiction is admissible. We show in this paper that a Smarandache’s de…nition of neutrosophic paraconsistent topology is not a generalization of Çoker’s intuitionistic fuzzy topology or Smarandache’s general neutrosophic topology. .Design /methodology/ approach- The possible relations between the intuitionistic fuzzy topology and the neutrosophic paraconsistent topology are studied.

Findings-Relations on intuitionistic fuzzy topology and neutrosophic paraconsistent topology.

Research limitations/implications- Clearly, the paper is con…ned to IFSs and NSs. Practical implications-The main applications are in the mathematical …eld.

Originality/value- The paper shows original results on fuzzy sets and Topology.

Keywords: Logic, Set-Theory,Topology, Atanassov’s IFSs, neutrosophic sets, paraconsistent sets.

Article type: Research paper

1. Introduction.

In various recent papers, F. Smarandache (1998, 2002, 2003, 2005a) generalizes intuitionistic fuzzy sets (IFSs) and other kinds of sets to neutrosophic sets

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(NSs). In (Smarandache, 2005a) some distinctions between NSs and IFSs are underlined.

The notion of intuitionistic fuzzy set de…ned by K.T. Atanassov (1983) has been applied by Çoker (1997) for study intuitionistic fuzzy topological spaces.

This concept has been developed by many authors (Bayhan and Çoker,2003; Çoker, 1996, 1997; Çoker and E¸s,1995; E¸s and Çoker,1996; Gürçay, Çoker and E¸s, 1997; Hanafy, 2003; Hur, Kim and Ryou, 2004; Lee and Lee, 2000; Lupiáñez, 2004a, 2004b, 2006a, 2006b, 2007, 2008;Turanh and Çoker, 2000).

F. Smarandache also de…ned the general neutrosophic topology on a neutrosophic set (Smarandache, 2005b).

On the other hand, various authors (Priest and other, 1989) worked on "paraconsistent Logics", that is, logics where some contradiction is admissible.

We remark the theories exposed in (Da Costa 1958, Routley and other 1982, and Peña 1987).

Smarandache de…ned also the neutrosophic paraconsistent sets (Smarandache 2005a) and he proposed a natural de…nition of neutrosophic paraconsistent topology.

A problem that we consider is the possible relation between this concept of neutrosophic paraconsistent topology and the previous notions of general neutrosophic topology and intuitionistic fuzzy topology. We show in this paper that neutrosophic paraconsistent topology is not an extension of intuitionistic fuzzy topology.

2. Basic de…nitions.

First, we present some basic de…nitions: De…nition 1 Let X be a non-empty set. An intuitionistic fuzzy set (IFS for short) A, is an object having the form A = f< x; A ; A > =x 2 Xg where the functions A : X ! I and A : X ! I denote the degree of membership (namely A (x)) and the degree of nonmembership (namely A (x)) of each element x 2 X to the set A, respectively, and 0 1 for each x 2 X. A (x) + A (x) (Atanassov, 1983).

De…nition 2 Let X be a non-empty set, and the IFSs A = f< x; A ; Xg, B = f< x; B ; B > jx 2 Xg. Let A = f< x; A ; A > jx 2 Xg A \ B = f< x; A ^ B ; A _ B > jx 2 Xg A [ B = f< x; A _ B ; A ^ B > jx 2 Xg:(Atanassov, 1988).

A

> jx 2

De…nition 3 Let X be a non-empty set. Let 0s = f< x; 0; 1 > jx 2 Xg and 1s = f< x; 1; 0 > jx 2 Xg:(Çoker, 1997).

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De…nition 4 An intuitionistic fuzzy topology (IFT for short) on a non-empty set X is a family of IFSs in X satisfying: (a) 0s ,1s 2 ; (b) G1 \ G2 2 for any G1 ; G2 2 , (c) [Gj 2 for any family fGj jj 2 Jg : In this case the pair (X; ) is called an intuitionistic fuzzy topological space (IFTS for short) and any IFS in is called an intuitionistic fuzzy open set (IFOS for short) in X. (Çoker, 1997).

De…nition 5 Let T , I,F be real standard or non-standard subsets of the nonstandard unit interval ] 0; 1+ [, with sup T = tsup , inf T = tinf sup I = isup , inf I = iinf sup F = fsup , inf F = finf and nsup = tsup + isup + fsup ninf = tinf + iinf + finf ; T , I,F are called neutrosophic components. Let U be an universe of discourse, and M a set included in U . An element x from U is noted with respect to the set M as x(T; I; F ) and belongs to M in the following way: it is t% true in the set, i% indeterminate (unknown if it is) in the set, and f % false, where t varies in T , i varies in I, f varies in F: The set M is called a neutrosophic set (NS). (Smarandache, 2005a).

Remark. All IFS is a NS. De…nition 6 Let M be a non-empty set. A general neutrosophic topology on M is a family of neutrosophic sets in M satisfying the following axioms: (a) 0s = x(0; 0; 1) ,1s = x(1; 0; 0) 2 (b) If A; B 2 , then A \ B 2 (c) If a family fAj jj 2 Jg ;then [Aj 2 : (Smarandache,2005b) De…nition 7 A neutrosophic set x(T; I; F ) is called paraconsistent if inf (T )+ inf (I) + inf (F ) > 1:(Smarandache,2005a) De…nition 8 For neutrosophic paraconsistent sets 0_ = x(0; 1; 1) and 1_ = x(1; 1; 0):(Smarandache).

Remark. If we use the unary neutrosophic negation operator for neutrosophic sets (Smarandache 2005b), nN (x(T; I; F )) = x(F; I; T ) by interchanging the thuth T and falsehood F components, we have that nN (0_) = 1_ .

De…nition 9 Let X be a non-empty set. A family of neutrosophic paraconsistent sets in X will called a neutrosophic paraconsistent topology if: (a) 0_ and 1_ 2 (b) If A; B 2 , then A \ B 2 (c) Any union of a subfamily of paraconsistent sets of is also in : (Smarandache).

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3. Results.

Proposition 1. The neutrosophic paraconsistent topology is not an extension of intuitionistic fuzzy topology.

Proof. We have that 0s =< x; 0; 1 > and 1s =< x; 1; 0 > are members of all intuitionistic fuzzy topology, but x(0; 0; 1) 2 j(0s ) 6= 0_, and, x(1; 0; 0) 2 j(1s ) 6= 1_: Proposition 2. A neutrosophic paraconsistent topology is not a general neutrosophic topology.

Proof. Let the family f1_; 0_g . Clearly it is a neutrosophic paraconsistent topology., but 0s ,1s are not in this family.

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