Journal of Intelligent & Fuzzy Systems 26 (2014) 2993–3004 DOI:10.3233/IFS-130965 IOS Press 2993 Neutrosophic deductive filters on BL-algebras R.A. Borzooei ∗ , H. Farahani and M. Moniri Faculty of Mathematical Sciences, Shahid Beheshti University, G.C., Evin, Tehran, Iran Abstract. In this paper, we introduce the notions of neutrosophic deductive filter, Boolean neutrosophic deductive filter (BNDF) and implicative neutrosophic deductive filter (INDF) on BL-algebras as generalizations of the fuzzy corresponding versions. We also investigate some properties of these filters and drive some characterizations of them. The relation between BNDF and INDF is investigated and it is proved that every BNDF is an INDF, but the converse is true when certain condition is satisfied. Furthermore, we construct a quotient structure related to the neutrosophic deductive filter and prove certain isomorphism theorems. Keywords: BL-algebra, neutrosophic deductive filter, quotient structure 1. Introduction Fuzzy set theory was introduced by Zadeh in 1965 [11]. A fuzzy subset A of a set X is a function μ A : X → [0, 1], where for each x ∈ X, μ A (x) represents the grade of membership of the element x ∈ X to A. In [1], Atanassov introduced the intuitionistic fuzzy sets as a generalization of fuzzy sets. The intuition- istic fuzzy sets consider both membership degree and nonmembership degree. In 1998, neutrosophy has been proposed by Smaran- dache [9] as a new branch of philosophy in order to formally represent neutralities. The fundamental thesis of neutrosophy is that every idea has not only a cer- tain degree of truth and a certain degree of falsity but also an indeterminacy degree that have to be consid- ered independently from each other. In neutrosophic set theory, indeterminacy is measured explicitly and inde- pendently. This assumption is very important in many applications such as information fusion in which we try to combine the data from different sensors. As an exam- ple, suppose there are 10 voters during a voting process. One possible situation is that there are three yes votes, two no votes and five undecided ones. We note that this ∗ Corresponding author. R.A. Borzooei, Faculty of Mathemati- cal Sciences, Shahid Beheshti University, G.C., Evin, Tehran, Iran. E-mail: borzooei@sbu.ac.ir. example is beyond the scope of intuitionistic fuzzy set theory. In 1960 Abraham Robinson introduced non-standard analysis as a formalization of analysis and a branch of mathematical logic. In non-standard analysis a nonzero number ε is said to be infinitely small, or infinitesi- mal if and only if for all positive integers n, |ε|≤ 1/n. In this case the reciprocal δ = 1/ε will be infinitely large, or simply infinite, meaning that for all positive integers n, we have |δ| >n. The set of hyper-real num- bers is an extension of the set of real numbers which includes the class of infinite numbers and the class of infinitesimal numbers. The non-standard unit interval is ]0 - , 1 + [= 0 - ∪ [0, 1] ∪ 1 + . Here 0 - is the set of all hyper-real numbers 0 - ε, and 1 + is the set of all hyper- real numbers 1 + λ, where ε and λ are infinitesimal. If U is a set, a neutrosophic set defined on the universe U assigns to each element x ∈ U, a triple (T (x),I (x),F (x)), where T (x),I (x) and F (x) are stan- dard or non-standard elements of ]0 - , 1 + [. T is the degree of membership in the set U, I is the degree of indeterminacy-membership in the set U and F is the degree of nonmembership in the set U. In this paper we work with special netrosophic sets that their neutro- sophic elements are standard real numbers in [0,1]. Neutrosophy has laid the foundation for a whole fam- ily of new mathematical theories, such as neutrosophic 1064-1246/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved