Neutrosophic Crisp Set Theory

—The focus of this paper is to propose a new notion of neutrosophic crisp sets via neutrosophic crisp ideals and to study some basic operations and results in neutrosophic crisp topological spaces. Also, neutrosophic crisp L-openness and neutrosophic crisp L- continuity are considered as a generalizations for a crisp and fuzzy concepts. Relationships between the above new neutrosophic crisp notions and the other relevant classes are investigated. Finally, we define and study two different types of neutrosophic crisp functions. Index Terms —Neutrosophic Crisp Set; Neutrosophic Crisp Ideals; Neutrosophic Crisp L-open Sets; Neutrosophic Crisp L- Continuity; Neutrosophic Sets. I. I NTRODUCTION The fuzzy set was introduced by Zadeh [20] in 1965, where each element had a degree of membership. In 1983 the intuitionstic fuzzy set was introduced by K. Atanassov [1, 2, 3] as a generalization of fuzzy set, where besides the degree of membership and the degree of non- membership of each element. Salama et al [11] defined intuitionistic fuzzy ideal and neutrosophic ideal for a set and generalized the concept of fuzzy ideal concepts, first initiated by Sarker [19]. Smarandache [16, 17, 18] defined the notion of neutrosophic sets, which is a generalization of Zadeh's fuzzy set and Atanassov's intuitionistic fuzzy set. Neutrosophic sets have been investigated by Salama et al. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In this paper is to introduce and study some new neutrosophic crisp notions via neutrosophic crisp ideals. Also, neutrosophic crisp L-openness and neutrosophic crisp L- continuity are considered. Relationships between the above new neutrosophic crisp notions and the other relevant classes are investigated. Recently, we define and study two different types of neutrosophic crisp functions. The paper unfolds as follows. The next section briefly introduces some definitions related to , implies neutrosophic set theory and some terminologies of neutrosophic crisp set and neutrosophic crisp ideal. Section 3 presents neutrosophic crisp L- open and neutrosophic crisp L- closed sets. Section 4 presents neutrosophic crisp L–continuous functions. Conclusions appear in the last section. II. P

The focus of this paper is to propose a new notion of neutrosophic crisp sets via neutrosophic crisp ideals and to study some basic operations and results in neutrosophic crisp topological spaces. Also, neutrosophic crisp L-openness and neutrosophic crisp Lcontinuity are considered as a generalizations for a crisp and fuzzy concepts. Relationships between the above new neutrosophic crisp notions and the other relevant classes are investigated. Finally, we define and study two different types of neutrosophic crisp functions. Index Terms—Neutrosophic Crisp Set; Neutrosophic Crisp Ideals; Neutrosophic Crisp L-open Sets; Neutrosophic Crisp L- Continuity; Neutrosophic Sets. I. INTRODUCTION The fuzzy set was introduced by Zadeh [20] in 1965, where each element had a degree of membership. In 1983 the intuitionstic fuzzy set was introduced by K. Atanassov [1, 2, 3] as a generalization of fuzzy set, where besides the degree of membership and the degree of non- membership of each element. Salama et al [11] defined intuitionistic fuzzy ideal and neutrosophic ideal for a set and generalized the concept of fuzzy ideal concepts, first initiated by Sarker [19]. Smarandache [16, 17, 18] defined the notion of neutrosophic sets, which is a generalization of Zadeh's fuzzy set and Atanassov's intuitionistic fuzzy set. Neutrosophic sets have been investigated by Salama et al. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In this paper is to introduce and study some new neutrosophic crisp notions via neutrosophic crisp ideals. Also, neutrosophic crisp L-openness and neutrosophic crisp L- continuity are considered. Relationships between the above new neutrosophic crisp notions and the other relevant classes are investigated. Recently, we define and study two different types of neutrosophic crisp functions. The paper unfolds as follows. The next section briefly introduces some definitions related to neutrosophic set theory and some terminologies of neutrosophic crisp set and neutrosophic crisp ideal. Section 3 presents neutrosophic crisp L- open and neutrosophic crisp Lclosed sets. Section 4 presents neutrosophic crisp L– continuous functions. Conclusions appear in the last section.

The purpose of this paper is to introduce a new types of crisp sets are called the neutrosophic crisp set with three types 1, 2, 3. After given the fundamental definitions and operations, we obtain several properties, and discussed the relationship between neutrosophic crisp sets and others. Finally, we introduce and study the notion of neutrosophic crisp relations.

Our work announces novel concepts called neutrosophic crisp i-open sets, neutrosophic crisp inter-open sets and neutrosophic crisp ii-open sets by generalizing neutrosophic crisp open set in neutrosophic crisp topological space. Numerous properties and characterizations of these concepts are studied, and the relationships of these concepts with various other concepts of neutrosophic crisp open sets are illustrated.

In this paper, the neutrosophic crisp minimal structure which is a more general structure than the neutrosophic minimal structure is built on neutrosophic crisp sets. The necessary arguments which are neutrosophic minimal crisp open set, neutrosophic minimal crisp closed set, neutrosophic crisp minimal closure, and neutrosophic crisp minimal interior are defined and their basic properties are presented. Also, the neutrosophic crisp minimal structure subspace of neutrosophic crisp minimal structure is defined and studied some of its properties. Finally, many examples are presented.

The purpose of this paper is to introduce a new types of crisp sets are called the neutrosophic crisp set with three types 1, 2, 3. After given the fundamental definitions and operations, we obtain several properties, and discussed the relationship between neutrosophic crisp sets and others. Finally, we introduce and study the notion of neutrosophic crisp relations.

This study reveals new concepts of neutrosophic crisp closed sets named neutrosophic crisp g-closed sets, neutrosophic crisp αg-closed sets, neutrosophic crisp gα-closed sets, and neutrosophic crisp gαg-closed sets. Furthermore, their ultimate features in neutrosophic crisp topological spaces are examined. Moreover, the consequent new concepts are introduced, such as neutrosophic crisp gαg-closure and neutrosophic crisp gαg-interior and finding some of their characteristics.

In this paper, we aim to apply the concepts of the neutrosophic crisp sets and its operations to the classical mathematical morphological operations, introducing what wecall Neutrosophic Crisp Mathematical Morphology. Sever-al operators are to be developed, including the neutrosophic crisp dilation, the neutrosophic crisp erosion, the neutrosoph-ic crisp opening and the neutrosophic crisp closing. Moreover, we extend the definition of some morphological filters using the neutrosophic crisp sets concept. For instance, we introduce the neutrosophic crisp boundary extraction, the neutrosophic crisp Top-hat and the neutrosophic crisp Bot-tom-hat filters. The idea behind the new introduced operators and filters is to act on the image in the neutrosophic crisp domain instead of the spatial domain.

In this paper we present a new neutrosophic crisp family generated from the three components’ neutrosophic crisp sets presented by Salama [4]. The idea behind Salam’s neutrosophic crisp set was to classify the elements of a universe of discourse with respect to an event ”A” into three classes: one class contains those elements that are fully supportive to A, another class contains those elements that totally against A, and a third class for those elements that stand in a distance from being with or against A. Our aim here is to study the elements of the universe of discourse which their existence is beyond the three classes of the neutrosophic crisp set given by Salama. By adding more components we will get a four components’ neutrosophic crisp sets called the Ultra Neutrosophic Crisp Sets. Four types of set’s operations is defined and the properties of the new ultra neutrosophic crisp sets are studied. Moreover, a definition of the relation between two ultra neutrosophic crisp sets is given.

Since the world is full of indeterminacy, the neutro- sophics found their place into contemporary research. The pur- pose of this paper is to introduce a new type of neutrosophic crisp set as the *- neutrosophic crisp sets as a generalization to star intuitionistic set due to Indira et al.(4 ), and study some of its properties. Finally we introduce and study the notion of *- neutrosophic relation and some of its properties. .

This paper deals with the application of Neutrosophic Crisp sets (which is a generalization of Crisp sets) on the classical probability, from the construction of the Neutrosophic sample space to the Neutrosophic crisp events reaching the definition of Neutrosophic classical probability for these events. Then we offer some of the properties of this probability, in addition to some important theories related to it. We also come into the definition of conditional probability and Bayes theory according to the Neutrosophic Crisp sets, and eventually offer some important illustrative examples. This is the link between the concept of Neutrosophic for classical events and the neutrosophic concept of fuzzy events. These concepts can be applied in computer translators and decision-making theory.

The main goal of this paper is to propose a new type of separation axioms via neutrosophic crisp semi open sets and neutrosophic crisp points in neutrosophic crisp topological spaces, namely neutrosophic crisp semi separation axioms. Finally, we examine the relationship between them in details. And also includes the study of the connections between these neutrosophic crisp semi separation axioms and the existing neutrosophic crisp separation axioms. Moreover, many examples are presented, to illustrate the concepts introduced in this paper. and investigate their fundamental properties, relationships and characterizations.

The general direction for generating any stable neutrosophic crisp topology is through base or the stable neutrosophic crisp interior concept, which is closed in the finite intersection process and not closed in the union process, likewise the stable neutrosophic crisp exterior is closed in the finite union process but not closed in finite intersection. Our research deals with finding necessary and sufficient condition for the finite union and finite intersection to be closed respectively using the concept of confused crisp sets.

Neutrosophic sets and Logic plays a significant role in approximation theory. It is a generalization of fuzzy sets and intuitionistic fuzzy set. Neutrosophic set is based on the neutrosophic philosophy in which every idea Z, has opposite denoted as anti(Z) and its neutral which is denoted as neut(Z). This is the main feature of neutrosophic sets and logic. This chapter is about the basic concepts of neutrosophic sets as well as some of their hybrid structures. This chapter starts with the introduction of fuzzy sets and intuitionistic fuzzy sets respectively. The notions of neutrosophic set are defined and studied their basic properties in this chapter. Then we studied neutrosophic crisp sets and their associated properties and notions. Moreover, interval valued neutrosophic sets are studied with some of their properties. Finally, we presented some applications of neutrosophic sets in the real world problems.

In this paper we introduce the notion of filter on the neutrosophic crisp set, then we consider a generalization of the filter’s studies. Afterwards, we present the important neutrosophic crisp filters. We also study several relations between different neutrosophic crisp filters and neutrosophic topologies. Possible applications to database systems are touched upon.