Introduction to Neutrosophic BCI/BCK-Algebras

Abstract

We introduce the concept of neutrosophic BCI/BCK-algebras. Elementary properties of neutrosophic BCI/BCK algebras are presented.

1. Introduction

Logic algebras are the algebraic foundation of reasoning mechanism in many fields such as computer sciences, information sciences, cybernetics, and artificial intelligence. In 1966, Imai and Iséki [1, 2] introduced the notions, called BCK-algebras and BCI-algebras. These notions are originated from two different ways: one of them is based on set theory; another is from classical and nonclassical propositional calculi. As is well known, there is a close relationship between the notions of the set difference in set theory and the implication functor in logical systems. Since then many researchers worked in this area and lots of literatures had been produced about the theory of BCK/BCI-algebra. On the theory of BCK/BCI-algebras, for example, see [2–6]. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. MV-algebras were introduced by Chang in [7], in order to show that Lukasiewicz logic is complete with respect to evaluations of propositional variables in the real unit interval . It is well known that the class of MV-algebras is a proper subclass of the class of BCK- algebras.

By a BCI-algebra we mean an algebra of type satisfying the following axioms, for all :(1),(2),(3),(4) and imply .We can define a partial ordering by if and only if .

If a BCI-algebra satisfies for all , then we say that is a BCK-algebra. Any BCK-algebra satisfies the following axioms for all :(1),(2),(3),(4), .Let be a BCK-algebra. Consider the following:(1) is said to be commutative if for all we have ;(2) is said to be implicative if for all , we have .

In 1995, Smarandache introduced the concept of neutrosophic logic as an extension of fuzzy logic; see [8–10]. In 2006, Kandasamy and Smarandache introduced the concept of neutrosophic algebraic structures; see [11, 12]. Since then, several researchers have studied the concepts and a great deal of literature has been produced. Agboola et al. in [13–17] continued the study of some types of neutrosophic algebraic structures.

Let be a nonempty set. A set generated by and is called a neutrosophic set. The elements of are of the form , where and are elements of .

In the present paper, we introduce the concept of neutrosophic BCI/BCK-algebras. Elementary properties of neutrosophic BCI/BCK-algebras are presented.

2. Main Results

Definition 1. Let be any BCI/BCK-algebra and let be a set generated by and . The triple is called a neutrosophic BCI/BCK-algebra. If and are any two elements of with , we define An element is represented by and represents the constant element in . For all , we define where is the negation of in .

Example 2. Let be any commutative neutrosophic group. For all define Then is a neutrosophic BCI-algebra.

Example 3. Let be a neutrosophic set and let and be any two nonempty subsets of . Define Then is a neutrosophic BCK-algebra.

Theorem 4. Every neutrosophic BCK-algebra is a neutrosophic BCI-algebra.

Proof. It is straightforward.

Theorem 5. Every neutrosophic BCK-algebra is a BCK-algebra and not the converse.

Proof. Suppose that is a neutrosophic BCK-algebra. Let , , and be arbitrary elements of . Then we have the following. (1)We have where Hence, Now, we obtain Also, we have This shows that and, consequently, .(2)We have where Then, Therefore, we obtain where Since , it follows that .(3)We have (4)Suppose that and . Then and from which we obtain and . These imply that and and therefore, , , , and from which we obtain and . Hence, ; that is, .(5)We have Items (1)–(5) show that is a BCK-algebra.

Lemma 6. Let be a neutrosophic BCK-algebra. Then , if and only if .

Proof. Suppose that . Then which implies that from which we obtain . The converse is obvious.

Lemma 7. Let be a neutrosophic BCI-algebra. Then for all ,(1);(2).

Theorem 8. Let be a neutrosophic BCK-algebra. Then for all , (1) implies that and ;(2);(3).

Proof. (1) Suppose that . Then from which we obtain Now, Hence, where This shows that and so . Similar computations show that .
(2) Put where Therefore, Now, we have Thus, Similarly, it can be shown that
(3) Put where Thus, we have Now, where Since , it follows that . Hence this completes the proof.

Theorem 9. Let be a neutrosophic BCI/BCK-algebra. Then (1) is not commutative even if is commutative;(2) is not implicative even if is implicative.

Proof. (1) Suppose that is commutative. Let . Then where Also, where This shows that and therefore is not commutative.
(2) Suppose that is implicative. Let . Then where Hence, and so is not implicative.

Definition 10. Let be a neutrosophic BCI/BCK-algebra. A nonempty subset is called a neutrosophic subalgebra of if the following conditions hold:(1);(2) for all ;(3) contains a proper subset which is a BCI/BCK-algebra.If does not contain a proper subset which is a BCI/BCK-algebra, then is called a pseudo neutrosophic subalgebra of .

Example 11. Any neutrosophic subgroup of the commutative neutrosophic group of Example 2 is a neutrosophic BCI-subalgebra.

Theorem 12. Let be a neutrosophic BCK-algebra and for let be a subset of defined by Then,(1) is a neutrosophic subalgebra of ;(2).

Proof. (1) Obviously, and contains a proper subset which is a BCK-algebra. Let . Then and from which we obtain , , , and . Since , we have . Now, This shows that and the required result follows.
(2) Follows.

Definition 13. Let and be two neutrosophic BCI/BCK-algebras. A mapping is called a neutrosophic homomorphism if the following conditions hold:(1), ;(2).In addition,(3)if is injective, then is called a neutrosophic monomorphism;(4)if is surjective, then is called a neutrosophic epimorphism;(5)if is a bijection, then is called a neutrosophic isomorphism. A bijective neutrosophic homomorphism from onto is called a neutrosophic automorphism.

Definition 14. Let be a neutrosophic homomorphism of neutrosophic BCK/BCI-algebras. Consider the following:(1);(2).

Example 15. Let be a neutrosophic BCI/BCK-algebra and let be a mapping defined by Then is a neutrosophic isomorphism.

Lemma 16. Let be a neutrosophic homomorphism from a neutrosophic BCI/BCK-algebra into a neutrosophic BCI/BCK-algebra . Then .

Proof. It is straightforward.

Theorem 17. Let be a neutrosophic homomorphism of neutrosophic BCK/BCI-algebras. Then is a neutrosophic monomorphism if and only if .

Proof. The proof is the same as the classical case and so is omitted.

Theorem 18. Let , , and be neutrosophic BCI/BCK-algebras. Let be a neutrosophic epimorphism and let be a neutrosophic homomorphism. If , then there exists a unique neutrosophic homomorphism such that . The following also hold:(1);(2);(3) is a neutrosophic monomorphism if and only if ;(4) is a neutrosophic epimorphism if and only if is a neutrosophic epimorphism.

Proof. The proof is similar to the classical case and so is omitted.

Theorem 19. Let , , be neutrosophic BCI/BCK-algebras. Let be a neutrosophic homomorphism and let be a neutrosophic monomorphism such that . Then there exists a unique neutrosophic homomorphism such that . Also,(1);(2);(3) is a neutrosophic monomorphism if and only if is a neutrosophic monomorphism;(4) is a neutrosophic epimorphism if and only if .

Proof. The proof is similar to the classical case and so is omitted.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.