Novel Concepts of Bipolar Fuzzy BCK-Submodules

Alghamdi, M. A.; Muthana, N. M.; Alshehri, N. O.

doi:https://doi.org/10.1155/2017/2084191

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2017 | Article ID 2084191 | https://doi.org/10.1155/2017/2084191

Novel Concepts of Bipolar Fuzzy BCK-Submodules

M. A. Alghamdi,^1,2N. M. Muthana,¹and N. O. Alshehri¹

Academic Editor: Juan R. Torregrosa

Received03 Sept 2016

Accepted07 Nov 2016

Published23 Jan 2017

Abstract

Translations and multiplications of bipolar fuzzy BCK-submodules are discussed. Extensions of bipolar fuzzy BCK-submodules are introduced. Relations between translations and multiplications of bipolar fuzzy BCK-submodules are presented.

1. Introduction

Fuzzy set theory was first proposed by Zadeh [1] as a means of handling uncertainty that is due to imprecision or vagueness rather than to randomness. It permits the gradual assessment of the membership of elements in a set, where every element is given a degree of membership between 0 and 1.

As an extension of the traditional fuzzy sets, Lee [2] introduced the concept of bipolar-valued fuzzy sets where the membership degree range is enlarged from to . Bipolarity is important to distinguish between the following: (i) positive information that shows how much elements satisfy an associated property to the fuzzy set and (ii) negative information that demonstrates the satisfaction degree of elements to a counterproperty of the concerned fuzzy set. Many researchers were interested in studying bipolar-valued fuzzy sets and they applied them on several algebraic structures such as the following: BCK-algebras [3], hemirings [4], and semirings [5].

The notion of BCK-modules was presented by Abujabal et al. [6] in 1994. As a sequel of this work in field of the fuzzy set theory, Bakhshi [7] introduced the concept of fuzzy BCK-submodules and investigated their properties. Later, generalized fuzzy BCK-submodules were formulated by other researchers [8] with some obtained basic properties.

Recently, M. A. Alghamdi, N. O. Alshehri, and N. M. Muthana introduced the notion of bipolar fuzzy BCK-submodules and discussed many related results.

In this paper, we study translations, extensions, and multiplications of bipolar fuzzy BCK-submodules and investigate relations between them.

2. Preliminaries

In this section we provide basic definitions and results regarding BCK-algebras (modules), bipolar fuzzy BCK-submodules, and BCK-module homomorphisms.

By a BCK-algebra, we mean an algebra of type satisfying the following axioms:(I)(II)(III)(IV)(V) and implies , for all

Let be a BCK-algebra. Then is a partially ordered set with the partial ordering defined on by if and only if . is said to be bounded if there is an element such that for all . is said to be commutative (implicative) if for all , where .

Definition 1 (see [6]). Let be a BCK-algebra. Then by a left -module (abbreviated -module) we mean an abelian group with an operation with satisfies the following axioms for all and :(1)(2)(3)Moreover, if is bounded and satisfies , for all , then is said to be unitary.

Example 2. Any bounded implicative BCK-algebra forms an -module, where “+” is defined as and .
A subgroup of an -module is called submodule of if is also an -module.

Theorem 3 (see [7]). A subset of a BCK-module is a BCK-submodule of if and only if and for all and .

Definition 4 (see [6]). Let be modules over a BCK-algebra . A mapping is called -homomorphism if(1);(2) for all and .

A BCK-module homomorphism is said to be monomorphism (epimorphism) if it is one to one (onto). If it is both one to one and onto, then we say that it is an isomorphism.

Let be the universe of discourse. A bipolar-valued fuzzy set (abbreviated bipolar fuzzy set) of is an object having the formwhere and are mappings. The positive membership degree denotes the satisfaction degree of an element to the property corresponding to a bipolar-valued fuzzy set , and the negative membership degree denotes the satisfaction degree of to some implicit counterproperty of .

We shall use the symbol for the bipolar fuzzy set .

For a bipolar fuzzy set and , we definewhich are called the positive -level cut of and the negative -level cut of , respectively. For , the setis called the -level cut of (see [3]).

Definition 5 (see [9]). Let and be bipolar fuzzy sets of . If and for all , then we say that is a bipolar fuzzy extension of (simply is subset of ) and we write .

Definition 6. A bipolar fuzzy set of a BCK-module is said to be a bipolar fuzzy BCK-submodule if it satisfies(BFS1) and(BFS2) and ,(BFS3) and

For the sake of simplicity, we shall use the symbols and for the set of all bipolar fuzzy sets of and the set of all bipolar fuzzy BCK-submodules of , respectively.

Theorem 7. A bipolar fuzzy set if and only if(i) and ,(ii) and for all and

Theorem 8. Let . Then if and only if for all .

Definition 9. Let be a BCK-module homomorphism and let be a bipolar fuzzy set on Then the image of of under is defined as follows:We shall call the homomorphic image of under .

Theorem 10. Let be a BCK-module epimorphism and let be a bipolar fuzzy BCK-submodule of Then the homomorphic image of under is a bipolar fuzzy BCK-submodule on

Definition 11. Let be a homomorphism of BCK-modules and a bipolar fuzzy set of . Then the inverse image of , , is the bipolar fuzzy set on given by and for all .

Theorem 12. Let be a homomorphism of BCK-modules and , and then the inverse image .

Theorem 13. Let be an epimorphism of BCK-modules. If is a bipolar fuzzy set on such that the inverse image , then .

3. Fuzzy Translations and Fuzzy Multiplications of

Newly, many researchers have studied fuzzy translations and fuzzy multiplications and their effects on several types of fuzzy sets. Lee and others [10] discussed fuzzy translations and multiplications in the case of fuzzy subalgebras. Bipolar fuzzy translations in BCK/BCI algebras were introduced by Jun et al. [9]. Later, Jun [11] investigated translations of fuzzy ideals of BCK-algebras. In this section, we present fuzzy translations and multiplications of bipolar fuzzy BCK-submodules and discuss related properties.

In what follows, and are considered to be modules over some BCK-algebra , unless otherwise specified.

For any bipolar fuzzy set on , we denote

Definition 14. Let be a bipolar fuzzy set on and By a bipolar fuzzy -translation of we mean a bipolar fuzzy set , where

Theorem 15. Let and Then the -translation

Proof. Let and ThenAnalogously, Hence is a bipolar fuzzy BCK-submodule on

Theorem 16. Let such that the bipolar fuzzy -translation of is a bipolar fuzzy BCK-submodule on for some Then

Proof. Assume that for some For any and , we havethat is, Moreover, which implies that By similar argument, we have and Hence

Definition 17. Let and be bipolar fuzzy sets on Then is called a bipolar fuzzy -extension of if the following assertions hold:(i) is a bipolar fuzzy extension of (ii)If is a bipolar fuzzy BCK-submodule of , then so is

By means of the definition of -translation, we know that and for all Hence, according to Theorem 15, we have the following theorem.

Theorem 18. Let be a bipolar fuzzy BCK-submodule of and Then the bipolar fuzzy -translation is a bipolar fuzzy -extension of

The converse of Theorem 18 is not true in general as seen in the following example.

Example 19. Let along with a binary operation defined in (14), and then forms an unbounded commutative BCK-algebra. Let be a subset of with binary operation + defined by . Then is a commutative group as shown in (15) (see [12]). Define scalar multiplication by for all and that is given in (16). Consequently, forms an -module.Define a bipolar fuzzy set on as shown inThen is a bipolar fuzzy BCK-submodule on . Let be a bipolar fuzzy set on given byThen is also a bipolar fuzzy BCK-submodule on . Since and for all , hence is a bipolar fuzzy -extension of . But it is not a bipolar fuzzy -translation of for all .
For a bipolar fuzzy set on , consider the following two sets:where and
If is a bipolar fuzzy BCK-submodule on , then and are submodules of for all with and However, if we do not give the condition that is a bipolar fuzzy BCK-submodule on , then at least one of and may not be a submodule of as seen in the following example.

Example 20. Let be a commutative BCK-algebra with the binary operation defined by (20). Let be a subset of along with an operation + defined by (21). Then is a commutative group. Define the action of on by that is shown in (22). Consequently, a routine exercise of calculations shows that forms an -module (see [12]).Equation (23) define a bipolar fuzzy set on .For and , is not a submodule of since . Note that is not a bipolar fuzzy BCK-submodule on since .

Theorem 21. Let be a bipolar fuzzy set on and Then the -translation of is a bipolar fuzzy BCK-submodule of if and only if and are submodules of for all with and

Proof. Proof is immediately by Theorems 8, 15, and 16.

Theorem 22. Let be a bipolar fuzzy BCK-submodule of and let and If and , then the -translation of is a bipolar fuzzy -extension of the -translation .

Proof. It is straightforward.

Theorem 23. Let be a bipolar fuzzy BCK-submodule of and For every bipolar fuzzy -extension of the -translation , there exists such that and , and is a bipolar fuzzy -extension of the -translation

Proof. Assume that is a bipolar fuzzy -extension of the -translation Then and Pick such that and Then is an extension of and since is a bipolar fuzzy BCK-submodule of , both of and are bipolar fuzzy BCK-submodules of Therefore is a bipolar fuzzy -extension of the -translation .

Theorem 23 is illustrated by the following example.

Example 24. Let be a set along with binary operation defined by (24). Then is a bounded commutative BCK-algebra. Let and define + on as which is given in (25). Then is a commutative group. Quick calculations reveal that is an -module under the operation shown in (26) (see [12]):Define a bipolar fuzzy set on byThen is a bipolar fuzzy BCK-submodule on , and , . If we take and , then (28) shows the bipolar fuzzy -translation of Let be a bipolar fuzzy set on defined by Clearly is a bipolar fuzzy BCK-submodule on which is a bipolar fuzzy extension of . But is not a bipolar fuzzy -translation of for all . Take , and then , and the fuzzy -translation of is given by Note that and for all , and so is a bipolar fuzzy -extension of the bipolar fuzzy -translation of .

Definition 25. Let be a bipolar fuzzy set on and . By a bipolar fuzzy -multiplication of we mean a bipolar fuzzy set , wherefor all

Theorem 26. If , then the bipolar fuzzy -multiplication for all .

Proof. Assume that is a bipolar fuzzy BCK-submodule on , and let and . ThenMoreover, Therefore is a bipolar fuzzy BCK-submodule on .

Theorem 27. For any bipolar fuzzy set on , the following are equivalent:(i) is a bipolar fuzzy BCK-submodule on .(ii) is a bipolar fuzzy BCK-submodule on .

Proof. By Theorem 27, (i) implies (ii). Now let be such that is a bipolar fuzzy BCK-submodule on and let . Thenand since and , then we have and . Nowwhich means that .
Similarly, . Hence is a bipolar fuzzy BCK-submodule on .

The following theorem provides a strong relationship between -translation and -multiplication of any bipolar fuzzy BCK-submodule.

Theorem 28. Let be a bipolar fuzzy set on , , and . Then is a bipolar fuzzy -extension of .

Proof. For every , we haveand so is a bipolar fuzzy extension of .
Now let be a bipolar fuzzy BCK-submodule on , where . Then is a bipolar fuzzy BCK-submodule on by Theorem 27. It follows from Theorem 15 that the -translation of is a bipolar fuzzy BCK-submodule on for all . Therefore every -translation of is a bipolar fuzzy -extension of the -multiplication of .

The following example shows that Theorem 28 is not valid for (or ).

Example 29. Consider a BCK-algebra and -module that are defined in Example 19.
Equation (37) defines a bipolar fuzzy set on If we take , then the -multiplication is a bipolar fuzzy BCK-submodule on () sincefor all and . But if we take and , then for all and so . Hence is not a bipolar fuzzy -extension of for all , .

Corollary 30. Let be a homomorphism of BCK-modules and a bipolar fuzzy set on . If is a bipolar fuzzy BCK-submodule, then so is for any -translation of with

Proof. Proof is directly by Theorems 12 and 15.

Joining Theorems 13 and 16 we have the following corollary.

Corollary 31. Let be an epimorphism of BCK-modules and a bipolar fuzzy set on . If the inverse image of the -translation is a bipolar fuzzy BCK-submodule for some , then so is .

Corollary 32. Let be an epimorphism of BCK-modules and . Then for any -translation of , the homomorphic image .

Using Theorems 10, 12, 13, and 27, we deduce the following results.

Corollary 33. Let be a homomorphism of BCK-modules and a bipolar fuzzy BCK-submodule on , and then the inverse image for any -multiplication of with .

Corollary 34. Let be an epimorphism of BCK-modules. If for a bipolar fuzzy set and some , then .

Corollary 35. Let be an epimorphism of BCK-modules and , and then for any .

4. Conclusion

Translations and multiplications are types of operators attractive to researchers who are interested in fuzzy set theory. Generally, main objective of studying translations and multiplications is to inspect whether a certain form of fuzzy set remains invariant under such operators. In this paper, we showed that a bipolar fuzzy BCK-submodule keeps its property under translations and multiplications. Further, we presented a relation between translations and multiplications of bipolar fuzzy BCK-submodules. More properties and results were provided.

Competing Interests

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

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Copyright © 2017 M. A. Alghamdi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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