<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 60.4051 16.7875" width="60.4051pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g><g transform="matrix(.017,0,0,-0.017,14.998,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,21.787,0)"><use xlink:href="#g117-173"/></g><g transform="matrix(.017,0,0,-0.017,35.641,0)"><path id="g117-64" d="M574 548H512L295 69H293L76 548H14L263 0H325L574 548Z"/></g><g transform="matrix(.017,0,0,-0.017,45.753,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g><g transform="matrix(.017,0,0,-0.017,54.178,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-Fuzzy<i> BCK</i>-Submodules

Altaiary, M. M.; Bashammakh, S. A.; Alshehri, N. O.

doi:https://doi.org/10.1155/2016/8746275

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2016 | Article ID 8746275 | https://doi.org/10.1155/2016/8746275

-Fuzzy BCK-Submodules

M. M. Altaiary,¹S. A. Bashammakh,²and N. O. Alshehri²

Academic Editor: Chris Goodrich

Received25 Sept 2015

Revised15 Jan 2016

Accepted27 Jan 2016

Published22 Feb 2016

Abstract

We introduce the concept of generalized fuzzy module of BCK-algebras and present some fundamental properties. Also, we introduce the concept of generalized interval-valued fuzzy BCK-submodule and present some fundamental properties.

1. Introduction

In 1966, Iseki and Imai [1, 2] introduced BCK-algebra. This notion was originated from two different ways: () set theory and () classical and no classical propositional calculi. Certain algebraic structures, for example, Boolean-algebra and MV-algebras, are introduced as BCK-algebras [3]. Every module is an action of ring on certain group. This is, indeed, a source of motivation to study the action of certain algebraic structures on groups. BCK-module is an action of BCK-algebra on commutative group. In 1994, the notion of BCK-module was introduced by Abujabal et al. [4]. They established isomorphism theorems and studied some properties of BCK-modules. The theory of BCK-modules was further developed by Perveen et al. [5].

After the introduction of fuzzy sets by Zadeh [6], there have been a number of generalizations of this fundamental concept. In 1991, Xi [7] applied fuzzy set theory to BCK-algebras and introduced the notion of fuzzy subalgebras (ideals) of the BCK-algebras, and since then some authors studied fuzzy subalgebras and fuzzy ideals. A new type of fuzzy subgroup, that is, the -fuzzy subgroup, was introduced in an earlier paper of Bhakat and Das [8]. In fact, the -fuzzy subgroup is an important generalization of Rosenfeld’s fuzzy subgroup. It is now natural to investigate similar type of generalizations of the existing fuzzy subsystems with other algebraic structures. With this objective in view, Jun [9] introduced the concept of -fuzzy ideals of a BCK/BCI-algebra and investigated related results. In [10] Zadeh made an extension of the concept of fuzzy set by an interval-valued fuzzy set. In 2000, Jun [11] applied interval-valued fuzzy set theory to BCK-algebras and introduced the notion of interval-valued fuzzy subalgebras (ideals) of the BCK-algebras. In this paper, we introduce the concept of generalized fuzzy BCK-submodules and some basic properties are obtained and we define the concept of generalized interval-valued fuzzy BCK-submodules and some basic properties are obtained.

2. Preliminaries

Definition 1 (see [8]). By a BCK-algebra one means an algebra of type satisfying the following axioms: (BCK-1) , (BCK-2) , (BCK-3) , (BCK-4) , (BCK-5) and imply , for all .

A partial ordering “” is defined on by . A BCK-algebra is said to be bounded if there is an element such that , for all , commutative if it satisfies the identity , where , for all , and implicative if , for all .

Definition 2 (see [12]). Let be a BCK-algebra. Then by a left -module (abbreviated -module), one means that an abelian group with an operation with satisfies the following axioms for all and :(i),(ii),(iii).

Moreover, if is bounded and satisfies , for all , then is said to be unitary.

A mapping is called a fuzzy set in a BCK-algebra . For any fuzzy set in and any , we define set , which is called upper -level cut of .

Definition 3 (see [12]). A fuzzy subset of is said to be a fuzzy BCK-submodule if for all and , the following axioms hold: (FBCKM1) , (FBCKM2) , (FBCKM3) .

Definition 4 (see [13]). A fuzzy set in a set of the formis said to be a fuzzy point with support and value and is denoted by ; we say that a fuzzy point belongs to a fuzzy set and write , if . A fuzzy point is quasi-coincident with a fuzzy set , if In this case we write . means that or means that and .

Theorem 5 (see [12]). Let , where is the set of all fuzzy subsets of BCK-module . Then is a fuzzy BCK-submodule of if and only if(i),(ii).

3. ()-Fuzzy BCK-Submodules

Definition 6. A fuzzy subset of is said to be an -fuzzy BCK-submodule if for all , and the following axioms hold:(i),(ii),(iii)

Example 7. Let and consider the following equation:Then is a bounded implicative BCK-algebra and so is a BCK-module over itself. Now let be such that . Define by , , and . Then is an -fuzzy BCK-submodule of .

Theorem 8. A fuzzy subset of is an -fuzzy BCK-submodule if and only if(i),(ii).

Proof. From the definition of -fuzzy BCK-submodule, (i) , (ii) . But we know that . Then . Conversely, we have . Since , then . And . This proves that is an -fuzzy BCK-submodule.

Theorem 9. Let , where is the set of all fuzzy subsets of BCK-module . Then is an -fuzzy BCK-submodule of if and only if(i),(ii).

Proof. Let be an -fuzzy BCK-submodule: (i) Consider , put and then . Consider . Hence, . Conversely, we have and . This proves that is an -fuzzy BCK-submodule.

Theorem 10. A fuzzy set in is an -fuzzy BCK-submodule if and only if , is BCK-submodule.

Proof. Let , , , and ; let ; then . Since is an -fuzzy BCK-submodule, , and , then ; let ; then ; then ; then . Hence, and from (i) and (ii) we get that is BCK-submodule.
Conversely, let , ; since is BCK-submodule, then ; this implies that . Hence, ; let ; let ; then ; this implies that . Hence, proving is an -fuzzy BCK-submodule.

Theorem 11. Let be a fuzzy set in . Then is BCK-submodule for all if and only if satisfies(i),(ii).

Proof. Assume that is BCK-submodule of for all if there is such that condition (i) is not valid implying that ; then and . But implies , a contradiction. Hence, (i) is valid. Suppose that for some . Then and but since . This is a contradiction, and therefore (ii) is valid.
Conversely, assume that satisfies conditions (i) and (ii). Let for any ; we have , so ; thus, ; let be such that and ; then , and thus implies that . Hence, is BCK-submodule.

Theorem 12. Every fuzzy BCK-submodule is an -fuzzy BCK-submodule.

Proof. Let be fuzzy BCK-submodule ; then we have ; hence, (i) and ; then (ii); from (i) and (ii) we get that is an -fuzzy BCK-submodule.

The following example shows that the converse of Theorem 12 is not true in general.

Example 13. Let be the set along with binary operation defined on it by (3); then forms a bounded commutative, nonimplicative BCK-algebra. Now for the subset of , define an operation + as . By (4), it follows that forms a commutative group. Define scaler multiplication by for all and (see (5)):Then forms an -module. Now let be such that . Define by , , and . Then is an -fuzzy BCK-submodule of , but it is not fuzzy BCK-submodule of if .

Theorem 14. Let be an -fuzzy BCK-submodule of such that for all . Then is a fuzzy BCK-submodule of .

Proof. Since is an -fuzzy BCK-submodule of , then it satisfies these two conditions: (i) , (ii) . Now we want to show that is an -fuzzy BCK-submodule of . Since , then Hence, , and since , then . Hence, . Then is a fuzzy BCK-submodule of .

Lemma 15. Let be a -fuzzy BCK-submodule of module . Let , such that ; then(i) if ,(ii) if .

Proof. Since is a fuzzy submodule of , we have . (i) Let ; then since , since . Then . (ii) If , then . But . This implies that . If , then , but , and . Then .

Definition 16. Let and be two -fuzzy BCK-submodules and the intersection of and is defined as follows:

Proposition 17. Let be a nonempty family of an -fuzzy BCK-submodule of . Then is an -fuzzy BCK-submodule of .

Proof. Since is an -fuzzy BCK-submodule of , then satisfies conditions of -fuzzy BCK-submodule. Let ; let and be an -fuzzy BCK-submodule; then and satisfy conditions of -fuzzy BCK-submodule.
Now we want to prove that is an -fuzzy BCK-submodule. (i) Consider . Hence, (ii) Consider , , . Hence, From (i) and (ii) we get that is an -fuzzy BCK-submodule. Furthermore, is an -fuzzy BCK-submodule.

Similarly, we can prove the generalization of previous proposition.

Theorem 18. Let be a family of an -fuzzy BCK-submodule of . Then is an -fuzzy BCK-submodule of .

The following example shows that the union of two -fuzzy BCK-submodules of may not be an -fuzzy BCK-submodule of .

Example 19. Let be BCK-algebra which is given in Example 7 and let be an -fuzzy BCK-submodule of which is . Let be a fuzzy set in defined by . Then is an -fuzzy BCK-submodule of . The union of and is given by . Hence, . Then Suppose that and ; then :a contradiction; hence, is not -fuzzy BCK-submodule of .

Definition 20. Let ; is an -fuzzy BCK-submodule for and . Then define as follows:

Proposition 21. Let be an -fuzzy BCK-submodule. Then .

Proof. One has

Theorem 22. Let be an -fuzzy BCK-submodule, for ; then is an -fuzzy BCK-submodule.

Proof. Since is an -fuzzy BCK-submodule, then the following conditions hold: (i) , for , (ii) , for . Now we want to prove that satisfies conditions (i) and (ii). We know that Now we want to prove that is an -fuzzy BCK-submodule. Prove (i): Since is an -fuzzy BCK-submodule then (i) holds. Prove (ii): Since is an -fuzzy BCK-submodule, then (ii) holds. This implies that is an -fuzzy BCK-submodule.

4. Some Kinds of ()-Interval-Valued Fuzzy BCK-Submodule

For any and we define , for all . In particular, if , we write . Let and . An interval-valued fuzzy set of a BCK-algebra is said to be an interval-valued fuzzy point , with support and interval-valued , iffor all ; we say belongs to (resp., is quasi-coincident with) an interval-valued fuzzy set , written by (resp., ), if (resp., ); if or , then we write .

Definition 23 (see [7]). An interval-valued fuzzy set of is , where one denotes, for each , .

Definition 24 (see [7]). Let be an interval-valued fuzzy set of . Then for every , the crisp set is called the level subset of .

Definition 25. An interval-valued fuzzy subset of is said to be an -interval-valued fuzzy BCK-submodule of for all and for all , and the following axioms hold:(i),(ii),(iii).

Example 26. Consider the BCK-algebra as in Example 7. Define an interval-valued fuzzy set of by , , and . Hence, is an -interval-valued fuzzy BCK-submodule of .

Theorem 27. Let , where is the set of all interval-valued fuzzy subsets of BCK-module . Then is an -interval-valued fuzzy BCK-submodule of if and only if(i),(ii).

Proof. Let be an -interval-valued fuzzy BCK-submodule: (i) , and implies that , . Then . Conversely, we have . Consider . Hence, is an -interval-valued fuzzy BCK-submodule.

Theorem 28. An interval-valued fuzzy subset of is an -interval-valued fuzzy BCK-submodule if and only if(i),(ii).

Proof. From the definition of -interval-valued fuzzy BCK-submodule, (i) , (ii) . But we know that . Then . Conversely, we have . Use Theorem 27, since , and . Hence, is an -interval-valued fuzzy BCK-submodule.

Theorem 29. An interval-valued fuzzy subset of is an -interval-valued fuzzy BCK-submodule if and only if is BCK-submodule of for all .

Proof. Let be an -interval-valued fuzzy BCK-submodule of and for any ; then . Since is an -interval-valued fuzzy BCK-submodule, then , which implies that . Hence, ; let , ; then . Furthermore, , and then . Hence, ; hence is BCK-submodule.
Conversely, let ; then ; since is BCK-submodule, then implies that ; then ; let , and let ; then implies that ; then .
Hence, is an -interval-valued fuzzy BCK-submodule.

5. Conclusions

In this paper, we introduced the concept of generalized fuzzy BCK-submodules and some basic properties were obtained and we defined the concept of generalized interval-valued fuzzy BCK-submodules and some basic properties were obtained. Other topics in this area which could be a further research are -fuzzy BCK-submodules, translations of -fuzzy BCK-submodule, and other generalizations of fuzzy set theory to BCK-submodules.

Conflict of Interests

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

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Copyright

Copyright © 2016 M. M. Altaiary 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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