The Scientific World Journal

Volume 2013, Article ID 678586, 7 pages

http://dx.doi.org/10.1155/2013/678586

## A Novel Definition of -Fuzzy Lattice Based on Fuzzy Set

Department of Basic Science, Beijing University of Agriculture, Beijing 102206, China

Received 2 April 2013; Accepted 12 May 2013

Academic Editors: F. Başar and A. Favini

Copyright © 2013 Jun-Fang Zhang. 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.

#### Abstract

The concept of -fuzzy lattice is presented by means of an -fuzzy partially ordered set. An -fuzzy partially ordered set is an -fuzzy lattice if and only if one of , , and is a lattice.

#### 1. Introduction

Many concepts of fuzzy algebra were presented, since the concept of fuzzy subgroups was introduced by Rosenfeld. The concept of fuzzy lattice was also introduced by Tepavević and Trajkovski [1]. But the authors in [1] defined an -fuzzy lattice based on a fuzzy set of a crisp lattice, so the scope is very limited, and they gave its characterizations by only one kind of its cut sets. To overcome this shortcoming, in this paper we try to present a new definition of an -fuzzy lattice in an -fuzzy subset of a general set in terms of an -poset. Furthermore, its many characterizations are given using the theory of -sets and -sets proposed by Shi Fu-Gui in 1995. In this way, we obtain more generalized conclusion about -fuzzy lattice in order to make their applications becoming comprehensive.

#### 2. Preliminaries

Throughout this paper denotes a completely distributive lattice, and denotes the set of all nonzero -irreducible elements in . denotes the set of all nonunit prime elements in . denotes a nonempty usual sets. is the set of all -fuzzy sets on . We will not differ a crisp set from its character function. For empty set , we define and . According to [2], each element in has a greatest maximal family and a greatest minimal family which we, respectively, denote by and . From [2] we know that is a maximal family of , , and is a minimal family of , .

Now we recall some basic concepts and results.

*Definition 1 (see [3]). *Let and . Define

From [3] we know that implies and implies . If , then and .

*Definition 2 (see [3–5]). *For each -fuzzy set *A* in , we have(1),
(2).

*Definition 3 (see [6]). *Let and . Define an -fuzzy set on by
is called the product of and .

*Definition 4 (see [6]). *Let . is called an -fuzzy relation from (of) to if .

*Definition 5 (see [7]). *Let , . An -fuzzy relation from to is called an -fuzzy mapping from (of) into if is a mapping from (of) into , and then we write it as .

Theorem 6 (see [7]). *Let , and , and then the following conditions are equivalent:*(1)* is an -fuzzy mapping from into ;*(2)*for each is a mapping from into ;*(3)*for each is a mapping from into .*

Theorem 7 (see [8]). *Let , and . If for every , , then the following conditions are equivalent:*(1)* is an -fuzzy mapping from into ;*(2)*for every is a mapping from into .*

*Definition 8 (see [9]). *Let , , and . For , we define . Then is called the image of under .

Theorem 9 (see [9]). *Let , and , and then for we have*(1)*for each ,*(2)*,
*(3)*for each ,*(4)*. *

*Definition 10 (see [9]). *Let , and . For , we define . Then is called the inverse image of under .

Theorem 11 (see [9]). *Let , and . Then for , we have*(1)*for each ,*(2)*,
*(3)*for each ,*(4)*. *

*Definition 12 (see [10]). *Let be a set, , and . An -fuzzy relation from to is called an -fuzzy partial order on if satisfies the following conditions:(1)for all ,(2),(3)for all .When is an -fuzzy partial order on , we call an -fuzzy partial order set or -poset for short.

Theorem 13 (see [10]). *For an -fuzzy relation on , the following implications and are true.*(1)* is an -fuzzy partial order on .*(2)*For each , if is not empty set, then it is a partial order on .*(3)*For each , if is not empty set, then it is a partial order on .*(4)*For each , if is not empty set, then it is a partial order on .*(5)*For each , if is not empty set, then it is a partial order on .*(6)*For each , if is not empty set, then it is a partial order on .*(7)*For each , if is not empty set, then it is a partial order on .*

*Remark 14 (see [10]). *In general, in the previous theorem is not true. This can be seen from the following example.

*Example 15 (see [10]). *Let , where are different and is an interval. We define the order in as follow.

For all . The order in is as usual. Then is a completely distributive lattice. Take such that
Obviously we have
Thus is an -fuzzy partial order on . But it is easy to check that
This shows that is not a partial order on .

#### 3. -Fuzzy Lattice

*Definition 16. *Let , , be an -fuzzy partial order on . is called an -fuzzy supremum of if the following conditions are true:()(),
().

*Definition 17. *Let , , be an -fuzzy partial order on . is called an -fuzzy infimum of if the following conditions are true:(),
(),
().

*Definition 18. *An -fuzzy partially ordered set is called an -fuzzy lattice on if for any , both -fuzzy supremum and -fuzzy infimum of exist.

Theorem 19. *Let be an -fuzzy partially ordered set. Then the following conditions are equivalent.*(1)* is an -fuzzy lattice on .*(2)*For any is a lattice.*(3)*For any is a lattice.*(4)*For any is a lattice.*(5)*For any is a lattice.*(6)*For any is a lattice.*

*Proof. *. Since above-mentioned sets have been posets, we only need to prove that supremum and infimum of exist.

. For any , let and , and then . By Definitions 1 and 2 we know that there exist such that
So . By (S2) we know that and . Therefore, . Analogously we can prove that . For any , we have that , hence . Analogously, for any , we can prove that . Thereby, are, respectively, supremum and infimum of with respect to in . Then it is proved that is a lattice.

is obvious.

. For any , let . Since , there exist such that . Take , then we have and (since a is a prime element). By (3) we know that for any is a lattice. Then there exist such that

Thus is supremum with respect to of in . Analogously we can prove that infimum exists in too. Hence is a lattice.

. Let and . Then ; that is, . From (6) there exist and in ; that is, and . So we have
and by (6) we obtain that and . This shows that . Therefore,
By (6) we have that for each ; that is, , and we can obtain that ; that is, . Therefore . As before we can prove that
Let . Then from Definition 1 we know that are -supremum and -infimum of respectively.

Then we prove .

. For any , let ; that is, . From (1) there exist such that and . This implies that and ; that is, . From (1) we know that and . Then we obtain that
And by we have that for each ; that is, , and then ; that is, . Analogously we prove that for any . Hence is supremum of with respect to in . Similarly we can prove that is infimum of with respect to in . So is a lattice.

is obvious.

. For any , let and ; that is, . From (5) there exist ; that is, . This shows that
From (5) we have that and ; that is, and . Therefore,
And by (5), for any ; that is, . Then we have ; that is, . Hence . Analogously we can conclude that
Let , and then are -supremum and -infimum of , respectively.

Theorem 20. *Let be an -fuzzy partially ordered set. If for each , is a lattice, and then is an -fuzzy lattice on .*

*Proof. *For any , let and ; that is, . There exist and ; that is, and . Thereby
Since is a lattice, we have that and ; that is, and . This implies that
Analogously we can conclude that for all ; that is, , and we have ; that is, . Then we obtain that . Similarly, we can prove that
Let , so it is proved that are -supremum and -infimum of , respectively.

*Remark 21. *Inversely the previous theorem is not true when is a poset. This can be seen from Remark 14.

*Definition 22. *Let be a nonempty set, and let be -fuzzy lattices of . is called an -fuzzy sublattice of if .

*Definition 23. *Let be nonempty sets, and let be -fuzzy lattices of . An -fuzzy mapping is called an -fuzzy lattice homomorphism if for any is a lattice homomorphism.

From the corresponding theorems in [9] and knowledge in general algebra we can easily obtain the following theorem.

Theorem 24. *Let be nonempty sets, let be -fuzzy lattices of , let be an -fuzzy lattice homomorphism, and then the following propositions are true.*(1)*If is an -fuzzy sublattice of , then is an -fuzzy sublattice of .*(2)*If is an -fuzzy sublattice of , then is an -fuzzy sublattice of .*

#### 4. Fuzzy Sublattice

In [1], the author gave an -valued fuzzy lattice. From the following analysis we can see that -valued fuzzy lattice is a special case of bifuzzy lattices in fact.

*Definition 25 (see [1]). *Let be a complete lattice with the greatest element and the least element , and let be a lattice, is called a lattice-valued fuzzy lattice if all the -cuts of are sublattices of .

*Remark 26. *Here the -cut of indicates the case of in fact.

Theorem 27 (see [1]). *Let be a complete lattice with the greatest element and the least element , let be a lattice, , and then is called an L-valued fuzzy lattice if and only if for all , the following conditions are true:*()*,
*()*. *

*Remark 28. *When is a completely distributive lattice, is a lattice, here is a relation on . In the Definitions 16–18 of -fuzzy lattice, let -fuzzy partial order , , and then the conditions (R1)–(R4) in Definition 25 are true. Thus it can be seen that the *L*-valued fuzzy lattice is a special case of -fuzzy lattice. Since is an fuzzy subset of , we call this *L*-valued fuzzy lattice as fuzzy sublattice of in the following application.

Now we study the relation between fuzzy sublattice and crisp lattice by means of their four level cut sets, furthermore we give the definition of fuzzy lattice homomorphism and corresponding theorems.

*Definition 29. *Let be a completely distributive lattice, let be a lattice, and let be a relation on , , if for all , there exist such that(),(), then we call as a fuzzy sublattice of .

Theorem 30. *Let be a completely distributive lattice, let be a lattice, and let be a relation on , . Then we can obtain that are equivalent, and is true.*(1)* is a fuzzy sublattice of .*(2)*For each is a sublattice of .*(3)*For each is a sublattice of .*(4)*For each is a sublattice of .*(5)*For each is a sublattice of .*(6)*For each is a sublattice of .*(7)*For each is a sublattice of .*(8)*For each is a sublattice of .*

*Proof. *. For each , let . Then we have . From (1), there exist such that and . Therefore . Then it is proved that is a sublattice of .

is obvious.

. For each , let . There exist such that . Hence . Since is a prime element, we have that . From (3), we have , therefore . So we obtain that is a sublattice of .

. For each , let . Hence for all , and is a prime element. We know that . From is a sublattice of , so we have . Thus . Therefore, is a sublattice of .

is obvious.

. For each , let and ; that is, . From (5) there exist ; that is, . Thus . Therefore, is a fuzzy lattice of .

is obvious.

. For each , let and ; that is, . From (7), there exist ; that is, . So it is proved that and . This shows that is a fuzzy lattice of .

*Remark 31. *Generally, in the previous theorem is not true. This can be seen from the following example.

*Example 32. *, where , where (see Figure 1).

Then , and . This implies for all is a sublattice of ; that is, is a fuzzy lattice of . While for is not a sublattice of . This implies that .

*Remark 33. *The following condition makes be true.

Theorem 34. *If for all , then in the previous theorem is true.*

*Proof. *(1)(6). For each , and then . Therefore, . From (1) we know that and . So we have that and . This shows that is a sublattice of .

*Definition 35. *Let be a nonempty set, let be fuzzy lattices of , and if , we call as a fuzzy sublattice of .

*Definition 36. *Let be nonempty sets, and let be fuzzy sublattices of , respectively. -fuzzy mapping is named a fuzzy lattice homomorphism if it satisfies that for all is a lattice homomorphism.

The following theorems are easily obtained by the corresponding theorem in [9] and the knowledge in general algebra.

Theorem 37. *Let be nonempty sets, and let be fuzzy sublattices of , respectively. is a fuzzy lattice homomorphism, and then the following conditions are true:*(1)*if is a fuzzy sublattice of , then is a fuzzy sublattice of ;*(2)*if is a fuzzy sublattice of , then is a fuzzy sublattice of .*

Theorem 38. *Let be nonempty sets, and let be fuzzy sublattices of , respectively. is a fuzzy mapping, then the following conditions are true:*(1)* is a fuzzy lattice homomorphism;*(2)*for all is a lattice homomorphism from to .*

*Proof. *. Let and , and then . For any , we have . From (1) we know that is a fuzzy lattice homomorphism. Thus there exist such that ; that is, . From we obtain that
Take such that , and then . And take such that and ; in this way we have . From (1) we know that . Then take ; we obtain that , and . Therefore
Similarly, it is easy to prove that . This implies that is a lattice homomorphism.

. Let and , then . Since is a prime element we obtain . Take such that and , and then we have . From (2) we know that is a lattice homomorphism, so there exist such that and . Hence
Similarly, ; that is, is a lattice homomorphism.

Similar to [11], we can easily prove the following theorems.

Theorem 39. *Let be nonempty sets, and let be fuzzy sublattices of , respectively. is an -fuzzy mapping, if for all , and then the following conditions are equivalent:*(1)* is a fuzzy lattice homomorphism;*(2)*for all is a lattice homomorphism from to . *

Theorem 40. *Let be nonempty sets, and let be fuzzy sublattices of , respectively. is an -fuzzy mapping, if for all is a lattice homomorphism from to , and then is a fuzzy lattice homomorphism.*

Theorem 41. *Let be nonempty sets, and let be fuzzy sublattices of , respectively. is an -fuzzy mapping, if for all , and then the following conditions are equivalent:*(1)* is a fuzzy lattice homomorphism;*(2)*for all is a lattice homomorphism from to .*

#### 5. Fuzzy Lattice

In Definitions 16, 17, and 18, let , and then the conditions are true obviously. So we can get another special case of bi-fuzzy lattice-fuzzy lattice. Now we give its definition and corresponding theorems.

*Definition 42. *Let be a completely distributive lattice, and let be an -fuzzy relation on , if satisfies for any , there exist such that(),
(),
(),
(). then we call as supremum and infimum of *x, y* with respect to , respectively.

*Definition 43. *Let be a completely distributive lattice, and let be an -fuzzy relation on , if for any , both supremum and infimum of *x*, *y* with respect to exist, and then we call as a fuzzy lattice with respect to .

Same to the corresponding theorems in last section,we have the following theorem.

Theorem 44. *Let be a completely distributive lattice, and let be an -fuzzy relation; then , (), (), (), (), and () of the following conditions are equivalent, and is true.*(1)* is a fuzzy lattice with respect to .*(2)*For each is a lattice.*(3)*For each is a lattice.*(4)*For each is a lattice.*(5)*For each is a lattice.*(6)*For each is a lattice.*(7)*For each is a lattice.*(8)*For each is a lattice.*

#### Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 61171195).

#### References

- A. Tepavčević and G. Trajkovski, “$L$-fuzzy lattices: an introduction,”
*Fuzzy Sets and Systems*, vol. 123, no. 2, pp. 209–216, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G.-J. Wang,
*Theory of L-Fuzzy Topological Spaces*, Shanxi Normal University Press, Xi'an, China, 1988. - F.-G. Shi, “The theory and applications of ${L}_{\beta}$-nested sets and ${L}_{\alpha}$-nested sets,”
*Fuzzy Systems and Mathematics*, vol. 9, no. 4, pp. 65–72, 1995 (Chinese). View at Google Scholar · View at MathSciNet - F.-G. Shi, “$L$-fuzzy sets and prime element nested sets,”
*Journal of Mathematical Research and Exposition*, vol. 16, no. 3, pp. 398–402, 1996 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F.-G. Shi, “Theory of molecular nested sets and its applications,”
*Journal of Yantai Teachers University (Natural Science)*, no. 1, pp. 33–36, 1996 (Chinese). View at Google Scholar - F.-G. Shi, “$L$-fuzzy relations and $L$-fuzzy subgroups,”
*Journal of Fuzzy Mathematics*, vol. 8, no. 2, pp. 491–499, 2000. View at Google Scholar · View at MathSciNet - F.-G. Shi, “$L$-fuzzy mapping of $L$-fuzzy sets,”
*Fuzzy Systems and Mathematics*, vol. 14, no. 3, pp. 16–24, 2000. View at Google Scholar · View at MathSciNet - X.-D. Zhang, “Characterizations of $L$-fuzzy mappings,”
*Fuzzy Systems and Mathematics*, vol. 16, no. 2, pp. 48–51, 2002. View at Google Scholar · View at MathSciNet - X.-D. Zhang, “Representations of image and inverse of $L$-fuzzy mappings,”
*Fuzzy Systems and Mathematics*, vol. 16, no. 1, pp. 24–28, 2002. View at Google Scholar · View at MathSciNet - J. Zhao, “$L$-fuzzy posets,”
*Fuzzy Systems and Mathematics*, vol. 15, no. 3, pp. 36–39, 2001. View at Google Scholar · View at MathSciNet - J. F. Zhang, “$L$-fuzzy subgroups and $L$-fuzzy group homomorphisms,”
*Fuzzy Systems and Mathematics*, vol. 17, no. 3, pp. 61–67, 2003. View at Google Scholar · View at MathSciNet