Categories of -Fuzzy Greedoids

Abstract

The concepts of -greedoids, fuzzifying greedoids, and ()-fuzzy greedoids are introduced and feasibility preserving mappings between greedoids are defined. Then -feasibility preserving mappings, fuzzifying feasibility preserving mappings, and ()-fuzzy feasibility preserving mappings are given as generalizations of feasibility preserving mappings. We study the relations among greedoids, -greedoids, fuzzifying greedoids, and ()-fuzzy greedoids from a categorical point of view.

1. Introduction

Greedoids have been invented by Korte and Lovász in [1, 2]. Originally, the main motivation for proposing this generalization of the matroid concept came from combinatorial optimization. The optimality of the greedy algorithm could in several instances be traced back to be an underlying combinatorial structure that was not a matroid but a greedoid. Optimality of the greedy solution for a broad class of objective mappings characterizes these structures. Many algorithmic approaches in different areas of combinatorics and other fields of numerical mathematics define the structure of a greedoid. Examples are scheduling under precedence constraints, breadth first search, shortest path, Gaussian elimination, shellings of trees, chordal graphs and convex sets, line and point search, series-parallel decomposition, retracting and dismantling of posets and graphs, and bisimplicial elimination.

The fuzzification of matroids was first investigated by Goetschel and Voxman [3] and the concept of fuzzy matroids was introduced, where a family of independent fuzzy sets was defined as a crisp family of fuzzy subsets of a finite set satisfying certain set of axioms. Subsequently many authors investigated Goetschel-Voxman fuzzy matroids (see [3–12]). The concept of -fuzzifying matroids was introduced as a new approach to the fuzzification of matroids by Shi [13], and his approach to the fuzzification of matroids preserves many basic properties of crisp matroids (see [14–17]). Particularly, the categorical relations among matroids, fuzzy matroids, and fuzzifying matroids are studied [18], and the main results are shown as follows:where , in the diagram mean, respectively, reflective and coreflective.

In [19], the concepts of -matroids and -fuzzy matroids are introduced as generations of matroids and were widely investigated (see [20–23]). In [22], the relations among matroids, -prematroids, fuzzifying matroids, and -fuzzy prematroids were studied from a categorical viewpoint. The main results are summarized as follows:where , in the diagram mean, respectively, reflective and coreflective. In order to see the relations clearly, we give the following equivalent diagram:

The aim of this paper is to introduce the concepts of -greedoids, fuzzifying greedoids, and -fuzzy greedoids and study the relations among greedoids, -greedoids, fuzzifying greedoids, and -fuzzy greedoids from a categorical point of view. It is easy to prove that greedoids and feasibility preserving mappings form a category, fuzzifying greedoids and fuzzifying feasibility preserving mappings form a category, -greedoids and -feasibility preserving mappings form a category, and -fuzzy greedoids and -fuzzy feasibility preserving mappings form a category. In what follows, they are denoted by , , , and , respectively. denotes the category of closed and perfect -greedoids and -feasibility preserving mappings as morphisms.

The paper is organized as follows. In Sections 3 and 4, the concepts of -greedoids, fuzzifying greedoids, and -fuzzy greedoids are introduced, respectively. In Section 5, we show that is isomorphic to and is a concretely coreflective full subcategory of . In Section 6, we show that can be embedded in as a simultaneously concretely reflective and coreflective full subcategory and is a simultaneously concretely reflective and coreflective full subcategory of . In Section 7, can be embedded in as a concretely coreflective full subcategory and is a reflective full subcategory of . In summary, we show that where , in the diagram mean, respectively, reflective and coreflective.

2. Preliminaries

Throughout this paper, and is a nonempty finite set. We denote the set of all subsets of by and the set of all fuzzy subsets of by .

For , .

For and , define a fuzzy set as follows: A fuzzy set is called a fuzzy point and denoted by .

For and , we define

Definition 1 (see [24]). If is a nonempty subset of , then the pair is called a (crisp) set system. A set system is called a matroid if it satisfies the following conditions: (I1).(I2)If , , and , then .(I3)If and , then there is such that .

Definition 2 (see [25]). A greedoid is a pair of , where is a set system satisfying the following conditions: (G1)For every there is an such that .(G2)For such that , there is an such that .The sets in are called feasible (rather than “independent”).

Remark 3. In [25], () and (G2) together define greedoids as well (G1) and (G2), where () . Obviously, (G1) in Definition 2 could be replaced by the weaker axiom () and greedoids are defined as generalizations of matroids.

Definition 4. Let be greedoids. A mapping is called a feasibility preserving mapping from to if for all .

Remark 5. In [26], a function between two convex structures, called a convexity preserving function, inverts convex sets into convex sets. Here, similarly, we give a mapping between two greedoids, which inverts feasible sets into feasible sets and is called a feasibility preserving mapping.

Definition 6 (see [13]). A mapping is called an -fuzzy family of independent sets on if it satisfies the following conditions: (FI1).(FI2)For any , .(FI3)For any , if , then .If is an -fuzzy family of independent sets on , then the pair is called an -fuzzifying matroid. For , can be regarded as the degree to which is an independent set.

Definition 7 (see [19]). A subfamily of is called a family of independent -fuzzy sets on if it satisfies the following conditions: (LI1).(LI2), , and .(LI3)If and for some , then there exists such that , where .If is a family of independent -fuzzy sets on , then the pair is called an -matroid.

Definition 8 (see [19]). A mapping is called an -fuzzy family of independent -fuzzy sets on if it satisfies the following conditions:  (LMFI1) . (LMFI2) For any , . (LMFI3) If for and for some , then If is an -fuzzy family of independent -fuzzy sets on , then the pair is called an -fuzzy matroid.

Remark 9. (1) In [13, 19], and denote completely distributive lattices. In [22], when , an -fuzzy matroid is also called -fuzzy prematroid, an -matroid is called -prematroid, and an -fuzzifying matroid is called a fuzzifying matroid.
(2) When is replaced by the interval , it is easy to see .

Definition 10 (see [19]). Let be an -fuzzy set on a finite set . Then the mapping defined by, , is called the -fuzzy cardinality of .

Lemma 11 (see [19]). For a finite set , it holds that for any and any .

Lemma 12 (see [19]). Let . Then for any and for any .

3. -Greedoids

Based on Definitions 2 and 7, we give the following definition.

Definition 13. A subfamily of is called a family of feasible fuzzy sets on if it satisfies the following conditions: (IG1).(IG2)If and for some , then there exists such that , where If is a family of feasible fuzzy sets on , then the pair is called an -greedoid.

Theorem 14. Let ; , define . If is an -greedoid, then is a greedoid for any .

Proof. Suppose that is an -greedoid. Now we prove that, , is a greedoid.(1)Obviously, for any .(2)If and , then there are such that , , and . Next we prove , .If , then it is obviously .
If , let . Since , it is easy to see that there exists such that . By (IG2), there exists such that , where . We get and .
Since , it is easy to see that there exists such that . By (IG2), there exists such that , where . We get and .
Continue the above process and we can obtain .
Analogously, we can obtain .
Since , take such that . Then by Lemma 12, . Since , there is such that and . It is easy to see that , , and . This implies that , , and .
Therefore is a greedoid for any .

Corollary 15. Let be an -greedoid. If , then , .

Remark 16. If is an -prematroid, it is obvious to see that Corollary 15 holds by (LI2). An -greedoid need not satisfy (LI2). However, by Theorem 14, Corollary 15 still holds. This is an important property for -greedoids. Many results of -greedoids in this paper are based on this property.

Definition 17. An -greedoid is called a perfect -greedoid if it satisfies the following condition: , if, , , then .

By Corollary 15 and Definition 17, we can easily obtain the following corollary.

Corollary 18. An -greedoid is a perfect -greedoid if it satisfies the following condition: , if, , , then .

Example 19. Let . Define byand by . It is easy to verify that is an -greedoid but it is not perfect, since for all but .

Lemma 20. Let be an -greedoid. If , then .

Proof. Let . By the definition of , there exists such that . By Corollary 15, we have . Then .

Theorem 21. Let be an -greedoid. Define . Then is an -greedoid. Moreover, if is perfect, then .

Proof. (1) Obviously, for any . Thus .
(2) Let , and for some . Then . By Lemma 11, we know that . Since and is a greedoid, there exists such that . In this case, . It is obvious that for every and for every . This implies that .
(3) It is obvious that . Since is perfect, by Corollary 18, . Hence .

Corollary 22. Let be an -greedoid. Then is perfect if and only if .

Definition 23. Let be -greedoids. A mapping is called an -feasibility preserving mapping from to if for all , where .

Theorem 24. Let be -greedoids and let be an -feasibility preserving mapping from to . If is a perfect -greedoid, then the following conditions are equivalent:(1) is an -feasibility preserving mapping from to .(2) is a feasibility preserving mapping from to for each .

Proof. Let . Then there exists such that . Since is an -feasibility preserving mapping from to , . We have . This implies that is a feasibility preserving mapping from to for each .
Assume that is a feasibility preserving mapping from to for each . Let . Then for all . Thus for all . Since is a perfect -greedoid, by Corollary 18, . Therefore is an -feasibility preserving mapping from to .

Theorem 25. Let be an -greedoid. Then, , is a greedoid. Since is a set finite, there is at most a finite number of greedoids on . Thus there is a finite sequence such that(1)if , then ;(2)if , then .The sequence is called the fundamental sequence for .

Proof. We define an equivalence relation on by . Since is a finite set, the number of greedoids on is finite. Thus there exist at most finitely many equivalence classes which are, respectively, denoted by . Next we prove that each is an interval. We only need to show that, with , if , then . Since , by Lemma 20, we know that . As , . Thus by the definition of . This implies that is an interval. Let and . Obviously, the sequence is the fundamental sequence for .

Definition 26. An -greedoid with the fundamental sequence is called a closed -greedoid if whenever , then .

Theorem 27. Let be an -greedoid with the fundamental sequence . Then is a closed -greedoid if and only if it satisfies the following condition:  and , if for all , then .

Proof. Suppose that satisfies ; , then . Let for all . Then for all ; thus for all . Since satisfies , . Thus . This implies that for all . Therefore, for all ; that is, is closed. Conversely, assume that is a closed -greedoid. Let , , and for all . Since , for some . Take . Then and thus . Since is closed, . Hence . By Corollary 15, we have . This means that satisfies .

4. -Fuzzy Greedoids and Fuzzifying Greedoids

Definition 28. A mapping is called a fuzzy family of feasible fuzzy sets on if it satisfies the following conditions:  (IIFG1) . (IIFG2) If for and for some , then where .If is a fuzzy family of feasible fuzzy sets on , then the pair is called an -fuzzy greedoid.

Definition 29. A mapping is called a fuzzy family of feasible sets on if it satisfies the following conditions:  (FYG1) . (FYG2) For any , if , then .If is a fuzzy family of feasible sets on , then the pair is called a fuzzifying greedoid. For , can be regarded as the degree to which is a feasible set.

Theorem 30. Let be a mapping. Then the following conditions are equivalent:(1) is an -fuzzy greedoid.(2)For each , is an -greedoid.(3)For each , is an -greedoid.

Proof. (IG1) It is obvious that for any .
(IG2) If and for some , then, by (IIFG2), we have . Further by , we obtain . Then there exists such that ; that is, . This shows that satisfies (IG2). Therefore is an -greedoid for each .
(IIFG1) For any , . We have .
(IIFG2) Suppose that and for some . In order to prove (IIFG2), take and . Then and . This implies . Hence, by (IG2), there exists such that . This implies . Further we have . By the arbitrariness of , we obtain .
Analogously, we can obtain .

Corollary 31. Let be a map. Then the following conditions are equivalent:(1) is a fuzzifying greedoid.(2) is a greedoid for each .(3) is a greedoid for each .

Definition 32. Let be -fuzzy greedoids. A mapping is called an -fuzzy feasibility preserving mapping from to if for all , where .

Definition 33. Let be fuzzifying greedoids. A mapping is called a fuzzifying feasibility preserving mapping from to if for all .

Theorem 34. Let be -fuzzy greedoids and let be an -fuzzy feasibility preserving mapping from to . Then the following conditions are equivalent:(1) is an -fuzzy feasibility preserving mapping from to .(2) is an -feasibility preserving mapping from to for each .(3) is an -feasibility preserving mapping from to for each .

Proof. Let be an -fuzzy feasibility preserving mapping from to and ; , we have and . In other words, is a fuzzy feasibility preserving mapping from to for each .
Next we need to prove that for any . Let . Then . Since is an -feasibility preserving mapping from to for each , ; that is, . By the arbitrariness of , we have . Therefore is an -fuzzy feasibility preserving mapping from to .
Analogously, we can obtain .

Corollary 35. Let be fuzzifying greedoids and let be a fuzzifying feasibility preserving mapping from to . Then the following conditions are equivalent:(1) is a fuzzifying feasibility preserving mapping from to .(2) is a feasibility preserving mapping from to for each .(3) is a feasibility preserving mapping from to for each .

Theorem 36. Let be a fuzzifying greedoid. Then, , is a greedoid. Since is a finite set, there is at most a finite number of greedoids that can be defined on . Thus there is a finite sequence such that(1)if , then ;(2)if , then .The sequence is called the fundamental sequence for .

Proof. For , we can easily check that if , then . We define an equivalence relation on by . Since is a finite set, the number of greedoids on is finite. Thus there exist at most finitely many equivalence classes which are, respectively, denoted by . Next we prove that each is a left open and right closed interval. It is trivial that each is an interval. Suppose that and is the corresponding cut greedoid with respect to . Then holds. For all , and for all . It follows that and . Thus . Hence and . Obviously, the sequence is the fundamental sequence for .

5. FYG as a Subcategory of IG

In this section, we will study the relation between fuzzifying greedoids and fuzzy greedoids from the viewpoint of category theory.

Theorem 37. Let be a fuzzifying greedoid, and . Then (1);(2) is a closed and perfect -greedoid.

Proof. (1) , suppose that , where . Then by the definition of . This means that . Conversely, assume that . It is obvious that for every and for every . Hence . So . This implies that . Therefore .
(2) It is easy to see that satisfies (IG1). Now we prove that satisfies (IG2). Suppose that and for some . Then . We know that . Since and is a greedoid, there exists such that . In this case, . It is obvious that for every and for every . This implies that .
By (1) and the definition of , . By Corollary 22, is a perfect -greedoid. By Theorem 36, is a closed -greedoid.

Theorem 38. Let be fuzzifying greedoids. If is a fuzzifying feasibility preserving mapping from to , then is an -feasibility preserving mapping from to .

Proof. Since is a fuzzifying feasibility preserving mapping from to , is a feasibility preserving mapping from to for each by Corollary 35. By Theorem 37, is a feasibility preserving mapping from to for each . Since is perfect, by Theorem 24, is an -feasibility preserving mapping from to .

Theorem 39. Let be a closed and perfect -greedoid. Define a map by Then(1) for all ;(2) is a fuzzifying greedoid.

Proof. (1) , suppose that . Then by the definition of . We have . This means that . Conversely, assume that ; that is, . Since is a closed -greedoid, there exists such that and . By , . This means that . Therefore for all .
(2) By (1) and Corollary 31, is a fuzzifying greedoid.

Theorem 40. Let be closed and perfect -greedoids. If is an -feasibility preserving mapping from to , then is a fuzzifying feasibility preserving mapping from to .

Proof. Since is an -feasibility preserving mapping from to , is a feasibility preserving mapping from to for each by Theorem 24. By Theorem 37, is a feasibility preserving mapping from to for each . By Corollary 35, is a fuzzifying feasibility preserving mapping from to .

Theorem 41. (1) If is a fuzzifying greedoid, then .
(2) If is a closed and perfect -greedoid, then .

Proof. (1) By Theorem 37, is a closed and perfect -greedoid. Then for all by Theorems 37 and 39. Hence .
(2) By Theorems 37 and 39, is a closed and perfect -greedoid. Then . Since is a closed and perfect -greedoid, for all by Theorem 39. Therefore since is a closed and perfect -greedoid.

By Theorems 37–41, we have the following.

Theorem 42. is isomorphic to .

Theorem 43. Let be an -greedoid with the fundamental sequence . , define , where . Let . Then(1) is an -greedoid;(2) for all ;(3) is a closed and perfect -greedoid.

Proof. (1) It is easy to see that satisfies (IG1). Now we prove that satisfies (IG2). Suppose that and for some . Then . We know that . Since and is a greedoid, there exists such that . In this case, . It is obvious that for every and for every . This implies that .
(2) Suppose that . Then there exists such that and for each by the definition of . This implies that . Conversely, assume that . Then there exists such that and . We have for every and for every . This implies that . Hence . We have . Therefore for all .
(3) By (1) and (2), is a closed -greedoid. By (2) and the definition of , . Hence is a perfect -greedoid.

Theorem 44. If is a closed and perfect -greedoid, then .

Proof. Since is a closed -greedoid, by Theorem 43, for all . By Theorem 43, is a perfect -greedoid. Since is a perfect -greedoid, then .

Theorem 45. Let be -greedoids. If is an -feasibility preserving mapping from to , then is an -feasibility preserving mapping from to .

Proof. Since is an -feasibility preserving mapping from to , is a feasibility preserving mapping from to for each by Theorem 24. Let and be the fundamental sequences for and , respectively. , there exist and such that . Let . Then . It is easy to verify that and . Thus is an -feasibility preserving mapping from to for all . By Theorem 43, is a perfect -greedoid. By Theorem 24, is an -feasibility preserving mapping from to .

Therefore, there is a functor from to sending every -greedoid to , in symbols .

By Theorems 43–45, we have the following.

Theorem 46. Let be the inclusion functor. Then(1) for any -greedoid ;(2) for any closed and perfect -greedoid .

By Theorem 46, we have the following.

Theorem 47. is a coreflective full subcategory of .

By Theorems 47 and 42, we have the following.

Corollary 48. can be regarded as a coreflective full subcategory of .

6. as a Subcategory of FYG and IG as a Subcategory of IIFG

In the following, we will study the relation between -greedoids and -fuzzy greedoids and the relation between greedoids and fuzzifying greedoids from the viewpoint of category theory.

Let be an -greedoid. Then is an -fuzzy greedoid. Therefore, we define the inclusion mapping by .

By Theorem 30, Corollary 31, Theorem 34, and Corollary 35, we will define two functors from to in the following theorem.

Theorem 49. (1) , is a functor.
(2) , is a functor.

Theorem 50. (1) For any -greedoid , .
(2) For any -fuzzy greedoid , and .

Proof. The proof is trivial and straightforward.

Since both and are concrete categories, we have the following.

Theorem 51. Both and are Galois correspondences between and , and is the right inverse of both and . Hence can be embedded in as a simultaneously reflective and coreflective full subcategory.

Corollary 52. is a simultaneously reflective and coreflective full subcategory of .

7. as a Subcategory of IG and FYG as a Subcategory of IIFG

In this section, we will study the relation between fuzzifying greedoids and -fuzzy greedoids and the relation between greedoids and -greedoids from the viewpoint of category theory.

Theorem 53. Let be a fuzzifying greedoid. Define by Then is an -fuzzy greedoid.

Proof. It is easy to see that satisfies (IIFG1). Now we prove that satisfies (IIFG2). Suppose that , and for some . Then . We know that . This implies , since is a fuzzifying greedoid. It is obvious that for every and for every . This implies where . Therefore is an -fuzzy greedoid.

Corollary 54. Let be a greedoid. Define by . Then is an -greedoid.

Theorem 55. Let be fuzzifying greedoids. If is a fuzzifying feasibility preserving mapping from to , then is an -fuzzy feasibility preserving mapping from to .

Proof. , for all . Since is a fuzzifying feasibility preserving mapping from to , for all . By the definition of and , we haveThis implies that is an -fuzzy feasibility preserving mapping from to .

Corollary 56. Let be greedoids. If is a feasibility preserving mapping from to , then is an -feasibility preserving mapping from to .

Theorem 57. Let be an -fuzzy greedoid. Define by Then is a fuzzifying greedoid.

Proof. It is easy to see that satisfies (FYG1). Now we prove that satisfies (FYG2). Suppose that and . Then , , and for each . Since is an -fuzzy greedoid, that is, HenceThis implies that (FYG2) holds. Therefore is a fuzzifying greedoid.

Corollary 58. Let be an -greedoid. Define by for some . Then is a greedoid.

Theorem 59. Let be -fuzzy greedoids. If is an -fuzzy feasibility preserving mapping from to , then is a fuzzifying feasibility preserving mapping from to .

Proof. For each , by the definition of and being an -fuzzy feasibility preserving mapping from to , we have This implies that is a fuzzifying feasibility preserving mapping from to .

Corollary 60. Let be fuzzy greedoids. If is an -feasibility preserving mapping from to , then is a feasibility preserving mapping from to .

Theorem 61. (1) If is an -fuzzy greedoid, then .
(2) If is a fuzzifying greedoid, then .

Proof. (1) For each , by the definitions of and , we haveThis implies that .
(2) For each , by the definitions of and , we have This implies that .

Corollary 62. (1) If is an -greedoid, then .
(2) If is a greedoid, then .

Based on the theorems and corollaries in this section, we have the following.

Theorem 63. (1)   is a coreflective full subcategory of  .
(2) is a coreflective full subcategory of .

8. Conclusion

In this paper, we introduce the concepts of -greedoids, fuzzifying greedoids, and -fuzzy greedoids as different approaches of the fuzzification of greedoids and study the relations among greedoids, fuzzy greedoids, fuzzifying greedoids, and -fuzzy greedoids from a categorical point of view. The main results in this paper are in the following diagram, where , mean, respectively, reflective and coreflective: Such facts will be useful to help further investigations and it is possible that the fuzzification of greedoids would be applied to some combinatorial optimization problems in the future.

Conflict of Interests

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

Acknowledgments

The project is supported by the National Natural Science Foundation of China (11371002) and Specialized Research Fund for the Doctoral Program of Higher Education (20131101110048).