Categories of ( I , I )-Fuzzy Greedoids

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


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][4][5][6][7][8][9][10][11][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][15][16][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.

Preliminaries
Throughout this paper,  = [0, 1] and  is a nonempty finite set.We denote the set of all subsets of  by 2  and the set of all fuzzy subsets of  by   .
For  ∈ (0, 1] and  ⊆ , define a fuzzy set  ∧  as follows: A fuzzy set  ∧ {} is called a fuzzy point and denoted by   .
For  ∈ (0, 1] and  ∈   , we define Definition 1 (see [24]).If I is a nonempty subset of 2  , then the pair (, I) is called a (crisp) set system.A set system (, I) is called a matroid if it satisfies the following conditions: Definition 2 (see [25]).A greedoid is a pair of (, F), where F ⊆ 2  is a set system satisfying the following conditions: (G1) For every  ∈ F there is an  ∈  such that  − {} ∈ F.
The sets in F are called feasible (rather than "independent").
Definition 4. Let (  , F  ) ( = 1, 2) be greedoids.A mapping  :  1 →  2 is called a feasibility preserving mapping from 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 I : 2  →  is called an -fuzzy family of independent sets on  if it satisfies the following conditions: If I is an -fuzzy family of independent sets on , then the pair (, I) is called an -fuzzifying matroid.For  ∈ 2  , I() can be regarded as the degree to which  is an independent set.
Definition 7 (see [19]).A subfamily I of   is called a family of independent -fuzzy sets on  if it satisfies the following conditions: (LI1)  0 ∈ I.
If I is an -fuzzy family of independent -fuzzy sets on , then the pair (, I) is called an (, )-fuzzy matroid.
If F is a fuzzy family of feasible fuzzy sets on , then the pair (, F) is called an (, )-fuzzy greedoid.
Definition 29.A mapping F : 2  →  is called a fuzzy family of feasible sets on  if it satisfies the following conditions: If F is a fuzzy family of feasible sets on , then the pair (, F) is called a fuzzifying greedoid.For  ∈ 2  , F() can be regarded as the degree to which  is a feasible set.
This implies that  is a fuzzifying feasibility preserving mapping from ( This implies that   ∘   (F) ≥ F.
Based on the theorems and corollaries in this section, we have the following.

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.