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In the application of fuzzy reasoning, researchers usually choose the membership function optionally in some degree. Even though the membership functions may be different for the same concept, they can generally get the same (or approximate) results. The robustness of the membership function optionally chosen has brought many researchers' attention. At present, many researchers pay attention to the structural interpretation (definition) of a fuzzy concept, and find that a hierarchical quotient space structure may be a better tool than a fuzzy set for characterizing the essential of fuzzy concept in some degree. In this paper, first the uncertainty of a hierarchical quotient space structure is defined, the information entropy sequence of a hierarchical quotient space structure is proposed, the concept of isomorphism between two hierarchical quotient space structures is defined, and the sufficient condition of isomorphism between two hierarchical quotient space structures is discovered and proved also. Then, the relationships among information entropy sequence, hierarchical quotient space structure, fuzzy equivalence relation, and fuzzy similarity relation are analyzed. Finally, a fast method for constructing a hierarchical quotient space structure is presented.

Since the fuzzy set theory was proposed by Zadeh in 1965 [

The isomorphic fuzzy equivalence relations have the same hierarchical quotient space structure [

The paper is organized as follows. Some relevant preliminary concepts are reviewed briefly in Section

For convenience, some preliminary concepts are reviewed or defined at first. Let

Let

for all

for all

Let

for all

for all

for all

Let

According to Proposition

Let

Proposition

Let

Let

In addition, a pyramid model can be established based on the number of block in each layer of the hierarchical quotient space structure, which is shown in Figure

However, the different fuzzy equivalence relations may become into the same hierarchical quotient space structure. In Example

The hierarchical quotient space structure

The pyramid model of a hierarchical quotient space structure.

Let

Let

Let

Let

A fuzzy equivalence relation

Let

From the viewpoints of both clustering and classification, the isomorphic fuzzy similarity relations can induce the same hierarchical quotient space structure, and have the same ability of clustering (or classification) for objects in

A hierarchical quotient space structure can uncover the essential characteristics of a fuzzy concept better than a fuzzy set. However, how to measure uncertainty of the hierarchical quotient space structure is still an open question. Information entropy is a very useful tool for measuring the uncertainty of vague information, and it has been studied based on the rough set and fuzzy set in the literature. For instance, Liang et al. [

Let

Let

Each layer of the hierarchical quotient space structure

Let

Obviously, any one partition sequence of the hierarchical quotient space structure

Let

Let

It follows from the definition of the hierarchical quotient space structure that

Theorem

Let

Assume that

Obviously, two different

Let

A

Let

If a subblock sequence

Let

Let

Let

Let

Therefore,

Theorem

Let

Firstly we prove that the conclusion in Theorem

When

Given a hierarchical quotient space structure

Let

Assume that

From the viewpoint of classification (clustering) analysis, if two hierarchical quotient space structures are isomorphic, they have the same classification abilities of the objects in the set of

Let

Since

In Theorem

Let

According to Definitions

The relationships between fuzzy relations and hierarchical quotient space structure.

Isogeny but not Similarity ( | Similarity ( | Isomorphism ( | Same ( |

Similarity but not Isomorphism ( | Isomorphism ( | Same ( | |

Isomorphism but not Same | Same ( | ||

Same ( |

In Section

Assume that

Because

Let the relation matrix of the fuzzy similarity relation

If

Let the matrix of fuzzy similarity relation

Based on Theorem

Let

relation derived from

can be obtained.

{If

in the different subblock respectively. A quotient space

{if

There is an example to illuminate Algorithm

Let

Because the similarity relation matrix

When

When

When

When

When

Since

Due to human various subjective ideas and evaluation criteria, the same fuzzy concept might have different memberships which leads to different fuzzy similarity relations. However, if these fuzzy similarity relations are isomorphic, then they will induce to the same hierarchical quotient space structure. If different hierarchical quotient space structures have the same information entropy sequence and the same subblock sequence, then they have the same classification (clustering) ability and contain the same information. Therefore, the information entropy sequence is a very useful attribute for measuring the uncertainty of a hierarchical quotient space structure. In this paper, the information entropy sequence of a hierarchical quotient space structure is discussed, through the analysis of the relationships among information entropy sequence, hierarchical quotient space structure, fuzzy equivalence relation, and fuzzy similarity relation. A fast-constructing hierarchical quotient space structure method is presented. This further reveals the essential of a hierarchical quotient space structure. In real world, “fuzziness” and “crispness” are relative and can be transformed into each other with different granularity. The hierarchical quotient space structure is just a bridge between the fuzzy granule world and the clear granule world, and the research on uncertainty of the hierarchical quotient space structure will contribute to develop both granular computing theory and information entropy theory.

This work is supported by the National Natural Science Foundation of China (no. 61073146 and no. 60773113), Science & Technology Research Program of the Municipal Education Committee of Chongqing of China (no. KJ060517), Natural Science Foundation Project of CQ CSTC (no. 2008BA2017), and Doctor Foundation of CQUPT (no. 2010-06).