Obviously, for each , X can be divided into more than two classes, which satisfy in various degrees. We then end up with a general tree structure of X.

In reality, this kind of granulations is used extensively.

Assume that is a partition of Y. Define:

is a partition of X.

Define: .

X₄ is the set of examinees admitted to key universities, X₃ is the set of examinees admitted to general universities, etc. In a word, based on the partition of a subset [0,700] of real numbers, we have a corresponding partition of examinees.

Granulation by attribute values is extensively used in rough set theory (Pawlak, 1982). Assume that , denoted by in rough set, is a data table (information system), where . A_i is the quotient set corresponding to . Granulation by attribute values is sometimes called the quantification of attribute values.

Define . X_i is a quotient set of X, where is the granulation of f. If X is simultaneously granulated by and , the corresponding quotient space obtained is denoted by . is the supremum of and . Using all combinations of the quantification of attribute values, the corresponding quotient spaces (sets) gained are all possible quotient spaces that can be obtained by the granulation based on the attribute values. One of the main goals in rough set analysis is to choose a proper one from among all the possible quotient spaces so that the recognition or classification task can be effectively achieved.

Based on attribute values we have the following quotient spaces.

and have

wheredenotes the supremum operation.

If all quotient spaces in a semi-order lattice, which order is decided by the inclusion relation of subsets of attributes, are given, the so-called ‘attribute reduction’ in rough set theory is to find the simplest supremum within the semi-order lattice, where the ‘simplest’ means the minimal number of attributes.

In rough set theory, given a quotient space (knowledge base) and a set S, if S can be entirely represented by the union of elements in the quotient space, S is called crisp or discernible, otherwise, indiscernible. The indiscernible set can be represented by the upper and lower approximation sets.

‘Fuzziness’ is an inevitable outcome of his/her observation when he/she watches the world at a coarse grain-size. So the concept of fuzziness is closely related to granularity and can be described by quotient spaces with different granularities. Using the elements of a set of quotient spaces to depict ‘fuzziness’, the cost is greatly reduced since the potential of quotient spaces is much less than that of the original space. The description of fuzziness by membership functions in fuzzy set theory (Zadeh, 1965) is very expensive. The description of fuzziness in rough set theory is less expensive but still much more expensive than the quotient space description.

When ‘fuzziness’ appears in our observation, this means that we are lacking detail. If we use an elaborate tool to describe a ‘fuzzy’ object in detail, it seems unreasonable. Thus, the representation of fuzziness by membership functions in fuzzy set theory is not necessarily an effective method.

The geometrical interpretation of the projection-based method is that observing the same object from different view-points. For example, the three perspective drawings of a mechanical part are based on the projection-based method.

From quotient set X₁ defined by R, we have a quotient space . Since structure T₁ is coarser than T, is a quotient space of . Through coarsening structure T, we have a new coarse space which may not necessarily be obtained from domain granulation or granulation by attributes.

Let , T₁<T.

Then, we have X₁ and quotient space , where X₁ = X.

Since the quotient topology [T] of X₁ is T, cannot be obtained from domain granulation.

An example is given below to show how to use the structure-based granulation to problem solving.

Assume that X is a set of events in question. All time intervals that events happened in are regarded as attribute functions, the 13 temporal relations as structure T. Then, temporal planning is transformed into solving problem .

Coarsening structure T, e.g., 13 temporal relations are simplified to eight relations T₁ as follows.

Quotient structure T₁ is obtained from structure T by merging both and into , and both and into .

Define quotient set , where is a set of temporal relations between events x and y.

Quotient space is coarser than the original space . In Chapter 6, we will show the problem solving based on the coarsening structure and by using the corresponding falsity preserving, etc. properties to reduce the computational complexity in problem solving.

It should be noted that , doesn't always hold here. In order to distinguish the classification with overlapped elements, we use angle brackets < > instead of square brackets [ ]. Here the symbol A_i is used for representing both subsets of X denoted by , and elements of denoted by .

In Chapter 2, we will discuss one specific case, i.e., tolerant relations, of classification with overlapped elements.