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the 
operator less (<) is called the lower diagonal partition (Figure 
1). 
Using this scheme partitions corresponding to the fuzzy 
modifications of the operators equal, greater, and less , may be 
assigned in the search space as shown in Figure 2. This is 
possible because, by common sense reasoning, the fuzzy 
expression about  modifies the expression equal to by 
generating a narrow band around the value x. The width of the 
band or size of the partition generated also depends on the 
magnitude of the crisp value x. By common sense the partition 
generated by "about x", "more or less x" , and "roughly equal 
to x" are radial partitions stretching over both sides of the line of 
equality (Figure 2). Naturally the partition induced by "roughly 
equal to x" must be somewhat wider than those generated by 
"more or less x", or "about x". 
When the fuzzy modifier much is applied to the operators great 
and /ess it induces partitions which exclude all values close to 
the line of equality. It is logical, therefore, to place the lower 
boundary of the partition of much greater than x as far away as 
possible in the upper diagonal space. This will create a radial 
partition enclosing a "very wide" angle with the line of equality. 
Similarly the partition for much less than x encloses a "very 
wide" angle with the line of equality in the lower diagonal 
space. Note that the partitions generated by greater than x and 
less than x are supersets of the partitions induced by much 
greater than x and much less than x respectively. 
The fuzzy expressions slightly greater than x and slightly less 
than x give rise to asymmetric narrow, radial partitions "very 
close" to the line of equality. Naturally, the partition induced by 
slightly less than x lies in the lower diagonal space. The 
partition induced by slightly more than x lies in the upper 
diagonal space, very close to the line of equality. 
The common sense interpretation of the fuzzy expressions 
introduced above, must now be defined mathematically, in 
order to form the basis for the proposed fuzzy geometric 
partitions based representation of fuzzy objects and comparison 
of fuzzy database objects for retrieval purposes. 
2.1 Specification of the Fuzzy Partitions Induced 
by Fuzzy Restrictions. 
Fuzzy expressions, such as about x and more or less x where 
x is a number, are said to constitute elastic constraints on the 
set of admissible real numbers (Dubois and Prade, 1980, 
Kandel, 1986). An arbitrary real number x which satisfies the 
elastic constraint is said to be a generic value of the fuzzy 
expression (Kandel, 1986; Dubois and Prade, 1980). 
The term generic value is used, in this study, to characterize 
crisp values lying within a vague interval or the fuzzy partition 
associated with a fuzzy number or fuzzy restriction. The basic 
idea upon which the concept of fuzzy geometric partitions is 
founded, is the simple, intuitive, idea that in common sense 
reasoning a vague expression, such as about 5, invokes a 
mental band of uncertain but narrow width around the crisp 
number 5 as explained in the previous section. The main 
assumption is that the human mind realizes this vague interval 
by a process in which values picked out from the domain of real 
numbers are subconsciously compared with the crisp value 5 
and rejected if they differ "too much" from it. In this respect the 
vague expression, about 5, is equivalent to the generic binary 
relation, about(x, 5), where x is an arbitrary number which 
may or may not be an acceptable member of the vague set about 
5, depending on its "distance" from the crisp value 5 (see 
Figures 2 and 6). 
Let the collection of the fuzzy restrictions: equal to, greater than, 
less than, much greater than, much less than, slightly greater 
than, slightly less than, about, more or less, roughly equal to, 
be denoted by PRED. 
Using the idea of the diagonal subset generated by the equality 
operator in Dowsing et al. (1986), the definition of the equality 
operator can be extended and generalized to a general 
177 
>< 
  
   
  
  
line of equality 
  
>>, 
e greater than 
C mo less than o. greater than 
Figure 1: Basic partitions of the search space. 
  
  
Y 
A 
line of equality 
  
  
  
€ roughly 
slightly less than 
about 
e» more or less 
Figure 2: Partitions induced by some common 
fuzzy expressions. 
  
  
slightly greater than 
comparison operator R e PRED. Let the partition induced by 
the general fuzzy comparison operator, R, in a two dimensional 
search space be denoted by Pg. The generalized extension of 
the definition of equality operator (Dowsings et al., 1986) is as 
follows: 
For an interpretation I, with the universe X, the set Ry on 
which R is to be true must be a radial or sectoral subset: 
{xy)lxe X,ye X,xRy}of XxX 
Based on this generalized definition, individual members of the 
PRED set can now be defined. If we let R = much greater than 
we have: 
much greater than, = {(x,y)|x € X, y € X, x >> y} for all 
x,y) € X x X. (1)