
 
variable, material> based on the engineering knowl-
edge; for each category the classification is built on 
a specialisation/generalisation relations (i.e. inclu-
sion/coverage relations), i.e. moving to the next 
lower level of the directed graph each category 
(<event, system, function, variable,  material>) is 
specified in subclasses (and over sub-sub classes etc 
down to specific concepts; i.e. “system elements” if 
we follow the “System” category) and moving to the 
next higher level of the directed graph sub-classes 
(or individual concepts) are generalised to the next 
level of sub-classes (or a class).  
Keywords are going to be used to quickly find 
documents through queries of (very) large databases;  
this should be possible by building keyword combi-
nations without following the predefined structure of 
the classification but using the relations  
We superimpose a partonomy on the keyword 
classification, or more precisely a fuzzy partonomy; 
this will allow us to find keywords which are partly 
the same for a query regardless of where they are 
defined in the underlying keyword classification (or 
where they are located in the directed graph).  
A partonomy that is built on part-of relationships 
is a primitive of the formal theory of parthood rela-
tions; parthood relations specify part-of and overlap 
within a whole; part-of is reflexive, anti-symmetric 
and transitive (the transitivity is sometimes difficult 
to justify) and overlap between x and y is defined as 
O(x, y) := {z │ z  
x and z  y} where the symbol 
“”  now denotes part-of. 
The fuzzy keyword classification and partonomy 
are built on inclusion and coverage,  which are un-
derstood to be relations between fuzzy subsets. The 
classifications and part-of relations are collected in 
matrices of coverage/inclusion of keywords; the 
cells of the matrix are numbers [0, 1] which show 
the degree of coverage and inclusion.  
A fuzzy ontology is a relation on fuzzy sets, i.e. a 
relation associated with a membership function; let 
K
i
 be a finite fuzzy set of keywords identified with a 
level of the directed graph and a category <event, 
system, function, variable,  material>, hence i = 1, 
…, 5; a membership function is a mapping of K
i 
 x 
K
j
 on L, a lattice or a partially ordered set; the set of 
linguistic labels {negligible, weak, moderate, strong, 
perfect} is a lattice which means that a relation be-
tween two sets of keywords can be stated and de-
scribed with a linguistic label. 
4.2  Fuzzy Reasoners  
We need to find a way to combine linguistic labels 
and numbers for the following reasoning schemes so 
that we can use them to get numbers for the inclu-
sion/coverage matrix; this can be done in the follow-
ing way (the linguistic labels can be defined accord-
ing to the context; the labels can also be overlap-
ping; cf. Carlsson et al (2010b) for details). Let us 
consider a domain  of keywords that have 
been classified based on some property with real 
numbers in [0, 1]; we will consider three fuzzy sub-
sets A, B and C of keywords (similar to K
i
) in the 
domain D; we will first work with the fuzzy subsets 
A and B. We say that A is a fuzzy subset of B (both 
defined in the domain D) and write  
 
   
       
 
(1)
We can then define the two concepts inclusion and 
coverage in terms of these fuzzy subsets (as both are 
defined in the same domain) by following the 
intuitive understanding we have in Figure 3
;  it 
should be noted that the min-operator is one of a 
class of t-norms that can be used to express the 
combinations (cf. Carlsson et al (2010b)). 
Degree of subsethood (inclusion) of  in  
,
min
,
/
 
(2)
Degree of supersethood (coverage) 
,
min
,
/
 
(3)
Now we can combine the two concepts as a 
categorisation of the two subsets which can be used 
to order the subsets of keywords – for this we have 
several possibilities but we can use the following 
simple characterisation:  
Degree of similarity 
,
min
,
/max
,
 
(4) 
It is clear that ,  ,.  
We will get a similar representation of the fuzzy 
subset C as it is fully a subset of A (cf. Figure 3). 
We can now illustrate these concepts with some 
numerical examples; the numbers would be similar 
to those used in Figure 4.  
Let 
  0.4,0.6,0.8,0.3 and  
 0.5,0.4,0.8,0.6.  
Then A is almost a subset of B since 
 for   1,3,4,5 but not quite since 
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