ALGEBRAIC STRUCTURES FOR ROUGH SETS
GIANPIERO CATTANEO AND DAVIDE CIUCCI
Abstract. Using as example an incomplete information system with support
a set of objects X , we discuss a possible algebrization of the concrete algebra
of the power set of X through quasi BZ lattices. This structure enables us
to define two rough approximations based on a similarity and on a preclusive
relation, with the second one always better that the former. Then, we turn our
attention to Pawlak rough sets and consider some of their possible algebraic
structures. Finally, we will see that also Fuzzy Sets are a model of the same
algebras. Particular attention is given to HW algebra which is a strong and
rich structure able to characterize both rough sets and fuzzy sets.
1. Introduction
Rough structures describe the behavior of concepts, properties, data, abstract
objects in general, that may present some intrinsic vague, ambiguous, unsharp
features. For the sake of simplicity we will simply speak of vague objects. To these
objects it is usually associated some data which characterize them and which can
be used to classify them. Data that are often organized as a table associating to
each object some attributes values. As observed by Pawlak [Paw98]:
Data are often presented as a table, columns of which are labelled
by attributes, rows by objects of interest and entries of the table
are attributes values.
For example, in a table containing information about patients
suffering from a certain disease objects are patients (strictly speak-
ing their ID’s), attributes can be for example blood pressure, body
temperature etc., whereas the entry corresponding to object Smiths
and the attribute blood pressure can be normal.
Such tables are known as information systems, attribute-value
tables or information tables.
A natural abstract definition, that reflects this behavior, is the following one.
Definition 1.1. An Information System is a structure
K(X) = hX, Att(X), val(X), F i
where:
• X (called the universe) is a non empty set of objects (situations, entities,
states);
• Att(X) is a non empty set of attributes, which assume values for the objects
belonging to the set X;
• val(X) is the set of all possible values that can be observed for an attribute
a from Att(X) in the case of an object x from X;
1
2 G. CATTANEO AND D. CIUCCI
• F (called the information map) is a mapping F : X × Att(X) → val(X)
which associates to any pair, consisting of an object x ∈ X and of an
attribute a ∈ Att(X), the value F (x, a) ∈ val(X) assumed by a for the
object x.
In general we assume that an information system satisfies the following two
conditions of coherence:
(1) F must be surjective; this means that if there exists a value v ∈ val(X)
which is not the result of the application of the information map F to any
pair (x, a) ∈ X × Att(X), then this value has no interest with respect to
the knowledge stored in the information system.
(2) For any attribute a ∈ Att(X) there exist at least two objects x
1
and x
2
such that F (x
1
, a) 6= F (x
2
, a), otherwise this attribute does not supply any
knowledge and can be suppressed.
Example 1.2. Imagine you want to rent a flat, and you start to collect information
about some apartments. The features you are interested in are: the price of the flat;
its location, i.e., if it is down-town or not; the number of rooms and if it has furniture
or not. But you are not interested in, for example, at which floor it is located. So,
when organizing the data in your possession, you will consider just the first four
attributes and omit the floor number attribute. The result is a situation similar to
the one presented in Table 1, where the set of objects is X = {f
1
, f
2
, f
3
, f
4
, f
5
, f
6
},
the family of attributes is Att(X)={Price, Rooms, Down-Town, Furniture} and the
set of all possible values is val(X)={high, low, medium, 1, 2, yes, no}.
Flat Price Rooms Down-Town Furniture
f
1
high 2 yes no
f
2
high 1 yes no
f
3
high 2 yes no
f
4
low 1 no no
f
5
low 1 no no
f
6
medium 1 yes yes
Table 1. Flats information system.
A generalization of such a concept are incomplete information systems, i.e., in-
formation systems in which not all values are available for all objects. Such a
generalization is justified by empirical observations. Quoting [ST99]:
An explicit hypothesis done in the classic rough set theory is that
all available objects are completely described by the set of available
attributes. [...]
Such a hypothesis, although sound, contrasts with several empirical
situations where the information concerning the set A is only partial
either because it has not been possible to obtain the attribute values
or because it is definitely impossible to get a value for some object
on a given attribute.
ALGEBRAIC STRUCTURES FOR ROUGH SETS 3
So, let K(X) = hX, Att(X), V al(X), F i be an incomplete information system. In
order to denote the fact that the value possessed by an object x
i
with respect to
the attribute a
j
is unknown, we introduce a null value and write F (x
i
, a
j
) = ∗.
Example 1.3. As a concrete example, let us consider the information system
described in Table 1. It can happen that given a flat we do not know all its
features, for instance because some information was missing on the advertisement.
The result is some missing values in the information system, as shown in Table 2.
Flat Price Rooms Down-Town Furniture
f
1
high 2 yes *
f
2
high * yes no
f
3
* 2 yes no
f
4
low * no no
f
5
low 1 * no
f
6
* 1 yes *
Table 2. Flats incomplete information system.
Given an information system, complete or not, if we consider pairs of objects
belonging to the universe X, we can describe their relationship through a binary
relation R. A classification and logical–algebraic characterization of such binary
relations can be found in literature (for an overview see [Orl98b]). Generally, these
relations are divided into two groups: indistinguishability and distinguishability
relations. In our analysis, we are dealing with a tolerance (or similarity) relation,
i.e., a reflexive and symmetric relation, and its opposite, a preclusivity relation,
i.e., an irreflexive and symmetric relation. From the intuitive point of view, two
individuals are similar when they have an “indistinguishable role” with respect to
the intended application, even if they are not equivalent. On the other side, they
are preclusive if they have a “distinguishable role”, even if they are not totally
different.
These considerations lead to the definition of similarity space.
Definition 1.4. A similarity space is a structure S = hX, Ri, where X (called the
universe of the space) is a non empty set of objects and R (called the similarity
relation of the space) is a reflexive and symmetric binary relation defined on X. In
other words:
(i) ∀x ∈ X : xRx (reflexivity);
(ii) ∀x, y ∈ X : xRy implies yRx (symmetry).
Let us stress that, in general, a similarity relation is not required to be transi-
tive. This feature strongly differentiate its behavior with respect to the equivalence
relation, which is at the basis of classical rough sets theory.
In the context of an incomplete information system K(X), for a fixed set of
attributes D ⊆ Att(X) a natural similarity relation is that two objects are similar
if they possess the same values with respect to all known attributes inside D. In a
4 G. CATTANEO AND D. CIUCCI
more formal way:
(1.1) ∀x, y ∈ X : xR
D
y iff ∀a
i
∈ D ⊆ Att(X),
either F (x, a
i
) = F (y, a
i
) or F (x, a
i
) = ∗ or F (y, a
i
) = ∗
This is the approach introduced by Kryszkiewicz in [Kry98] which has the advantage
that the possibility of null values “corresponds to the idea that such values are just
missing, but they do exist. In other words, it is our imperfect knowledge that
obliges us to work with a partial information table”[ST99].
Example 1.5. Considering the information system in Table 2, we have, for exam-
ple, that f
4
is similar to f
5
whatever be the chosen subset D of attributes. The
same applies to f
5
and f
6
. On the other hand, the pair of objects f
4
and f
6
are not
similar relatively to any subset of attributes which contains the attribute Down–
Town; indeed F (Down − T own, f
4
) = no whereas F (Down − T own, f
6
) = yes.
In this way, it is also verified that the similarity relation induced from any set of
attributes D which contains Down–Town is not transitive since f
4
is similar to f
5
and f
5
is similar to f
6
, but f
4
is not similar to f
6
relatively to D.
Given a similarity space hX, Ri, the similarity class generated by the element
x ∈ X is the collection of all objects similar to x, i.e.,
S(x) := {y ∈ X : xRy}
Thus, the similarity class generated by x consists of all the elements which are
indistinguishable from x with respect to the similarity relation R. In this way this
class constitute a granule of similarity knowledge about x and is also called the
granule generated by x.
Trivially, being R a reflexive relation, it holds that x ∈ S(x) and so no similarity
class is empty. Further, as a consequence of the non–transitivity of the relation,
the similarity classes are not necessarily disjoint, i.e., there may exist x, y such that
S(x) 6= S(y) and S(x) ∩ S(y) 6= ∅.
Example 1 .6. Making reference to example 1.3, if one considers the set of all
attributes (i.e., D = Att(X)) and the induced similarity relation according to (1)
we have that S(f
4
) = {f
4
, f
5
} and S(f
6
) = {f
2
, f
5
, f
6
}. Thus, S(f
4
) 6= S(f
6
) and
S(f
4
) ∩ S(f
6
) = {f
5
}.
Using this notion of similarity class, it is possible to define in a natural way
a rough approximation by similarity of any set of objects ([Vak91, SS96, Ste98,
ST01]).
Definition 1.7. Given a similarity space hX, Ri, and a set of objects A ⊆ X,
the rough approximation of A by similarity is defined as the pair hL
R
(A), U
R
(A)i,
where
L
R
(A) := {x ∈ X : S(x) ⊆ A} = {x ∈ X : ∀z (xRz ⇒ z ∈ A)}(1.2a)
U
R
(A) := {x ∈ X : S(x) ∩ A 6= ∅} = {x ∈ X : ∃z (xRz and z ∈ A)}(1.2b)
It is easy to verify that the following chain of inclusions holds:
(1.3) L
R
(A) ⊆ A ⊆ U
R
(A).
L
R
(A) is called the similarity lower approximation of A (since it is constituted
by all objects whose granule is contained in A) and U
R
(A) the similarity upper
ALGEBRAIC STRUCTURES FOR ROUGH SETS 5
approximation of A (since it is constituted by all objects whose granule has at least
a point in common with A).
Example 1 .8. Consider again the information system in Table 2 with similarity
relation (1) relative to D = Att(X). Choosing the set of objects A = {f
4
, f
5
, f
6
}, we
have L
R
(A) = {f
4
, f
5
} and U
R
(A) = {f
2
, f
4
, f
5
, f
6
} with the obvious satisfaction
of the chain (3).
As said before, the opposite of a similarity relation is a preclusive relation: two
objects are in a preclusive relation iff it is possible to distinguish one from the other.
Using such a relation it is possible to define a notion dual to the one of similarity
space.
Definition 1.9. A preclusivity space is a structure S = hX, #i, where X (called
the universe of the space) is a non empty set and # (called the preclusivity relation
of the space) is an irreflexive and symmetric relation defined on X. In other words:
(i) ∀x ∈ X : not x#x (irreflexivity);
(ii) ∀x, y ∈ X : x#y implies y#x (symmetry).
Needless to stress, any similarity space hX, Ri determines a corresponding preclu-
sivity space hX, #
R
i with x#
R
y iff ¬(xRy), and vice versa any preclusivity space
hX, #i determines a similarity space hX, R
#
i with xR
#
y iff ¬(x#y). In this case
we will say that we have a pair of correlated similarity–preclusive relations.
Suppose, now, a preclusivity space hX, #i. The preclusive relation # permits us
to introduce for all H ∈ P(X) (where we denote by P(X) the power set of X) its
preclusive complement defined as
H
#
:= {x ∈ X : ∀y ∈ H (x#y)}.
In other words, H
#
contains all and only the elements of X that are distinguishable
from all the elements of H. Whenever x ∈ H
#
we will also write: x#H and we will
say that x is preclusive to the set H. Further, we will say that two subsets H and
K of X are mutually preclusive (H#K) iff all the elements of H are preclusive to
K, and all the elements of K are preclusive to H. We remark that, in the context
of modal analysis of rough approximation spaces, the operation
#
is a sufficiency
operator [DO01].
The following result has been proved in [Cat97].
Proposition 1.10. Let S = hX, #i be a preclusivity space and let us consider the
structure
P(X), ∩, ∪,
c
,
#
, ∅, X
. Then, the following hold:
(1) The substructure hP(X), ∩, ∪, ∅, Xi is a distributive (complete) lattice,
with respect to standard set theoretic intersection ∩ and union ∪, bounded
by the least element ∅ and the greatest element X. The partial order relation
induced from this lattice structure is the standard set theoretic inclusion ⊆.
(2) The operation
c
: P(X) → P(X) associating to any su bset H of X its set
theoretic complement H
c
= X \ H is a standard complementation, i.e., it
satisfies:
(C-1) H = (H
c
)
c
(involution)
(C-2a) H ⊆ K iff K
c
⊆ H
c
(contraposition)
(C-2b) H
c
∩ K
c
= (H ∪ K)
c
( ∩ de Morgan)
(C-2c) H
c
∪ K
c
= (H ∩ K)
c
( ∪ de Morgan)
