We address in principle the notion of a boundary and we propose a version of mereology better adapted to rough set theory than the original version. We discuss the motivation and di erences between the original and proposed now versions of rough mereology and we show that the Pawlak notion of a boundary in rough set theory is a particular case of the more general notion of a boundary in the rough mereological theory proposed in this work.
It is evident to all who study rough set idea that the most important notion is
the notion of a boundary and most important things that conform to that notion
are boundaries of concepts as they witness the uncertainty of the concept. The
notion of a boundary has been the subject of investigation by philosophers,
logicians, topologists from ancient times to now. The basic problem with the
notion of a boundary stems of course from the fact that our perception of the
world is continuous whereas the world has a discrete structure. Whence follow
the philosophical dilemmas like Bolzano's paradox of the two touching balls A
and B. The question is where A ends and B begins? If q is the last point on A
which must exists by closedness of A, then if p is the rst point on B and p is not q
then A and B do not touch and there is no boundary between them but we know
they touch as we are not able to push A or B any further. Similar are problems
by Leonardo of air and water: what is the boundary between water in the river
and air? Those and other similar questions have occupied philosophers since
times of antiquity. One needs not to reach for real physical phenomena in order
to nd problems and di culties with boundaries, e.g., time imagined continuity
have caused similar problems: what is the last moment an event occurs? See
Varzi [
Mathematicians resolved the problem by postulating the completeness of
the real line which cannot be dissected into two open disjoint non{empty sets;
returning to philosophical aspects of the notion of a boundary, this implies that
when an event has the last moment p in which it happens, we are not able to
point to the rst moment in which it does not happen. They also approached the
problem of a boundary from a local point of view with the idea of a neighborhood
and closeness: the boundary of a set is the set of points `in nitely close' to the set
in the sense that each neighborhood of a boundary point must needs intersect
the set and its complement, see Engelking [
This point of view prevails in Pawlak's idea of a boundary see Pawlak [
Objects are then coded as information sets of the form Inf (u) = fa(u) : a 2 Ag. The crux of the rough set approach is in identi cation of objects having the same information sets: Ind(u; v) if and only if Inf (u) = Inf (v), where Ind(u; v) is the indiscernibility relation . From that moment on, objects loose their real names and become visible only by their information sets which allows for making some of them indiscernible. The equivalence relation of indiscernibility partitions the universe U into classes which are also regarded as primitive granules of knowledge [u] or black boxes.
One addresses the problem of concepts understood as subsets of the universe U and distinguishes among them certain ones as those which can be assembled from granules by the union of sets operator i.e. a concept C U is certain if and only if C = Sf[u] : [u] Cg. Other concepts are not certain, commonly called rough. A rough concept R then must have an object u such that the class [u] is not contained in it but it does intersect the complement U n R. Such objects in R constitute the boundary of R, BdR, i.e. BdR = fu 2 U : [u] \ (U n R) 6= ; 6= [u] \ Rg. One may say that the boundary of R consists of objects which have their copies both in R and in U n R. This is clearly a topological approach as the class [u] is the least neighborhood of u in the partition topology induced by the indiscernibility relation Ind.
Let us point to advantages of this approach: rst it is objective as the shape of
the boundary follows by the data without any intervention of subjective factors
which are so essential in e.g. fuzzy set theory see Zadeh [
Mereology and its extensions into domain of
uncertainty
Mereology due to Stanislaw Lesniewski, see Lesniewski [
M1 For each x 2 U it is not true that part(x; x);
M2 For each triple x; y; z of things in U if part(x; y) and part(y; z), then part(x; z).
The notion of an element is de ned as the relation el(x; y) which holds true if part(x; y) or x = y. It follows that the relation of being an element is a partial order on U and el(x; y) and el(y; x) are true simultaneously if and only if x = y. Clearly, part(x; y) if and only if el(x; y) and x 6= y.
The last important notion is that of the class understood as the object which represents a collective entity i.e. a property: for a property on U which is not void, the class pf , Cls , is the object such that:
C1 If (u) holds true, then el(u; Cls );
C2 For each u, if el(u; Cls ) then there are objects p; q such that el(p; u), el(p; q), (q) hold true. 2.1
We proposed a scheme which extended mereology based on the part relation
to the version based on the `part to a degree' relation, see Polkowski [
RM3 If (x; y; 1) and (z; x; r) hold true then (z; y; r) holds true; RM4 If (x; y; r) and s < r hold true then (x; y; s) holds true.
This approach has required the a priori mereology on the universe U of which
the relation was a di usion or fuzzi cation, cf. Varzi [
We say for a pre{norm f that relation is f {transitive if and only if from true conditions (x; y; r) and (y; z; s) the true condition (x; z; f (r; s)) follows. Proposition 1. If the relation on the universe U is f {transitive then the relation (x; y) which holds true if and only if (x; y; 1) and (y; x; 1) hold true is an equivalence relation on U .
Indeed, (x; x1) holds true by RM1; symmetry is evident by de nition; transitivity follows by f transitivity of .
An attempt to apply this version of in the rough set context is handicapped by the fact that by RM2, the relation should be the identity which does not capture the full scope of rough set cases. We need something more exible to accomodate equivalence relations of indiscernibility. 2.2
As stated below, we need a mereology which may account for rough set theoretic contexts. We assume an information system (U; A; val; V ) as our playground.
We de ne a truly rough inclusion, tr(x; y; r) as a relation which satis es on U the following assumptions: TRM1 tr(x; x; 1); TRM2 There is a partition P on U such that tr(x; y; 1) if and only if x and y are in the same partition class [x]P ; TRM3 If tr(x; y; 1) and tr(z; x; r) then tr(z; y; r);
TRM5 The truly rough inclusion tr(x; y; r) is f {transitive for some pre{norm f . The predicate el(x; y) if tr(x; y; 1) de nes x as an element of y.
We sum up basic consequences of our assumptions.
Proposition 2. The following are true by conditions TRM1-TRM5. 1. tr(x; y; 1) implies tr(y; x; 1) i.e. tr is symmetric.
2. The relation el(x; y) is symmetric and el(x; y) and el(y; x) imply that [x]P = [y]P .
3. fy : tr(y; x; 1)g=[x]P .
4. The relation tr(x; y; 1) is transitive in the sense that tr(z; x; 1) and tr(x; y; 1) imply tr(z; y; 1). 3
Boundaries in rough mereology for rough sets We use the language of predicates on the universe U in de nitions of boundaries by means of a truly rough inclusions tr. 3.1
For a truly rough inclusion tr, and x 2 U , r 2 [0; 1], we de ne a new predicate N (x; r)(z) if there exists an s r such that (z; x; s). N (x; r) is the neighborhood granular predicate about x of radius r.
Consider a predicate on U having a non{empty meaning [ ]. The complement to is the predicate such that (x) if and only if not (x). We de ne the upper extension of of radius r, denoted r+ by letting r+(x) if there exists z such that (z) and N (x; r)(z). Similarly, we de ne the lower restriction of of radius r, denoted r by letting r (x) if and only if not ( )r+(x). Proposition 3. 1. Predicates r+ and r are disjoint in the sense that there is no z 2 U such that r+(z) and r (z) hold true. 2. If r+(x) holds true then r+(y) holds true for each y such that tr(y; x; 1). 3. If r (x) holds true then r (y) holds true for each y such that tr(y; x; 1).
Proof. Claim 1 follows by de nitions of the two predicates. For Claim 2, consider x; y such that r+(x) and tr(y; x; 1). There exists z such that (z), N (x; s)(z) hold true with some s r so tr(z; x; s) holds true. By symmetry of tr, we have tr(x; y; 1) true and f {transitivity of tr for an adequate pre{norm f implies that tr(z; y; f (1; s)) holds true i.e. tr(z; y; s) holds true which means that N (y; r)(z) holds true and nally r+(y) holds true. For Claim 3, assume that r (x) and tr(y; x; 1) hold true i.e. which is equvalent to :9z; s
r: tr(z; x; s) ^ : (z) tr(z; x; s) ! tr(z; y; s) ! (z): (z); As tr(y; x; 1) is equivalent to tr(x; y; 1), we have by f {transitivity of tr that which is equivalent to the thesis r (y).
We will say that a predicate is el{saturated if and only if true formulas (x) and el(y; x) imply that (y). A corollary to Claim 3 in Proposition 3 says that for each r 2 [0; 1], predicates r+ and r are el{saturated.
the predicate boundary of with respect to a truly rough inclusion tr, denoted Bd tr as follows:
Bd tr
$ (: 1+) ^ (: 1 ):
Arguing like in proof of Proposition 3, we prove the following Proposition 4. 1. Bd tr is el{saturated 2. For no z 2 U , Bd tr (z) ^ 1+(z) is true and for no z 2 U , Bd tr (z) ^ 1 (z) is true.
Proposition 5. For each x 2 U , Bd tr (x) holds true if and only if there exist z; y 2 U such that (z), (y), tr(z; x; 1), tr(y; x; 1).
A predicate Open is de ned on predicates on U and a predicate open, Open( ) in symbols if and only if it is el{saturated. on U is (1) (2) (3) (4) Corollary 1. Open( r+) and Open( r ) hold true for each r 2 [0; 1].
Open(Bd tr ) holds true.
Proposition 6. For a nite collection of predicates f 1; 2; : : : ; kg if Open( i) holds true for each i k, then Open(Wi i) holds true.
A predicate Closed holds true for a predicate holds true. if and only if Open( ) Corollary 2. Closed( r+) and Closed( r ) hold true for each r 2 [0; 1].
Closed(Bd tr ) holds true.
For the mereotopological notion of boundary see also Polkowski and Semeniuk{
Polkowska [
We return to an information system (U; A; val; V ). We derive a truly rough inclusion from any Archimedean t{norm. There exist two non{isomorphic Archimedean t{norms: { the Lukasiewicz t{norm L(x; y) = maxf0; x + y { the product t{norm P (x; y) = x y. 1g; Both these t{norms admit a Hilbert{style representation
t(x; y) = g(f (x) + f (y)); where f : [0; 1] ! [0; 1] is a continuous decreasing function with f (0) = 1, and g : [0; 1] ! [0; 1] is the inverse to f . In case of the t{norm L, f (x) = 1 x and g(y) = 1 y. We let for an Archimedean t{norm t: t(x; y; r) if and only if g( card(Dis(x; y)) card(A) ) where Dis(x; y) = fa 2 A : a(x) 6= a(y)g.
In particular, as for the Lukasiewicz t{norm we have g(y) = 1 Lukasiewicz truly rough inclusion can be de ned as tLr(x; y; r) if and only if card(Ind(x; y)) card)A) where Ind(x; y) = A n Dis(x; y). In particular, L is L{transitive.
The predicate of element el(x; y) holds true if and only if tLr(x; y; 1) holds true if and only if Ind(x; y) i.e. x; y are indiscernible. Hence, a predicate is el{ saturated if and only if its meaning is the union of a family of indiscernibility classes and rough mereological notions of 1+ and 1 become, respectively, the notions of the upper and the lower approximations of the meaning of and the meaning of the boundary predicate Bd L is the boundary of the meaning of . r; r; y, the (5) (6)
We have proposed a new version of rough mereology suitable for rough set theory and we show that the rough set theory is a particular case of an abstract rough mereotopological theory. 5
The author would like to dedicate this work to the memory of Professor Zdzislaw Pawlak (1926-2006).