Z. Yu et al. / International Journal of Approximate Reasoning 103 (2018) 383–393 385
Let C be a covering of U and x ∈ U . Denote Friends(x) =
{K ∈ C | x ∈ K }, N(x) =
{K ∈ C | x ∈ K } and
Md(x) ={K ∈ C | x ∈ K ∧ (∀S ∈ C ∧ x ∈ S ∧ S ⊆ K ⇒ K = S)}.
MD(x) ={K ∈ C | x ∈ K ∧ (∀S ∈ C ∧ x ∈ S ∧ S ⊇ K ⇒ K = S)}.
Remark 1. Md(x) and MD(x) were called the minimal description neighborhood system and the maximal description neigh-
borhood
system of x, respectively [2], [37]. It is easy to check that Friends(x) =
MD(x), and
MD(x) was also denoted
as n
1
(MD(C , x)) in [37].
Definition
2.1. (Reflexive and symmetric 1-NS) Let (U , C) be a covering-based approximation space. For each element x
of U ,
one associates it with a subset n(x) of U and n(x) is called a neighborhood of x. A neighborhood system N
x
of x is a
nonempty family of neighborhoods of x . The collection {N
x
: x ∈ U } is called a neighborhood system on U . A neighborhood
system on U in which each N
x
has exactly one neighborhood N
x
is called an 1-neighborhood system (1-NS for short)[35]
on
U . An 1-NS on U is said to be reflexive and symmetric if x ∈ N
x
for each x ∈ U and for every x, y ∈ U , x ∈ N
y
if and only
if y ∈ N
x
.
Remark
2. 1-SN, reflexive and symmetric 1-NS were discussed in [35,39].
Definition
2.2. (Base) Suppose A, B are two families in U and B ⊆A. Then B is called a base for A if every non-empty
O ∈ A can be represented as the union of a subfamily of B.
Let
C be a covering of U and K ∈ C. If K is the union of some sets in C −{K}, then we say K is reducible in C ;
Otherwise, K is irreducible. If every element in C is irreducible, then we say that C is an irreducible covering; Otherwise,
we say that C is a reducible covering. If C is a reducible covering, then by [53], after deleting all reducible elements of C,
the remainder is still a covering of U and we denote this remainder as reduct(C).
Definition
2.3. (Dual operators) Assume that H, L : P (U ) → P (U) are two operators on U . If ∀ X ⊆ U , H(X) =∼ L(∼ X),
then H, L are called dual operators or H is the dual operators of L.
Definition
2.4. (Lower approximation operators CL and EL). Let C be a covering of U . The operators
CL, EL : P(U) → P(U) are defined as follows: ∀X ∈ P (U),
CL(X) =
{K ∈ C : K ⊆ X},
EL(x) =
{Friends(x) : Friends(x) ⊆ X}.
Definition 2.5. (Upper approximation operators FH, SH, TH, RH and EH). Let C be a covering of U . The operators
FH(X) = CL(X) ∪ (
{
Md(x) : x ∈ (X − CL(X))}),
SH(X) =
{K ∈ C : K ∩ X = ∅},
TH(X) =
{
Md(x) : x ∈ X},
RH( X) = CL(X) ∪ (
{K ∈ C : K ∩ (X − CL(X)) = ∅}), and
EH
(X) ={z :∀y(z ∈ Friends( y) ⇒ Friends( y) ∩ X = ∅)}.
Remarks 3. (a). The notations of operators CL, FH, SH, TH and RH in the above definitions are consistent with those in
[52], [55] and [56]. In [45], SH and RH were denoted as FH and TH respectively. Same as in [52], [55] and [56], we call
FH, SH, TH and RH the first, the second, the third and the fourth type of covering-based upper approximation operators
respectively in the sequel. In [27], EL and EH were denoted as P
2
and P
2
respectively, and in [37], they were denoted as
apr
N
4
and apr
N
4
respectively.
(b). It is easy to check that EL, EH are dual operators.
Let
H : P (U ) → P(U ) be an operator on U . Denote I ={A ⊆ U : H(A) = A} and we call I the family of invariant subsets
(under H).
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