Pawlak [1, 2] introduced rough set theory as a formal tool to deal with imprecision and uncertainty in data analysis. Hájek [3] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Radzikowska and Kerre [4] developed fuzzy rough sets in complete residuated lattice. Bělohlávek [5] investigated information systems and decision rules in complete residuated lattices. Lai and Zhang [6, 7] introduced Alexandrov L-topologies induced by fuzzy rough sets. Kim [8, 9] investigate relations between lower approximation operators as a generalization of fuzzy rough set and Alexandrov L-topologies. Algebraic structures of fuzzy rough sets are developed in many directions [4, 8, 10]
In this paper, we investigate the properties of L-lower approximation operators as a generalization of fuzzy rough set in complete residuated lattices. We study relations lower (upper, join meet, meet join) approximation operators and Alexandrov L-topologies. Moreover, we give their examples as approximation operators induced by various L-fuzzy relations.
Definition 1.1. [3, 5] An algebra (L,∧,∨,⊙,→,⊥,⊤) is called a complete residuated lattice if it satisfies the following conditions:
(C1) L = (L,≤,∨,∧,⊥,⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥;
(C2) (L,⊙,⊤) is a commutative monoid;
C3) x ⊙ y ≤ z iff x ≤ y → z for x, y, z ∈ L
Remark 1.2. [3, 5] (1) A completely distributive lattice L = (L,≤,∨,∧ = ⊙,→, 1, 0) is a complete residuated lattice defined by
(2) The unit interval with a left-continuous t-norm ⊙,
is a complete residuated lattice defined by
In this paper, we assume (L,∧,∨,⊙,→,* ⊥,⊤) is a complete residuated lattice with the law of double negation;i.e. x** = x. For 𝛼 ∈ L, A,⊤_{x} ∈ L^{X},
and
Lemma 1.3. [3, 5] For each x, y, z, x_{i}, y_{i} ∈ L, we have the following properties.
Definition 1.4. [8, 9]
(1) A map H : L^{X} → L^{X} is called an L-upper approximation operator iff it satisfies the following conditions
(2) A map 𝒥 : L^{X} → L^{X} is called an L-lower approximation operator iff it satisfies the following conditions
(3) A map K : L^{X} → L^{X} is called an L-join meet approximation operator iff it satisfies the following conditions
(4) A map M : L^{X} → L^{X} is called an L-meet join approximation operator iff it satisfies the following conditions
Definition 1.5. [6, 9] A subset 𝜏 ⊂ L^{X} is called an Alexandrov L-topology if it satisfies:
Theorem 1.6. [8, 9]
(1) 𝜏 is an Alexandrov topology on X iff 𝜏_{⁎} = {A* ∈ L^{X} | A ∈ 𝜏} is an Alexandrov topology on X.
(2) If H is an L-upper approximation operator, then 𝜏_{H} = {A ∈ L^{X} | H(A) = A} is an Alexandrov topology on X.
(3) If 𝒥 is an L-lower approximation operator, then 𝜏_{𝒥} = {A ∈ L^{X} | 𝒥 (A) = A} is an Alexandrov topology on X.
(4) If K is an L-join meet approximation operator, then 𝜏_{K} = {A ∈ L^{X} | K(A) = A*} is an Alexandrov topology on X.
(5) If M is an L-meet join operator, then 𝜏_{M} = {A ∈ L^{X} | M(A) = A*} is an Alexandrov topology on X.
Definition 1.7. [8, 9] Let X be a set. A function R : X × X → L is called:
If R satisfies (R1) and (R3), R is called a L-fuzzy preorder.
If R satisfies (R1), (R2) and (R3), R is called a L-fuzzy equivalence relation
Theorem 2.1. Let 𝒥 : L^{X} → L^{X} be an L-lower approximation operator. Then the following properties hold.
(1) For A ∈ L^{X},.
(2) Define H_{J} (B) = ∧{A | B ≤ 𝒥 (A)}. Then H_{J} : L^{X} → L^{X} with
is an L-upper approximation operator such that (H_{J} ,𝒥 )
is a residuated connection;i.e.,
HJ (B) ≤ A iff B ≤ 𝒥 (A).
Moreover, 𝜏_{𝒥} = 𝜏_{HJ} .
(3) If 𝒥 (𝒥 (A)) = 𝒥 (A) for A ∈ L^{X}, then H_{J} (H_{J} (A)) = H_{J} (A) for A ∈ L^{X} such that 𝜏_{𝒥} = 𝜏_{HJ} with
(4) If 𝒥 (𝒥 *(A)) = 𝒥 *(A) for A ∈ L^{X}, then 𝒥 (𝒥 (A)) = 𝒥 (A) such that
(5) Define H_{s}(A) = 𝒥 (A*)*. Then Hs : L^{X} → L^{X} with
is an L-upper approximation operator. Moreover, 𝜏_{Hs} = (𝜏_{𝒥} )_{*} = (𝜏_{HJ} )_{*}.
(6) If 𝒥 (𝒥 (A)) = 𝒥 (A) for A ∈ L^{X}, then
Hs(Hs(A)) = Hs(A)
for A ∈ L^{X} such that 𝜏_{Hs} = (𝜏_{𝒥} )_{*} = (𝜏_{HJ} )_{*}. with
(7) If 𝒥 (𝒥 *(A)) = 𝒥 *(A) for A ∈ L^{X}, then
such that
(8) Define K_{J} (A) = 𝒥 (A*). Then K_{J} : L^{X} → L^{X} with
is an L-join meet approximation operator.
(9) If 𝒥 (𝒥 (A)) = 𝒥 (A) for A ∈ L^{X}, then
for A ∈ L^{X} such that 𝜏_{KJ} = (𝜏_{𝒥} )_{*} with
(10) If 𝒥 (𝒥 *(A)) = 𝒥 *(A) for A ∈ L^{X}, then
such that
(11) Define M_{J} (A) = (𝒥 (A))*. Then M_{J} : L^{X} → L^{X} with
is an L-meet join approximation operator. Moreover, 𝜏_{MJ} = 𝜏_{𝒥} .
(12) If 𝒥 (𝒥 (A)) = 𝒥 (A) for A ∈ L^{X}, then for A ∈ L^{X} such that 𝜏_{MJ} = (𝜏_{𝒥} )_{*} with
(13) If 𝒥 (𝒥 *(A)) = 𝒥 *(A) for A ∈ L^{X}, then
such that
(14) Define K_{HJ} (A) = (H_{J} (A))*. Then K_{HJ} : L^{X} → L^{X} with
is an L-meet join approximation operator. Moreover, 𝜏_{KHJ} = 𝜏_{𝒥} .
(15) If 𝒥 (𝒥 (A)) = 𝒥 (A) for A ∈ L^{X}, then
for A ∈ L^{X} such that 𝜏_{KHJ} = (𝜏_{𝒥} )_{*} with
(16) If for A ∈ L^{X}, then
such that
(17) Define M_{HJ} (A) = H_{J} (A*). Then M_{HJ} : L^{X} → L^{X}
with
is an L-join meet approximation operator. Moreover, 𝜏_{MHJ} = (𝜏_{𝒥} )_{*}.
(18) If 𝒥 (𝒥 (A)) = 𝒥 (A) for A ∈ L^{X}, then
for A ∈ L^{X} such that 𝜏_{MHJ} = (𝜏_{𝒥} )_{*} with
(19) If for A ∈ L^{X}, then
such that
(20) (K_{HJ} ,K_{J} ) is a Galois connection;i.e,
A ≤ KHJ (B) iff B ≤ KJ (A).
Moreover, 𝜏_{KJ} = (𝜏_{KHJ} )_{*}.
(21) (M_{J} ,M_{HJ} ) is a dual Galois connection;i.e,
MHJ (A) ≤ B iff MJ (B) ≤ A.
Moreover, 𝜏_{MJ} = (𝜏_{MHJ} )_{*}.
Proof.
(1) Since , by (J2) and (J3),
(2) Since
iff , we have
(H1) Since H_{J} (A) ≤ H_{J} (A) iff A ≤ 𝒥 (H_{J} (A)), we have A ≤ 𝒥 (H_{J} (A)) ≤ H_{J} (A).
(H3) By the definition of H_{J} , since H_{J} (A) ≤ H_{J} (B) for B ≤ A, we have
Since 𝒥 (∨_{i∈𝚪} H_{J} (A_{i})) ≥ 𝒥 (H_{J} (A_{i})) ≥ A_{i}, then
𝒥 (∨_{i∈𝚪} H_{J} (A_{i})) ≥ ∨_{i∈𝚪} A_{i}. Thus
Thus H_{J} : L^{X} → L^{X} is an L-upper approximation operator. By the definition of H_{J} , we have
HJ (B) ≤ A iff B ≤ 𝒥 (A).
Since A ≤ 𝒥 (A) iff H_{J} (A) ≤ A, we have 𝜏_{HJ} = 𝜏_{𝒥} .
(3) Let 𝒥 (𝒥 (A)) = 𝒥 (A) for A ∈ L^{X}. Since 𝒥 (B) ≥ H_{J} (A) iff 𝒥 (𝒥 (B)) = 𝒥 (B) ≥ A from the definition of H_{J} , we have
(4) Let 𝒥 *(A) ∈ 𝜏_{𝒥} . Since 𝒥 (𝒥 *(A)) = 𝒥 *(A),
𝒥 (𝒥 (A)) = 𝒥 (𝒥 *(𝒥 *(A))) = (𝒥 (𝒥 *(A)))* = 𝒥 (A).
Hence 𝒥 (A) ∈ 𝜏_{𝒥} ; i.e. 𝒥 *(A) ∈ (𝜏_{𝒥} )_{*}. Thus, 𝜏_{𝒥} ⊂ (𝜏_{𝒥} )_{*}.
Let A ∈ (𝜏_{𝒥} )_{*}. Then A* = 𝒥 (A*). Since 𝒥 (A) = 𝒥 (𝒥 *(A*)) = 𝒥 *(A*) = A, then A ∈ 𝜏_{𝒥} . Thus, (𝜏_{𝒥} )_{*} ⊂ 𝜏_{𝒥} .
(5) (H1) Since 𝒥 (A*) ≤ A*, H_{s}(A) = 𝒥 (A*)* ≥ A.
Hence H_{s} is an L-upper approximation operator such that
Moreover, 𝜏_{Hs} = (𝜏_{𝒥} )_{*} from:
A = Hs(A) iff A = 𝒥 (A*)* iff A* = 𝒥 (A*).
(6) Let 𝒥 (𝒥 (A)) = 𝒥 (A) for A ∈ L^{X}. Then
Hence
(7) Let 𝒥 (𝒥 *(A)) = 𝒥 *(A) for A ∈ L^{X}. Then
Hence
By a similar method in (4), 𝜏_{Hs} = (𝜏_{Hs} )_{*}.
(8) It is similarly proved as (5).
(9) If 𝒥 (𝒥 (A)) = 𝒥 (A) for A ∈ L^{X}, then
(10) If 𝒥 (𝒥 *(A)) = 𝒥 *(A) for A ∈ L^{X}, then
Since ,
Hence 𝜏_{KJ} = {K_{J} (A) | A ∈ L^{X}} = (𝜏_{KJ} )_{*}.
(11) , (12), (13) and (14) are similarly proved as (5), (9), (10) and (5), respectively.
(15) If 𝒥 (𝒥 (A)) = 𝒥 (A) for A ∈ L^{X}, then H_{J} (H_{J} (A)) = H_{J} (A). Thus,
Since 𝒥 (A) = A iff H_{J} (A) = A iff K_{HJ} (A) = A*, 𝜏_{KHJ} = (𝜏_{𝒥} )_{*} with
(16) If for A ∈ L^{X}, then
(17) , (18) and (19) are similarly proved as (14), (15) and (16), respectively.
(20) (K_{HJ} ,K_{J} ) is a Galois connection;i.e,
A ≤ KHJ (B) iff A ≤ (HJ (B))*
iff HJ (B) ≤ A* iff B ≤ 𝒥 (A*) = KJ (A)
Moreover, since A* ≤ K_{J} (A) iff A ≤ K_{HJ} (A*), 𝜏_{KJ} = (𝜏_{KHJ} )_{*}.
(21) (M_{J} ,M_{HJ} ) is a dual Galois connection;i.e,
MHJ (A) ≤ B iff HJ (A*) ≤ B
iff A* ≤ 𝒥 (B) iff MJ (B) = (𝒥 (B))* ≤ A.
Since M_{HJ} (A*) ≤ A iff M_{J} (A) ≤ A*, 𝜏_{MJ} = (𝜏_{MHJ} )_{*}.
Let R ∈ L^{X × X} be an L-fuzzy relation. Define operators as follows
Example 2.2. Let R be a reflexive L-fuzzy relation. Define 𝒥_{R} : L^{X} → L^{X} as follows:
(1) (J1) 𝒥_{R}(A)(y) ≤ R(y, y) → A(y) = A(y): 𝒥_{R} satisfies the conditions (J1) and (J2) from:
Hence 𝒥_{R} is an L-lower approximation operator.
(2) Define H_{JR}(B) = ∨ {A | B ≤ 𝒥_{R}(A)}. Since
then
By Theorem 2.1(2), H_{JR} = H_{R-1} is an L-upper approximation operator such that (H_{JR},𝒥_{R}) is a residuated connection;i.e.,
HJR(A) ≤ B iff A ≤ 𝒥R(B).
Moreover, 𝜏_{HJR} = 𝜏_{𝒥R}.
(3) If R is an L-fuzzy preorder, then R^{-1} is an L-fuzzy preorder. Since
By Theorem 2.1(3), H_{JR}(H_{JR}(A)) = H_{JR}(A): By Theorem 2.1(3), 𝜏_{HJR} = 𝜏_{𝒥R} with
(4) Let R be a reflexive and Euclidean L-fuzzy relation. Since R(x, z) ⊙ R(y, z) ⊙ A*(x) ≤ R(x, y) ⊙ A*(x) iff R(x, z) ⊙ A*(x) ≤ R(y, z) → R(x, y) ≤ A*(x),
Thus, .
By Theorem 2.1(4), 𝒥_{R}(𝒥_{R}(A)) = 𝒥_{R}(A) for A ∈ L^{X}.
Thus, 𝜏_{𝒥R} = (𝜏_{𝒥R})_{*} with
(5) Define H_{s}(A) = 𝒥_{R}(A*)*. By Theorem 2.1(5), H_{s} = H_{R} is an L-upper approximation operator such that
Moreover, 𝜏_{Hs} = 𝜏_{HR} = (𝜏_{HJR} )_{*}.
(6) If R is an L-fuzzy preorder, then 𝒥_{R}(𝒥_{R}(A)) = 𝒥_{R}(A) for A ∈ L^{X}. By Theorem 2.1(6), then H_{s}(H_{s}(A)) = H_{s}(A) for A ∈ L^{X} such that 𝜏H_{s} = (𝜏_{𝒥R})_{*} = (𝜏_{HJR} )_{*} with
(7) If R is a reflexive and Euclidean L-fuzzy relation, then
(8) Define K_{JR}(A) = 𝒥_{R}(A*). Then K_{JR} : L^{X} → L^{X} with
is an L-join meet approximation operator. Moreover, 𝜏_{KJR} = (𝜏_{𝒥R})_{*}.
(9) R is an L-fuzzy preorder, then 𝒥_{R}(𝒥_{R}(A)) = 𝒥_{R}(A) for A ∈ L^{X}. By Theorem 2.1(9), for A ∈ L^{X} such that 𝜏_{KJR} = (𝜏_{𝒥R})_{*} with
(10) If R is a reflexive and Euclidean L-fuzzy relation, then for A ∈ L^{X}. By Theorem 2.1(10), such that
(11) Define M_{JR}(A) = (𝒥_{R}(A))*. Then M_{JR} : L^{X} → L^{X} with
is an L-join meet approximation operator. Moreover, 𝜏_{MJR} = 𝜏_{𝒥R}.
(12) If R is an L-fuzzy preorder, then 𝒥_{R}(𝒥_{R}(A)) = 𝒥_{R}(A) for A ∈ L^{X}. By Theorem 2.1(12), for A ∈ L^{X} such that 𝜏_{MJR} = 𝜏_{𝒥R} with
(13) If R is a reflexive and Euclidean L-fuzzy relation, then for A ∈ L^{X}. By Theorem 2.1(13), such that
(14) Define K_{HJR} (A) = (H_{JR}(A))*. Then
KHJR : LX → LX
with
is an L-join meet approximation operator. Moreover, 𝜏_{KR-1} = 𝜏_{𝒥R} = 𝜏_{HR-1} .
(15) If R is an L-fuzzy preorder, then 𝒥_{R}(𝒥_{R}(A)) = 𝒥_{R}(A) for A ∈ L^{X}. By Theorem 2.1(15), for A ∈ L^{X} such that 𝜏_{KR-1} = 𝜏_{𝒥R} = 𝜏_{HR-1} with
(16) Let R^{-1} be a reflexive and Euclidean L-fuzzy relation. Since
we have
Thus,
Hence
By (K1), such that
(17) Define M_{HJR} (A) = H_{JR}(A*). Then
MHJR : LX → LX
is an L-meet join approximation operator as follows:
Moreover, 𝜏_{MHJR} = (𝜏_{𝒥R})_{*}.
(18) If R is an L-fuzzy preorder, then 𝒥_{R}(𝒥_{R}(A)) = 𝒥_{R}(A) for A ∈ L^{X}. By Theorem 2.1(18),
for A ∈ L^{X} such that 𝜏_{MHJR} = (𝜏_{𝒥} )_{*} with
(19) Let R^{-1} be a reflexive and Euclidean L-fuzzy relation.
Since
then (R(y, x) → A(x)) ⊙ R(z, y) ≤ R(z, x) → A(x).
Thus,
By (M1), such that
(20) (K_{HJR} = K_{R-1*} ,K_{JR} = K_{R*} ) is a Galois connection; i.e, A ≤ K_{HJR} (B) iff B ≤ K_{JR}(A): Moreover, 𝜏_{KJR} = (𝜏_{KHJR} )_{*}.
(21) (M_{JR} = M_{R},M_{HJR} = M_{R-1} ) is a dual Galois connection; i.e, M_{HJR} (A) ≤ B iff M_{JR}(B) ≤ A. Moreover, 𝜏_{MJR} = (𝜏_{MHJR} )_{*}.