# (

Authors: M. M. Altaiary; S. A. Bashammakh; N. O. Alshehri
Journal: Discrete Dynamics in Nature and Society
 (2016)

---

## Abstract

We introduce the concept of generalized fuzzy module ofBCK-algebras and present some fundamental properties. Also, we introduce the concept of generalized interval-valued fuzzyBCK-submodule and present some fundamental properties.


---

## Body

## 1. Introduction

In 1966, Iseki and Imai [1, 2] introducedBCK-algebra. This notion was originated from two different ways: (1) set theory and (2) classical and no classical propositional calculi. Certain algebraic structures, for example, Boolean-algebra andMV-algebras, are introduced asBCK-algebras [3]. Every module is an action of ring on certain group. This is, indeed, a source of motivation to study the action of certain algebraic structures on groups.BCK-module is an action ofBCK-algebra on commutative group. In 1994, the notion ofBCK-module was introduced by Abujabal et al. [4]. They established isomorphism theorems and studied some properties ofBCK-modules. The theory ofBCK-modules was further developed by Perveen et al. [5].After the introduction of fuzzy sets by Zadeh [6], there have been a number of generalizations of this fundamental concept. In 1991, Xi [7] applied fuzzy set theory toBCK-algebras and introduced the notion of fuzzy subalgebras (ideals) of theBCK-algebras, and since then some authors studied fuzzy subalgebras and fuzzy ideals. A new type of fuzzy subgroup, that is, the (
∈
,
∈
∨
q
)-fuzzy subgroup, was introduced in an earlier paper of Bhakat and Das [8]. In fact, the (
∈
,
∈
∨
q
)-fuzzy subgroup is an important generalization of Rosenfeld’s fuzzy subgroup. It is now natural to investigate similar type of generalizations of the existing fuzzy subsystems with other algebraic structures. With this objective in view, Jun [9] introduced the concept of (
α
,
β
)-fuzzy ideals of aBCK/BCI-algebra and investigated related results. In [10] Zadeh made an extension of the concept of fuzzy set by an interval-valued fuzzy set. In 2000, Jun [11] applied interval-valued fuzzy set theory toBCK-algebras and introduced the notion of interval-valued fuzzy subalgebras (ideals) of theBCK-algebras. In this paper, we introduce the concept of generalized fuzzyBCK-submodules and some basic properties are obtained and we define the concept of generalized interval-valued fuzzyBCK-submodules and some basic properties are obtained.


## 2. Preliminaries

Definition 1 (see [8]).
By aBCK-algebra one means an algebra (
X
,
∗
,
0
) of type (
2,0
) satisfying the following axioms:(BCK-1) (
(
x
∗
y
)
∗
(
x
∗
z
)
)
∗
(
z
∗
y
)
=
0,
(BCK-2) (
x
∗
(
x
∗
y
)
)
∗
y
=
0,
(BCK-3) x
∗
x
=
0,
(BCK-4) 0
∗
x
=
0,
(BCK-5) x
∗
y
=
0 and y
∗
x
=
0 imply x
=
y, for all x
,
y
,
z
∈
X.A partial ordering “≤” is defined on X by x
≤
y
⇔
x
∗
y
=
0. ABCK-algebra X is said to be bounded if there is an element 1
∈
X such that x
≤
1, for all x
∈
X, commutative if it satisfies the identity x
∧
y
=
y
∧
x, where x
∧
y
=
y
∗
(
y
∗
x
), for all x
,
y
∈
X, and implicative if x
∗
(
y
∗
x
)
=
x, for all x
,
y
∈
X.Definition 2 (see [12]).
LetX be aBCK-algebra. Then by a left X-module (abbreviated X-module), one means that an abelian group M with an operation X
×
M
→
M with (
x
,
m
)
↦
x
m satisfies the following axioms for all x
,
y
∈
X and m
,
n
∈
M:(i)
(
x
∧
y
)
m
=
x
(
y
m
),
(ii)
x
(
m
+
n
)
=
x
m
+
x
n,
(iii)
0
m
=
0.Moreover, ifX is bounded and M satisfies 1
m
=
m, for all m
∈
M, then M is said to be unitary.A mappingμ
:
X
→
[
0,1
] is called a fuzzy set in aBCK-algebra X. For any fuzzy set μ in X and any t
∈
[
0,1
], we define set U
(
μ
;
t
)
=
μ
t
=
{
x
∈
X
∣
μ
(
x
)
≥
t
}, which is called upper t-level cut of μ.Definition 3 (see [12]).
A fuzzy subsetμ of M is said to be a fuzzyBCK-submodule if for all m
,
m
1
,
m
2
∈
M and x
∈
X, the following axioms hold:(FBCKM1)μ
(
m
1
+
m
2
)
≥
m
i
n
{
μ
(
m
1
)
,
μ
(
m
2
)
},
(FBCKM2)μ
(
-
m
)
=
μ
(
m
),
(FBCKM3)μ
(
x
m
)
≥
μ
(
m
).Definition 4 (see [13]).
A fuzzy setμ in a set X of the form(1)
μ
y
=
t
∈
0,1
,
i
f
y
=
x
0
,
i
f
y
≠
x
,is said to be a fuzzy point with support x and value t and is denoted by x
t; we say that a fuzzy point x
t belongs to a fuzzy set μ and write x
t
∈
μ, if μ
(
x
)
≥
t. A fuzzy point x
t is quasi-coincident with a fuzzy set μ, if μ
(
x
)
+
t
>
1
. In this case we write x
t
q
μ. x
t
∈
∨
q
μ means that x
t
∈
μ or x
t
q
μ
,
x
t
∈
∧
q
μ means that x
t
∈
μ and x
t
q
μ.Theorem 5 (see [12]).
Letμ
∈
F
(
M
), where F
(
M
) is the set of all fuzzy subsets ofBCK-module M. Then μ is a fuzzyBCK-submodule of M if and only if(i)
μ
(
0
)
≥
μ
(
m
),
(ii)
μ
(
x
m
-
y
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
}.


## 3. (∈
,
∈
∨
q)-FuzzyBCK-Submodules

Definition 6.
A fuzzy subsetμ of M is said to be an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule if for all m
,
m
1, m
2
∈
M and x
∈
X the following axioms hold:(i)
μ
(
m
1
+
m
2
)
≥
m
i
n
{
μ
(
m
1
)
,
μ
(
m
2
)
,
0.5
},
(ii)
μ
(
-
m
)
≥
m
i
n
{
μ
(
m
)
,
0.5
},
(iii)
μ
(
x
m
)
≥
m
i
n
{
μ
(
m
)
,
0.5
}
.Example 7.
LetX
=
{
0
,
a
,
b
,
c
} and consider the following equation:(2)
∗
0
a
b
c
0
0
0
0
0
a
a
0
a
0
b
b
b
0
0
c
c
b
a
0Then (
X
,
∗
) is a bounded implicativeBCK-algebra and so is aBCK-module over itself. Now let t
0
,
t
1
,
t
2
∈
[
0,1
] be such that t
0
>
t
1
>
t
2. Define μ
:
X
→
[
0,1
] by μ
(
0
)
=
t
0, μ
(
a
)
=
t
1, and μ
(
b
)
=
μ
(
c
)
=
t
2. Then μ is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of X.Theorem 8.
A fuzzy subsetμ of M is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule if and only if(i)
μ
(
x
m
)
≥
m
i
n
{
μ
(
m
)
,
0.5
},
(ii)
μ
(
m
1
-
m
2
)
≥
m
i
n
{
μ
(
m
1
)
,
μ
(
m
2
)
,
0.5
}.Proof.
From the definition of(
∈
,
∈
∨
q
)-fuzzyBCK-submodule,(i)μ
(
x
m
)
≥
m
i
n
{
μ
(
m
)
,
0.5
},
(ii)μ
(
m
1
-
m
2
)
=
μ
(
m
1
+
(
-
m
2
)
)
≥
m
i
n
{
μ
(
m
1
)
,
μ
(
-
m
2
)
,
0.5
}.
But we know thatμ
(
-
m
)
=
μ
(
m
).
Thenμ
(
m
1
-
m
2
)
≥
m
i
n
{
μ
(
m
1
)
,
μ
(
m
2
)
,
0.5
}.
Conversely, we haveμ
(
x
m
)
=
μ
(
x
m
-
y
0
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
0
)
,
0.5
}.
Sinceμ
(
0
)
≥
m
i
n
{
μ
(
m
)
,
0.5
},
thenμ
(
x
m
)
≥
m
i
n
{
μ
(
m
)
,
0.5
}.
Andμ
(
m
1
-
m
2
)
=
μ
(
1
·
m
1
-
1
·
m
2
)
≥
m
i
n
{
μ
(
m
1
)
,
μ
(
m
2
)
,
0.5
}.
This proves thatμ is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.Theorem 9.
Letμ
∈
F
(
M
), where F
(
M
) is the set of all fuzzy subsets ofBCK-module M. Then μ is an ∈
,
∈
∨
q-fuzzyBCK-submodule of M if and only if(i)
μ
(
0
)
≥
m
i
n
{
μ
(
m
)
,
0.5
},
(ii)
μ
(
x
m
-
y
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
,
0.5
}.Proof.
Letμ be an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule:(i) Considerμ
x
m
≥
min
⁡
μ
m
,
0.5, put x
=
0
, and then μ
0
≥
min
⁡
μ
m
,
0.5.
Considerμ
(
x
m
-
y
n
)
≥
m
i
n
{
μ
(
x
m
)
,
μ
(
y
n
)
,
0.5
}
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
,
0.5
}.
Hence,μ
(
x
m
-
y
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
,
0.5
}.
Conversely, we haveμ
(
x
m
-
y
0
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
0
)
,
0.5
}
≥
m
i
n
{
μ
(
m
)
,
0.5
}
andμ
(
m
-
n
)
=
μ
(
1
·
m
-
1
·
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
,
0.5
}.
This proves thatμ is an ∈
,
∈
∨
q-fuzzyBCK-submodule.Theorem 10.
A fuzzy setμ in X is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule if and only if μ
t
≠
ϕ, t
∈
(
0,0.5
] isBCK-submodule.Proof.
Letμ
t
≠
ϕ, t
∈
(
0,0.5
], t
<
0.5, and μ
t
=
{
x
∈
X
:
μ
(
x
)
≥
t
}; let m
,
n
∈
μ
t; then μ
(
m
)
,
μ
(
n
)
≥
t. Since μ is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule, μ
(
m
-
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
,
0.5
}
=
t, and μ
(
m
-
n
)
≥
t, then m
-
n
∈
μ
t
→
(
i
); let m
∈
μ
t
,
x
∈
X; then μ
(
m
)
≥
t; then μ
(
x
m
)
≥
m
i
n
{
μ
(
m
)
,
0.5
}
≥
m
i
n
{
t
,
0.5
}
=
t; then μ
(
x
m
)
≥
t. Hence, x
m
∈
μ
t
→
(
i
i
) and from (i) and (ii) we get that μ
t isBCK-submodule.
Conversely, lett
=
m
i
n
{
μ
(
x
)
,
μ
(
y
)
,
0.5
}
,
x
,
y
∈
μ
t, μ
(
x
)
,
μ
(
y
)
≥
t; since μ
t isBCK-submodule, then x
-
y
∈
μ
t; this implies that μ
(
x
-
y
)
≥
t
=
m
i
n
{
μ
(
x
)
,
μ
(
y
)
,
0.5
}. Hence, μ
(
x
-
y
)
≥
m
i
n
{
μ
(
x
)
,
μ
(
y
)
,
0.5
}; let s
=
m
i
n
{
μ
(
m
)
,
0.5
}; let x
∈
μ
t
,
m
∈
μ
s; then x
m
∈
μ
s; this implies that μ
(
x
m
)
≥
s
=
m
i
n
{
μ
(
m
)
,
0.5
}. Hence, μ
(
x
m
)
≥
m
i
n
{
μ
(
m
)
,
0.5
} proving μ is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.Theorem 11.
Letμ be a fuzzy set in X. Then μ
t
≠
ϕ isBCK-submodule for all t
∈
(
0.5,1
] if and only if μ satisfies(i)
∀
x
∈
X
,
m
∈
M
,
m
a
x
{
μ
(
x
m
)
,
0.5
}
≥
μ
(
m
),
(ii)
∀
x
∈
X
,
m
,
n
∈
M
,
m
a
x
{
μ
(
m
-
n
)
,
0.5
}
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
}.Proof.
Assume thatμ
t isBCK-submodule of X for all t
∈
(
0.5,1
] if there is a
∈
X such that condition (i) is not valid implying that (
∃
a
∈
X
)
(
m
a
x
{
μ
(
x
m
)
,
0.5
}
<
μ
(
a
)
); then μ
(
a
)
∈
(
0.5,1
] and a
∈
U
(
μ
,
μ
(
a
)
)
=
μ
μ
a. But μ
(
x
m
)
≤
μ
(
a
) implies x
m
∉
μ
μ
a, a contradiction. Hence, (i) is valid. Suppose that m
a
x
{
μ
(
a
-
b
)
,
0.5
}
<
m
i
n
{
μ
(
a
)
,
μ
(
b
)
}
=
s for some a
,
b
∈
X. Then s
∈
(
0.5,1
] and a
,
b
∈
μ
s but a
-
b
∉
μ
s since μ
(
a
-
b
)
<
s. This is a contradiction, and therefore (ii) is valid.
Conversely, assume thatμ satisfies conditions (i) and (ii). Let t
∈
(
0.5,1
] for any x
∈
μ
t; we have m
a
x
{
μ
(
x
m
)
,
0.5
}
≥
μ
(
m
)
≥
t
>
0.5, so μ
(
x
m
)
≥
t; thus, x
m
∈
μ
t; let m
,
n
∈
X be such that m
∈
μ
t and n
∈
μ
t; then m
a
x
{
μ
(
m
-
n
)
,
0.5
}
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
}
≥
t
>
0.5, and thus μ
(
m
-
n
)
≥
t implies that m
-
n
∈
μ
t. Hence, μ
t isBCK-submodule.Theorem 12.
Every fuzzyBCK-submodule is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.Proof.
Letμ be fuzzyBCK-submodule (
m
∈
M
,
x
,
y
∈
X
,
t
∈
(
0,1
]
); then we have μ
(
0
)
≥
μ
(
m
)
≥
m
i
n
{
μ
(
m
)
,
0.5
}; hence, μ
(
0
)
≥
m
i
n
{
μ
(
m
)
,
0.5
} (i) and μ
(
x
m
-
y
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
}
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
,
0.5
}; then μ
(
x
m
-
y
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
,
0.5
} (ii); from (i) and (ii) we get that μ is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.The following example shows that the converse of Theorem12 is not true in general.Example 13.
LetX
=
{
0
,
a
,
b
,
c
,
d
,
1
} be the set along with binary operation ∗ defined on it by (3); then (
X
,
∗
,
0
) forms a bounded commutative, nonimplicativeBCK-algebra. Now for the subset M
=
{
0
,
a
,
c
,
d
} of X, define an operation + as x
+
y
=
(
x
∗
y
)
∨
(
y
∗
x
). By (4), it follows that (
M
,
+
) forms a commutative group. Define scaler multiplication (
X
,
M
)
→
M by x
m
=
x
∧
m for all x
∈
X and m
∈
M(see (5)):(3)
∗
0
a
b
c
d
1
0
0
0
0
0
0
0
a
a
0
0
a
0
0
b
b
a
0
b
a
0
c
c
c
c
0
0
0
d
d
c
c
a
0
0
1
1
d
c
b
a
0
(4)
+
0
a
c
d
0
0
a
c
d
a
a
0
d
c
c
c
d
0
a
d
d
c
a
0
(5)
∧
0
a
c
d
0
0
0
0
0
a
0
a
0
a
b
0
a
0
a
c
0
0
c
c
d
0
a
c
d
1
0
a
c
dThen (
M
,
+
) forms an X-module. Now let t
0
,
t
1
,
t
2
∈
[
0,1
] be such that t
0
>
t
1
>
t
2. Define μ
:
M
→
[
0,1
] by μ
(
0
)
=
t
0, μ
(
a
)
=
t
1, and μ
(
c
)
=
μ
(
d
)
=
t
2. Then μ is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of X, but it is not fuzzyBCK-submodule of X if μ
<
0.5.Theorem 14.
Letμ be an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of X such that μ
(
m
)
,
μ
(
n
)
<
0.5 for all m
,
n
∈
X. Then μ is a fuzzyBCK-submodule of X.Proof.
Sinceμ is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of X, then it satisfies these two conditions:(i)μ
(
0
)
≥
m
i
n
{
μ
(
m
)
,
0.5
},
(ii)μ
(
x
m
-
y
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
,
0.5
}.
Now we want to show thatμ is an ∈
,
∈-fuzzyBCK-submodule of X.
Sinceμ
(
m
)
<
0.5,
thenμ
(
0
)
≥
m
i
n
{
μ
(
m
)
,
0.5
}
≥
μ
(
m
)
.
Hence,μ
(
0
)
≥
μ
(
m
),
and sinceμ
(
m
)
,
μ
(
n
)
<
0.5,
thenμ
(
x
m
-
y
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
,
0.5
}
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
}.
Hence,μ
(
x
m
-
y
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
}.
Thenμ is a fuzzyBCK-submodule of X.Lemma 15.
Letμ be a (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of module M. Let m
,
n
∈
M, x
,
y
∈
X such that μ
(
m
)
<
μ
(
n
); then(i)
μ
(
x
m
-
y
n
)
≥
μ
(
m
) if μ
(
m
)
<
0.5,
(ii)
μ
(
x
m
)
,
μ
(
x
n
)
≥
0.5 if μ
(
m
)
≥
0.5.Proof.
Sinceμ is a fuzzy submodule of M, we have μ
(
x
m
)
≥
m
i
n
{
μ
(
m
)
,
0.5
}.(i) Letμ
(
m
)
<
0.5; then μ
(
x
m
-
y
n
)
≥
m
i
n
{
μ
(
m
)
,
μ
(
n
)
,
0.5
}
≥
m
i
n
{
μ
(
m
)
,
0.5
} since μ
(
m
)
<
μ
(
n
)
=
μ
(
m
), since μ
(
m
)
<
0.5.
Thenμ
(
x
m
-
y
n
)
≥
μ
(
m
).
(ii) Ifμ
(
m
)
≥
0.5, then μ
(
x
m
)
≥
m
i
n
{
μ
(
m
)
,
0.5
}
=
μ
(
m
). But μ
(
m
)
≥
0.5.
This implies thatμ
(
x
m
)
≥
0.5.
Ifμ
(
m
)
≥
0.5, then μ
(
x
n
)
≥
m
i
n
{
μ
(
n
)
,
0.5
}, but μ
(
m
)
<
μ
(
n
)
=
μ
(
m
), and μ
(
m
)
≥
0.5.
Thenμ
(
x
n
)
≥
0.5.Definition 16.
Letμ and ν be two (
∈
,
∈
∨
q
)-fuzzyBCK-submodules and the intersection of μ and ν is defined as follows: (6)
μ
x
∩
ν
x
=
μ
∩
ν
x
=
min
⁡
μ
x
,
ν
x
,
0.5
.Proposition 17.
Let{
μ
i
:
i
∈
∧
} be a nonempty family of an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of M. Then ∩
i
∈
∧
μ
i is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of M.Proof.
Sinceμ
i is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of M, then μ
i satisfies conditions of (
∈
,
∈
∨
q
)-fuzzyBCK-submodule. Let i
=
1,2
,
…
,
n; let μ
1 and μ
2 be an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule; then μ
1 and μ
2 satisfy conditions of (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.
Now we want to prove that(
μ
1
∩
μ
2
) is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.(i) Consider(
μ
1
∩
μ
2
)
(
0
)
=
m
i
n
{
μ
1
(
0
)
,
μ
2
(
0
)
,
0.5
}
≥
m
i
n
{
μ
1
(
m
)
,
μ
2
(
m
)
,
0.5
}
=
(
μ
1
∩
μ
2
)
(
m
).
Hence,(7)
μ
1
∩
μ
2
0
≥
μ
1
∩
μ
2
m
.
(ii) Consider(
μ
1
∩
μ
2
)
(
x
m
-
y
n
)
=
m
i
n
{
μ
1
(
x
m
-
y
n
)
,
μ
2
(
x
m
-
y
n
)
,
0.5
}
≥
m
i
n
{
m
i
n
{
μ
1
(
m
)
,
μ
1
(
n
)
,
0.5
}, m
i
n
{
μ
2
(
m
)
,
μ
2
(
n
)
,
0.5
}
,
0.5
}
=
m
i
n
{
m
i
n
{
μ
1
(
m
)
,
μ
2
(
m
)
,
0.5
}, m
i
n
{
μ
1
(
n
)
,
μ
2
(
n
)
,
0.5
}
,
0.5
}
=
m
i
n
{
(
μ
1
∩
μ
2
)
(
m
)
,
(
μ
1
∩
μ
2
)
(
n
)
,
0.5
}.
Hence,(8)
μ
1
∩
μ
2
x
m
-
y
n
≥
min
⁡
μ
1
∩
μ
2
m
,
μ
1
∩
μ
2
n
,
0.5
.
From (i) and (ii) we get that(
μ
1
∩
μ
2
) is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.
Furthermore,∩
i
∈
∧
μ
i is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.Similarly, we can prove the generalization of previous proposition.Theorem 18.
Let{
μ
i
:
i
∈
∧
} be a family of an (
∈
,
∈
∨
q
k
)-fuzzyBCK-submodule of M. Then ∩
i
∈
∧
μ
i is an (
∈
,
∈
∨
q
k
)-fuzzyBCK-submodule of M.The following example shows that the union of two(
∈
,
∈
∨
q
)-fuzzyBCK-submodules of X may not be an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of X.Example 19.
LetX
=
{
0
,
a
,
b
,
c
} beBCK-algebra which is given in Example 7 and let μ be an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of X which is μ
(
0
)
=
0.6
,
μ
(
a
)
=
0.5
,
μ
(
b
)
=
μ
(
c
)
=
0.3. Let ν be a fuzzy set in X defined by ν
(
0
)
=
0.5
,
ν
(
a
)
=
0.4
,
ν
(
b
)
=
ν
(
c
)
=
0.2. Then ν is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of X. The union of μ and ν is given by μ
(
x
)
∪
ν
(
x
)
=
(
μ
∪
ν
)
(
x
)
=
m
a
x
{
μ
(
x
)
,
ν
(
x
)
}. Hence, (
μ
∪
ν
)
(
0
)
=
0.6
,
(
μ
∪
ν
)
(
a
)
=
0.5
,
(
μ
∪
ν
)
(
b
)
=
(
μ
∪
ν
)
(
c
)
=
0.3. Then (9)
μ
∪
ν
m
1
+
m
2
≥
m
i
n
μ
∪
ν
m
1
,
μ
∪
ν
m
2
,
0.5
.Suppose that m
1
=
0 and m
2
=
b; then 0
+
b
=
b:(10)
μ
∪
ν
b
≥
min
⁡
μ
∪
ν
0
,
μ
∪
ν
b
,
0.5
0.3
≥
min
⁡
max
⁡
μ
0
,
μ
b
,
max
⁡
ν
0
,
ν
b
,
0.5
0.3
≥
min
⁡
max
⁡
0.6,0.3
,
max
⁡
0.5,0.2
,
0.5
0.3
≥
min
⁡
0.6,0.5,0.5
0.3
≥
0.5
,a contradiction; hence, (
μ
∪
ν
) is not (
∈
,
∈
∨
q
)-fuzzyBCK-submodule of X.Definition 20.
Letk
∈
N; μ
i is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule for i
∈
N
k
=
{
1,2
,
…
,
k
} and x
∈
X. Then define ∑
i
∈
N
k
μ
i as follows:(11)
∑
i
∈
N
k
μ
i
m
=
sup
m
=
∑
i
∈
N
k
a
i
⁡
min
⁡
μ
1
a
1
,
μ
2
a
2
,
…
,
μ
k
a
k
,
0.5
.Proposition 21.
Letμ
i be an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule. Then ∑
i
∈
N
k
μ
i
+
∑
i
∈
N
k
μ
i
≤
∑
i
∈
N
k
μ
i.Proof.
One has(12)
∑
i
∈
N
k
μ
i
+
∑
i
∈
N
k
μ
i
=
∑
i
∈
N
k
μ
i
+
μ
i
=
sup
m
=
∑
i
∈
N
k
a
i
⁡
m
i
n
μ
1
+
μ
1
a
1
,
μ
2
+
μ
2
a
2
,
…
,
μ
k
+
μ
k
a
k
,
0.5
≤
sup
m
=
∑
i
∈
N
k
a
i
⁡
m
i
n
μ
1
a
1
,
μ
2
a
2
,
…
,
μ
k
a
k
,
0.5
≤
∑
i
∈
N
k
μ
i
.Theorem 22.
Letμ
i be an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule, for i
∈
N
k; then ∑
i
∈
N
k
μ
i is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.Proof.
Sinceμ
i is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule, then the following conditions hold:(i)μ
i
(
0
)
≥
m
i
n
{
μ
i
(
m
)
,
0.5
}, for i
∈
N
k,
(ii)μ
i
(
x
m
-
y
n
)
≥
m
i
n
{
μ
i
(
m
)
,
μ
i
(
n
)
,
0.5
}, for i
∈
N
k.
Now we want to prove that∑
i
∈
N
k
μ
i satisfies conditions (i) and (ii).
We know that(13)
∑
i
∈
N
k
μ
i
m
=
s
u
p
m
=
∑
i
∈
N
k
a
i
min
⁡
μ
1
a
1
,
μ
2
a
2
,
…
,
μ
k
a
k
,
0.5
.
Now we want to prove that∑
i
∈
N
k
μ
i is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.
Prove (i):(14)
∑
i
∈
N
k
μ
i
0
=
s
u
p
m
=
∑
i
∈
N
k
0
i
min
⁡
μ
1
0
1
,
μ
2
0
2
,
…
,
μ
k
0
k
,
0.5
≥
min
⁡
μ
1
x
1
,
μ
2
x
2
,
…
,
μ
k
x
k
,
0.5
.
Sinceμ
i is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule
then (i) holds.
Prove (ii):(15)
∑
i
∈
N
k
μ
i
x
m
-
y
n
=
s
u
p
x
m
-
y
n
=
∑
i
∈
N
k
a
i
m
i
n
μ
1
a
1
,
μ
2
a
2
,
…
,
μ
k
a
k
,
0.5
≥
min
⁡
μ
1
m
1
,
μ
1
n
1
,
μ
2
m
2
,
μ
2
n
2
,
…
,
μ
k
m
k
,
μ
k
n
k
,
0.5
.
Sinceμ
i is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule, then (ii) holds.
This implies that∑
i
∈
N
k
μ
i is an (
∈
,
∈
∨
q
)-fuzzyBCK-submodule.


## 4. Some Kinds of (∈
,
∈
∨
q)-Interval-Valued FuzzyBCK-Submodule

For anyF
¯
(
x
)
=
[
F
-
(
x
)
,
F
+
(
x
)
] and t
¯
=
[
t
-
,
t
+
] we define F
¯
(
x
)
+
t
¯
=
[
F
-
(
x
)
+
t
-
,
F
+
(
x
)
+
t
+
], for all x
∈
X. In particular, if F
-
(
x
)
+
t
-
>
1, we write F
¯
(
x
)
+
t
¯
>
[
1,1
]. Let x
∈
X and t
¯
∈
D
[
0,1
]. An interval-valued fuzzy set G
¯ of aBCK-algebra X is said to be an interval-valued fuzzy point x
t
¯, with support x and interval-valued t
¯, if(16)
G
¯
y
=
t
¯
≠
0,0
i
f
y
=
x
0,0
i
f
y
≠
xfor all y
∈
X; we say x
t
¯ belongs to (resp., is quasi-coincident with) an interval-valued fuzzy set F
¯, written by x
t
¯
∈
F
¯ (resp., x
t
¯
q
F
¯), if F
¯
(
x
)
≥
t
¯ (resp., F
¯
(
x
)
+
t
¯
>
[
1,1
]); if x
t
¯
∈
F
¯ or x
t
¯
q
F
¯, then we write x
t
¯
∈
∨
q
F
¯.Definition 23 (see [7]).
An interval-valued fuzzy set ofX is F
¯
:
X
→
D
[
0,1
], where one denotes, for each x
∈
X, F
¯
(
x
)
=
[
F
-
(
x
)
,
F
+
(
x
)
]
∈
D
[
0,1
].Definition 24 (see [7]).
LetF
¯ be an interval-valued fuzzy set of X. Then for every [
0,0
]
<
t
¯
≤
[
1,1
], the crisp set F
¯
t
¯
=
{
x
∈
X
∣
F
¯
(
x
)
≥
t
¯
} is called the level subset of F
¯.Definition 25.
An interval-valued fuzzy subsetF
¯ of M is said to be an (
∈
,
∈
∨
q
)-interval-valued fuzzyBCK-submodule of X for all [
0,0
]
<
t
¯
≤
1,1 and for all m
,
m
1, m
2
∈
M and x
∈
X the following axioms hold:(i)
F
¯
(
m
1
+
m
2
)
≥
r
m
i
n
{
F
¯
m
1
,
F
¯
m
2
,
[
0.5,0.5
]
},
(ii)
F
¯
(
-
m
)
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
},
(iii)
F
¯
(
x
m
)
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
}.Example 26.
Consider theBCK-algebra X
=
{
0
,
a
,
b
,
c
} as in Example 7. Define an interval-valued fuzzy set F
¯ of X by F
¯
(
0
)
=
[
0.4,0.5
], F
¯
(
a
)
=
[
0.3,0.2
], and F
¯
(
b
)
=
F
¯
(
c
)
=
[
0.1,0.2
]. Hence, F
¯ is an (
∈
,
∈
∨
q
)-interval-valued fuzzyBCK-submodule of X.Theorem 27.
LetF
¯
∈
F
(
M
), where F
(
M
) is the set of all interval-valued fuzzy subsets ofBCK-module M. Then F
¯ is an ∈
,
∈
∨
q-interval-valued fuzzyBCK-submodule of M if and only if(i)
F
¯
(
0
)
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
},
(ii)
F
¯
(
x
m
-
y
n
)
≥
r
m
i
n
{
F
¯
m
,
F
¯
n
,
[
0.5,0.5
]
}.Proof.
LetF
¯ be an ∈
,
∈
∨
q-interval-valued fuzzyBCK-submodule:(i)F
¯
(
x
m
)
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
}, and x
=
0 implies that
F
¯
(
0
)
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
},
F
¯
(
x
m
-
y
n
)
≥
r
m
i
n
{
F
¯
x
m
,
F
¯
y
n
,
[
0.5,0.5
]
}
≥
r
m
i
n
{
F
¯
m
,
F
¯
n
,
[
0.5,0.5
]
}.
ThenF
¯
(
x
m
-
y
n
)
≥
r
m
i
n
{
F
¯
m
,
F
¯
n
,
[
0.5,0.5
]
}.
Conversely, we haveF
¯
(
x
m
)
=
F
¯
(
x
m
-
y
0
)
≥
r
m
i
n
{
F
¯
m
,
F
¯
0
,
[
0.5,0.5
]
}
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
}.
ConsiderF
¯
(
m
-
n
)
=
F
¯
(
1
·
m
-
1
·
n
)
≥
r
m
i
n
{
F
¯
m
,
F
¯
n
,
[
0.5,0.5
]
}.
Hence,F
¯ is an (
∈
,
∈
∨
q
)-interval-valued fuzzyBCK-submodule.Theorem 28.
An interval-valued fuzzy subsetF
¯ of M is an (
∈
,
∈
∨
q
)-interval-valued fuzzyBCK-submodule if and only if(i)
F
¯
(
x
m
)
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
},
(ii)
F
¯
(
m
1
-
m
2
)
≥
r
m
i
n
{
F
¯
m
1
,
F
¯
m
2
,
[
0.5,0.5
]
}.Proof.
From the definition of(
∈
,
∈
∨
q
)-interval-valued fuzzyBCK-submodule,(i)F
¯
(
x
m
)
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
},
(ii)F
¯
(
m
1
-
m
2
)
=
F
¯
(
m
1
+
(
-
m
2
)
)
≥
r
m
i
n
{
F
¯
m
1
,
F
¯
-
m
2
,
[
0.5,0.5
]
}.
But we know thatF
¯
(
-
m
)
=
F
¯
(
m
).
ThenF
¯
(
m
1
-
m
2
)
≥
r
m
i
n
{
F
¯
m
1
,
F
¯
m
2
,
[
0.5,0.5
]
}.
Conversely, we haveF
¯
(
x
m
)
=
F
¯
(
x
m
-
y
0
)
≥
r
m
i
n
{
F
¯
m
,
F
¯
0
,
[
0.5,0.5
]
}
,
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
}.
Use Theorem27, since F
¯
(
0
)
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
},
andF
¯
(
m
1
-
m
2
)
=
F
¯
(
1
·
m
1
-
1
·
m
2
)
≥
r
m
i
n
{
F
¯
m
1
,
F
¯
m
2
,
[
0.5,0.5
]
}.
Hence,F
¯ is an (
∈
,
∈
∨
q
)-interval-valued fuzzyBCK-submodule.Theorem 29.
An interval-valued fuzzy subsetF
¯ of M is an (
∈
,
∈
∨
q
)-interval-valued fuzzyBCK-submodule if and only if F
¯
t
¯
(
≠
ϕ
) isBCK-submodule of X for all [
0,0
]
<
t
¯
≤
[
0.5,0.5
].Proof.
LetF
¯ be an (
∈
,
∈
∨
q
)-interval-valued fuzzyBCK-submodule of X and [
0,0
]
<
t
¯
≤
[
0.5,0.5
] for any m
,
n
∈
F
¯
t
¯; then F
¯
(
m
)
,
F
¯
(
n
)
≥
t
¯. Since F
¯ is an (
∈
,
∈
∨
q
)-interval-valued fuzzyBCK-submodule, then F
¯
(
m
-
n
)
≥
r
m
i
n
{
F
¯
m
,
F
¯
n
,
[
0.5,0.5
]
}
=
t
¯, which implies that F
¯
(
m
-
n
)
≥
t
¯. Hence, m
-
n
∈
F
¯
t
¯; let m
∈
F
¯
t
¯, x
∈
X; then F
¯
(
m
)
≥
t
¯. Furthermore, F
¯
(
x
m
)
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
}
≥
r
m
i
n
{
t
¯
,
[
0.5,0.5
]
}
=
t
¯, and then F
¯
(
x
m
)
≥
t
¯. Hence, x
m
∈
F
¯
t
¯; hence F
¯
t
¯ isBCK-submodule.
Conversely, lett
¯
=
r
m
i
n
{
F
¯
m
,
F
¯
n
,
[
0.5,0.5
]
}
,
m
,
n
∈
F
¯
t
¯; then F
¯
(
m
)
,
F
¯
(
n
)
≥
t
¯; since F
¯
t
¯ isBCK-submodule, then m
-
n
∈
F
¯
t
¯ implies that F
¯
(
m
-
n
)
≥
t
¯
=
r
m
i
n
{
F
¯
m
,
F
¯
n
,
[
0.5,0.5
]
}; then F
¯
(
m
-
n
)
≥
r
m
i
n
{
F
¯
m
,
F
¯
n
,
[
0.5,0.5
]
}; let s
=
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
}, and let x
∈
F
¯
t
¯
,
m
∈
F
¯
s
¯; then x
m
∈
F
¯
s
¯ implies that F
¯
(
x
m
)
≥
s
=
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
}; then F
¯
(
x
m
)
≥
r
m
i
n
{
F
¯
m
,
[
0.5,0.5
]
}.
Hence,F
¯ is an (
∈
,
∈
∨
q
)-interval-valued fuzzyBCK-submodule.


## 5. Conclusions

In this paper, we introduced the concept of generalized fuzzyBCK-submodules and some basic properties were obtained and we defined the concept of generalized interval-valued fuzzyBCK-submodules and some basic properties were obtained. Other topics in this area which could be a further research are (
∈
¯
,
∈
¯
∨
q
¯
)-fuzzyBCK-submodules, translations of (
∈
,
∈
∨
q
)-fuzzyBCK-submodule, and other generalizations of fuzzy set theory toBCK-submodules.



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*Source: 8746275-2016-02-22.xml*