Table Of ContentAnnals of Fuzzy Mathematics and Informatics
FMI
@
Volume 14, No. 1, (July 2017), pp. 87–97
ISSN: 2093–9310 (print version) (cid:13)c Kyung Moon Sa Co.
ISSN: 2287–6235 (electronic version) http://www.kyungmoon.com
http://www.afmi.or.kr
Neutrosophic subalgebras of BCK/BCI-algebras
based on neutrosophic points
A. Borumand Saeid, Young Bae Jun
Received 27 March 2017; Revised 28 April 2017; Accepted 20 May 2017
Abstract. Properties on neutrosophic ∈∨q-subsets and neutrosophic
q-subsets are investigated. Relations between an (∈, ∈∨q)-neutrosophic
subalgebraanda(q,∈∨q)-neutrosophicsubalgebraareconsidered. Char-
acterizationofan(∈,∈∨q)-neutrosophicsubalgebrabyusingneutrosophic
∈-subsets are discussed. Conditions for an (∈, ∈∨q)-neutrosophic subal-
gebra to be a (q, ∈∨q)-neutrosophic subalgebra are provided.
2010 AMS Classification: 06F35, 03B60, 03B52.
Keywords: Neutrosophic set, neutrosophic ∈-subset, neutrosophic q-subset, neu-
trosophic ∈∨q-subset, neutrosophic T -point, neutrosophic I -point, neutrosophic
Φ Φ
F -point.
Φ
Corresponding Author: Y. B. Jun ([email protected])
1. Introduction
T
he concept of neutrosophic set (NS) developed by Smarandache [17, 18, 19]
is a more general platform which extends the concepts of the classic set and fuzzy
set (see [20], [21]), intuitionistic fuzzy set (see [1]) and interval valued intuitionistic
fuzzy set (see [2]). Neutrosophic set theory is applied to various part (see [4], [5],
[8], [9], [10], [11], [12], [13], [15], [16]). For further particulars, we refer readers to
the site http://fs.gallup.unm.edu/neutrosophy.htm. Barbhuiya [3] introduced and
studied the concept of (∈, ∈ ∨q)-intuitionistic fuzzy ideals of BCK/BCI-algebras.
Jun [7] introduced the notion of neutrosophic subalgebras in BCK/BCI-algebras
with several types. He provided characterizations of an (∈,∈)-neutrosophic sub-
algebra and an (∈,∈ ∨q)-neutrosophic subalgebra. Given special sets, so called
neutrosophic ∈-subsets, neutrosophic q-subsets and neutrosophic ∈∨q-subsets, he
considered conditions for the neutrosophic ∈-subsets, neutrosophic q-subsets and
neutrosophic ∈∨q-subsets to be subalgebras. He discussed conditions for a neutro-
sophic set to be a (q, ∈∨q)-neutrosophic subalgebra.
A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97
Inthispaper,wegiverelationsbetweenan(∈,∈∨q)-neutrosophicsubalgebraand
a (q, ∈∨q)-neutrosophic subalgebra. We discuss characterization of an (∈, ∈∨q)-
neutrosophic subalgebra by using neutrosophic ∈-subsets. We provide conditions
foran(∈,∈∨q)-neutrosophicsubalgebratobea(q,∈∨q)-neutrosophicsubalgebra.
We investigate properties on neutrosophic q-subsets and neutrosophic ∈∨q-subsets.
2. Preliminaries
ByaBCI-algebrawemeananalgebra(X,∗,0)oftype(2,0)satisfyingtheaxioms:
(a1) ((x∗y)∗(x∗z))∗(z∗y)=0,
(a2) (x∗(x∗y))∗y =0,
(a3) x∗x=0,
(a4) x∗y =y∗x=0 ⇒ x=y,
for all x,y,z ∈X. If a BCI-algebra X satisfies the axiom
(a5) 0∗x=0 for all x∈X,
thenwesaythatX isaBCK-algebra. AnonemptysubsetSofaBCK/BCI-algebra
X is called a subalgebra of X if x∗y ∈S for all x,y ∈S.
We refer the reader to the books [6] and [14] for further information regarding
BCK/BCI-algebras.
For any family {a |i∈Λ} of real numbers, we define
i
(cid:26)
(cid:95) max{a |i∈Λ} if Λ is finite,
{a |i∈Λ}:= i
i sup{a |i∈Λ} otherwise.
i
(cid:26)
(cid:94) min{a |i∈Λ} if Λ is finite,
{a |i∈Λ}:= i
i inf{a |i∈Λ} otherwise.
i
(cid:87)
If Λ = {1,2}, we will also use a ∨a and a ∧a instead of {a | i ∈ Λ} and
1 2 1 2 i
(cid:86)
{a |i∈Λ}, respectively.
i
Let X be a non-empty set. A neutrosophic set (NS) in X (see [18]) is a structure
of the form:
A:={(cid:104)x;A (x),A (x),A (x)(cid:105)|x∈X}
T I F
where A : X → [0,1] is a truth membership function, A : X → [0,1] is an
T I
indeterminate membership function, and A : X → [0,1] is a false membership
F
function. For the sake of simplicity, we shall use the symbol A = (A ,A ,A ) for
T I F
the neutrosophic set
A:={(cid:104)x;A (x),A (x),A (x)(cid:105)|x∈X}.
T I F
88
A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97
3. Neutrosophic subalgebras of several types
GivenaneutrosophicsetA=(A ,A ,A )inasetX,α,β ∈(0,1]andγ ∈[0,1),
T I F
we consider the following sets:
T (A;α):={x∈X |A (x)≥α},
∈ T
I (A;β):={x∈X |A (x)≥β},
∈ I
F (A;γ):={x∈X |A (x)≤γ},
∈ F
T (A;α):={x∈X |A (x)+α>1},
q T
I (A;β):={x∈X |A (x)+β >1},
q I
F (A;γ):={x∈X |A (x)+γ <1},
q F
T (A;α):={x∈X |A (x)≥α or A (x)+α>1},
∈∨q T T
I (A;β):={x∈X |A (x)≥β or A (x)+β >1},
∈∨q I I
F (A;γ):={x∈X |A (x)≤γ or A (x)+γ <1}.
∈∨q F F
WesayT (A;α),I (A;β)andF (A;γ)areneutrosophic∈-subsets;T (A;α),I (A;β)
∈ ∈ ∈ q q
andF (A;γ)areneutrosophic q-subsets; andT (A;α), I (A;β)andF (A;γ)
q ∈∨q ∈∨q ∈∨q
are neutrosophic ∈∨q-subsets. For Φ∈{∈,q,∈∨q}, the element of T (A;α) (resp.,
Φ
I (A;β) and F (A;γ)) is called a neutrosophic T -point (resp., neutrosophic I -
Φ Φ Φ Φ
point and neutrosophic F -point) with value α (resp., β and γ) (see [7]).
Φ
It is clear that
(3.1) T (A;α)=T (A;α)∪T (A;α),
∈∨q ∈ q
(3.2) I (A;β)=I (A;β)∪I (A;β),
∈∨q ∈ q
(3.3) F (A;γ)=F (A;γ)∪F (A;γ).
∈∨q ∈ q
Definition3.1([7]). GivenΦ,Ψ∈{∈,q,∈∨q},aneutrosophicsetA=(A ,A ,A )
T I F
inaBCK/BCI-algebraX iscalleda(Φ, Ψ)-neutrosophic subalgebraofX ifthefol-
lowing assertions are valid.
x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∧α ),
Φ x Φ y Ψ x y
(3.4) x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∧β ),
Φ x Φ y Ψ x y
x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∨γ )
Φ x Φ y Ψ x y
for all x,y ∈X, α ,α , β ,β ∈(0,1] and γ ,γ ∈[0,1).
x y x y x y
Lemma 3.2 ([7]). A neutrosophic set A = (A ,A ,A ) in a BCK/BCI-algebra
T I F
X is an (∈, ∈∨q)-neutrosophic subalgebra of X if and only if it satisfies:
(cid:86)
A (x∗y)≥ {A (x),A (y),0.5}
T T T
(3.5) (∀x,y ∈X) A (x∗y)≥(cid:86){A (x),A (y),0.5} .
I I I
(cid:87)
A (x∗y)≤ {A (x),A (y),0.5}
F F F
Theorem 3.3. A neutrosophic set A=(A ,A ,A ) in a BCK/BCI-algebra X is
T I F
an (∈, ∈∨q)-neutrosophic subalgebra of X if and only if the neutrosophic ∈-subsets
T (A;α), I (A;β) and F (A;γ) are subalgebras of X for all α,β ∈ (0,0.5] and
∈ ∈ ∈
γ ∈[0.5,1).
89
A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97
Proof. Assume that A = (A ,A ,A ) is an (∈, ∈∨q)-neutrosophic subalgebra of
T I F
X. For any x,y ∈X, let α∈(0,0.5] be such that x,y ∈T (A;α). Then A (x)≥α
∈ T
and A (y)≥α. It follows from (3.5) that
T
(cid:94)
A (x∗y)≥ {A (x),A (y),0.5}≥α∧0.5=α
T T T
andsothatx∗y ∈T (A;α). ThusT (A;α)isasubalgebraofX forallα∈(0,0.5].
∈ ∈
Similarly, I (A;β) is a subalgebra of X for all β ∈ (0,0.5]. Now, let γ ∈ [0.5,1) be
∈
such that x,y ∈F (A;γ). Then A (x)≤γ and A (y)≤γ. Hence
∈ F F
(cid:95)
A (x∗y)≤ {A x),A (y),0.5}≤γ∨0.5=γ
F F F
by (3.5), and so x ∗ y ∈ F (A;γ). Thus F (A;γ) is a subalgebra of X for all
∈ ∈
γ ∈[0.5,1).
Conversely, letα,β ∈(0,0.5]andγ ∈[0.5,1)besuchthatT (A;α), I (A;β)and
∈ ∈
F (A;γ) are subalgebras of X. If there exist a,b∈X such that
∈
(cid:94)
A (a∗b)< {A (a),A (b),0.5},
I I I
then we can take β ∈(0,1) such that
(cid:94)
(3.6) A (a∗b)<β < {A (a),A (b),0.5}.
I I I
Thus a,b∈I (A;β) and β <0.5, and so a∗b∈I (A;β). But, the left inequality in
∈ ∈
(3.6) induces a∗b∈/ I (A;β), a contradiction. Hence
∈
(cid:94)
A (x∗y)≥ {A (x),A (y),0.5}
I I I
for all x,y ∈X. Similarly, we can show that
(cid:94)
A (x∗y)≥ {A (x),A (y),0.5}
T T T
for all x,y ∈X. Now suppose that
(cid:95)
A (a∗b)> {A (a),A (b),0.5}
F F F
for some a,b∈X. Then there exists γ ∈(0,1) such that
(cid:95)
A (a∗b)>γ > {A (a),A (b),0.5}.
F F F
It follows that γ ∈(0.5,1) and a,b∈F (A;γ). Since F (A;γ) is a subalgebra of X,
∈ ∈
we have a∗b∈F (A;γ) and so A (a∗b)≤γ. This is a contradiction, and thus
∈ F
(cid:95)
A (x∗y)≤ {A (x),A (y),0.5}
F F F
for all x,y ∈ X. Using Lemma 3.2, A = (A ,A ,A ) is an (∈, ∈∨q)-neutrosophic
T I F
subalgebra of X. (cid:3)
Using Theorem 3.3 and [7, Theorem 3.8], we have the following corollary.
Corollary 3.4. For a neutrosophic set A = (A ,A ,A ) in a BCK/BCI-algebra
T I F
X,ifthenonemptyneutrosophic∈∨q-subsetsT (A;α),I (A;β)andF (A;γ)
∈∨q ∈∨q ∈∨q
are subalgebras of X for all α,β ∈ (0,1] and γ ∈ [0,1), then the neutrosophic ∈-
subsets T (A;α), I (A;β) and F (A;γ) are subalgebras of X for all α,β ∈ (0,0.5]
∈ ∈ ∈
and γ ∈[0.5,1).
90
A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97
Theorem 3.5. Given neutrosophic set A = (A ,A ,A ) in a BCK/BCI-algebra
T I F
X, the nonempty neutrosophic ∈-subsets T (A;α), I (A;β) and F (A;γ) are sub-
∈ ∈ ∈
algebras of X for all α,β ∈ (0.5,1] and γ ∈ [0,0.5) if and only if the following
assertion is valid.
A (x∗y)∨0.5≥A (x)∧A (y)
T T T
(3.7) (∀x,y ∈X) A (x∗y)∨0.5≥A (x)∧A (y) .
I I I
A (x∗y)∧0.5≤A (x)∨A (y)
F F F
Proof. Assume that the nonempty neutrosophic ∈-subsets T (A;α), I (A;β) and
∈ ∈
F (A;γ) are subalgebras of X for all α,β ∈ (0.5,1] and γ ∈ [0,0.5). Suppose
∈
that there are a,b ∈ X such that A (a∗b)∨0.5 < A (a)∧A (b) := α. Then
T T T
α∈(0.5,1] and a,b∈T (A;α). Since T (A;α) is a subalgebra of X, it follows that
∈ ∈
a∗b∈T (A;α), that is, A (a∗b)≥α which is a contradiction. Thus
∈ T
A (x∗y)∨0.5≥A (x)∧A (y)
T T T
for all x,y ∈ X. Similarly, we know that A (x ∗ y) ∨ 0.5 ≥ A (x) ∧ A (y) for
I I I
all x,y ∈ X. Now, if A (x∗y)∧0.5 > A (x)∨A (y) for some x,y ∈ X, then
F F F
x,y ∈ F (A;γ) and γ ∈ [0,0.5) where γ = A (x)∨A (y). But, x∗y ∈/ F (A;γ)
∈ F F ∈
which is a contradiction. Hence A (x∗y)∧0.5≤A (x)∨A (y) for all x,y ∈X.
F F F
Conversely, let A = (A ,A ,A ) be a neutrosophic set in X satisfying the con-
T I F
dition (3.7). Let x,y,a,b ∈ X and α,β ∈ (0.5,1] be such that x,y ∈ T (A;α) and
∈
a,b∈I (A;β). Then
∈
A (x∗y)∨0.5≥A (x)∧A (y)≥α>0.5,
T T T
A (a∗b)∨0.5≥A (a)∧A (b)≥β >0.5.
I I I
It follows that A (x∗y) ≥ α and A (a∗b) ≥ β, that is, x∗y ∈ T (A;α) and
T I ∈
a∗b ∈ I (A;β). Now, let x,y ∈ X and γ ∈ [0,0.5) be such that x,y ∈ F (A;γ).
∈ ∈
Then A (x ∗ y) ∧ 0.5 ≤ A (x) ∨ A (y) ≤ γ < 0.5 and so A (x ∗ y) ≤ γ, i.e.,
F F F F
x∗y ∈F (A;γ). This completes the proof. (cid:3)
∈
We consider relations between a (q, ∈∨q)-neutrosophic subalgebra and an (∈,
∈∨q)-neutrosophic subalgebra.
Theorem 3.6. In a BCK/BCI-algebra, every (q, ∈∨q)-neutrosophic subalgebra is
an (∈, ∈∨q)-neutrosophic subalgebra.
Proof. LetA=(A ,A ,A )bea(q,∈∨q)-neutrosophicsubalgebraofaBCK/BCI-
T I F
algebra X and let x,y ∈ X. Let α ,α ∈ (0,1] be such that x ∈ T (A;α ) and
x y ∈ x
y ∈T (A;α ). ThenA (x)≥α andA (y)≥α . Supposex∗y ∈/ T (A;α ∧α ).
∈ y T x T y ∈∨q x y
Then
(3.8) A (x∗y)<α ∧α ,
T x y
(3.9) A (x∗y)+(α ∧α )≤1.
T x y
It follows that
(3.10) A (x∗y)<0.5.
T
91
A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97
Combining (3.8) and (3.10), we have
(cid:94)
A (x∗y)< {α ,α ,0.5}
T x y
and so
(cid:94)
1−A (x∗y)>1− {α ,α ,0.5}
T x y
(cid:95)
= {1−α ,1−α ,0.5}
x y
(cid:95)
≥ {1−A (x),1−A (y),0.5}.
T T
Hence there exists α∈(0,1] such that
(cid:95)
(3.11) 1−A (x∗y)≥α> {1−A (x),1−A (y),0.5}.
T T T
The right inequality in (3.11) induces A (x)+α > 1 and A (y)+α > 1, that is,
T T
x,y ∈ T (A;α). Since A = (A ,A ,A ) is a (q, ∈∨q)-neutrosophic subalgebra
q T I F
of X, we have x∗y ∈ T (A;α). But, the left inequality in (3.11) implies that
∈∨q
A (x∗y)+α≤1, i.e., x∗y ∈/ T (A;α), and A (x∗y)≤1−α<1−0.5=0.5<α,
T q T
i.e., x∗y ∈/ T (A;α). Hence x∗y ∈/ T (A;α), a contradiction. Thus x∗y ∈
∈ ∈∨q
T (A;α ∧α ). Similarly, we can show that if x ∈ I (A;β ) and y ∈ I (A;β )
∈∨q x y ∈ x ∈ y
for β ,β ∈ (0,1], then x∗y ∈ I (A;β ∧β ). Now, let γ ,γ ∈ [0,1) be such
x y ∈∨q x y x y
that x ∈ F (A;γ ) and y ∈ F (A;γ ). A (x) ≤ γ and A (y) ≤ γ . If x∗y ∈/
∈ x ∈ y F x F y
F (A;γ ∨γ ), then
∈∨q x y
(3.12) A (x∗y)>γ ∨γ ,
F x y
(3.13) A (x∗y)+(γ ∨γ )≥1.
F x y
It follows that
(cid:95)
A (x∗y)> {γ ,γ ,0.5}
F x y
and so that
(cid:95)
1−A (x∗y)<1− {γ ,γ ,0.5}
F x y
(cid:94)
= {1−γ ,1−γ ,0.5}
x y
(cid:94)
≤ {1−A (x),1−A (y),0.5}.
F F
Thus there exists γ ∈[0,1) such that
(cid:94)
(3.14) 1−A (x∗y)≤γ < {1−A (x),1−A (y),0.5}.
F F F
Itfollows fromtherightinequalityin(3.14)that A (x)+γ <1and A (y)+γ <1,
F F
that is, x,y ∈ F (A;γ), which implies that x ∗ y ∈ F (A;γ). But, we have
q ∈∨q
x∗y ∈/ F (A;γ) by the left inequality in (3.14). This is a contradiction, and so
∈∨q
x∗y ∈F (A;γ ∨γ ). Therefore A=(A ,A ,A ) is an (∈, ∈∨q)-neutrosophic
∈∨q x y T I F
subalgebra of X. (cid:3)
The following example shows that the converse of Theorem 3.6 is not true.
92
A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97
Table 1. Cayley table of the operation ∗
∗ 0 1 2 3 4
0 0 0 0 0 0
1 1 0 0 1 1
2 2 1 0 2 2
3 3 3 3 0 3
4 4 4 4 4 0
X A (x) A (x) A (x)
T I F
0 0.6 0.8 0.3
1 0.2 0.3 0.6
2 0.2 0.3 0.6
3 0.7 0.1 0.7
4 0.4 0.4 0.9
Example 3.7. Consider a BCK-algebra X ={0,1,2,3,4} with the following Cay-
ley table.
Let A=(A ,A ,A ) be a neutrosophic set in X defined by
T I F
Then
{0,3} if α∈(0.4,0.5],
T (A;α)= {0,3,4} if α∈(0.2,0.4],
∈
X if α∈(0,0.2],
{0} if β ∈(0.4,0.5],
{0,4} if β ∈(0.3,0.4],
I (A;β)=
∈ {0,1,2,4} if β ∈(0.1,0.3],
X if β ∈(0,0.1],
X if γ ∈(0.9,1),
{0,1,2,3} if γ ∈[0.7,0.9),
F (A;γ)=
∈ {0,1,2} if γ ∈[0.6,0.7),
{0} if γ ∈[0.5,0.6),
which are subalgebras of X for all α,β ∈ (0,0.5] and γ ∈ [0.5,1). Using Theorem
3.3, A = (A ,A ,A ) is an (∈, ∈∨q)-neutrosophic subalgebra of X. But it is not
T I F
a (q, ∈∨q)-neutrosophic subalgebra of X since 2 ∈ T (A;0.83) and 3 ∈ T (A;0.4),
q q
but 2∗3=2∈/ T (A;0.4).
∈∨q
Weprovideconditionsforan(∈,∈∨q)-neutrosophicsubalgebratobea(q,∈∨q)-
neutrosophic subalgebra.
Theorem 3.8. Assume that any neutrosophic T -point and neutrosophic I -point
Φ Φ
has the value α and β in (0,0.5], respectively, and any neutrosophic F -point has
Φ
the value γ in [0.5,1) for Φ ∈ {∈,q,∈ ∨q}. Then every (∈, ∈ ∨q)-neutrosophic
subalgebra is a (q, ∈∨q)-neutrosophic subalgebra.
Proof. Let X be a BCK/BCI-algebra and let A = (A ,A ,A ) be an (∈, ∈∨q)-
T I F
neutrosophic subalgebra of X. For x,y,a,b ∈ X, let α ,α ,β ,β ∈ (0,0.5] be
x y a b
93
A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97
such that x ∈ T (A;α ), y ∈ T (A;α ), a ∈ I (A;β ) and b ∈ T (A;β ). Then
q x q y q a q b
A (x)+α > 1, A (y)+α > 1, A (a)+β > 1 and A (b)+β > 1. Since
T x T y I a I b
α ,α ,β ,β ∈(0,0.5], it follows that A (x)>1−α ≥α , A (y)>1−α ≥α ,
x y a b T x x T y y
A (a)>1−β ≥β andA (b)>1−β ≥β ,thatis,x∈T (A;α ),y ∈T (A;α ),
I a a I b b ∈ x ∈ y
a ∈ I (A;β ) and b ∈ I (A;β ). Also, let x ∈ F (A;γ ) and y ∈ F (A;γ ) for
∈ a ∈ b q x q y
x,y ∈ X and γ ,γ ∈ [0.5,1). Then A (x)+γ < 1 and A (y)+γ < 1, and so
x y F x F y
A (x)<1−γ ≤γ andA (y)<1−γ ≤γ sinceγ ,γ ∈[0.5,1). Thisshowsthat
F x x F y y x y
x∈F (A;γ )andy ∈F (A;γ ). Itfollowsfrom(3.4)thatx∗y ∈T (A;α ∧α ),
∈ x ∈ y ∈∨q x y
a∗b∈I (A;β ∧β ),andx∗y ∈F (A;γ ∨γ ). Consequently,A=(A ,A ,A )
∈∨q a b ∈∨q x y T I F
is a (q, ∈∨q)-neutrosophic subalgebra of X. (cid:3)
Theorem 3.9. Both (∈, ∈)-neutrosophic subalgebra and (∈∨q, ∈∨q)-neutrosophic
subalgebra are an (∈, ∈∨q)-neutrosophic subalgebra.
Proof. It is clear that (∈, ∈)-neutrosophic subalgebra is an (∈, ∈∨q)-neutrosophic
subalgebra. Let A = (A ,A ,A ) be an (∈∨q, ∈∨q)-neutrosophic subalgebra of
T I F
X. For any x,y,a,b ∈ X, let α , α , β , β ∈ (0,1] be such that x ∈ T (A;α ),
x y a b ∈ x
y ∈ T (A;α ), a ∈ I (A;β ) and b ∈ I (A;β ). Then x ∈ T (A;α ), y ∈
∈ y ∈ a ∈ b ∈∨q x
T (A;α ), a ∈ I (A;β ) and b ∈ I (A;β ) by (3.1) and (3.2). It follows
∈∨q y ∈∨q a ∈∨q b
that x∗y ∈ T (A;α ∧α ) and a∗b ∈ I (A;β ∧β ). Now, let x,y ∈ X and
∈∨q x y ∈∨q a b
γ ,γ ∈[0,1) be such that x∈F (A;γ ) and y ∈F (A;γ ). Then x∈F (A;γ )
x y ∈ x ∈ y ∈∨q x
and y ∈ F (A;γ ) by (3.3). Hence x∗y ∈ F (A;γ ∨γ ). Therefore A =
∈∨q y ∈∨q x y
(A ,A ,A ) is an (∈, ∈∨q)-neutrosophic subalgebra of X. (cid:3)
T I F
The converse of Theorem 3.9 is not true in general. In fact, the (∈, ∈ ∨q)-
neutrosophic subalgebra A = (A ,A ,A ) in Example 3.7 is neither an (∈, ∈)-
T I F
neutrosophic subalgebra nor an (∈∨q, ∈∨q)-neutrosophic subalgebra.
Theorem 3.10. For a neutrosophic set A=(A ,A ,A ) in a BCK/BCI-algebra
T I F
X, if the nonempty neutrosophic q-subsets T (A;α), I (A;β) and F (A;γ) are sub-
q q q
algebras of X for all α,β ∈(0.5,1] and γ ∈(0,0.5), then
x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∨α ),
∈ x ∈ y q x y
(3.15) x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∨β ),
∈ x ∈ y q x y
x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∧γ )
∈ x ∈ y q x y
for all x,y ∈X, α ,α , β ,β ∈(0.5,1] and γ ,γ ∈(0,0.5).
x y x y x y
Proof. Let x,y,a,b,u,v ∈ X and α , α , β , β ∈ (0.5,1] and γ ,γ ∈ (0,0.5) be
x y a b u v
such that x∈T (A;α ), y ∈T (A;α ), a∈I (A;β ), b∈I (A;β ), u∈F (A;γ )
∈ x ∈ y ∈ a ∈ b ∈ u
and v ∈ F (A;γ ). Then A (x) ≥ α > 1−α , A (y) ≥ α > 1−α , A (a) ≥
∈ v T x x T y y I
β >1−β , A (b)≥β >1−β , A (u)≤γ <1−γ and A (v)≤γ <1−γ .
a a I b b F u u F v v
It follows that x,y ∈T (A;α ∨α ), a,b∈I (A;β ∨β ) and u,v ∈F (A;γ ∧γ ).
q x y q a b q u v
Sinceα ∨α ,β ∨β ∈(0.5,1]andγ ∧γ ∈(0,0.5),wehavex∗y ∈T (A;α ∨α ),
x y a b u v q x y
a∗b∈I (A;β ∨β ) and u∗v ∈F (A;γ ∧γ ) by hypothesis. This completes the
q a b q u v
proof. (cid:3)
The following corollary is by Theorem 3.10 and [7, Theorem 3.7].
Corollary 3.11. Every (∈, ∈∨q)-neutrosophic subalgebra A = (A ,A ,A ) in a
T I F
BCK/BCI-algebra X satisfies the condition (3.15).
94
A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97
Corollary 3.12. Every (q, ∈∨q)-neutrosophic subalgebra A = (A ,A ,A ) in a
T I F
BCK/BCI-algebra X satisfies the condition (3.15).
Proof. It is by Theorem 3.6 and Corollary 3.11. (cid:3)
Theorem 3.13. For a neutrosophic set A=(A ,A ,A ) in a BCK/BCI-algebra
T I F
X, if the nonempty neutrosophic q-subsets T (A;α), I (A;β) and F (A;γ) are sub-
q q q
algebras of X for all α,β ∈(0,0.5] and γ ∈(0.5,1), then
x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∨α ),
q x q y ∈ x y
(3.16) x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∨β ),
q x q y ∈ x y
x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∧γ )
q x q y ∈ x y
for all x,y ∈X, α ,α , β ,β ∈(0,0.5] and γ ,γ ∈(0.5,1).
x y x y x y
Proof. Let x,y,a,b,u,v ∈ X and α , α , β , β ∈ (0,0.5] and γ ,γ ∈ (0.5,1)
x y a b u v
be such that x ∈ T (A;α ), y ∈ T (A;α ), a ∈ I (A;β ), b ∈ I (A;β ), u ∈
q x q y q a q b
F (A;γ ) and v ∈ F (A;γ ). Then x,y ∈ T (A;α ∨α ), a,b ∈ I (A;β ∨β ) and
q u q v q x y q a b
u,v ∈ F (A;γ ∧γ ). Since α ∨α ,β ∨β ∈ (0,0.5] and γ ∧γ ∈ (0.5,1), it
q u v x y a b u v
follows from the hypothesis that x∗y ∈ T (A;α ∨α ), a∗b ∈ I (A;β ∨β ) and
q x y q a b
u∗v ∈F (A;γ ∧γ ). Hence
q u v
A (x∗y)>1−(α ∨α )≥α ∨α , that is, x∗y ∈T (A;α ∨α ),
T x y x y ∈ x y
A (a∗b)>1−(β ∨β )≥β ∨β , that is, a∗b∈I (A;β ∨β ),
I a b a b ∈ a b
A (u∗v)<1−(γ ∧γ )≤γ ∧γ , that is, u∗v ∈F (A;γ ∧γ ).
F u v u v ∈ u v
Consequently, the condition (3.16) is valid for all x,y ∈ X, α ,α , β ,β ∈ (0,0.5]
x y x y
and γ ,γ ∈(0.5,1). (cid:3)
x y
Theorem 3.14. Given a neutrosophic set A = (A ,A ,A ) in a BCK/BCI-
T I F
algebra X, if the nonempty neutrosophic ∈∨q-subsets T (A;α), I (A;β) and
∈∨q ∈∨q
F (A;γ) are subalgebras of X for all α,β ∈ (0,0.5] and γ ∈ [0.5,1), then the
∈∨q
following assertions are valid.
x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∨α ),
q x q y ∈∨q x y
(3.17) x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∨β ),
q x q y ∈∨q x y
x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∧γ )
q x q y ∈∨q x y
for all x,y ∈X, α ,α , β ,β ∈(0,0.5] and γ ,γ ∈[0.5,1).
x y x y x y
Proof. Let x,y,a,b,u,v ∈ X and α , α , β , β ∈ (0,0.5] and γ ,γ ∈ [0.5,1) be
x y a b u v
such that x ∈ T (A;α ), y ∈ T (A;α ), a ∈ I (A;β ), b ∈ I (A;β ), u ∈ F (A;γ )
q x q y q a q b q u
and v ∈ F (A;γ ). Then x ∈ T (A;α ), y ∈ T (A;α ), a ∈ I (A;β ),
q v ∈∨q x ∈∨q y ∈∨q a
b ∈ I (A;β ), u ∈ F (A;γ ) and v ∈ F (A;γ ). It follows that x,y ∈
∈∨q b ∈∨q u ∈∨q v
T (A;α ∨α ), a,b ∈ I (A;β ∨β ) and u,v ∈ F (A;γ ∧γ ) which imply
∈∨q x y ∈∨q a b ∈∨q u v
from the hypothesis that x∗y ∈ T (A;α ∨α ), a∗b ∈ I (A;β ∨β ) and
∈∨q x y ∈∨q a b
u∗v ∈F (A;γ ∧γ ). This completes the proof. (cid:3)
∈∨q u v
Corollary 3.15. Every (∈, ∈∨q)-neutrosophic subalgebra A = (A ,A ,A ) of a
T I F
BCK/BCI-algebra X satisfies the condition (3.17).
Proof. It is by Theorem 3.14 and [7, Theorem 3.9]. (cid:3)
95
A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97
Theorem 3.16. Given a neutrosophic set A = (A ,A ,A ) in a BCK/BCI-
T I F
algebra X, if the nonempty neutrosophic ∈∨q-subsets T (A;α), I (A;β) and
∈∨q ∈∨q
F (A;γ) are subalgebras of X for all α,β ∈ (0.5,1] and γ ∈ [0,0.5), then the
∈∨q
following assertions are valid.
x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∨α ),
q x q y ∈∨q x y
(3.18) x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∨β ),
q x q y ∈∨q x y
x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∧γ )
q x q y ∈∨q x y
for all x,y ∈X, α ,α , β ,β ∈(0.5,1] and γ ,γ ∈[0,0.5).
x y x y x y
Proof. It is similar to the proof Theorem 3.14. (cid:3)
Corollary 3.17. Every (q, ∈∨q)-neutrosophic subalgebra A = (A ,A ,A ) of a
T I F
BCK/BCI-algebra X satisfies the condition (3.18).
Proof. It is by Theorem 3.16 and [7, Theorem 3.10]. (cid:3)
Combining Theorems 3.14 and 3.16, we have the following corollary.
Corollary 3.18. Given a neutrosophic set A = (A ,A ,A ) in a BCK/BCI-
T I F
algebra X, if the nonempty neutrosophic ∈∨q-subsets T (A;α), I (A;β) and
∈∨q ∈∨q
F (A;γ) are subalgebras of X for all α,β ∈(0,1] and γ ∈[0,1), then the following
∈∨q
assertions are valid.
x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∨α ),
q x q y ∈∨q x y
x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∨β ),
q x q y ∈∨q x y
x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∧γ )
q x q y ∈∨q x y
for all x,y ∈X, α ,α , β ,β ∈(0,1] and γ ,γ ∈[0,1).
x y x y x y
Conclusions
We have considered relations between an (∈, ∈∨q)-neutrosophic subalgebra and
a (q, ∈∨q)-neutrosophic subalgebra. We have discussed characterization of an (∈,
∈∨q)-neutrosophic subalgebra by using neutrosophic ∈-subsets, and have provided
conditions for an (∈, ∈∨q)-neutrosophic subalgebra to be a (q, ∈∨q)-neutrosophic
subalgebra. We have investigated properties on neutrosophic q-subsets and neutro-
sophic ∈∨q-subsets. Our future research will be focused on the study of generaliza-
tion of this paper.
4. Acknowledgements
The authors wish to thanks the anonymous reviewers for their valuable sugges-
tions.
References
[1] K.Atanassov,Intuitionisticfuzzysets,FuzzySetsandSystems20(1986)87–96.
[2] K. Atanassov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989)
343–349.
[3] S.R.Barbhuiya,(∈,∈∨q)-intuitionisticfuzzyidealsofBCK/BCI-algebras,NotesonIFS21
(1)(2015)24–42.
96
