2. Preliminaries
A
fuzzy set in a set
X is defined to be a function
For
, denote by
the collection of all fuzzy sets in a set
Define a relation ≤ on
as follows:
The complement of
denoted by
is defined by
For a family
of fuzzy sets in
X, we define the join (∨) and meet (∧) operations as follows:
An
interval-valued intuitionistic fuzzy set (briefly, IVIF set)
A over a set
X (see [
12]) is an object having the form
where
and
are intervals and for every
,
Especially, if
then the IVIF set
is reduced to an intuitionistic fuzzy set (see [
13]).
Given two closed subintervals
and
of
, we define the order “≪” as follows:
We also define the refined minimum (briefly, rmin) and refined maximum (briefly, rmax) as follows:
For a family
of closed subintervals of
, we define rinf (refined infimum) and rsup (refined supermum) as follows:
In this paper we use the interval-valued intuitionistic fuzzy set
over
X in which
and
are closed subintervals of
for all
, that is,
and
are interval-valued fuzzy (briefly, IVF) sets in
X where
is the set of all closed subintervals of
. Also, we use the notations
and
to mean the left end point and the right end point of the interval
, respectively, and so we have
. The interval-valued intuitionistic fuzzy set
over
X is simply denoted by
for
or
.
An algebra of type is called a -algebra if it satisfies the following axioms:
- (I)
- (II)
- (III)
- (IV)
If a BCI-algebra X satisfies the following identity:
- (V)
then
X is called a
-algebra. We can define a partial ordering ≤ on
X by
if and only if
Any BCK/BCI-algebra
X satisfies the following conditions:
A nonempty subset S of a -algebra X is called a subalgebra of X if for all A subset I of a BCK/BCI-algebra X is called an ideal of X if and the following condition is valid.
We refer the reader to the books [
14,
15] and the paper [
16] for further information regarding BCK/BCI-algebras.
3. Cubic Interval-Valued Intuitionistic Fuzzy Sets
Definition 1. Let X be a nonempty set. By a cubic interval-valued intuitionistic fuzzy set (briefly, cubic IVIF set) in X we mean a structurein which μ is a fuzzy set in X and is an interval-valued intuitionistic fuzzy set in X. A cubic IVIF set is simply denoted by
Let
be a cubic IVIF set in a nonempty set
X. Given
and
, we consider the sets
Definition 2. Let X be a nonempty set. A cubic IVIF set is said to be
α-internal if for all ,
β-internal if for all ,
internal if it is both α-internal and β-internal.
α-external if for all .
β-external if for all .
external if it is both α-external and β-external.
It is clear that if is a cubic IVIF set in X with for some , then cannot be internal.
Example 1. Let be a cubic IVIF set in I in which and for all .
- (1)
If , then is α-internal which is not β-internal.
- (2)
If , then is β-internal which is not α-internal.
- (3)
If , then is internal.
- (4)
If either or , then is external.
- (5)
If either or , then is α-external but may not be β-external.
- (6)
If either or , then is β-external but may not be α-external.
Proposition 1. Let be a cubic IVIF set in a nonempty set X in which for all . If is α-internal, then it is β-internal and so internal. Also, if is β-external, then it is α-external and so external.
Proof. Straightforward. ☐
Proposition 2. If is a cubic IVIF set in X which is not external, then there exist such that or .
Proof. Straightforward. ☐
Theorem 1. Let be a cubic IVIF set in X in which the right end point of (or ) is equal to the left end point of (or ) for all . If we define μ bythen is external. Proof. Straightforward. ☐
For any cubic IVIF set
in
X, let
Theorem 2. Let be a cubic IVIF set in X. If is both α-internal and α-external (resp., both β-internal and β-external), then (resp., ) for all .
Proof. Assume that is both -internal and -external. Then and for all . It follows that or , that is, or . Hence for all . Similarly, if is both -internal and -external, then for all . ☐
Given an IVIF set
in
X, the
complement of
is denoted by
and is defined as follows:
where
and
with
,
,
, and
.
Definition 3. For any cubic IVIF sets and in X, we definewhere means that and , that is, It is clear that the set of all cubic IVIF sets in X forms a poset under the P-order and the R-order .
For any
where
, we define
and
where
,
,
, and
.
Definition 4. Given a family of cubic IVIF sets in X, we define
- (1)
(P-union)
- (2)
(R-union)
- (3)
(P-intersection)
- (4)
(R-intersection)
Proposition 3. For any cubic IVIF sets , , and in X, we have
- (1)
.
- (2)
.
- (3)
.
- (4)
.
- (5)
.
- (6)
.
Proof. Straightforward. ☐
Theorem 3. Let be a cubic IVIF set in X. If is α-internal (resp., α-external), then so is the complement of .
Proof. Assume that
is
-internal. Then
, that is,
for all
. Thus
, that is,
for all
. Hence
is
-internal. If
is
-external, then
for all
, and so
or
. It follows that
or
, and so that
Therefore is -external. ☐
Theorem 4. Let be a cubic IVIF set in X. If is β-internal (resp., β-external), then so is the complement of .
Proof. It is similar to the proof of Theorem 3. ☐
Corollary 1. If a cubic IVIF set in X is internal, then so is .
Theorem 5. If is a family of α-internal cubic IVIF sets in X, then the P-union and the P-intersection of are also α-internal cubic IVIF sets in X.
Proof. Since
is
-internal, we have
, that is,
for all
and
. It follows that
and
Therefore and are -internal cubic IVIF sets in X. ☐
The following example shows that the R-union and R-intersection of -internal cubic IVIF sets need not be -internal.
Example 2. Let and be cubic IVIF sets in with and for all . Then and , where If and for all , then and are α-internal. The R-union of and isand it is not α-internal. The R-intersection of and iswhich is not α-internal. We provide a condition for the R-union of two -internal cubic IVIF sets to be -internal.
Theorem 6. Let and be cubic IVIF sets in X such that If and are α-internal, then so is the R-union of and .
Proof. Assume that
and
are
-internal cubic IVIF sets in
X that satisfies the condition (
17). Then
and
for all
, which imply that
. It follows from (
17) that
for all
. Therefore
is an
-internal cubic IVIF set in
X. ☐
We provide a condition for the R-intersection of two -internal cubic IVIF sets to be -internal.
Theorem 7. Let and be cubic IVIF sets in X such that If and are α-internal, then so is the R-intersection of and .
Proof. Let
and
be
-internal cubic IVIF sets in
X which satisfy the condition (
18). Then
and
for all
. It follows from (
18) that
for all
. Therefore
is an
-internal cubic IVIF set in
X. ☐
The following example shows that the P-union and the P-intersection of -external cubic IVIF sets need not be -external.
Example 3. Let and be cubic IVIF sets in X which are given in Example 2. If and for all , then and are α-external. The P-union of and isand it is not α-external. The P-intersection of and iswhich is not α-external. We consider conditions for the P-union and P-intersection of two -external cubic IVIF sets to be -external.
Theorem 8. Let and be cubic IVIF sets in X such thatand If and are α-external, then so are the P-union and P-intersection of and .
Proof. If
and
are
-external, then
and
, that is,
and
for all
. It follows from conditions (
19) and (
20) that
or
The conditions (
21) and (
22) induce
and
respectively, for all
. Hence
for all
, and therefore
is an
-external cubic IVIF set in
X. Also, (
21) and (
22) imply that
and
respectively, for all
. Therefore
is an
-external cubic IVIF set in
X. ☐
The following example shows that the R-union and R-intersection of -external cubic IVIF sets in X may not be -external.
Example 4. Let and be cubic IVIF sets in with and for all . Then and , where If and for all , then and are both α-external. The R-union of and isand it is not α-external. If and for all , then and are α-external. The R-intersection of and iswhich is not α-external. We discuss conditions for the R-union and R-intersection of two cubic IVIF sets to be -external.
Let
and
be cubic IVIF sets in
X such that
If
for all
, then
and so
.
If
for all
, then
and thus
.
If
for all
, then
which implies that
.
If
for all
, then
which induces
.
Therefore we have the following theorem.
Theorem 9. Let and be cubic IVIF sets in X satisfying the condition (23). If μ and λ satisfies any one of the following conditions.then the R-union of and is α-external. If μ and λ satisfies any one of the following conditions.then the R-intersection of and is α-external. Let
and
be cubic IVIF sets in
X such that
If
for all
, then
and so
.
If
for all
, then
and thus
.
If
, then
which induces
for all
.
If
, then
and so
for all
.
Therefore we have the following theorem.
Theorem 10. Let and be cubic IVIF sets in X satisfying the condition (28). If μ and λ satisfiesorthen the R-union of and is α-external. Also, iforthen the R-intersection of and is α-external. 4. Cubic IVIF Subalgebras and Ideals
In what follows, let X be a -algebra unless otherwise specified.
Definition 5. A cubic IVIF set in X is called a cubic IVIF subalgebra of X if the following conditions are valid. Example 5. Let be a -algebra with the Cayley table (Table 1). Define a cubic IVIF set in X by the tabular representation in Table 2. It is routine to verify that is a cubic IVIF subalgebra of X.
Proposition 4. If is a cubic IVIF subalgebra of X, then , and for all .
Proof. Since
for all
, we have
This completes the proof. ☐
Proposition 5. If is a cubic IVIF subalgebra of a -algebra X, then , and for all .
Proof. It follows from Definition 5 and Proposition 4. ☐
Proposition 6. For a cubic IVIF subalgebra of X, if there exists a sequence in X such that , and , then , and .
Proof. Using Proposition 4, we have
,
and
. It follows from hypothesis that
Hence , and . ☐
Theorem 11. If is a cubic IVIF subalgebra of X, then the sets , and , are subalgebras of X for all and .
Proof. For any
and
, let
be such that
Then
,
,
,
,
and
. It follows that
that is,
,
and
. Therefore
,
and
are subalgebras of
X for all
and
. ☐
Corollary 2. If is a cubic IVIF subalgebra of X, then is a subalgebra of X for all and .
The following example shows that the converse of Corollary 2 is not true in general.
Example 6. Let be a set with the Cayley table (Table 3). Then X is a -algebra (see [15]). Define a cubic IVIF set in X by the tabular representation in Table 4. It is routine to verify that is a subalgebra of X for all and . But, is not a cubic IVIF subalgebra of X since .
We provide conditions for a cubic IVIF set in X to be a cubic IVIF subalgebra of X.
Theorem 12. Let be a cubic IVIF set in X such that , and are subalgebras of X for all , and . Then is a cubic IVIF subalgebra of X.
Proof. For any
, putting
imply that
,
,
,
, and
,
. Hence
,
,
,
,
, and
. It follows from hypothesis that
,
,
, and so that
Therefore is a cubic IVIF subalgebra of X. ☐
The following theorem gives us a way to establish a new cubic IVIF subalgebra from old one in -algebras.
Theorem 13. Given a cubic IVIF subalgebra of a -algebra X, let be a cubic IVIF set in X defined by , and for all . Then is a cubic IVIF subalgebra of X.
Proof. Since
for all
, it follows that
and
for all
. Therefore
is a cubic IVIF subalgebra of
X. ☐
Definition 6. A cubic IVIF set is called a cubic IVIF ideal of X if the following conditions are valid. Example 7. Let be a -algebra with the Cayley table in Table 5. Define a cubic IVIF set in X by Table 6. Then is a cubic IVIF ideal of X.
Proposition 7. Every cubic IVIF ideal of X satisfies:for all with . Proof. Let
be such that
. Then
, and so
This completes the proof. ☐
We provide conditions for a cubic IVIF set to be a cubic IVIF ideal.
Theorem 14. Let be a cubic IVIF set in X in which conditions (
31)
and (
32)
are valid. If satisfies the condition (
35)
for all with , then is a cubic IVIF ideal of X. Proof. Since
for all
, it follows from (
35) that
Therefore is a cubic IVIF ideal of X. ☐
Lemma 1. Every cubic IVIF ideal in X satisfies the following condition. Proof. Assume that
for all
. Then
, and so
by (
31)–(
34). ☐
Theorem 15. In a -algebra X, every cubic IVIF ideal is a cubic IVIF subalgebra.
Proof. Let
be a cubic IVIF ideal of a
-algebra
X. Since
for all
, we have
,
,
by Lemma 1. It follows from (
33) and (
34) that
Therefore is a cubic IVIF subalgebra of X. ☐
The converse of Theorem 15 is not true in general as seen in the following example.
Example 8. The cubic IVIF subalgebra in Example 5 is not a cubic IVIF ideal of X since .
We establish a characterization of a cubic IVIF ideal in a -algebra.
Theorem 16. For a cubic IVIF set in a -algebra X, the following assertions are equivalent.
- (1)
is a cubic IVIF ideal of X.
- (2)
is a cubic IVIF subalgebra of X satisfying (
35)
for all with .
Proof. (1) ⇒ (2). It follows from Proposition 7 and Theorem 15.
(2) ⇒ (1). It is by Proposition 4 and Theorem 14. ☐
5. Conclusions
We have introduced a cubic interval-valued intuitionistic fuzzy set which is an extension of a cubic set, and have applied it to -algebra. We have investigated P-union, P-intersection, R-union and R-intersection of -internal and -external cubic IVIF sets. We have defined cubic IVIF subalgebra and ideal in -algebra, and have investigated related properties. We have considered relations between cubic IVIF subalgebra and cubic IVIF ideal. We have discussed characterizations of cubic IVIF subalgebra and cubic IVIF ideal. In the consecutive research, we will discuss P-union, P-intersection, R-union and R-intersection of -internal and -external cubic IVIF sets. We will apply this notion to several kind of ideals in -algebras, for example, (positive) implicative ideal, commutative ideal, associative ideal, q-ideal, etc. We will also apply this notion to several algebraic substructures in various algebraic structures, for example, -algebra, -algebra, -algebra, residuated lattice, -algebra, semigroup, semiring, near-ring, etc. We will try to study this notion in the neutrosophic environment.