In this section, we will introduce the basic notions and results of int N-soft subsemigroup and int N-soft left [right] ideals of S. The parameter set of an N-soft set, which we will use in this section is a semigroup, whereas the universe set is any set. We discuss the notions and properties of int N-soft product and int N-soft characteristic function. We also discuss the -generalized int N-soft subsemigroup and investigate several properties.
3.1. Int N-Soft Subsemigroup
Let U be an initial universe and S be a semigroup considered as a set of parameters. Let be an N-soft set.
Definition 10. Let A and B be any subsets of the semigroup S. Then, the multiplication of A and B is defined by Definition 11. For any subset γ of such that for each there exists a unique number n ≤ N-1, for which (u,n) ∈γ we write , the γ-inclusive set of an N-soft set over U is denoted and defined by By γ⊆ we always mean it satisfies the condition given above.
Definition 12. An N-soft set over U is called an int N-soft subsemigroup of S over U if for all and , we have Example 1. Let S = {a,b,c,d} be a semigroup as defined in the Cayley Table:S | a | b | c | d |
a | a | a | a | a |
b | a | b | a | a |
c | a | a | c | a |
d | a | b | a | d |
and let U = {u,u,u,u,u} be the universe. Consider the 5-soft set over U, as given in Table 1. Consider γ = . Then, = .
Simple calculations show that is an int 5-soft subsemigroup over U.
Lemma 1. An N-soft set over U is an int N-soft subsemigroup over U if and only if the non-empty γ-inclusive set of is a subsemigroup of S for all γ⊆.
Proof. Assume an N-soft set
over
U is an int N-soft subsemigroup over
U. Then, we show that the non-empty
-inclusive set of
is a subsemigroup of
S for all
. For this, let
. Then, by definition of
-inclusive, we have
and
Since,
and
, we have
Since
is an int N-soft subsemigroup over
U, we have for all
and
,
As , we have
Hence, the non empty -inclusive set of is a subsemigroup of S.
Conversely, assume that
is a subsemigroup of
S for all
. Let
and
be such that
Construct a subset
of
such that
Then but is a contradiction.
Hence, □
Theorem 1. If and are two int N-soft subsemigroups of a semigroup S over U. Then, their restricted (extended) intersection is an int N-soft subsemigroup of S.
Proof. Let
=
, where for
and
, we have
As
and
are int N-soft subsemigroups of
S over
U, we have for all
and
Hence, is an int N-soft subsemigroup of S over U. □
The converse of the above theorem may not be true which is explained in the following example.
Example 2. Let S = {a,b,c,d} be a semigroup as defined in the Cayley Table:S | a | b | c | d |
a | a | a | a | a |
b | a | b | a | a |
c | a | a | c | c |
d | a | a | d | d |
and let U = {u,u,u,u} be the universe. Consider and any two N-soft sets over U, as given in Table 2 and Table 3. Hence, and are not int N-soft subsemigroups over U. We know thatas given in Table 4. Simple calculations show that is an int 3-soft subsemigroup over U, where and are not int N-soft subsemigroups over U.
In general, the N-soft restricted union of two int N-soft subsemigroups over U is not an int N-soft subsemigroup over U, which is explained in the following example.
Example 3. Let S = {a,b,c} be a semigroup as defined in the Cayley Table: and let U = {u,u,u,u} be the universe. Consider the N-soft sets and over U, as given in Table 5 and Table 6. Simple calculations show that and are int N-soft subsemigroups over U. We know thatgiven in Table 7. Since,
Hence, is not an int 4-soft subsemigroup over U.
3.2. Int N-Soft Left [Right] Ideals of S
In this subsection, we define the int N-soft left [right] ideal of semigroups and study their basic properties.
Definition 13. An N-soft set over U is called an int N-soft left [right] ideal of S over U if for all and , we have An N-soft set over U is called an int N-soft two-sided ideal or simply an int N-soft ideal of S over U if it is both an int N-soft left ideal and an int N-soft right ideal of S over U.
Example 4. Let S = {a,b,c,d} be a semigroup, as defined in the Cayley Table:S | a | b | c | d |
a | a | a | a | a |
b | a | b | a | a |
c | a | a | c | a |
d | a | b | a | d |
and let U = {u,u,u,u,u} be the universe. Consider the 5-soft set over U, as given in Table 8. Simple calculations show that is an int N-soft subsemigroup over U.
Moreover, we can check that is an int 5-soft two-sided ideal of S over U. Hence, is an int N-soft two-sided ideal over U.
Remark 1. Obviously, every int N-soft left [right] ideal over U is an int N-soft subsemigroup over U.
However, the converse is not true, which is explained in the following example.
Example 5. Let S = {a,b,c} be a semigroup as defined in the Cayley Table:and let U = {u,u,u,u} be the universe. Consider the 4-soft set over U, as given in Table 9. Simple calculations show that is an int 4-soft subsemigroup over U. Moreover, we can calculate that Thus is not an int 4-soft left ideal nor int 4-soft right ideal over U.
Theorem 2. An N-soft set over U is an int N-soft left [right] ideal of S over U if and only if the non-empty γ-inclusive set of is a left [right] ideal of S for all subsets γ of .
Proof. Assume an N-soft set
over
U is an int N-soft left ideal over
U. Then, we show that the non-empty
-inclusive set of
is a left ideal of
S for all
. For this, let
. Then, by definition of
-inclusive, we have
Since
is an int N-soft left ideal over
U, we have for all
and
,
As , we have
Hence, the non empty -inclusive set of is a left ideal of S.
Conversely, assume that
is a left ideal of
S for all
. Let
and
be such that
Construct a subset
of
such that
Then but , a contradiction.
Hence,
Similarly an N-soft set over U is an int N-soft right ideal of S over U if and only if the non-empty -inclusive set of is a right ideal of S for all subsets of . □
Corollary 1. An N-soft set over U is an int N-soft two-sided ideal of S over U if and only if the non-empty γ-inclusive set of is a two-sided ideal of S for all subsets γ of .
Theorem 3. If and are any two int N-soft left [right] ideals of S over U. Then, their restricted (extended) intersection is an int N-soft left [right] ideal of S.
Proof. Let
=
, where for
and
, we have
As
and
are int N-soft left ideals of
S over
U, we have for all
and
,
Hence, is an int N-soft left ideal of S over U.
Similarly, if and are two int N-soft right ideals of S over U, then their restricted (extended) intersection is an int N-soft right ideal of S. □
Corollary 2. If and are any two int N-soft two sided ideals of S over U. Then, their restricted (extended) intersection is an int N-soft two-sided ideal of S.
Theorem 4. If and are any two int N-soft left [right] ideals of S over U. Then, their restricted (extended) union is an int N-soft left [right] ideal of S.
Proof. Let
=
, where for
and
, we have
As
and
are int N-soft left ideals of
S over
U, we have for all
and
,
Hence, is an int N-soft left ideal of S over U.
Similarly, if and are two int N-soft right ideals of S over U, then their restricted (extended) union is an int N-soft right ideal of S. □
Corollary 3. If and are any two int N-soft two-sided ideals of S over U, then their restricted (extended) union is an int N-soft two-sided ideal of S.
3.3. Int N-Soft Product and Int N-Soft Characteristic Function
In this subsection, we define int N-soft product and int N-soft characteristic function and study their properties.
Definition 14. The int N-soft product of any two N-soft sets and over the common universe U is denoted by = ∘ and is defined by for all x ∈S and .
Example 6. Let S = {a,b,c,d} be a semigroup as defined in the Cayley Table:S | a | b | c | d |
a | a | a | a | a |
b | a | a | a | a |
c | a | a | b | a |
d | a | a | b | b |
and let U = {u,u,u,u,u} be the universe. Consider the N-soft sets and over U, as given in Table 10 and Table 11. Hence, int N-soft product of and , is given in Table 12. Proposition 1. Let , , and be any N-soft sets over U. Ifthen Proof. Let
. If
x is not expressed as
for
∈
S, then clearly
Hence,
Suppose that there exists
∈
S such that
. Then,
Therefore □
Theorem 5. An N-soft set over U is an int N-soft subsemigroup over U if and only if Proof. Assume that
. Let
. Then we have for all
Thus, over U is an int N-soft subsemigroup over U.
Conversely, suppose that
is an int N-soft subsemigroup over
U. Then, for all
, we have
for all
with
. Thus
for all
. Hence
□
Theorem 6. Let and be any two N-soft sets over U. If is an int N-soft left ideal over U, then so is the int N-soft product
Proof. Let
. If
for some
∈
S, then
and for all
, we have
If
y is not expressible as
for all
∈
S, then for all
, we have
Thus for all , and so is an int N-soft left ideal over U. □
Corollary 4. Let and be any two N-soft sets over U. If is an int N-soft right ideal over U, then so is the int N-soft product
Theorem 7. If is an int N-soft right ideal over U and is an int N-soft left ideal over U, then Proof. Let
, if
x is not expressible as
for
∈
S. Then
Assume that there exists
∈
S such that
. Then
in any case, we have
□
Definition 15. For a non-empty subset A of S, defines N-soft characteristic function as follows, For each and , we have Then is an N-soft set over U, which is called N-soft characteristic set.
Example 7. Let S = {a,b,c,d} be a semigroup as defined in the Cayley Table:S | a | b | c | d |
a | a | a | a | a |
b | a | b | a | a |
c | a | a | c | a |
d | a | b | a | d |
Let A = {a,b} be a non-empty subset of S and let U = {u,u,u,u,u} be the universe. Then, the 5-soft characteristic set over U, is given in Table 13. Theorem 8. For any non-empty subset A of S, the following are equivalent,
- (1)
A is a left [right] ideal of S.
- (2)
An N-soft characteristic set is an int N-soft left [right] ideal over U.
Proof. Assume that
A is a left ideal of
S. For any
. If y ∉
A then for all
, we have
If
, then x
since
A is a left ideal of
S. Thus, for all
, we have
Therefore, ( is an int N-soft left ideal over U.
Conversely, suppose that (
is an int N-soft left ideal over
U. Let
and
. Then, for all
, we have
Hence,
That is
Thus and therefore A is a left ideal of S. □
Corollary 5. For any non-empty subset A of S, the following are equivalent,
- (1)
A is a two-sided ideal of S.
- (2)
An N-soft characteristic set is an int N-soft two sided ideal over U.
Theorem 9. For a non-empty subset T of S, the following are equivalent,
- (1)
T is a subsemigroup of S.
- (2)
An N-soft characteristic set is an int N-soft subsemigroup over U.
Proof. The proof is similar to the proof of Theorem 8. □
Theorem 10. Let and be any N-soft characteristic sets over U, where A and B are non-empty subsets of S. Then, the following properties hold
- (1)
- (2)
Proof. Let
, if
. Then
and
. Thus, we have
If
x∉
, then
x∉
A or
x∉
B. Hence, we have
Therefore,
- (2)
For any
, suppose
. Then, there exist
a∈
A and
b∈
B such that
. Thus, we have
Since , we obtain .
Now, suppose
x∉
, then
x≠
for all
a∈
A and
b∈
B. If
for some
∈
S then
y∉
A or
z∉
B. Thus,
If
x≠
for all
, then
3.4. -Generalized Int N-Soft Subsemigroup
In this subsection, we define -generalized int N-soft subsemigroup and study their several properties.
Definition 16. For a subset θ of U such that for each there exists a unique number m ∈ {}, for which , we write , an N-soft set is called a θ-generalized int N-soft subsemigroup over U, if for all and , we have By θ⊆ we always mean that it satisfies the condition given above.
Obviously, every int N-soft subsemigroup is a -generalized int N-soft subsemigroup but the converse is not true, which is shown in the following example.
Example 8. Let S = {a,b,c,d} be a semigroup as defined in the Cayley Table:S | a | b | c | d |
a | a | a | c | c |
b | b | b | d | d |
c | a | a | c | c |
d | b | b | d | d |
and let U = {u, u, u, u} be the universe. Consider N-soft sets over U, as given in Table 14. Hence is not an int N-soft subsemigroups over U.
Consider θ = {(u,1),(u,1),(u,2),(u,3)}. The simple calculations show that is a θ-generalized int N-soft subsemigroup over U.
Theorem 11. An N-soft set over U is a θ-generalized int N-soft subsemigroups over U if and only if the non-empty γ-inclusive set of is a subsemigroup of S for all γ with ≤.
Proof. Assume that
is a
-generalized int N-soft subsemigroups over
U. Let
where
≤
. Then, by definition of the
-inclusive set, we have
Since the int N-soft set over
U is a
-generalized int N-soft subsemigroups over
U, we have for all u
As , we have .
Hence, -inclusive set of is a subsemigroup of S for all with ≤.
Conversely, assume that
is a subsemigroup of
S for all
with
≤
. Let
and
be such that
Construct a subset of such that Then,
Then but , a contradiction.
Hence,
Therefore, is a -Generalized int N-soft subsemigroups over U. □
Theorem 12. If and are two θ-Generalized int N-soft subsemigroups of a semigroup S over U. Then, their restricted (extended) intersection is a θ-Generalized int N-soft subsemigroup of S.
Proof. Let = ,
where for
and
, we have
As
and
are
-Generalized int N-soft subsemigroups of
S over
U, we have for all
and
Hence, is a -Generalized int N-soft subsemigroup of S over U. □
In general, the N-soft restricted union of two -Generalized int N-soft subsemigroups over U is not a -Generalized int N-soft subsemigroup over U, which is explained in the following example.
Example 9. Let S = {a,b,c} be a semigroup as defined in the Cayley Table:and let U = {u,u,u} be the universe. Consider and N-soft sets over U, as given in Table 15 and Table 16. Consider θ = .
Simple calculations show that and are θ-Generalized int N-soft subsemigroups over U. However,
Hence, is not a θ-Generalized int 4-soft subsemigroup over U.
Theorem 13. For every θ, ϑ∈, if ≤ then every θ-Generalized int N-soft subsemigroup is a ϑ-Generalized int N-soft subsemigroup.
Proof. Let
,
∈
be such that
≤
. Let
be a
-generalized int N-soft subsemigroup over
U. For any
, we have
Therefore, is a -Generalized int N-soft subsemigroup over U. □
Theorem 14. If over U is a θ-generalized int N-soft subsemigroup over U, then the setis a subsemigroup of S for all a ∈S. Proof. Assume that
is a
-generalized int N-soft subsemigroup over
U. Then, for any
and
u, we have
Let
, we have
and
it follows that,
Thus, by definition xy ∈.
Hence, is a subsemigroup of S for all a ∈S. □
3.5. -Generalized Int N-Soft Left [Right] Ideals of S
In this subsection, we define -generalized int N-soft left [right] ideals of S and study their several properties.
Definition 17. For a subset θ of U such that for each there exists a unique number m ∈ {}, for which (u,m) ∈θ, we write , an N-soft set is called a θ-generalized int N-soft left [right] ideal of S over U, if for all and , we have An N-soft set over U is called a θ-generalized int N-soft two-sided ideal or simply a θ-generalized int N-soft ideal of S over U if it is both a θ-generalized int N-soft left ideal and a θ-generalized int N-soft right ideal of S over U.
By θ⊆we always mean it satisfies the condition given above.
Obviously, every int N-soft left [right] ideal of S is a -generalized int N-soft left [right] ideal of S, but the converse is not true, which is shown in the following example.
Example 10. Let S = {a,b,c} be a semigroup as defined in the Cayley Table:and let U = {u,u,u} be the universe. Consider an N-soft set over U, as given in Table 18. Hence, is not an int N-soft left [right] ideal of S over U.
Consider θ = {(u,1),(u,1),(u,2)}. The simple calculations show that is a θ-generalized int N-soft two-sided ideal of S over U.
Theorem 15. An N-soft set over U is a θ-generalized int N-soft left [right] ideal of S over U if and only if the non-empty γ-inclusive set of is a left [right] ideal of S for all γ with ≤.
Proof. Assume that
is a
-generalized int N-soft left ideal of
S over
U. Let
where
≤
. Then, by definition of the
-inclusive set, we have
Since an int N-soft set of
S over
U is a
-generalized int N-soft left ideal of
S over
U, we have for all u
As we have .
Hence, the non-empty -inclusive set of is a left ideal of S for all with .
Conversely, assume that
is a left ideal of
S for all
with
≤
. Let
and
be such that
Construct a subset of such that Then
Then but , a contradiction.
Hence,
Therefore, is a -Generalized int N-soft left ideal of S over U.
Similarly, an N-soft set over U is a -generalized int N-soft right ideal of S over U if and only if the non-empty -inclusive set of is a right ideal of S for all with ≤. □
Corollary 6. An N-soft set over U is a θ-generalized int N-soft two sided ideal of S over U if and only if the non-empty γ-inclusive set of is a two sided ideal of S for all γ with ≤.
Theorem 16. If and are any two θ-Generalized int N-soft left [right] ideals of S over U. Then, their restricted (extended) intersection is a θ-Generalized int N-soft left [right] ideal of S.
Proof. Let
=
, where for
and
, we have
As
and
are
-Generalized int N-soft left ideals of
S over
U, we have for all
and
,
Hence, is a -Generalized int N-soft left ideal of S over U.
Similarly, if and are two -Generalized int N-soft right ideals of a semigroup S over U, then their restricted (extended) intersection is a -Generalized int N-soft right ideal of S. □
Corollary 7. If and are any two θ-Generalized int N-soft two-sided ideals of S over U, then their restricted (extended) intersection is a θ-Generalized int N-soft two-sided ideal of S.
Theorem 17. If and are any two θ-Generalized int N-soft left [right] ideals of S over U. Then, their restricted (extended) union is a θ-Generalized int N-soft left [right] ideal of S.
Proof. Let
=
, where for
and
, we have
As
and
are
-Generalized int N-soft left ideals of
S over
U, we have for all
and
,
Hence, is a -Generalized int N-soft left ideal of S over U.
Similarly, if and are two -Generalized int N-soft right ideals of a semigroup S over U, then their restricted (extended) union is a -Generalized int N-soft right ideal of S. □
Corollary 8. If and are any two θ-Generalized int N-soft two sided ideals of S over U, then their restricted (extended) union is a θ-Generalized int N-soft two-sided ideal of S.
Theorem 18. If over U is a θ-generalized int N-soft left [right] ideal of S over U, then the set is a left [right] ideal of S for all a ∈S.
Proof. Assume that
is a
-generalized int N-soft left ideal of
S over
U. Then for any
and
u, we have
So by definition xy ∈.
Hence, is a left ideal of S for all a ∈S.
Similarly, if
over
U is a
-generalized int N-soft right ideal of
S over
U, then the set
is a right ideal of
S for all a ∈
S. □
Corollary 9. If over U is a θ-generalized int N-soft two-sided ideal of S over U, then the set is a two-sided ideal of S for all a ∈S.