2. Preliminaries
In this section, we give definitions and examples that are used in this paper. By a subsemigroup of a semigroup
S we mean a non-empty subset
A of
S such that
, and by a left (right) ideal of
S we mean a non-empty subset
A of
S such that
. By a two-sided ideal or simply an ideal, we mean a non-empty subset of a semigroup
S that is both a left and a right ideal of
S. A non-empty subset
A of
S is called an
interior ideal of
S if
, and a
quasi-ideal of
S if
. A subsemigroup
A of
S is called a
bi-ideal of
S if
. A non-empty subset
A is called a
generalized bi-ideal of
S if
[
9].
By the definition of a left (right) ideal of a semigroup S, it is easy to see that every left (right) ideal of S is a quasi-ideal of S.
Definition 1. A semigroup S is called regular if for all there exists such that .
Theorem 1. For a semigroup S, the following conditions are equivalent.
- (1)
S is regular.
- (2)
for every right ideal R and every left ideal L of S.
- (3)
for every quasi-ideal A of S.
Definition 2. Let X be a set; a fuzzy set (or fuzzy subset) f on X is a mapping , where is the usual interval of real numbers.
The symbols
and
will denote the following fuzzy sets on
S:
for all
.
A product of two fuzzy sets
f and
g is denoted by
and is defined as
Definition 3. Let S be a non-empty set. A BF set f on S is an object having the following form:where and . Remark 1. For the sake of simplicity we use the symbol for the BF set .
Definition 4. Given a BF set , , the setsandare called the positive -cut and negative -cut of f, respectively. The set is called the bipolar -cut of f. We give the generalization of a BF subsemigroup, which is defined by Kim et al. (2011).
Definition 5. A BF set on S is called a -BF subsemigroup on S, where if it satisfies the following conditions:
- (1)
- (2)
for all .
We note that every BF subsemigroup is a -BF subsemigroup.
The following examples show that is a -BF subsemigroup on S but is not a BF subsemigroup on S.
Example 1. The set is a semigroup under the usual multiplication. Let be a BF set on S defined as follows:for all . Let . ThenandThus, . Therefore is not a BF subsemigroup on S. Let , , and . Thus for all ,and Hence is a -BF subsemigroup on S.
We note that is a -BF subsemigroup on S for all and .
Definition 6. A BF set on S is called a -BF left (right) ideal on S, where and if it satisfies the following conditions:
- (1)
()
- (2)
()
for all .
A BF set on S is called a -BF ideal on S () if it is both a -BF left ideal and a -BF right ideal on S.
By Definition 6, every -BF ideal on a semigroup S is a -BF subsemigroup on S.
We note that a -BF left (right) ideal is a BF left (right) ideal.
Definition 7. A -BF subsemigroup on a subsemigroup S is called a -BF bi-ideal on S, where and if it satisfies the following conditions:
- (1)
- (2)
for all .
We note that every -BF bi-ideal on a semigroup is a -BF subsemigroup on the semigroup.
3. Generalized Bi-Ideal and Quasi-Ideal
In this section, we introduce a product of BF sets and characterize a regular semigroup by generalized BF subsemigroups.
We let
and
be two BF sets on a semigroup
S and let
and
. We define two fuzzy sets
and
on
S as follows:
for all
.
We define two operations
and
on
S as follows:
for all
, and we define products
and
as follows:
Then it is a BF set.
We note that
- (1)
,
- (2)
,
- (3)
,
- (4)
and .
Definition 8. A BF set on S is called a -BF generalized bi-ideal on S, where and if it satisfies the following conditions:
- (1)
- (2)
for all .
Definition 9. A BF set on S is called a -BF quasi-ideal on S, where and if it satisfies the following conditions:
- (1)
- (2)
for all .
In the following theorem, we give a relation between a bipolar -cut of f and a -BF generalized bi-ideal on S.
Theorem 2. Let be a BF set on a semigroup S with and . Then is a generalized bi-ideal of S for all and if and only if f is a -BF generalized bi-ideal on S for all and .
Proof. Let
. Suppose on the contrary that
f is not a
-BF generalized bi-ideal on
S. Then there exists
such that
Let
and
. Then
. By assumption, we have
. By Equation (
1),
or
. Thus,
. This is a contradiction. Therefore
f is a
-BF generalized bi-ideal on
S.
Conversely, let and , and suppose that . Let and . Then and By assumption, f is a -BF generalized bi-ideal on S, and thus and . Then and . Hence, . Therefore is a generalized bi-ideal of S. ☐
Corollary 1. Let be a BF set on a semigroup. Then the following statements hold:
- (1)
f is a -BF generalized bi-ideal on S for all and if and only if is a generalized bi-ideal of S for all and ;
- (2)
f is a -BF generalized bi-ideal on S for all and , if and only if is a generalized bi-ideal of S for all and .
Proof. (1) Set and , and apply Theorem 2.
(2) Set and , and apply Theorem 2. ☐
Lemma 1. Every -BF generalized bi-ideal on a regular semigroup S is a -BF bi-ideal on S.
Proof. Let S be a regular semigroup and be a -BF generalized bi-ideal on S. Let ; then there exists such that . Thus we have and . This shows that f is a -BF subsemigroup on S, and thus f is a -BF bi-ideal on S. ☐
Let
S be a semigroup and
. A positive characteristic function and a negative characteristic function are respectively defined by
and
Remark 2. - (1)
For the sake of simplicity, we use the symbol for the BF set. That is, . We call this a bipolar characteristic function.
- (2)
If , then . In this case, we denote .
In the following theorem, some necessary and sufficient conditions of -BF generalized bi-ideals are obtained.
Theorem 3. Let be a BF set on a semigroup S. Then the following statements are equivalent:
- (1)
f is a -BF generalized bi-ideal on S.
- (2)
and .
Proof. Let
a be any element of
S. In the case for which
, it is clear that
. Otherwise, there exist
such that
and
. Because
f is a
-BF generalized bi-ideal on
S, we have
and
. Consider
Hence .
Similarly, we can show that .
Conversely, let
such that
. Then we have
Similarly, we can show that for all . Therefore f is a -BF generalized bi-ideal on S for all and . ☐
Theorem 4. Let be a BF set on a semigroup S. Then the following statements are equivalent:
- (1)
f is a -BF bi-ideal on S .
- (2)
and .
Proof. The proof is similar to the proof of Theorem 3. ☐
In the following theorem, we give a relation between a bipolar -cut of f and a -BF quasi-ideal on S.
Theorem 5. Let be a BF set on a semigroup S with and . Then is a quasi-ideal of S for all and if and only if f is a -BF quasi-ideal on S for all and .
Proof. Let
and
. Suppose on the contrary that
f is not a
-BF quasi-ideal on
S. Then there exists
such that
or
Case 1:
. Let
. Then
,
. This implies that there exist
such that
. Then
Let . Then and .
Thus , and so and . Hence and , and it follows that . By hypothesis, .
Case 2:
. Let
. Then
and
. This implies that there exist
such that
. Then
Let
. Then
and
. Thus
, and so
and
. Hence
and
, and it follows that
. By hypothesis,
. Therefore
. By Equation (
2),
or
and it follows that
. This is a contradiction. Therefore
f is a
-BF quasi-ideal on
S.
Conversely, let and , and suppose that . Let be such that . Then and . Thus there exist and such that and .
By assumption,
f is a
-BF quasi-ideal on
S, and thus
Because , we have and . Then . Similarly, we can show that . Hence, . Therefore is a quasi-ideal of S. ☐
Corollary 2. Let be a BF set on a semigroup S. Then
- (1)
f is a -BF quasi-ideal on S for all and if and only if is a quasi-ideal of S for all and ;
- (2)
f is a -BF quasi-ideal on S for all and if and only if is a quasi-ideal of S for all and .
Proof. (1) Set and , and apply Theorem 5.
(2) Set and , and apply Theorem 5. ☐
In the following theorem, we discuss a quasi-ideal of a semigroup S in terms of the bipolar characteristic function being a -BF quasi-ideal on S.
Theorem 6. Let S be a semigroup. Then a non-empty subset I is a quasi-ideal of S if and only if the bipolar characteristic function is a -BF quasi-ideal on S for all and .
Proof. Let I be a quasi-ideal of S and . Let and .
Case 2:
. Then
or
. If
, then
and
. Thus
Therefore is a -BF quasi-ideal on S.
Conversely, let be a -BF quasi-ideal on S for all and . Let . Then there exist and such that . Then and . Hence , and so and . Hence and , and it follows that . By Corollary 2, is a quasi-ideal. Thus , and so . This implies that . Therefore I is a quasi-ideal on S. ☐
Theorem 7. Let S be a semigroup. Then I is a generalized bi-ideal of S if and only if the bipolar characteristic function is a -BF generalized bi-ideal on S for all and .
Proof. Let I be a generalized bi-ideal of S and . Let and .
Case 1:
. Then
; thus
and
Case 2:
or
. Then
and
Therefore is a -BF generalized bi-ideal on S.
Conversely, let be a -BF generalized bi-ideal on S for all and . Let and . Then and . Hence, . By Corollary 1, is a generalized bi-ideal. Thus , and so . This implies that . Therefore I is a generalized bi-ideal on S. ☐
Theorem 8. Every -BF left (right) ideal on a semigroup S is a -BF quasi-ideal on S.
Proof. Let
be a
-BF left ideal on
S and
. Then
Thus Hence Similarly, we can show that . Therefore f is a -BF quasi-ideal on S. ☐
Lemma 2. Every -BF quasi-ideal on a semigroup S is a -BF bi-ideal on S.
Proof. Let
be a
-BF quasi-ideal on
S and
. Then
Hence,
. Additionally,
Hence, . Similarly, we can show that and . Therefore f is a -BF bi-ideal on S. ☐
Lemma 3. Let A and B be non-empty subsets of a semigroup S. Then the following conditions hold:
- (1)
.
- (2)
.
- (3)
.
- (4)
.
Lemma 4. If is a -BF left ideal and is a -BF right ideal on a semigroup S, then and .
Theorem 9. For a semigroup S, the following are equivalent.
- (1)
S is regular.
- (2)
and for every -BF right ideal and every -BF left ideal on S.
Next, we characterize a regular semigroup by generalizations of BF subsemigroups.
Theorem 10. For a semigroup S, the following are equivalent.
- (1)
S is regular.
- (2)
and for every -BF right ideal , every -BF generalized bi-ideal and every -BF left ideal on S.
- (3)
and for every -BF right ideal , every -BF bi-ideal and every -BF left ideal on S.
- (4)
and for every -BF right ideal , every -BF quasi-ideal and every -BF left ideal on S.
Proof. . Let
and
g be a
-BF right ideal, a
-BF generalized bi-ideal and a
-BF left ideal on
S, respectively. Let
. Because
S is regular, there exists
such that
. Thus
Similarly, we can show that .
. This is straightforward, because every -BF bi-ideal is a -BF generalized bi-ideal and every -BF quasi-ideal is a -BF bi-ideal on S.
. Let
f and
g be any
-BF right ideal and
-BF left ideal on
S, respectively. Let
. By Theorem 8,
is a
-BF quasi ideal, and we have
Thus for every -BF right ideal f and every -BF left ideal g on S. Similarly, we can show that . By Lemma 4, and . Thus and . Therefore by Theorem 9, S is regular. ☐
Theorem 11. For a semigroup S, the following are equivalent.
- (1)
S is regular.
- (2)
and for every -BF generalized bi-ideal on S.
- (3)
and for every -BF bi-ideal on S.
- (4)
and for every -BF quasi-ideal on S.
Proof. . Let
f be a
-BF generalized bi-ideal on
S and
. Because
S is regular, there exists
such that
. Hence we have
Thus . Similarly, we can show that . By Theorem 3, and . Therefore, and .
. Obvious.
. Let
Q be any quasi-ideal of
S. By Theorem 6 and Lemma 3, we have
Thus, . Therefore it follows from Theorem 1 that S is regular. ☐
Theorem 12. For a semigroup S, the following are equivalent.
- (1)
S is regular.
- (2)
and for every -BF generalized bi-ideal and every -BF left ideal on S.
- (3)
and for every -BF bi-ideal and every -BF left ideal on S.
- (4)
and for every -BF quasi-ideal and every -BF left ideal on S.
Proof. . Let
f and
g be any
-BF generalized bi-ideal and any
-BF left ideal on
S, respectively. Let
. Because
S is regular, there exists
such that
. Thus we have
Hence . Similarly, we can show that .
. Obvious.
. Let f and g be any -BF right ideal and -BF left ideal on S, respectively. By Theorem 8, f is a -BF quasi ideal. Thus and . By Lemma 4, and . Thus and . Therefore by Theorem 9, S is regular. ☐