1. Introduction
Fuzzy set theory was firstly introduced by Zadeh [
1] and opened a new path of thinking to mathematicians, physicists, chemists, engineers and many others due to its diverse applications in various fields. Algebraic hyperstructure, which was introduced by the French mathematician Marty [
2], represents a natural extension of classical algebraic structures. Since then, many papers and several books have been written in this area. Nowadays, hyperstructures have a lot of applications in several domains of mathematics and computer science. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. The study of fuzzy hyperstructures is an interesting research area of fuzzy sets. As a generalization of fuzzy sets and interval-valued fuzzy sets, Ghosh and Samanta [
3] introduced the notion of hyperfuzzy sets, and applied it to group theory. Jun et al. [
4] applied the hyperfuzzy sets to
-algebras, and introduced the notion of
k-fuzzy substructures for
. They introduced the concepts of hyperfuzzy substructures of several types by using
k-fuzzy substructures, and investigated their basic properties. They also defined hyperfuzzy subalgebras of type
for
, and discussed relations between the hyperfuzzy substructure/subalgebra and its length. They investigated the properties of hyperfuzzy subalgebras related to upper- and lower-level subsets.
In this paper, we introduce the length-fuzzy subalgebra in -algebras based on hyperfuzzy structures, and investigate several properties.
2. Preliminaries
By a -algebra we mean a system in which the following axioms hold:
- (I)
- (II)
- (III)
- (IV)
for all
If a
-algebra
X satisfies
for all
then we say that
X is a
-algebra. We can define a partial ordering ≤ by
In a
-algebra
X, the following hold:
A non-empty subset S of a -algebra X is called a subalgebra of X if for all .
We refer the reader to the books [
5,
6] for further information regarding
-algebras.
An ordered pair of a nonempty set X and a fuzzy set on X is called a fuzzy structure over X.
Let
X be a nonempty set. A mapping
is called a
hyperfuzzy set over
X (see [
3]), where
is the family of all nonempty subsets of
. An ordered pair
is called a
hyper structure over
X.
Given a hyper structure
over a nonempty set
X, we consider two fuzzy structures
and
over
X in which
Given a nonempty set X, let and denote the collection of all -algebras and all -algebras, respectively. Also, .
Definition 1. [4] For any , a fuzzy structure over is called a fuzzy subalgebra of with type 1 (briefly, 1-fuzzy subalgebra of ) if fuzzy subalgebra of with type 2 (briefly, 2-fuzzy subalgebra of ) if fuzzy subalgebra of with type 3 (briefly, 3-fuzzy subalgebra of ) if fuzzy subalgebra of with type 4 (briefly, 4-fuzzy subalgebra of ) if
It is clear that every 3-fuzzy subalgebra is a 1-fuzzy subalgebra and every 2-fuzzy subalgebra is a 4-fuzzy subalgebra.
Definition 2. [4] For any and , a hyper structure over is called an -hyperfuzzy subalgebra of if is an i-fuzzy subalgebra of and is a j-fuzzy subalgebra of . 3. Length-Fuzzy Subalgebras
In what follows, let unless otherwise specified.
Definition 3. [4] Given a hyper structure over , we definewhich is called the length of . Definition 4. A hyper structure over is called a length 1-fuzzy (resp. 2-fuzzy, 3-fuzzy and 4-fuzzy) subalgebra of if satisfies the condition (
3)
(resp. (
4)–(
6)
). Example 1. Consider a -algebra with the binary operation ∗ which is given in Table 1 (see [6]). Let be a hyper structure over in which is given as follows: Then, the length of is given by Table 2. It is routine to verify that is a length 1-fuzzy subalgebra of .
Proposition 1. If is a length k-fuzzy subalgebra of for , then for all .
Proof. Let
be a length 1-fuzzy subalgebra of
. Then,
for all
. If
is a length 3-fuzzy subalgebra of
, then
for all
. ☐
Proposition 2. If is a length k-fuzzy subalgebra of for , then for all .
Proof. It is similar to the proof of Proposition 1. ☐
Theorem 1. Given a subalgebra A of and , let be a hyper structure over given by If , then is a length 1-fuzzy subalgebra of . Also, if , then is a length 4-fuzzy subalgebra of .
Proof. If
, then
and so
If
, then
and so
Assume that
. Then,
. Let
. If
, then
and so
If
, then
. Suppose that
and
(or,
and
). Then,
Therefore, is a length 1-fuzzy subalgebra of .
Assume that
. Then,
and so
for all
. If
or
, then
. Hence,
is a length 4-fuzzy subalgebra of
. ☐
It is clear that every length 3-fuzzy subalgebra is a length 1-fuzzy subalgebra and every length 2-fuzzy subalgebra is a length 4-fuzzy subalgebra. However, the converse is not true, as seen in the following example.
Example 2. Consider the -algebra in Example 1. Given a subalgebra of , let be a hyper structure over given by Then, is a length 1-fuzzy subalgebra of by Theorem 1. Sinceandwe have . Therefore, is not a length 3-fuzzy subalgebra of .
Give a subalgebra of , let be a hyper structure over given by Then, is a length 4-fuzzy subalgebra of by Theorem 1. However, it is not a length 2-fuzzy subalgebra of , since Theorem 2. A hyper structure over is a length 1-fuzzy subalgebra of if and only if the setis a subalgebra of for all with . Proof. Assume that
is a length 1-fuzzy subalgebra of
and let
be such that
is nonempty. If
, then
and
. It follows from (
3) that
and so
. Hence,
is a subalgebra of
.
Conversely, suppose that
is a subalgebra of
for all
with
. Assume that there exist
such that
If we take
, then
and so
. Thus,
, which is a contradiction. Hence,
for all
. Therefore,
is a length 1-fuzzy subalgebra of
. ☐
Corollary 1. If is a length 3-fuzzy subalgebra of , then the set is a subalgebra of for all with .
The converse of Corollary 1 is not true, as seen in the following example.
Example 3. Consider a -algebra with the binary operation ∗, which is given in Table 3 (see [6]). Let be a hyper structure over in which is given as follows: Then, the length of is given by Table 4.
Hence, we haveand so is a subalgebra of for all with . Since is not a length 3-fuzzy subalgebra of .
Theorem 3. A hyper structure over is a length 4-fuzzy subalgebra of if and only if the setis a subalgebra of for all with . Proof. Suppose that
is a length 4-fuzzy subalgebra of
and
for all
. Let
. Then,
and
, which implies from (
6) that
Hence, , and so is a subalgebra of .
Conversely, assume that
is a subalgebra of
for all
with
. If there exist
such that
then
by taking
. It follows that
, and so
, which is a contradiction. Hence,
for all
, and therefore
is a length 4-fuzzy subalgebra of
. ☐
Corollary 2. If is a length 2-fuzzy subalgebra of , then the set is a subalgebra of for all with .
The converse of Corollary 2 is not true, as seen in the following example.
Example 4. Consider the -algebra in Example 3 and let be a hyper structure over in which is given as follows: Then, the length of is given by Table 5. Hence, we haveand so is a subalgebra of for all with . However, is not a length 2-fuzzy subalgebra of since Theorem 4. Let be a hyper structure over in which satisfies the Condition (
4)
. If is a -hyperfuzzy subalgebra of for , then it is a length 1-fuzzy subalgebra of . Proof. Assume that
is a
-hyperfuzzy subalgebra of
for
in which
satisfies the Condition (
4). Then,
and
for all
, and
is a 1-fuzzy subalgebra of
X. It follows from (
3) that
for all
. Therefore
is a length 1-fuzzy subalgebra of
. ☐
Corollary 3. Let be a hyper structure over in which satisfies the Condition (
4)
. If is a -hyperfuzzy subalgebra of for , then it is a length 1-fuzzy subalgebra of . Corollary 4. For , every -hyperfuzzy subalgebra is a length 1-fuzzy subalgebra.
In general, any length 1-fuzzy subalgebra may not be a -hyperfuzzy subalgebra for , as seen in the following example.
Example 5. Consider a -algebra with the binary operation ∗, which is given in Table 6 (see [6]). Let be a hyper structure over in which is given as follows: The length of is given by Table 7 and it is routine to verify that is a length 1-fuzzy subalgebra of .
However, it is not a -hyperfuzzy subalgebra of X since We provide a condition for a length 1-fuzzy subalgebra to be a -hyperfuzzy subalgebra for .
Theorem 5. If is a length 1-fuzzy subalgebra of in which is constant on X, then it is a -hyperfuzzy subalgebra of for .
Proof. Assume that
is a length 1-fuzzy subalgebra of
in which
is constant on
X. It is clear that
is a
k-fuzzy subalgebra of
for
. Let
for all
. Then,
for all
. Thus,
is a 1-fuzzy subalgebra of
X. Therefore,
is a
-hyperfuzzy subalgebra of
for
. ☐
Corollary 5. If is a length 3-fuzzy subalgebra of in which is constant on X, then it is a -hyperfuzzy subalgebra of for .
Corollary 6. Let be a hyper structure over in which is constant on X. Then, is a -hyperfuzzy subalgebra of for if and only if is a length 1-fuzzy subalgebra of .
Theorem 6. If is a length 1-fuzzy subalgebra of in which is constant on X, then it is a -hyperfuzzy subalgebra of for .
Proof. Let
be a length 1-fuzzy subalgebra of
in which
is constant on
X. Clearly,
is a
k-fuzzy subalgebra of
for
. Let
for all
. Then,
for all
, and so
is a 4-fuzzy subalgebra of
. Therefore,
is a
-hyperfuzzy subalgebra of
for
. ☐
Theorem 7. Let be a hyper structure over in which satisfies the Condition (
5)
. For any , if is a -hyperfuzzy subalgebra of , then it is a length 1-fuzzy subalgebra of . Proof. Let
be a
-hyperfuzzy subalgebra of
for
in which
satisfies the Condition (
5). Then,
and
for all
, and
is a 4-fuzzy subalgebra of
. It follows from (
6) that
for all
. Hence,
is a length 1-fuzzy subalgebra of
. ☐
Corollary 7. Let be a hyper structure over in which satisfies the Condition (
5)
. For any , every -hyperfuzzy subalgebra is a length 1-fuzzy subalgebra. Theorem 8. Let be a hyper structure over in which satisfies the Condition (
5)
. If is a -hyperfuzzy subalgebra of for , then it is a length 4-fuzzy subalgebra of . Proof. Assume that
is a
-hyperfuzzy subalgebra of
for
in which
satisfies the Condition (
5). Then,
and
for all
, and
is a 4-fuzzy subalgebra of
X. Hence,
for all
, and so
is a length 4-fuzzy subalgebra of
. ☐
Corollary 8. Let be a hyper structure over in which satisfies the Condition (
5)
. If is a -hyperfuzzy subalgebra of for , then it is a length 4-fuzzy subalgebra of . Corollary 9. For , every -hyperfuzzy subalgebra is a length 4-fuzzy subalgebra.
Theorem 9. Let be a hyper structure over in which is constant. Then, every length 4-fuzzy subalgebra is a -hyperfuzzy subalgebra for .
Proof. Let
be a length 4-fuzzy subalgebra of
in which
is constant. It is obvious that
is a
k-fuzzy subalgebra of
for
. Let
for all
. Then,
for all
, and hence
is a 4-fuzzy subalgebra of
. Therefore,
is a
-hyperfuzzy subalgebra of
for
. ☐
Corollary 10. Let be a hyper structure over in which is constant. Then, every length 2-fuzzy subalgebra is a -hyperfuzzy subalgebra for .
Theorem 10. Let be a hyper structure over in which satisfies the Condition (
4)
. For every , every -hyperfuzzy subalgebra is a length 4-fuzzy subalgebra. Proof. For every
, let
be a
-hyperfuzzy subalgebra of
in which
satisfies the Condition (
4). Then,
and
for all
. Since
is a 1-fuzzy subalgebra of
, we have
for all
. Thus,
is a length 4-fuzzy subalgebra of
. ☐
Corollary 11. Let be a hyper structure over in which satisfies the Condition (
4)
. For every , every -hyperfuzzy subalgebra is a length 4-fuzzy subalgebra. Theorem 11. Let be a length 4-fuzzy subalgebra of . If is constant on X, then is a -hyperfuzzy subalgebra of for .
Proof. Assume that
is constant on
X in a length 4-fuzzy subalgebra
of
. Obviously,
is a
k-fuzzy subalgebra of
for
. Let
for all
. Then,
for all
, and so
is a 1-fuzzy subalgebra of
. Therefore,
is a
-hyperfuzzy subalgebra of
for
. ☐
Corollary 12. Let be a length 2-fuzzy subalgebra of . If is constant on X, then is a -hyperfuzzy subalgebra of for .