The concept of intervalvalued (
α, β
)fuzzy ideals, intervalvalued (
α, β
)fuzzy generalized biideals are introduced in LAsemigroups, using the ideas of belonging and quasicoincidence of an intervalvalued fuzzy point with an intervalvalued fuzzy set and some related properties are investigated. We define the lower and upper parts of intervalvalued fuzzy subsets of an LAsemigroup. Regular LAsemigroups are characterized by the properties of the lower part of intervalvalued (∈, ∈, ˅
q
)fuzzy left ideals, intervalvalued (∈, ∈, ˅
q
)fuzzy quasiideals and intervalvalued (∈, ∈, ˅
q
)fuzzy generalized biideals.
Main Facts
.
AMS Mathematics Subject Classification : 65H05, 65F10.
1. Introduction
The concept of fuzzy sets was first introduced by Zadeh
[16]
and then the fuzzy sets have been used in the reconsideration of classical mathematics. Fuzzy set theory has been shown to be a useful tool to describe situations in which data is imprecise or vague. Fuzzy sets handle such situations by attributing a degree to which a certain object belongs to a set. The fuzzy algebraic structures play a prominent role in mathematics with wide applications in many other branches such as theoratical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, group theory, real analysis, measure theory etc. The notion of fuzzy subgroups was defined by Rosen ed
[12]
. A systematic exposition of fuzzy semigroup was given by Mordeson et. al.
[7]
, and they have find theoratical results on fuzzy semigroups and their use in finite state machine, fuzzy languages and fuzzy coding. Using the notions "belong to" relation (∈) introduced by Pu and Liu
[11]
, in
[8]
Morali proposed the concept of a fuzzy point belonging to a fuzzy subset under natural equivalence on fuzzy subsets. Bhakat and Das introduced the concept of (
α, β
)fuzzy subgroups by using the
belong to
relation (∈) and
quasicoincident with
relation (q) between a fuzzy point and a fuzzy subgroup, and defined an (∈, ∈ ∨
q
)fuzzy subgroup of a group
[1]
. Kazanci and Yamak
[4]
studied generalized types fuzzy biideals of semigroups and defined
fuzzy biideals of semigroups. In
[14]
Shabir et. al. characterized regular semigroups by the properties of (
α, β
)fuzzy ideals, biideals and quasiideals. In
[15]
, Shabir and Yasir characterized regular semigroups by the properties of
fuzzy ideals, generalized biideals and quasiideals of a semigroup.
Intervalvalued fuzzy subsets were proposed about thirty years ago as a natural extension of fuzzy sets by Zadeh
[17]
. Intervalvalued fuzzy subsets have many applications in several areas such as the method of approximate inference. In
[10]
Al Narayanan and T. Manikantan introduced the notions of intervalvalued fuzzy ideals generated by an intervalvalued fuzzy subset in semigroups.
In this paper we introduced the concepts of intervalvalued (∈, ∈ ∨
q
)fuzzy ideals, intervalvalued (∈,∈ ∨
q
)fuzzy biideals and intervalvalue (∈,∈ ∨ ∧
q
)fuzzy quasiideals of an LAsemigroup, where
α, β
are any one of {∈,
q
, ∈ ∨
q
, ∈ ∧
q
} with
α
≠ ∈ ∧
q
, by using
belong to
relation (∈) and
quasicoincidence with
relation (
q
) between intervalvalued fuzzy set and an intervalvalued fuzzy point, and investigated related properties.
2. Preliminaries
We first recall some basic concepts. A groupoid (
S
, ∗) is called a left almost semigroup, abbreviated as an LAsemigroup, if it satisfies left invertive law:
A nonempty subset
A
of
S
is called a sub LAsemigroup of
S
if
AA
⊆
A
and is called left (resp. right) ideal of
S
if
SA
⊆
A
(
AS
⊆
A
). By a twosided ideal or simply an ideal we mean a nonempty subset of
S
which is both a left and a right ideal of
S
. A nonempty subset
B
of
S
is called a generalized biideal of
S
if (
BS
)
B
⊆
B
. A sub LAsemigroup
B
of
S
is called a biideal of
S
if (
BS
)
B
⊆
B
. A nonempty subset
Q
of
S
is called a quasiideal of
S
if
QS
∩
SQ
⊆
Q
. Obviously every onesided ideal of an LAsemigroup
S
is a quasiideal, every quasiideal is a biideal and every biideal is a generalized biideal. An LAsemigroup
S
is called regular if for each element
a
of
S
, there exists an element
x
in
S
such that
a
= (
ax
)
a
. It is well known that for a regular LAsemigroup the concepts of generalized biideal, biideal and quasiideal coincide.
We now review some concepts of fuzzy subsets. A fuzzy subset λ of an LAsemigroup
S
is a mapping λ :
S
→ [0, 1]. Throughout this paper
S
denotes an LAsemigroup.
Definition 1.
A fuzzy subset λ of
S
is called a fuzzy sub LAsemigroup of
S
if for all
x, y
∈
S
Definition 2.
A fuzzy subset λ of
S
is called a fuzzy left (resp. right) ideal of
S
if for all
x, y
∈
S
A fuzzy subset λ of
S
is called a fuzzy twosided ideal or simply fuzzy ideal of
S
if it is both a fuzzy left and a fuzzy right ideal of
S
.
Definition 3.
A fuzzy subsemigroup λ of
S
is called a fuzzy biideal of
S
if for all
x, y
∈
S
We will describe some results of an interval number. By an interval number on [0, 1], say ã is a closed subinterval of [0, 1], that is, ã = [a
^{−}
, a
^{+}
], where 0 ≤ a
^{−}
≤ a
^{+}
≤ 1. Let
D
[0, 1] denote the family of all closed subintervals of
Now we define ≤,=, ∧, ∨ in case of two elements in
D
[0, 1].
Consider two elements ã = [a
^{−}
, a
^{+}
], and
in
D
[0, 1]. Then,
if and only if a
^{−}
≤ b
^{−}
and a
^{+}
≤ b
^{+}
.
if and only if a
^{−}
= b
^{−}
and a
^{+}
= b
^{+}
.
= [min{a
^{−}
, b
^{−}
}, min{a
^{+}
, b
^{+}
}].
= [max{a
^{−}
, b
^{−}
}, max{a
^{+}
, b
^{+}
}].
Let
X
be a set. A mapping
:
X
→
D
[0, 1] is called an intervalvalued fuzzy subset ( briey, an iv fuzzy subset) of
X
, where
for all
x
∈
X
, where λ
^{−}
and λ
^{+}
are fuzzy subsets in
X
such that λ
^{−}
(
x
) ≤ λ
^{+}
(
x
) for all
x
∈
X
.
Let
be two intervalvalued fuzzy subsets of
X
. Define the relation ⊆ between λ and
μ
as follows:
if and only if
for all
x
∈
X
, that is, λ
^{−}
(
x
) ≤
μ
^{−}
(
x
) and λ
^{+}
(
x
) ≤
μ
^{+}
(
x
). An intervalvalued fuzzy subset
in a universe
X
of the form
for all
y
∈
X
, is said to be an intervalvalued fuzzy point with support
x
and interval value
and is denoted by
J. Zhan et al
[19]
gave meaning to the symbol
where
α
∈ {∈,
q
, ∈ ∨
q
, ∈ ∧
q
}. An intervalvalued fuzzy point
is said to
belongs to
(resp.
quasicoincident with
) an intervalvalued fuzzy set
written
if
and in this case
means that
To say that
means that
does not hold.
Let
be two intervalvalued fuzzy subsets of an LAsemigroup
S
. The product
is defined by
3. Main results
Definition 4.
An intervalvalued fuzzy subset
of an LAsemigroup
S
is called an intervalvalued fuzzy sub LAsemigroup of
S
if for all
x, y
∈
S
Definition 5.
An intervalvalued fuzzy subset
of an LAsemigroup
S
is called an intervalvalued fuzzy left (resp. right ) ideal of
S
if for all
x, y
∈
S
Definition 6.
An intervalvalued fuzzy subset
of an LAsemigroup
S
is called an intervalvalued fuzzy generalized biideal of
S
if for all
x, y, z
∈
S
Definition 7.
An intervalvalued fuzzy sub LAsemigroup
of
S
is called an intervalvalued fuzzy biideal of
S
if for all
x, y, z
∈
S
Definition 8.
An intervalvalued fuzzy subset
of an LAsemigroup
S
is called an intervalvalued fuzzy quasiideal of
S
if
where
:
S
→ [1, 1].
Theorem 1
(
[18]
).
Let S be an LA

semigroup with left identity e such that
(
xe
)
S
=
xS for all x
∈
S. Then, the following are equivalent

(1)Sis regular.

(2)R∩L=RLfor every right idealRand left idealLofS.

(3)A= (AS)Afor every quasiidealAofS.
4. INTERVALVALUED (α, β)FUZZY IDEALS OF LASEMIGROUPS
Let
S
be an LAsemigroup and
α, β
denote any one of ∈,
q
, ∈ ∨
q
, or ∈ ∧
q
unless otherwise specified.
Definition 9.
An intervalvalued fuzzy subsete
an LAsemigroup
S
is called an intervalvalued (
α, β
)fuzzy sub LAsemigroup of
S
, where
α
≠ ∈ ∧
q
, if
implies that
Let
be an intervalvalued fuzzy subset of
S
such that,
for all
x
∈
S
. Let
x
∈
S
and
∈
D
[0, 1] such that,
that is,
It follows that
This means that
Therefore, the case
α
=∈ ∧
q
in above definition is omitted.
Definition 10.
An intervalvalued fuzzy subset
of an LAsemigroup
S
is called an intervalvalued (
α, β
)fuzzy left (resp. right ) ideal of
S
, where
α
≠ ∈ ∧
q
, if it satisfies,
and
x
∈
S
implies that
for all
x, y
∈
S
.
An intervalvalued fuzzy subset
of an LAsemigroup
S
is called an intervalvalued (
α, β
)fuzzy ideal of
S
if it is both an intervalvalued (
α, β
)fuzzy left ideal and an intervalvalued (
α, β
)fuzzy right ideal of
S
.
Definition 11.
An intervalvalued fuzzy subset
of an LAsemigroup
S
is called an intervalvalued (
α, β
)fuzzy generalized biideal of
S
, where
α
≠ ∈ ∧
q
, if it satisfies
for all
x, y, z
∈
S
and for alle
where
implies that
Definition 12.
An intervalvalued fuzzy subset
of an LAsemigroup
S
is called an intervalvalued (
α, β
)fuzzy biideal of
S
, where
α
≠ ∈ ∧
q
, if it satisfies, the following two conditions.
(i) For all
x, y
∈
S
and for all
where
and
implies that
(ii) For all
x, y, z
∈
S
and for all
where
and
implies that
Lemma 1.
An interval

valued fuzzy subset
of an LA

semigroup S is an interval

valued fuzzy sub LA

semigroup of S if and only if it satisfies,
For all x, y
∈
S and for all
such that
implies that
Proof
. Suppose that
is an intervalvalued fuzzy sub LAsemigroup of an LAsemigroup
S
. Let
x, y
∈
S
and
where
such that,
Then
Since
is an intervalvalued fuzzy sub LAsemigroup of
S
. So
Hence,
Conversely, assume that
satisfies the given condition. We show that
On contrary, assume that there exist
x, y
∈
S
such that
Let
such that
Then
This contradicts our hypothesis. Thus,
Lemma 2.
An interval

valued fuzzy subset
of an LA

semigroup S is an intervalvalued fuzzy left
(
resp. right
)
ideal of S if and only if it satisfies
for all x, y
∈
S and for all
where
such that
implies that
Proof
. Suppose that
is an intervalvalued fuzzy left ideal of an LAsemigroup
S
. If
Since
is an intervalvalued fuzzy left ideal of
S
, so
Hence,
Conversely, suppose that
satisfies the given condition. We show that
On contrary, assume that there exist
x, y
∈
S
such that
Let
be such that
Then
Which contradicts our hypothesis. Hence,
Remark 1.
The above lemma shows that every fuzzy left (resp. right) ideal of
S
is an (∈, ∈)fuzzy left (resp. right) ideal of
S
.
Lemma 3.
An interval

valued fuzzy subset
of an LA

semigroup S is an interval

valued fuzzy generalized biideal of S if and only if it satisfies,
For all x, y, z
∈
S and for all
where
implies that
Proof
. Suppose that
is an intervalvalued fuzzy generalized biideal of
S
. Let
x, y, z
∈
S
and
where
such that
Then,
Since
is an intervalvalued fuzzy generalized biideal of
S
. So
Hence,
Conversely, suppose that
satisfies the given condition. We show that
Suppose contrary that
Let
be such that
Then,
Which contradicts our supposition. Hence,
Lemma 4.
An interval

valued fuzzy subset
of an LA

semigroup S is an interval

valued fuzzy bi

ideal of S if and only if it satisfy,
(1) for all x, y
∈
S and
where
implies that
(2) for all x, y, z
∈
S and for all