# Fuzzy Upper Bounds in Groupoids

Authors: Sun Shin Ahn; Young Hee Kim; J. Neggers
Journal: The Scientific World Journal
 (2014)

---

## Abstract

The notion of a fuzzy upper bound over a groupoid is introduced and some properties of it are investigated. We also define the notions of an either-or subset of a groupoid and a strong either-or subset of a groupoid and study some of their related properties. In particular, we consider fuzzy upper bounds inBin
(
X
), where Bin
(
X
) is the collection of all groupoids. Finally, we define a fuzzy-d-subset of a groupoid and investigate some of its properties.


---

## Body

## 1. Introduction

Iséki and Tanaka introduced two classes of abstract algebras:BCK-algebras and BCI-algebras [1, 2].Neggers and Kim [3] introduced the notion of d-algebras which is a useful generalization of BCK-algebras and then investigated several relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs. Han et al. [4] defined a variety of special d-algebras, such as strong d-algebras, (weakly) selective d-algebras, and others. The main assertion is that the squared algebra (
X
;
□
,
0
) of a d-algebra is a d-algebra if and only if the root (
X
;
*
,
0
) of the squared algebra (
X
;
□
,
0
) is a strong d-algebra. Recently, Kim et al. [5] explored properties of the set of d-units of a d-algebra. It was noted that many d-algebras are weakly associative, and the existence of nonweakly associative d
/
BCK-algebras was demonstrated. Moreover, we discussed the notions of a d-integral domain and a left-injectivity in d
/
BCK-algebras.The notion of the semigroup(
Bin
(
X
)
,
□
) was introduced by Kim and Neggers [6]. Fayoumi [7] introduced the notion of the center Z
Bin
(
X
) in the semigroup Bin
(
X
) of all binary systems on a set X and showed that if (
X
,
•
)
∈
Z
Bin
(
X
), then x
≠
y implies {
x
,
y
}
=
{
x
•
y
,
y
•
x
}. Moreover, she showed that a groupoid (
X
,
•
)
∈
Z
Bin
(
X
) if and only if it is a locally zero groupoid. Han et al. [8] introduced the notion of hypergroupoids (
H
Bin
(
X
)
,
□
) and showed that (
H
Bin
(
X
)
,
□
) is a supersemigroup of the semigroup (
Bin
(
X
)
,
□
) via the identification x
↔
{
x
}. They proved that (
H
Bin
*
(
X
)
,
⊖
,
[
∅
]
) is a BCK-algebra.


## 2. Preliminaries

Ad
-algebra [3] is a nonempty set X with a constant 0 and a binary operation “*” satisfying the following axioms:(D1)
x
*
x
=
0,
(D2)
0
*
x
=
0,
(D3)
x
*
y
=
0 and y
*
x
=
0 imply x
=
y for all x
,
y
∈
X.ABCK-algebra is a d-algebra X satisfying the following additional axioms:(D4)
(
(
x
*
y
)
*
(
x
*
z
)
)
*
(
z
*
y
)
=
0,
(D5)
(
x
*
(
x
*
y
)
)
*
y
=
0 for all x
,
y
,
z
∈
X.Example 1 (see [4]).
Consider the real numbersR, and suppose that (
R
;
*
,
e
) has the multiplication
(1)
x
*
y
=
(
x
-
y
)
(
x
-
e
)
+
e
.
Then x
*
x
=
e; e
*
x
=
e; x
*
y
=
y
*
x
=
e yield (
x
-
y
)
(
x
-
e
)
=
0, (
y
-
x
)
(
y
-
e
)
=
0 and x
=
y, or x
=
e
=
y; that is, x
=
y; that is, (
R
;
*
,
e
) is a d-algebra.
Given a nonempty setX, we let Bin
(
X
) the collection of all groupoids (
X
,
*
), where *
:
X
×
X
→
X is a map and where *
(
x
,
y
) is written in the usual product form. Given elements (
X
,
*
) and (
X
,
•
) of Bin
(
X
), define a product “□” on these groupoids as follows:
(2)
(
X
,
*
)
□
(
X
,
•
)
=
(
X
,
□
)
,
where
(3)
x
□
y
=
(
x
*
y
)
•
(
y
*
x
)
for any x
,
y
∈
X. Using the notion, Kim and Neggers [6] showed the following theorem.Theorem 2 (see [6]).
(
Bin
(
X
)
,
□
) is a semigroup; that is, the operation “□” as defined in general is associative. Furthermore, the left-zero semigroup is an identity for this operation.


## 3. Four Different Types of Fuzzy Subsets

In what follows, letX denote a groupoid unless otherwise specified.Definition 3.
Let(
X
,
*
,
0
) be a groupoid. A map μ of X is called afuzzy upper bound (or shortly, a fub) over (
X
,
*
,
0
) if μ
(
0
)
≥
μ
(
x
) for all x
∈
X. A map μ of X is called afuzzy lower bound (or shortly, a flb) over (
X
,
*
,
0
) if μ
(
0
)
≤
μ
(
x
) for all x
∈
X.Example 4.
Let(
X
,
*
) be any groupoid and let 0
∈
X such that μ
(
0
) is the greatest element of μ
(
X
)
:
=
{
μ
(
x
)
∣
x
∈
X
} where μ
:
X
→
[
0,1
]. Then μ
(
0
)
≥
μ
(
x
) for all x
∈
X; that is, μ is a fub over (
X
,
*
,
0
).
Given a fubμ
:
X
→
[
0,1
] over (
X
,
*
,
0
), we consider the following conditions: for any x
,
y
∈
X, (
A
)
μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
) implies μ
(
x
)
=
μ
(
y
),
(
B
)
μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
) implies x
=
y,
(
C
)
μ
(
x
*
y
)
=
μ
(
y
*
x
) implies μ
(
x
)
=
μ
(
y
),
(
D
)
μ
(
x
*
y
)
=
μ
(
y
*
x
) implies x
=
y. We have a diagram of implications as follows:
(4)
(
B
)
⟹
(
A
)
⇑
⇑
(
D
)
⟹
(
C
)
.
Thus we will look for counter examples to (
A
)
⇒
(
B
), (
B
)
⇒
(
D
), (
A
)
⇒
(
C
), and (
C
)
⇒
(
D
).Example 5.
(a) Let(
X
,
*
,
0
) be a groupoid with |
X
|
≥
2 and let μ
:
X
→
[
0,1
] be a constant map; that is, μ
(
x
)
=
α for some α
∈
[
0,1
] for all x
∈
X. Then (
A
) holds as does (
C
). On the other hand, x
≠
y implies (
B
) fails as does (
D
). Thus we have countered examples to (
A
)
⇒
(
B
) and (
C
)
⇒
(
D
).
(b) LetX
:
=
{
0,1
,
2,3
} be a d-algebra with the following table:
(5)
*
0
1
2
3
0
0
0
0
0
1
1
0
2
1
2
2
2
0
3
3
3
3
3
0
.
Define a map μ
:
X
→
[
0,1
] by
(6)
μ
(
x
)
:
=
{
1
,
if
x
=
0
,
0
,
if
x
≠
0
.
It is easy to check that μ is a fub over (
X
,
*
,
0
) having the condition (
B
). But μ does not have the condition (
D
), since μ
(
1
*
2
)
=
μ
(
2
)
=
μ
(
2
*
1
) and 1
≠
2. Hence (
B
)
⇒
(
D
) is false.
(c) LetX
=
2
A, where A
=
{
1,2
,
3,4
}. Define a binary operation “*” on X by x
*
y
:
=
{
t
∣
t
∈
x
,
t
∉
y
}
,
∀
x
,
y
∈
X. Thus, if x
⊆
y, then x
*
y
=
∅. Let μ
(
x
)
:
=
1
/
(
1
+
|
x
|
), where |
x
| is the cardinality of x. Assume μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
∅
). Since μ
(
∅
)
=
1, we have |
x
*
y
|
=
0
=
|
y
*
x
| and so x
*
y
=
∅
=
y
*
x. Therefore x
⊆
y, y
⊆
x. Hence x
=
y. On the other hand, if x
=
{
1,2
,
3
}, y
=
{
3,4
,
5
}, then x
*
y
=
{
1,2
}, y
*
x
=
{
4,5
}, and μ
(
x
*
y
)
=
1
/
3
=
μ
(
y
*
x
), but x
≠
y. This shows that (
B
) is true, but (
D
) is false.
(d) Let(
X
,
*
,
0
) be a left-zero-semigroup; that is, for any x
,
y
∈
X, x
*
y
=
x, and let μ
:
X
→
[
0,1
] be a fub over (
X
,
*
,
0
). Assume that μ
(
x
*
y
)
=
μ
(
y
*
x
) for any x
,
y
∈
X. Since (
X
,
*
,
0
) is a left-zero-semigroup, we have μ
(
x
)
=
μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
y
). Hence condition (
C
) holds for any fub μ over (
X
,
*
,
0
).
(e) Let(
X
,
*
,
0
)
=
(
R
,
-
,
0
), where R is the set of all real numbers and “−” is the usual subtraction on R. Also, if μ
(
x
)
=
e
-
x
2, then μ
(
0
)
=
1
≥
μ
(
x
) for all x
∈
R; that is, μ is a fub over (
R
,
-
,
0
). Note that μ
(
x
-
y
)
=
μ
(
y
-
x
)
=
μ
(
0
) implies x
-
y
=
y
-
x
=
0 and x
=
y, so that μ
(
x
)
=
μ
(
y
) as well; that is, the conditions (
B
) and (
A
) hold. Now if we let y
:
=
-
1 and x
:
=
2
+
y, then μ
(
x
-
y
)
=
μ
(
2
+
y
-
y
)
=
μ
(
2
)
=
e
-
4 and μ
(
y
-
x
)
=
μ
(
y
-
2
-
y
)
=
μ
(
-
2
)
=
e
-
4, so that μ
(
x
-
y
)
=
μ
(
y
-
x
). Since e
-
4
(
y
+
1
)
<
1 for any y
≠
-
1, we have μ
(
x
)
=
μ
(
2
+
y
)
=
e
-
(
2
+
y
)
2
=
e
-
y
2
e
-
4
(
y
+
1
)
<
e
-
y
2
=
μ
(
y
). Thus (
C
) fails to hold.
As a refinement of the condition(
A
), we give two conditions (
A
)
* and (
A
)
α
*, 0
≤
α
≤
1 as follows:(
A
)
*
μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
) implies μ
(
x
)
=
μ
(
y
)
=
μ
(
0
),
(
A
)
α
*
μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
) implies μ
(
x
)
=
μ
(
y
)
≥
α.Example 6.
(a) Ifμ
:
X
→
[
0,1
] is the constant map, that is, μ
(
x
)
=
α for all x
∈
X, then for any groupoid (
X
,
*
,
0
)  μ satisfies the condition (
A
)
*. Since (
A
)
*
⇒
(
A
), μ satisfies the condition (
A
).
(b) Let(
X
,
*
,
0
)
:
=
(
R
,
-
,
0
), where R is the set of all real numbers and “−” is the usual subtraction on R, and let μ
(
x
)
:
=
e
-
x
2. Then μ
(
0
)
=
1
≥
μ
(
x
)
>
0 for any x
∈
R, and μ
(
x
)
=
1 if and only if x
=
0. Assume that μ
(
x
-
y
)
=
μ
(
y
-
x
)
=
μ
(
0
). Then e
-
(
x
-
y
)
2
=
e
-
(
y
-
x
)
2
=
1 and so x
-
y
=
0. Hence x
=
y. Therefore μ
(
x
)
=
μ
(
y
); that is, (
A
) holds. Let x
=
y
≠
0 imply μ
(
y
)
=
μ
(
x
)
=
e
-
x
2
<
1
=
μ
(
0
). Thus (
A
)
* does not hold. Since lim
⁡
x
→
∞
μ
(
x
)
=
0, given α
>
0, there is an x such that μ
(
x
)
<
α; that is, (
A
)
α
* does not hold for α
>
0.
(c) For the groupoid(
R
,
-
,
0
), let μ
β
(
x
)
:
=
(
e
-
x
2
+
β
)
/
(
1
+
β
), where β
≥
0. Then μ
β
(
0
)
=
1
≥
μ
β
(
x
) for all x
∈
R and lim
⁡
x
→
∞
μ
β
(
x
)
=
β
/
(
1
+
β
)
≤
μ
β
(
x
) for all x
∈
X. If we take α
:
=
β
/
(
1
+
β
), then μ
β
(
x
)
≥
α for all x
∈
X. If we assume μ
β
(
x
-
y
)
=
μ
β
(
y
-
x
)
=
μ
β
(
0
), then (
x
-
y
)
2
=
(
y
-
x
)
2
=
0 and hence μ
β
(
x
)
=
μ
β
(
y
); that is, the condition (
A
)
α
* holds. If α
1
<
α
2, then (
A
)
α
2
*
⇒
(
A
)
α
1
* holds clearly. If we set β
1
:
=
α
1
/
(
1
-
α
1
) and β
2
:
=
α
2
/
(
1
-
α
2
), then β
1
<
β
2 and lim
⁡
x
→
∞
μ
β
1
(
x
)
=
lim
⁡
x
→
∞
(
(
e
-
x
2
+
β
1
)
/
(
1
+
β
1
)
)
=
(
β
1
)
/
(
1
+
β
1
)
=
α
<
α
2. Take an x
∈
R so that α
1
<
μ
β
1
(
x
)
<
α
2. Then e
-
x
2
+
β
1
<
α
2
(
1
+
β
1
). It follows that e
-
x
2
<
(
β
2
/
(
1
+
β
2
)
)
(
1
+
β
1
)
-
β
1
<
(
β
2
(
1
+
β
1
)
-
β
1
(
1
+
β
2
)
)
/
(
1
+
β
2
)
=
(
β
2
-
β
1
)
(
1
+
β
1
)
/
(
1
+
β
2
); that is, x
2
>
ln
⁡
[
(
1
+
β
2
)
/
(
β
2
-
β
1
)
(
1
+
β
1
)
]. Hence, if |
x
|
>
ln
⁡
[
(
1
+
β
2
)
/
(
β
2
-
β
1
)
(
1
+
β
1
)
], then (
A
)
α
1
*
⇒
(
A
)
α
2
* does not hold.Theorem 7.
Suppose thatμ
1 and μ
2 are f
u
bs over a groupoid (
X
,
*
,
0
) and that they both satisfy the condition (
A
). If μ
:
=
λ
μ
1
+
(
1
-
λ
)
μ
2, where 0
≤
λ
≤
1, then μ is also a f
u
b over (
X
,
*
,
0
) having the condition (
A
).Proof.
Let0
<
λ
<
1. For any x
∈
X, we have μ
(
0
)
=
λ
μ
1
(
0
)
+
(
1
-
λ
)
μ
2
(
0
)
≥
λ
μ
1
(
x
)
+
(
1
-
λ
)
μ
2
(
x
)
=
μ
(
x
). Hence μ is a fub over (
X
,
*
,
0
).
Assume thatμ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
), but μ
1
(
x
*
y
)
<
μ
1
(
0
) or μ
2
(
x
*
y
)
<
μ
2
(
0
). If μ
1
(
x
*
y
)
<
μ
1
(
0
), then λ
μ
1
(
x
*
y
)
<
λ
μ
1
(
0
) and thus (
1
-
λ
)
μ
2
(
x
*
y
)
=
μ
(
x
*
y
)
-
λ
μ
1
(
x
*
y
)
>
μ
(
x
*
y
)
-
λ
μ
1
(
0
)
= μ
(
0
)
-
λ
μ
1
(
0
)
=
(
1
-
λ
)
μ
2
(
0
). This shows that μ
2
(
x
*
y
)
>
μ
2
(
0
), which is a contradiction. Similarly, μ
2
(
x
*
y
)
>
μ
2
(
0
) implies μ
1
(
x
*
y
)
>
μ
1
(
0
), which is also a contradiction. Hence we obtain μ
1
(
x
*
y
)
=
μ
1
(
0
), μ
2
(
x
*
y
)
=
μ
2
(
0
). Similarly, we obtain μ
1
(
y
*
x
)
=
μ
1
(
0
), μ
2
(
y
*
x
)
=
μ
2
(
0
). Since μ
i has the condition (
A
), we have μ
i
(
x
)
=
μ
i
(
y
)
(
i
=
1,2
). Hence μ
(
x
)
=
λ
μ
1
(
x
)
+
(
1
-
λ
)
μ
2
(
x
)
=
λ
μ
1
(
y
)
+
(
1
-
λ
)
μ
2
(
y
)
=
μ
(
y
). Thus μ has the condition (
A
).Theorem 8.
Suppose thatμ
1 and μ
2 are f
u
bs over a groupoid (
X
,
*
,
0
) and that μ
2 has the condition (
A
)
* with μ
1
(
0
)
<
μ
2
(
0
). If μ
:
=
μ
1
∨
μ
2, where (
μ
1
∨
μ
2
)
(
x
)
:
=
max
⁡
{
μ
1
(
x
)
,
μ
2
(
x
)
}, then μ is a f
u
b over (
X
,
*
,
0
) having the condition (
A
)
*.Proof.
Ifμ
=
μ
1
∨
μ
2, then μ
(
0
)
=
μ
1
(
0
)
∨
μ
2
(
0
), μ
(
x
)
=
μ
1
(
x
)
∨
μ
2
(
x
). If μ
1
(
x
)
≥
μ
2
(
x
), then μ
(
x
)
=
μ
1
(
x
) and μ
(
0
)
≥
μ
1
(
0
)
≥
μ
1
(
x
)
=
μ
(
x
) implies μ
(
0
)
≥
μ
(
x
) for any x
∈
X. Similarly, if μ
2
(
x
)
≥
μ
1
(
x
), then μ
(
0
)
≥
μ
(
x
) for any x
∈
X. Hence μ is a fub over (
X
,
*
,
0
)
Assume thatμ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
). Then μ
1
(
x
*
y
)
∨
μ
2
(
x
*
y
)
=
μ
(
x
*
y
)
=
μ
(
0
)
= μ
1
(
0
)
∨
μ
2
(
0
)
=
μ
2
(
0
). If μ
2
(
x
*
y
)
<
μ
2
(
0
), then μ
1
(
x
*
y
)
=
μ
2
(
0
) and μ
1
(
0
)
≥
μ
1
(
x
*
y
)
=
μ
2
(
0
), which is a contradiction. Hence μ
2
(
x
*
y
)
≥
μ
2
(
0
); that is, μ
2
(
x
*
y
)
=
μ
2
(
0
). Similarly, μ
(
y
*
x
)
=
μ
(
0
) implies μ
2
(
0
)
=
μ
2
(
y
*
x
). Hence μ
2
(
x
*
y
)
=
μ
2
(
y
*
x
)
=
μ
2
(
0
). Since μ
2 has the condition (
A
)
*, we have μ
2
(
x
)
=
μ
2
(
y
)
=
μ
2
(
0
) and so μ
(
x
)
=
μ
(
0
). Similarly, we obtain μ
(
y
)
=
μ
(
0
). Thus μ has the condition (
A
)
*.Example 9.
(a) SupposeA and B are subsets of X such that A
∖
B, A
∩
B, and B
∖
A are nonempty. Let 0
∈
A
∩
B and let μ
1
:
=
χ
A, μ
2
:
=
χ
B, where x
∈
A
∖
B, y
∈
B
∖
A, and (
X
,
*
) is the left-zero-semigroup. It follows that μ
(
x
*
y
)
=
μ
1
(
x
*
y
)
∨
μ
2
(
x
*
y
)
=
μ
(
x
)
=
μ
1
(
x
)
∨
μ
2
(
x
)
=
1
∨
0
=
1 and that μ
(
y
*
x
)
=
1
=
μ
(
y
), μ
(
0
)
=
1
=
μ
1
(
1
)
=
μ
2
(
1
). Hence μ
(
x
)
=
μ
(
y
)
=
μ
(
0
); that is, the condition (
A
)
* holds.
(b) Let(
R
,
-
,
0
) be the set of all real numbers equipped with subtraction and the zero element 0. Let μ
1
(
x
)
:
=
1
+
x
-
⌈
x
⌉ and μ
2
(
x
)
:
=
1
+
⌊
x
⌋
-
x. If x
∈
Z, then x
=
⌈
x
⌉
=
⌊
x
⌋ and hence μ
1
(
x
)
=
μ
2
(
x
)
=
1
=
μ
(
0
). If x
=
n
+
α where n
∈
Z, 0
<
α
<
1
/
2, then μ
1
(
x
)
=
α and μ
2
(
x
)
=
1
-
α. Hence μ
(
x
)
=
μ
1
(
x
)
∨
μ
2
(
x
)
=
1
-
α. If x
=
n
+
β where n
∈
Z, 1
/
2
<
β
<
1, then μ
1
(
x
)
=
β and μ
2
(
x
)
=
1
-
β. Hence μ
(
x
)
=
μ
1
(
x
)
∨
μ
2
(
x
)
=
β. Assume μ
(
x
-
y
)
=
μ
(
y
-
x
)
=
μ
(
0
). Since μ
(
0
)
=
1, we have x
-
y, y
-
x
∈
Z, say y
-
x
=
n
∈
Z. If we let x
:
=
m
+
γ where m
∈
Z, 0
<
γ
<
1, then y
=
x
+
n
=
(
n
+
m
)
+
γ. It follows that μ
(
x
)
=
μ
(
y
) has the value either γ or 1
-
γ. Hence μ satisfies the condition (
A
), but not the condition (
A
)
*.Note that, in Example9 (b), we know that μ
1
(
0
)
=
μ
2
(
0
)
=
1. This shows that the condition μ
1
(
0
)
<
μ
2
(
0
) is a very necessary condition in Theorem 8.


## 4. Either-or in Groupoids

Definition 10.
Let(
X
,
*
) be a groupoid and let ∅
≠
U
⊆
X. Then U is said to beeither-or if x
*
y
∈
U, y
*
x
∈
U implies either {
x
,
y
}
⊆
U or {
x
,
y
}
⊆
U
c for any x
,
y
∈
X.Proposition 11.
Suppose thatμ is a f
u
b over a groupoid (
X
,
*
,
0
) and that μ satisfies the condition (
A
). Let Ker
⁡
μ
:
=
{
x
∈
X
∣
μ
(
x
)
=
μ
(
0
)
}. Then Ker
⁡
μ is an either-or subset of X.Proof.
Letx
*
y, y
*
x
∈
Ker
⁡
μ for any x
,
y
∈
X. Then μ
(
x
*
y
)
=
μ
(
0
)
=
μ
(
y
*
x
). Since μ satisfies the condition (
A
), we have μ
(
x
)
=
μ
(
y
). Thus, also, if x
∈
Ker
⁡
μ, then y
∈
Ker
⁡
μ, and if x
∉
Ker
⁡
μ, then y
∉
Ker
⁡
μ. Therefore Ker
⁡
μ is an either-or subset of X.Note that0
∈
Ker
⁡
μ in Proposition 11.Definition 12.
An either-or subsetU is said to bewith alternative if there are elements {
x
,
y
}
⊆
U
,
{
u
,
v
}
⊆
U
c such that {
x
*
y
,
y
*
x
}
⊆
U
,
{
u
*
v
,
v
*
u
}
⊆
U.Example 13.
If(
Z
,
+
) is the groupoid of all integers with respect to the usual addition and if U
:
=
2
Z is the set of all even integers, then U is an either-or subset of X. In fact, let x
+
y
=
y
+
x
=
2
u
∈
U and x
=
2
v, where u
,
v
∈
Z. Then y
=
2
u
-
2
v
=
2
(
u
-
v
)
∈
U. Thus, U is an either-or subset of (
Z
,
+
). On the other hand, U
c consists of all odd integers. Now if x
+
y
=
y
+
x is odd for any x
,
y
∈
Z and if x is odd, then y
=
(
x
+
y
)
-
x is even. If x is even, then y is odd. Hence U
c fails to be an either-or subset of (
Z
,
+
). The subset U
=
2
Z of (
Z
,
+
) is an either-or subset with alternative. Both Z and Z
c
=
∅ are either-or subsets, but without alternative.Proposition 14.
Let0
∈
U be a nonempty subset of a groupoid (
X
,
*
). Suppose that U is an either-or subset of the groupoid (
X
,
*
) and that μ
=
χ
U is the characteristic function of U; that is, μ
(
x
)
=
1 if x
∈
U and μ
(
x
)
=
0 otherwise. Then μ satisfies the condition (
A
).Proof.
Let0
∈
U be a “selected” element; that is, μ
(
0
)
=
1. Then μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
) implies x
*
y
,
y
*
x
∈
U. Since U is an either-or subset of the groupoid (
X
,
*
), we have either {
x
,
y
}
⊆
U or {
x
,
y
}
⊆
U
c; that is, μ
(
x
)
=
μ
(
y
)
=
1 or μ
(
x
)
=
μ
(
y
)
=
0. In any case μ
(
x
)
=
μ
(
y
) and the condition (
A
) holds.Let(
X
,
*
) be a groupoid and let ∅
≠
S
⊆
X. Let EO
(
S
) be the collection of all either-or subsets of the groupoid (
X
,
*
), sometimes denoted as EO
(
S
;
(
X
,
*
)
), which contain the subset S of X. Then, since (
X
,
*
) is an either-or subset of (
X
,
*
), it follows that EOR
(
S
)
:
=
∩
i
{
U
i
∣
U
i
∈
EO
(
S
;
(
X
,
*
)
)
}
⊇
S.Proposition 15.
Let(
X
,
*
) be a groupoid and let ∅
≠
S
⊆
U. Then E
O
R
(
S
;
(
X
,
*
)
) is an either-or subset of (
X
,
*
) containing S.Proof.
Letx
,
y
∈
X such that x
*
y
,
y
*
x
∈
EOR
(
S
;
(
X
,
*
)
). If {
x
,
y
}
⊆
U
c for some either-or subset U of (
X
,
*
) which contains S, then EOR
(
S
;
(
X
,
*
)
)
⊆
U implies U
c
⊆
EOR
(
S
,
(
X
,
*
)
)
c and {
x
,
y
}
⊆
EOR
(
S
,
(
X
,
*
)
)
c. If this is not the case, then {
x
,
y
}
⊈
U
c for any either-or subset U of (
X
,
*
) containing S, whence {
x
,
y
}
⊆
U. Thus {
x
,
y
}
⊆
EOR
(
S
;
(
X
,
*
)
). This shows that EOR
(
S
;
(
X
,
*
)
) is an either-or subset of (
X
,
*
).We have the following corollaries.Corollary 16.
Let(
X
,
*
) be a groupoid. If ∅
≠
S
⊆
T, then E
O
R
(
S
;
(
X
,
*
)
)
⊆
E
O
R
(
T
;
(
X
,
*
)
).Corollary 17.
Let(
X
,
*
) be a groupoid. Then we have
(7)
∅
=
E
O
R
(
∅
;
(
X
,
*
)
)
⊆
E
O
R
(
{
0
}
;
(
X
,
*
)
)
⊆
⋯
⊆
E
O
R
(
X
;
(
X
,
*
)
)
=
X
.Example 18.
(a) Let(
X
,
*
) be the left-zero-semigroup; that is, x
*
y
=
x for all x
,
y
∈
X, and let 0 be a fixed element for which μ
(
0
)
≥
μ
(
x
) for all x
∈
X, where μ
:
X
→
[
0,1
] is a fuzzy subset of X. Then μ
(
x
*
y
)
=
μ
(
x
)
=
μ
(
y
*
x
)
=
μ
(
y
)
=
μ
(
0
) implies μ
(
x
)
=
μ
(
y
)
=
μ
(
0
) and μ satisfies the condition (
A
)
*. If U is any nonempty subset of X, then x
*
y, y
*
x
∈
U implies {
x
,
y
}
⊆
U, so that U is an either-or subset without alternative. Hence EOR
(
U
;
(
X
,
*
)
)
=
U for any ∅
≠
U
⊆
X whatever. Of course, if (
X
,
*
,
0
) is the right-zero-semigroup, that is, x
*
y
=
y for any x
,
y
∈
X, then the same conclusions hold.
(b) Let(
Z
,
+
,
0
) be the group of all integers with respect to addition. We claim that n
Z
=
{
n
x
∣
x
∈
Z
} is an either-or subset of (
Z
,
+
,
0
). Given n
∈
Z, if n
=
0, and if x
+
y
,
y
+
x
∈
{
0
} and x
=
0, then y
=
0 and hence {
x
,
y
}
=
{
0
}
⊆
{
0
}. On the other hand, if x
≠
0, then y
=
-
x implies {
x
,
y
}
=
{
x
,
-
x
}
⊆
{
0
}
c
=
Z
-
{
0
}. Hence {
0
}
=
0
Z is an either-or subset of the groupoid (
Z
,
+
,
0
). Let n
≠
0. If x
+
y
=
y
+
x
=
n
u, and if x
=
n
v, then y
=
(
x
+
y
)
-
x
=
n
u
-
n
v
=
n
(
u
-
v
); that is, y
∈
n
Z as well. Thus {
x
,
y
}
⊆
n
Z. Assume that x
∉
n
Z; that is, x
=
n
u
′
+
α for some u
′
∈
Z and α
∈
Z such that 0
<
α
<
n. Then y
=
n
u
-
x
=
n
(
u
-
u
′
)
-
α
∉
n
Z. Hence {
x
,
y
}
⊆
(
n
Z
)
c.
We claim thatEOR
(
{
n
}
;
(
Z
,
+
)
)
=
n
Z for all n
∈
Z. Since {
0
} is the smallest either-or subset of (
Z
,
+
), EOR
(
{
0
}
;
(
Z
,
+
)
)
=
{
0
}
=
0
Z. Let n
≠
0 in Z. Then 0
∈
EOR
(
{
n
}
;
(
Z
,
+
)
). In fact, if U
i is an either-or subset of (
Z
,
+
) containing n, then either {
n
,
0
}
⊆
U
i or {
n
,
0
}
⊆
U
i
c, since n
+
0
=
0
+
n
=
n
∈
U
i. It follows that 0
∈
U
i for all either-or subset U
i of (
Z
,
+
); that is, 0
∈
EOR
(
{
n
}
;
(
Z
,
+
)
).
We claim that-
n
∈
EOR
(
{
n
}
;
(
Z
,
+
)
). If U
i is an either-or subset of (
Z
,
+
), either {
n
,
-
n
}
⊆
U
i or {
n
,
-
n
}
⊆
U
i
c for any either-or subset U
i of (
Z
,
+
), since n
+
(
-
n
)
=
(
-
n
)
+
n
=
0
∈
U
i. Since n
∈
U
i, we have -
n
∈
U
i for all either-or subset U
i. This proves that -
n
∈
EOR
(
{
n
}
;
(
Z
,
+
)
). Hence {
-
n
,
0
,
n
}
⊆
EOR
(
{
n
}
;
(
Z
,
+
)
).
Assume thatB is an either-or subset of (
Z
,
+
) containing n. Then it is easy to see that 0
,
-
n
∈
B. If we let x
:
=
n
,
y
:
=
-
2
n, then x
+
y
=
y
+
x
=
-
n
∈
B. Since x
=
n
∈
B and B is an either-or subset of (
Z
,
+
), we obtain -
2
n
=
y
∈
B. Similarly, if we let x
=
-
n and y
:
=
2
n, then we obtain 2
n
=
y
∈
B. Similarly, we obtain ±
3
n
,
±
4
n
,
…
,
∈
B. Thus n
Z
⊆
B; that is, B is an either-or subset of (
Z
,
+
) containing n
Z. It follows that n
Z
=
∩
{
U
i
∣
U
i
:
either-or
subset
containing
n
}
=
EOR
(
{
n
}
;
(
Z
,
+
)
).
(c) LetN
:
=
{
1,2
,
3
,
…
} and let a
∈
N. If x
+
y
=
a, then x
<
a, y
<
a and thus {
a
} is an either-or subset of N without alternative, since {
x
,
y
}
⊆
{
a
}
c. If U
a
k
:
=
{
a
,
2
a
,
…
,
k
a
} for any a
∈
N, let x
+
y
=
y
+
x
∈
U
a
k. Then there exists h
∈
N such that x
+
y
=
h
a, h
≤
k. If x
=
j
a, j
<
h, then y
=
h
a
-
x
=
(
h
-
j
)
a
∈
U
a
k. Hence {
x
,
y
}
⊆
U
a
k. If x
∉
U
a
k, then y
∉
U
a
k and {
x
,
y
}
⊆
(
U
a
k
)
c, so that U
a
k is an either-or subset of N with alternative if k
≥
2.Proposition 19.
Let(
X
,
*
) be a groupoid and let U
⊆
X such that x
*
y, y
*
x
∈
U implies x
=
y. Then U is an either-or subset of (
X
,
*
).Proof.
Letx
*
y, y
*
x
∈
U for any x
,
y
∈
X. By assumption, we have x
=
y and so {
x
,
y
}
=
{
x
}. Hence either {
x
}
⊆
U or {
x
}
⊆
U
c. Thus U is an either-or subset of X.We call such a subsetU of X astrong either-or subset of (
X
,
*
).Example 20.
(a) Let(
X
,
*
,
f
) be a leftoid, and let U
=
{
a
} be a singleton. If x
*
y
=
f
(
x
)
∈
U and y
*
x
=
f
(
y
)
∈
U, then f
(
y
)
=
f
(
x
)
=
a. Hence, if |
f
-
1
(
a
)
|
=
1, then x
=
y, and U is a strong either-or subset of (
X
,
*
).
(b) LetN be the set of all natural numbers and “+” be the usual addition on N. If U
:
=
{
2
}, then U is a strong either-or subset of N. In fact, if x
+
y
=
y
+
x
∈
U, then x
+
y
=
y
+
x
=
2 and hence x
=
y
=
1.Proposition 21.
Let(
X
,
*
,
0
) be a groupoid and let U be a strong either-or subset of X with 0
∈
U. If μ
:
=
χ
U is the characteristic function of U, then μ is a f
u
b over (
X
,
*
,
0
). Furthermore, μ satisfies the condition (
B
) and Ker
⁡
μ
=
U.Proof.
Since0
∈
U, we have μ
(
0
)
=
1
≥
μ
(
x
) for all x
∈
X and μ is a fub over (
X
,
*
,
0
). Let μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
) for any x
,
y
∈
X. Since μ
(
0
)
=
1, we have x
*
y, y
*
x
∈
U. Hence x
=
y, since U is a strong either-or subset of X. Therefore μ satisfies the condition (
B
) and in that case Ker
⁡
μ
=
μ
-
1
(
μ
(
0
)
)
=
U.Proposition 22.
Let(
X
,
*
,
e
) be a group all of whose elements have finite order and let μ
:
X
→
[
0,1
] be a fuzzy subgroup. Then μ has the condition (
A
) and Ker
⁡
μ is an either-or subset of X.Proof.
Letμ be a fuzzy subgroup of X. Then μ
(
x
*
y
)
≥
min
⁡
{
μ
(
x
)
,
μ
(
y
)
} for any x
,
y
∈
X. Thus μ
(
x
2
)
=
μ
(
x
*
x
)
≥
μ
(
x
) and μ
(
x
n
)
≥
min
⁡
{
μ
(
x
n
-
1
)
,
μ
(
x
)
}. By induction μ
(
x
n
-
1
)
≥
μ
(
x
) implies μ
(
x
n
)
≥
μ
(
x
), so that μ
(
e
)
=
μ
(
x
n
)
≥
μ
(
x
) implies μ
(
e
)
≥
μ
(
x
) for all x
∈
X; that is, μ is a fub over (
X
,
*
,
e
).
IfKer
⁡
μ
=
μ
-
1
(
μ
(
e
)
)
=
U, then μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
e
) implies μ
(
(
x
*
y
)
*
y
-
1
)
=
μ
(
x
)
≥
min
⁡
{
μ
(
x
*
y
)
,
μ
(
y
-
1
)
) and hence μ
(
x
)
≥
μ
(
y
-
1
)
≥
μ
(
y
), so that μ
(
x
)
=
μ
(
y
); the condition (
A
) holds for μ, and Ker
⁡
μ is an either-or subset of (
X
,
*
,
e
).


## 5. Fuzzy Upper Bounds inBin
(
X
)

Theorem 23.
Letμ be a f
u
b over (
X
,
*
,
0
) with the condition (
A
) and a f
u
b over (
X
,
•
,
0
) with the condition (
A
)
*. If (
X
,
□
)
:
=
(
X
,
*
)
□
(
X
,
•
), then μ is a f
u
b over (
X
,
□
,
0
) with the condition (
A
)
*.Proof.
Assume thatμ
(
x
□
y
)
=
μ
(
y
□
x
)
=
μ
(
0
) for any x
,
y
∈
X. Then μ
(
(
x
*
y
)
•
(
y
*
x
)
)
=
μ
(
(
y
*
x
)
•
(
x
*
y
)
)
=
μ
(
0
) implies μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
), so that μ
(
x
)
=
μ
(
y
)
=
μ
(
0
). Hence μ satisfies the condition (
A
)
* over (
X
,
□
,
0
).Corollary 24.
Letμ be a f
u
b over both (
X
,
*
,
0
) and (
X
,
•
,
0
) with the condition (
A
)
*. If (
X
,
□
)
:
=
(
X
,
*
)
□
(
X
,
•
), then μ is an l
u
b over (
X
,
□
,
0
) with the condition (
A
)
*.Proof.
Straightforward.LetX be a nonempty set. If 〈
μ
;
(
A
)
〉 and 〈
μ
;
(
A
)
*
〉 are the collections of groupoids (
X
,
*
,
0
) such that μ satisfies the condition (
A
) (resp., (
A
)
*), then 〈
μ
;
(
A
)
*
〉
⊆
〈
μ
;
(
A
)
〉 and
(8)
〈
μ
;
(
A
)
〉
□
〈
μ
;
(
A
)
*
〉
⊆
〈
μ
;
(
A
)
〉
,
〈
μ
;
(
A
)
*
〉
□
〈
μ
;
(
A
)
*
〉
⊆
〈
μ
;
(
A
)
*
〉
;
that is, (
〈
μ
;
(
A
)
*
〉
,
□
) is a subsemigroup of (
Bin
(
X
)
,
□
).Theorem 25.
Letμ be a f
u
b over (
X
,
*
,
0
) with the condition (
B
), and let μ be a fub over (
X
,
•
,
0
) with the condition (
A
)
*. If (
X
,
□
)
:
=
(
X
,
*
)
□
(
X
,
•
), then μ is a f
u
b over (
X
,
□
,
0
) with the condition (
B
).Proof.
Assume thatμ
(
x
□
y
)
=
μ
(
y
□
x
)
=
μ
(
0
) for any x
,
y
∈
X. Then μ
(
(
x
*
y
)
•
(
y
*
x
)
)
=
μ
(
(
y
*
x
)
•
(
x
*
y
)
)
=
μ
(
0
). Since μ satisfies the condition (
A
)
* over (
X
,
•
,
0
), we have μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
). Since μ has the condition (
B
) over (
X
,
*
,
0
), we obtain x
=
y, proving the theorem.


## 6. Fuzzy-d-Subsets in Groupoids

Given a groupoid(
X
,
*
,
0
), the following are interesting properties in fuzzy subgroupoids μ
:
X
→
[
0,1
]:(1)
μ
(
x
*
x
)
=
μ
(
0
) for all x
∈
X,
(2)
μ
(
0
*
x
)
=
μ
(
0
) for all x
∈
X,
(3)
ifμ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
), then μ
(
x
)
=
μ
(
y
) for all x
,
y
∈
X.Definition 26.
Afub  μ over a groupoid (
X
,
*
) is called afuzzy-
d
-subset of the groupoid (
X
,
*
,
0
) if it satisfies conditions (1), (2), and (3).Example 27.
(a) LetX
:
=
R be the set of all real numbers and let “+” be the usual addition on R. Then every fuzzy-d-subset μ of (
X
,
+
,
0
) is constant. In fact, for all x
∈
X, we have μ
(
0
)
=
μ
(
x
+
x
)
=
μ
(
2
x
). If we let y
:
=
2
x, then μ
(
y
)
=
μ
(
0
) for all y
∈
X; that is, μ is a constant function on X.
(b) Let(
X
,
*
,
0
) be a group with identity 0. Let μ be a fuzzy subset of X with (2); then μ
(
x
)
=
μ
(
0
*
x
)
=
μ
(
0
) for all x
∈
X, whence μ is a constant function. Therefore μ is a fuzzy-d-subset of (
X
,
*
,
0
).Proposition 28.
Let(
X
,
*
,
0
) be a d-algebra. If μ
=
χ
{
0
} is the characteristic function of {
0
}; that is, μ
(
0
)
=
1 and μ
(
x
)
=
0 otherwise, then μ is a fuzzy-d-subset of X and Ker
⁡
μ
=
{
0
}.Proof.
Letx
,
y
∈
X. Then μ
(
x
*
x
)
=
μ
(
0
), μ
(
0
*
x
)
=
μ
(
0
). If μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
), then x
*
y
=
y
*
x
=
0. Since (
X
,
*
,
0
) is a d-algebra, we obtain x
=
y. Hence μ
(
x
)
=
μ
(
y
). This shows that μ is a fuzzy-d-subset of (
X
,
*
,
0
) and Ker
⁡
μ
=
μ
-
1
(
μ
(
0
)
)
=
μ
-
1
(
1
)
=
{
0
}.Proposition 29.
LetS be an either-or subset of a groupoid (
X
,
*
,
0
) such that Δ
(
X
,
*
)
:
=
{
x
*
x
∣
x
∈
X
} and Δ
0
(
X
,
*
)
:
=
{
0
*
x
∣
x
∈
X
} are subsets of S. Let μ
:
=
χ
S be the characteristic function of S and 0
∈
S. Then μ is a fuzzy-d-subset of (
X
,
*
,
0
).Proof.
Since0
∈
S, we have μ
(
0
)
=
1
≥
μ
(
x
) for all x
∈
X. Since Δ
(
X
,
*
), Δ
0
(
X
,
*
)
⊆
S, we obtain μ
(
x
*
x
)
=
μ
(
0
), μ
(
0
*
x
)
=
μ
(
0
). Assume that μ
(
x
*
y
)
=
μ
(
y
*
x
)
=
μ
(
0
). Then x
*
y, y
*
x
∈
S. Since S is an either-or set over (
X
,
*
,
0
), either {
x
,
y
}
⊆
S or {
x
,
y
}
⊆
S
c. It follows that either μ
(
x
)
=
μ
(
y
)
=
1 or μ
(
x
)
=
μ
(
y
)
=
0. Hence μ is a fuzzy-d-subset of (
X
,
*
,
0
).



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*Source: 697012-2014-05-28.xml*