# On Abelian and Related Fuzzy Subsets of Groupoids

Authors: Seung Joon Shin; Hee Sik Kim; J. Neggers
Journal: The Scientific World Journal
 (2013)

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## Abstract

We introduce the notion of abelian fuzzy subsets on a groupoid, and we observe a variety of consequences which follow. New notions include, among others, diagonal symmetric relations, several types of quasi orders, convex sets, and fuzzy centers, some of whose properties are also investigated.


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## Body

## 1. Introduction

The notion of a fuzzy subset of a set was introduced by Zadeh [1]. His seminal paper in 1965 has opened up new insights and applications in a wide range of scientific fields. Rosenfeld [2] used the notion of a fuzzy subset to set down corner stone papers in several areas of mathematics. Mordeson and Malik [3] published a remarkable book, Fuzzy commutative algebra, presented a fuzzy ideal theory of commutative rings, and applied the results to the solution of fuzzy intersection equations. The book included all the important work that has been done on L-subspaces of a vector space and on L-subfields of a field.Kim and Neggers [4] introduced the notion of Bin(X) and obtained a semigroup structure. Fayoumi [5] introduced the notion of the center ZBin(X) in the semigroup Bin(X) of all binary systems on a set X and showed that a groupoid (X,•)∈ZBin(X) if and only if it is a locally zero groupoid.In this paper, we introduce the notion of abelian fuzzy subgroupoids on a groupoid and discuss diagonal symmetric relations, convex sets, and fuzzy centers onBin(X).


## 2. Preliminaries

Given a nonempty setX, we let Bin(X) denote the collection of all groupoids (X,*), where *:X×X→X is a map and *(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:
(1)(X,*)□(X,•)=(X,□),
where
(2)x□y=(x*y)•(y*x)
for any x,y∈X. Using that notion, Kim and Neggers proved the following theorem.Theorem 1 (see [4]).
(
B
i
n
(
X
)
,
□
) is a semigroup; that is, the operation “□” as defined in general is associative. Furthermore, the left-zero-semigroup is the identity for this operation.LetZBin(X) denote the collection of elements (X,•) of Bin(X) such that (X,*)□(X,•)=(X,•)□(X,*), for all (X,*)∈Bin(X); that is, ZBin(X)={(X,•)∈Bin(X)∣(X,*)□(X,•)=(X,•)□(X,*), for all (X,*)∈Bin(X)}. We call ZBin(X) the center of the semigroup Bin(X).Proposition 2 (see [5]).
If(X,•)∈ZBin(X), then x•x=x for all x∈X.Proposition 3 (see [5]).
Let(X,•)∈ZBin(X). If x≠y in X, then ({x,y},•) is either a left-zero-semigroup or a right-zero-semigroup.


## 3. Abelian Fuzzy Groupoids

Let(X,*)∈Bin(X). A map μ:X→[0,1] is said to be abelian fuzzy if μ(x*y)=μ(y*x) for all x,y∈X.Example 4.
Let(X,*) be a left-zero-semigroup; that is, x*y=x for all x,y∈X. Let μ:X→[0,1] be an abelian fuzzy subset of X. Then, μ(x)=μ(x*y)=μ(y*x)=μ(y) for all x,y∈X. It follows that μ is a constant map.Similarly, every abelian fuzzy subset of a right-zero-semigroup is also a constant function.Proposition 5.
If(X,*)∈ZBin(X), then every abelian fuzzy subset on (X,*) is a constant function.Proof.
Assume thatμ is an abelian fuzzy subset of (X,*). Then, μ(x*y)=μ(y*x) for all x,y∈X. Let (X,*)∈ZBin(X). By Proposition 3, if x≠y in X, then ({x,y},*) is either a left-zero-semigroup or a right-zero-semigroup. It follows that either x*y=x, y*x=y or x*y=y, y*x=x. In either cases, we obtain μ(x)=μ(y) for all x,y∈X, proving that μ is a constant function.Given a groupoid(X,*)∈Bin(X), we denote the set of all abelian fuzzy subgroupoids on (X,*) by A(X,*).Proposition 6.
Let(X,*)∈Bin(X). Then, (X,*) is commutative if and only if A(X,*)=[0,1]X.Proof.
Assume that(X,*) is not commutative; that is, there exist x,y∈X such that x*y  ≠y*x. If we let α:=x*y and let μ:=χα be the characteristic function of α, then μ(x*y)=1, μ(y*x)=0, proving that μ is not an abelian fuzzy subset of (X,*). The converse is straightforward.Given(X,*)∈Bin(X), we define a fuzzy subset μα:X→[0,1] by μα(x):=α for all x∈X, where α∈[0,1]. Denote by C(X,*):={μα|α∈[0,1]}. Then, C(X,*)⊆A(X,*) for all groupoids (X,*) whatsoever. Thus, the extreme of non-commutativity is the situation C(X,*)=A(X,*).Proposition 7.
Letμ∈A(X,*). If μ is one-one, then (X,*) is commutative.Proof.
Ifμ∈A(X,*), then μ(x*y)=μ(y*x) for all x,y∈X. Since μ is one-one, we have x*y=y*x for all x,y∈X.Given(X,*)∈Bin(X), we define a set (X,*)a:={(x,y)∈X×X∣x*y=y*x}. Note that (x,y)∈(X,*)a implies (y,x)∈(X,*)a as well. If we let ν:=χ(X,*)a be the characteristic function of (X,*)a, then ν is an abelian fuzzy subgroupoid on X×X.Proposition 8.
Letμ:X→[0,1] be a fuzzy subset of X. If we define (X,*)aμ:={(x,y)∈X×X∣μ(x*y)=μ(y*x)}, then  (i)
(X,*)a⊆(X,*)aμ,(ii)
ifμ is one-one, then (X,*)a=(X,*)aμ,(iii)
ifμ is constant, then (X,*)aμ=X×X.Proof.
It is straightforward.Theorem 9.
If(X,*)∈Bin(X), then there exists a fuzzy subset μ of X such that (X,*)a=(X,*)aμ.Proof.
Assume that there exists(X,*)∈Bin(X) such that (X,*)a≠(X,*)aμ for any fuzzy subset μ of X. Then, there exists an element (x,y)∈(X,*)aμ∖(X,*)a. It follows that μ(x*y)=μ(y*x), but x*y≠y*x for some x,y∈X. If we let α∶=x*y and let μ∶=χα be the characteristic function of α, then μ(x*y)=1, but μ(y*x)=0, which proves that (x,y)∉(X,*)aμ.Proposition 10.
Let(X,*)∈Bin(X) and let · be the usual product on the set of real numbers. If μ:(X,*)→([0,1],·) is a homomorphism, then μ∈A(X,*) and (X,*)aμ=X×X.Proof.
Ifμ:(X,*)→([0,1],·) is a homomorphism, then μ(x*y)=μ(x)·μ(y)=μ(y)·μ(x)=μ(y*x) for all x,y∈X; that is, μ∈A(X,*).
If(x,y)∈X×X, then μ(x*y)=μ(x)·μ(y)=μ(y*x), proving that (x,y)∈(X,*)aμ.Example 11.
Let(X,*)∈Bin(X). Define a binary operation “⋆” on [0,∞) by x⋆y:=(1/2)(x+y) for all x,y∈[0,∞). If μ:(X,*)→([0,∞),⋆) is a homomorphism, then μ(x*y)=μ(x)⋆μ(y)=(1/2)(x+y)=μ(y*x) for all x,y∈X, which proves that μ∈A(X,*) and (X,*)aμ=X×X.


## 4. Diagonal Symmetric Relations

Given(X,*)∈Bin(X), we denote △(X):={(x,x)∣x∈X}. Then, △(X)⊆(X,*)a. In particular, if (X,*) is a left-zero-semigroup, then △(X)=(X,*)a.LetX∶=R be the set of all real numbers, and let “−” be the usual subtraction on X. Then, (X,-)a={(x,y)∣x-y=y-x}={(x,y)|x=y}=△(X).LetX∶=R be the set of all real numbers, and let (X,*,f) be a leftoid; that is, x*y∶=f(x) for all x,y∈X, where f:R→R is an even function. If we denote △′(X)∶={(x,-x)∣x∈X}, then x*(-x)=f(x)=f(-x)=-x*x for all x∈X. It follows that △′(X)⊆(X,*,f).LetX be a nonempty set, and let S⊆X×X such that △(X)⊆S. S is said to be diagonal symmetric if (x,y)∈S, then (y,x)∈S as well. If (X,*)∈Bin(X), then (X,*)a is diagonal symmetric. Define a map Φa:Bin(X)→P(X×X) by Φa(X,*)∶=(X,*)a.Proposition 12.
IfS is a diagonal symmetric relation on X, then there exists (X,*)∈Bin(X) such that Φa(X,*)=S.Proof.
LetS be a diagonal symmetric relation on X. Define a binary operation “*” on X by
(3)x*y∶={xif(x,y)∉S,zotherwise,
where z is an element of X satisfying z=x*y=y*x. Then, (X,*)a=S. In fact, if (x,y)∈S, then x*y=y*x, and hence, (x,y)∈(X,*)a. Assume that (x,y)∈(X,*)a∖S. Then, since S is symmetric diagonal, we have x*y=x and (y,x)∉S. It follows that x*y=x, y*x=y. Since (x,y)∈(X,*)a, we have x=x*y=y*x=y, proving that (x,y)=(x,x)∈△(X)⊆S, a contradiction. Hence, Φa(X,*)=(X,*)a=S.IfS and T are diagonal symmetric relations on X, then the same is true for S∩T and for S∪T, while △(X) itself is also a diagonal symmetric relation on X. In the latter case, the left-zero-semigroup is among the possible groupoids for which Φa(X,*)=(X,*)a=△(X).Proposition 13.
Let(X,*),(X,•)∈Bin(X). If (X,□)=(X,*)□(X,•), then (X,*)a⊆(X,□)a.Proof.
If(x,y)∈(X,*)a, then x*y=y*x, and hence, x□y=(x*y)•(y*x)=(y*x)□(x*y)=y□x. Hence, (x,y)∈(X,□)a.Thus, if(X,*) and (X,□) are given and the question comes up whether (X,□)=(X,*)□(X,•) for some (X,•), then (X,*)a⊆(X,□)a is a necessary precondition for this to be the case. For example, if △(X)=(X,□)a, then (X,*)a=△(X) as well. Thus, we find that in (Bin(X),□), “the product does not decrease commutativity” as a general principle.Proposition 14.
Let(X,*)∈Bin(X). If (A,*) is a subgroupoid of (X,*), then (X,*)a∩(A×A)=(A,*)a.Proof.
The proof is straightforward.Proposition 15.
Let(X,*),(X,•)∈Bin(X). If (X,□)=(X,*)□(X,•), then A(X,•)⊆A(X,□).Proof.
Ifμ∈A(X,•), then μ(x•y)=μ(y•x) for all x,y∈X. It follows that
(4)μ(x□y)=μ((x*y)•(y*x))=μ((y*x)•(x*y))=μ(y□x).
Hence, μ∈A(X,□).Proposition 16.
Let(X,*)∈Bin(X) satisfy the condition: for any a,b∈X, there exist x,y∈X such that a=x*y, b=y*x. If (X,□)=(X,*)□(X,•), then A(X,•)=A(X,□).Proof.
Ifμ∈A(X,□), then μ(x□y)=μ(y□x) for all x,y∈X. Given a,b∈X, by assumption, we have x,y∈X such that a=x*y, b=y*x. It follows that
(5)μ(a•b)=μ((x*y)•(y*x))=μ(x□y)=μ(y□x)=μ((y*x)•(x*y))=μ(b•a).
Hence, μ∈A(X,•); that is, A(X,□)⊆A(X,•). By Proposition 15, we prove the proposition.Let(X,*),(X,•)∈Bin(X). Define a relation “≤” on Bin(X) by (X,*)≤(X,•) if and only if (X,*)aμ⊆(X,•)aμ for any fuzzy subset μ:X→[0,1].Proposition 17.
The relation≤ is a quasi order on Bin(X).Proof.
Since(X,*)aμ⊆(X,*)aμ for any fuzzy subset μ:X→[0,1], we have (X,*)≤(X,*) for all (X,*)∈Bin(X).
If(X,*)≤(X,•) and (X,•)≤(X,⋆), then (X,*)aμ⊆(X,•)aμ and (X,•)aμ⊆(X,⋆)aμ, and hence, (X,*)aμ⊆(X,⋆)aμ, for any fuzzy subset μ:X→[0,1]. It follows that (X,*)≤(X,⋆).Note that the relation≤ described in Proposition 17 need not be a partial order on Bin(X), since (X,*)aμ=(X,•)aμ for any fuzzy subset μ:X→[0,1] does not imply (X,*)=(X,•).Given(X,*)∈Bin(X), we define
(6)[(X,*)]∶={(X,•)∈Bin(X)∣(X,*)aμ=(X,•)aμ,∀μ:fuzzysubsetofX}.Let〈Bin(X)〉∶={[(X,*)]∣(X,*)∈Bin(X)}. If we define a relation ≤q on 〈Bin(X)〉 by
(7)[(X,*)]≤q[(X,•)]⟺(X,*)≤(X,•).
then it is easy to see that (〈Bin(X),≤q〉) is a partially ordered set. For partially ordered sets, we refer to [6].Proposition 18.
If[(X,*)]=[(X,•)] in 〈Bin(X)〉, then A(X,*)=A(X,•).Proof.
Assume that[(X,*)]=[(X,•)]. Then, (X,*)aμ=(X,•)aμ for any fuzzy subset μ:X→[0,1]. It follows that
(8)μ∈A(X,*)⟺μ(x*y)=μ(y*x)∀x,y∈X⟺(X,*)aμ=X×X=(X,•)aμ⟺μ(x•y)=μ(y•x)∀x,y∈X⟺μ∈A(X,•),
proving that A(X,*)=A(X,•).


## 5. Convex Sets

Proposition 19.
Letμ,ν be fuzzy subsets of (X,*). If λ:=tμ+(1-t)ν, where t∈[0,1], then (X,*)aμ∩(X,*)aν⊆(X,*)aλ.Proof.
If(x,y)∈(X,*)aμ∩(X,*)aν, then μ(x*y)=μ(y*x) and ν(x*y)=ν(y*x) for all x,y∈X. It follows that
(9)λ(x*y)=tμ(x*y)+(1-t)ν(x*y)=tμ(y*x)+(1-t)ν(y*x)=λ(y*x),
proving that (x,y)∈(X,*)aλ.In Proposition19, the equality does not hold in general. See the following example.Example 20.
LetX:=[0,1]. Define a binary operation “*” on X by x*y∶=x for all x,y∈X; that is, (X,*) is a left-zero-semigroup. Define two fuzzy subsets μ,ν on X by μ(x)∶=x,ν(x)∶=1-x for all x∈X. Then, μ and ν are one-one mappings. Let ∈ be a real number such that 0<∈<1/2, and let u∶=(1/2)-∈,v∶=(1/2)+∈. Then, μ(v)-μ(u)=ν(u)-ν(v)≠0. Let λ∶=(1/2)(μ+ν). Then,
(10)λ(u*v)-λ(v*u)=12[μ(u*v)+ν(u*v)]-12[μ(v*u)+ν(v*u)]=12[μ(u*v)-μ(v*u)]+12[ν(u*v)-ν(v*u)]=12[μ(u)-μ(v)]+12[ν(u)-ν(v)]=0,
which shows that (u,v)∈(X,*)aλ.Since(X,*) is a left-zero-semigroup and μ is one-one, we have (X,*)aμ={(x,y)∣μ(x*y)=μ(y*x)}={(x,y)∣μ(x)=μ(y)}=△(X)=(X,*)aν. This shows that (u,v)∉△(X)=(X,*)aμ∩(X,*)aν.Corollary 21.
Letμ,ν be fuzzy subsets of (X,*). If λ∶=tμ+(1-t)ν, where t∈[0,1], and if (X,*)aμ=(X,*)aν, then (X,*)aμ=(X,*)aν⊆(X,*)aλ.Theorem 22.
Letμ,ν be fuzzy subsets of (X,*). If λi∶=tiμ+(1-ti)ν, (i=1,2), where t1,t2∈[0,1] with t1≠t2, then (X,*)aλ1∩(X,*)aλ2⊆(X,*)aμ∩(X,*)aν.Proof.
If(x,y)∈(X,*)aλ1∩(X,*)aλ2, then
(11)λi(x*y)-λi(y*x)=0,
where i=1,2. It follows that
(12)t1[μ(x*y)-μ(y*x)]+(1-t1)×[ν(x*y)-ν(y*x)]=0t2[μ(x*y)-μ(y*x)]+(1-t2)×[ν(x*y)-ν(y*x)]=0.
Since |t11-t1t21-t2|=t1(1-t2)-t2(1-t1)=t1-t2≠0, by Cramer's rule, we have μ(x*y)-μ(y*x)=ν(x*y)-ν(y*x)=0, so that (x,y)∈(X,*)aμ∩(X,*)aν. This proves the theorem.Corollary 23.
Letμ,ν be fuzzy subsets of (X,*), and let λi∶=tiμ+(1-ti)ν, (i=1,2), where t1,t2∈[0,1] with t1≠t2. If (X,*)aμ=(X,*)aν, then (X,*)aλ1∩(X,*)aλ2=(X,*)aμ=(X,*)aν.Proof.
It follows immediately from Corollary21 and Theorem 22.We give a pause to find some examples of fuzzy subsetsμ,ν of X and groupoids (X,*) and (X,•) such that
(13)(I)(X,•)aμ⊆(X,*)aμ,(X,*)μ⊆(X,•)aν
or
(14)(II)(X,*)aν⊆(X,*)aμ,(X,•)μ⊆(X,•)aν
or both (I) and (II) simultaneously.Example 24.
LetX:={a,b,c} with the tables:
(15)*abcaabcbcabcbca•abcaabcbbbccccc
Define two fuzzy subsets μ,ν on X by μ(a)=1,μ(b)=1/2,μ(c)=1/3 and ν(a)=1,ν(b)=ν(c)=1/2. Then, it is easy to see that (X,*)a={(a,a),(b,b),(c,c)}=(X,*)aμ, (X,*)aν=(X,•)aν=(X,•)aν=X×X. Hence, (X,*)aμ⊆(X,•)aμ, (X,•)aν=(X,*)aν, and (X,*)aμ⊆(X,*)aν, (X,*)aν⊆(X,•)aν.


## 6. Fuzzy Center

Let(X,*)∈Bin(X), and let μ be fuzzy subsets of X. Define a set Zμ(X,*) by
(16)Zμ(X,*)∶={x∈X∣μ(x*y)=μ(y*x)}.
We call it a μ-center of (X,*). In Example 24, Zμ(X,*)=∅, Zν(X,*)=X.Proposition 25.
Let(X,*)∈Bin(X). Then, x∈Zμ(X,*) if and only if ({x}×X)∪(X×{x})⊆(X,*)aμ.Proof.
It follows that(17)x∈Zμ(X,*)⟺μ(x*y)=μ(y*x),∀y∈X⟺(x,y),(y,x)∈(X,*)aμ⟺({x}×X)∪(X×{x})⊆(X,*)aμ.Let(X,*)∈Bin(X), and let μ:X→[0,1] be a fuzzy subset of X. Define a set OZμ(X,*) by
(18)OZμ(X,*)∶={x∈Xμ(x*y)=μ(y*x)⟹μ(x)=μ(y)}.
With the notion of μ-center, we obtain the following.Proposition 26.
IfOZμ(X,*)∩Zμ(X,*)≠∅, then μ is a constant function.Proof.
Ifx∈OZμ(X,*)∩Zμ(X,*), then μ(x*y)=μ(y*x) for all y∈X. Since x∈OZμ(X,*), we have μ(x)=μ(y) for all y∈X, proving that μ is a constant function.Proposition 27.
Ifμ is a constant function, then OZμ(X,*)=X=Zμ(X,*).Proof.
μ is a constant function, then μ(x*y)=μ(y*x) for all x,y∈X. It follows that Zμ(X,*)=X.
Assume thatx∉OZμ(X,*). Then, there exists an y∈X such that μ(x*y)=μ(y*x), but μ(x)≠μ(y). Since μ is a constant function, we obtain μ(x)=μ(y)≠μ(x), a contradiction. Hence, OZμ(X,*)=X.



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*Source: 476057-2013-10-29.xml*