#### 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.

#### 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 -subspaces of a vector space and on -subfields of a field.

Kim and Neggers [4] introduced the notion of and obtained a semigroup structure. Fayoumi [5] introduced the notion of the center in the semigroup of all binary systems on a set and showed that a groupoid 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 on .

#### 2. Preliminaries

Given a nonempty set , we let denote the collection of all groupoids , where is a map and is written in the usual product form. Given elements and of , define a product “” on these groupoids as follows: where for any . Using that notion, Kim and Neggers proved the following theorem.

Theorem 1 (see [4]). * is a semigroup; that is, the operation “” as defined in general is associative. Furthermore, the left-zero-semigroup is the identity for this operation. *

Let denote the collection of elements of such that , for all ; that is, , for all . We call the * center* of the semigroup .

Proposition 2 (see [5]). *If , then for all . *

Proposition 3 (see [5]). *Let . If in , then is either a left-zero-semigroup or a right-zero-semigroup. *

#### 3. Abelian Fuzzy Groupoids

Let . A map is said to be * abelian fuzzy* if for all .

*Example 4. *Let be a left-zero-semigroup; that is, for all . Let be an abelian fuzzy subset of . Then, for all . It follows that is a constant map.

Similarly, every abelian fuzzy subset of a right-zero-semigroup is also a constant function.

Proposition 5. *If , then every abelian fuzzy subset on is a constant function. *

*Proof. *Assume that is an abelian fuzzy subset of . Then, for all . Let . By Proposition 3, if in , then is either a left-zero-semigroup or a right-zero-semigroup. It follows that either , or , . In either cases, we obtain for all , proving that is a constant function.

Given a groupoid , we denote the set of all abelian fuzzy subgroupoids on by .

Proposition 6. *Let . Then, is commutative if and only if . *

*Proof. *Assume that is not commutative; that is, there exist such that . If we let and let be the characteristic function of , then , , proving that is not an abelian fuzzy subset of . The converse is straightforward.

Given , we define a fuzzy subset by for all , where . Denote by . Then, for all groupoids whatsoever. Thus, the extreme of non-commutativity is the situation .

Proposition 7. *Let . If is one-one, then is commutative. *

*Proof. *If , then for all . Since is one-one, we have for all .

Given , we define a set . Note that implies as well. If we let be the characteristic function of , then is an abelian fuzzy subgroupoid on .

Proposition 8. *Let be a fuzzy subset of . If we define , then *(i), (ii)if is one-one, then , (iii)if is constant, then .

*Proof. *It is straightforward.

Theorem 9. *If , then there exists a fuzzy subset of such that . *

*Proof. *Assume that there exists such that for any fuzzy subset of . Then, there exists an element . It follows that , but for some . If we let and let be the characteristic function of , then , but , which proves that .

Proposition 10. *Let and let be the usual product on the set of real numbers. If is a homomorphism, then and . *

*Proof. *If is a homomorphism, then for all ; that is, .

If , then , proving that .

*Example 11. *Let . Define a binary operation “” on by for all . If is a homomorphism, then for all , which proves that and .

#### 4. Diagonal Symmetric Relations

Given , we denote . Then, . In particular, if is a left-zero-semigroup, then .

Let be the set of all real numbers, and let “−” be the usual subtraction on . Then, .

Let be the set of all real numbers, and let be a leftoid; that is, for all , where is an even function. If we denote , then for all . It follows that .

Let be a nonempty set, and let such that . is said to be * diagonal symmetric* if , then as well. If , then is diagonal symmetric. Define a map by .

Proposition 12. *If is a diagonal symmetric relation on , then there exists such that . *

*Proof. *Let be a diagonal symmetric relation on . Define a binary operation “” on by
where is an element of satisfying . Then, . In fact, if , then , and hence, . Assume that . Then, since is symmetric diagonal, we have and . It follows that , . Since , we have , proving that , a contradiction. Hence, .

If and are diagonal symmetric relations on , then the same is true for and for , while itself is also a diagonal symmetric relation on . In the latter case, the left-zero-semigroup is among the possible groupoids for which .

Proposition 13. *Let . If , then . *

*Proof. *If , then , and hence, . Hence, .

Thus, if and are given and the question comes up whether for some , then is a necessary precondition for this to be the case. For example, if , then as well. Thus, we find that in , “the product does not decrease commutativity” as a general principle.

Proposition 14. *Let . If is a subgroupoid of , then . *

*Proof. *The proof is straightforward.

Proposition 15. *Let . If , then . *

*Proof. *If , then for all . It follows that
Hence, .

Proposition 16. *Let satisfy the condition: for any , there exist such that , . If , then . *

*Proof. *If , then for all . Given , by assumption, we have such that , . It follows that
Hence, ; that is, . By Proposition 15, we prove the proposition.

Let . Define a relation “” on by if and only if for any fuzzy subset .

Proposition 17. *The relation is a quasi order on . *

*Proof. *Since for any fuzzy subset , we have for all .

If and , then and , and hence, , for any fuzzy subset . It follows that .

Note that the relation described in Proposition 17 need not be a partial order on , since for any fuzzy subset does not imply .

Given , we define

Let . If we define a relation on by then it is easy to see that is a partially ordered set. For partially ordered sets, we refer to [6].

Proposition 18. *If in , then . *

*Proof. *Assume that . Then, for any fuzzy subset . It follows that
proving that .

#### 5. Convex Sets

Proposition 19. *Let be fuzzy subsets of . If , where , then . *

*Proof. *If , then and for all . It follows that
proving that .

In Proposition 19, the equality does not hold in general. See the following example.

*Example 20. *Let . Define a binary operation “” on by for all ; that is, is a left-zero-semigroup. Define two fuzzy subsets on by for all . Then, and are one-one mappings. Let be a real number such that , and let . Then, . Let . Then,
which shows that .

Since is a left-zero-semigroup and is one-one, we have . This shows that .

Corollary 21. *Let be fuzzy subsets of . If , where , and if , then . *

Theorem 22. *Let be fuzzy subsets of . If , , where with , then . *

*Proof. *If , then
where . It follows that
Since , by Cramer's rule, we have , so that . This proves the theorem.

Corollary 23. *Let be fuzzy subsets of , and let , , where with . If , then . *

*Proof. *It follows immediately from Corollary 21 and Theorem 22.

We give a pause to find some examples of fuzzy subsets of and groupoids and such that or or both and simultaneously.

*Example 24. *Let with the tables:
Define two fuzzy subsets on by and . Then, it is easy to see that , . Hence, , , and , .

#### 6. Fuzzy Center

Let , and let be fuzzy subsets of . Define a set by
We call it a *-center* of . In Example 24, , .

Proposition 25. *Let . Then, if and only if . *

*Proof. *It follows that

Let , and let be a fuzzy subset of . Define a set by With the notion of -center, we obtain the following.

Proposition 26. *If , then is a constant function. *

*Proof. *If , then for all . Since , we have for all , proving that is a constant function.

Proposition 27. *If is a constant function, then . *

*Proof. * is a constant function, then for all . It follows that .

Assume that . Then, there exists an such that , but . Since is a constant function, we obtain , a contradiction. Hence, .