Research Article | Open Access
On Abelian and Related Fuzzy Subsets of Groupoids
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.
The notion of a fuzzy subset of a set was introduced by Zadeh . His seminal paper in 1965 has opened up new insights and applications in a wide range of scientific fields. Rosenfeld  used the notion of a fuzzy subset to set down corner stone papers in several areas of mathematics. Mordeson and Malik  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  introduced the notion of and obtained a semigroup structure. Fayoumi  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 .
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 ). 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 ). If , then for all .
Proposition 3 (see ). 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 .
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 .
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, .
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Copyright © 2013 Seung Joon Shin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.