Fuzzy Upper Bounds in Groupoids

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 in , where is the collection of all groupoids. Finally, we define a fuzzy--subset of a groupoid and investigate some of its properties.

1. Introduction

Iséki and Tanaka introduced two classes of abstract algebras: -algebras and -algebras [1, 2].

Neggers and Kim [3] introduced the notion of -algebras which is a useful generalization of -algebras and then investigated several relations between -algebras and -algebras as well as several other relations between -algebras and oriented digraphs. Han et al. [4] defined a variety of special -algebras, such as strong -algebras, (weakly) selective -algebras, and others. The main assertion is that the squared algebra of a -algebra is a -algebra if and only if the root of the squared algebra is a strong -algebra. Recently, Kim et al. [5] explored properties of the set of -units of a -algebra. It was noted that many -algebras are weakly associative, and the existence of nonweakly associative -algebras was demonstrated. Moreover, we discussed the notions of a -integral domain and a left-injectivity in -algebras.

The notion of the semigroup was introduced by Kim and Neggers [6]. Fayoumi [7] introduced the notion of the center in the semigroup of all binary systems on a set and showed that if , then implies . Moreover, she showed that a groupoid if and only if it is a locally zero groupoid. Han et al. [8] introduced the notion of hypergroupoids and showed that is a supersemigroup of the semigroup via the identification . They proved that is a -algebra.

2. Preliminaries

A -algebra [3] is a nonempty set with a constant and a binary operation “” satisfying the following axioms:(D1),(D2),(D3) and imply for all .

A -algebra is a -algebra satisfying the following additional axioms:(D4),(D5) for all .

Example 1 (see [4]). Consider the real numbers , and suppose that has the multiplication Then ; ; yield , and , or ; that is, ; that is, is a -algebra.
Given a nonempty set , we let the collection of all groupoids , where is a map and where is written in the usual product form. Given elements and of , define a product “” on these groupoids as follows: where for any . Using the notion, Kim and Neggers [6] showed the following theorem.

Theorem 2 (see [6]). 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, let denote a groupoid unless otherwise specified.

Definition 3. Let be a groupoid. A map of is called a fuzzy upper bound (or shortly, a ) over if for all . A map of is called a fuzzy lower bound (or shortly, a ) over if for all .

Example 4. Let be any groupoid and let such that is the greatest element of where . Then for all ; that is, is a over .
Given a fub over , we consider the following conditions: for any ,    implies ,   implies ,   implies ,   implies . We have a diagram of implications as follows: Thus we will look for counter examples to , , , and .

Example 5. (a) Let be a groupoid with and let be a constant map; that is, for some for all . Then holds as does . On the other hand, implies fails as does . Thus we have countered examples to and .
(b) Let be a -algebra with the following table: Define a map by It is easy to check that is a over having the condition . But does not have the condition , since and . Hence is false.
(c) Let , where . Define a binary operation “” on by . Thus, if , then . Let , where is the cardinality of . Assume . Since , we have and so . Therefore , . Hence . On the other hand, if , , then , , and , but . This shows that is true, but is false.
(d) Let be a left-zero-semigroup; that is, for any , , and let be a over . Assume that for any . Since is a left-zero-semigroup, we have . Hence condition holds for any fub over .
(e) Let , where is the set of all real numbers and “−” is the usual subtraction on . Also, if , then for all ; that is, is a over . Note that implies and , so that as well; that is, the conditions and hold. Now if we let and , then and , so that . Since for any , we have . Thus fails to hold.
As a refinement of the condition , we give two conditions and , as follows:   implies ,   implies .

Example 6. (a) If is the constant map, that is, for all , then for any groupoid    satisfies the condition . Since , satisfies the condition .
(b) Let , where is the set of all real numbers and “−” is the usual subtraction on , and let . Then for any , and if and only if . Assume that . Then and so . Hence . Therefore ; that is, holds. Let imply . Thus does not hold. Since , given , there is an such that ; that is, does not hold for .
(c) For the groupoid , let , where . Then for all and for all . If we take , then for all . If we assume , then and hence ; that is, the condition holds. If , then holds clearly. If we set and , then and . Take an so that . Then . It follows that ; that is, . Hence, if , then does not hold.

Theorem 7. Suppose that and are s over a groupoid and that they both satisfy the condition . If , where , then is also a over having the condition .

Proof. Let . For any , we have . Hence is a over .
Assume that , but or . If , then and thus  . This shows that , which is a contradiction. Similarly, implies , which is also a contradiction. Hence we obtain , . Similarly, we obtain , . Since has the condition , we have . Hence . Thus has the condition .

Theorem 8. Suppose that and are s over a groupoid and that has the condition with . If , where , then is a over having the condition .

Proof. If , then , . If , then and implies for any . Similarly, if , then for any . Hence is a over
Assume that . Then  . If , then and , which is a contradiction. Hence ; that is, . Similarly, implies . Hence . Since has the condition , we have and so . Similarly, we obtain . Thus has the condition .

Example 9. (a) Suppose and are subsets of such that , , and are nonempty. Let and let , , where , , and is the left-zero-semigroup. It follows that and that , . Hence ; that is, the condition holds.
(b) Let be the set of all real numbers equipped with subtraction and the zero element . Let and . If , then and hence . If where , , then and . Hence . If where , , then and . Hence . Assume . Since , we have , , say . If we let where , , then . It follows that has the value either or . Hence satisfies the condition , but not the condition .

Note that, in Example 9 (b), we know that . This shows that the condition is a very necessary condition in Theorem 8.

4. Either-or in Groupoids

Definition 10. Let be a groupoid and let . Then is said to be either-or if , implies either or for any .

Proposition 11. Suppose that is a over a groupoid and that satisfies the condition . Let . Then is an either-or subset of .

Proof. Let , for any . Then . Since satisfies the condition , we have . Thus, also, if , then , and if , then . Therefore is an either-or subset of .

Note that in Proposition 11.

Definition 12. An either-or subset is said to be with alternative if there are elements such that .

Example 13. If is the groupoid of all integers with respect to the usual addition and if is the set of all even integers, then is an either-or subset of . In fact, let and , where . Then . Thus, is an either-or subset of . On the other hand, consists of all odd integers. Now if is odd for any and if is odd, then is even. If is even, then is odd. Hence fails to be an either-or subset of . The subset of is an either-or subset with alternative. Both and are either-or subsets, but without alternative.

Proposition 14. Let be a nonempty subset of a groupoid . Suppose that is an either-or subset of the groupoid and that is the characteristic function of ; that is, if and otherwise. Then satisfies the condition .

Proof. Let be a “selected” element; that is, . Then implies . Since is an either-or subset of the groupoid , we have either or ; that is, or . In any case and the condition holds.

Let be a groupoid and let . Let be the collection of all either-or subsets of the groupoid , sometimes denoted as , which contain the subset of . Then, since is an either-or subset of , it follows that .

Proposition 15. Let be a groupoid and let . Then is an either-or subset of containing .

Proof. Let such that . If for some either-or subset of which contains , then implies and . If this is not the case, then for any either-or subset of containing , whence . Thus . This shows that is an either-or subset of .

We have the following corollaries.

Corollary 16. Let be a groupoid. If , then .

Corollary 17. Let be a groupoid. Then we have

Example 18. (a) Let be the left-zero-semigroup; that is, for all , and let be a fixed element for which for all , where is a fuzzy subset of . Then implies and satisfies the condition . If is any nonempty subset of , then , implies , so that is an either-or subset without alternative. Hence for any whatever. Of course, if is the right-zero-semigroup, that is, for any , then the same conclusions hold.
(b) Let be the group of all integers with respect to addition. We claim that is an either-or subset of . Given , if , and if and , then and hence . On the other hand, if , then implies . Hence is an either-or subset of the groupoid . Let . If , and if , then ; that is, as well. Thus . Assume that ; that is, for some and such that . Then . Hence .
We claim that for all . Since is the smallest either-or subset of , . Let in . Then . In fact, if is an either-or subset of containing , then either or , since . It follows that for all either-or subset of ; that is, .
We claim that . If is an either-or subset of , either or for any either-or subset of , since . Since , we have for all either-or subset . This proves that . Hence .
Assume that is an either-or subset of containing . Then it is easy to see that . If we let , then . Since and is an either-or subset of , we obtain . Similarly, if we let and , then we obtain . Similarly, we obtain . Thus ; that is, is an either-or subset of containing . It follows that .
(c) Let and let . If , then , and thus is an either-or subset of without alternative, since . If for any , let . Then there exists such that , . If , , then . Hence . If , then and , so that is an either-or subset of with alternative if .

Proposition 19. Let be a groupoid and let such that , implies . Then is an either-or subset of .

Proof. Let , for any . By assumption, we have and so . Hence either or . Thus is an either-or subset of .

We call such a subset of a strong either-or subset of .

Example 20. (a) Let be a leftoid, and let be a singleton. If and , then . Hence, if , then , and is a strong either-or subset of .
(b) Let be the set of all natural numbers and “+” be the usual addition on . If , then is a strong either-or subset of . In fact, if , then and hence .

Proposition 21. Let be a groupoid and let be a strong either-or subset of with . If is the characteristic function of , then is a over . Furthermore, satisfies the condition and .

Proof. Since , we have for all and is a over . Let for any . Since , we have , . Hence , since is a strong either-or subset of . Therefore satisfies the condition and in that case .

Proposition 22. Let be a group all of whose elements have finite order and let be a fuzzy subgroup. Then has the condition and is an either-or subset of .

Proof. Let be a fuzzy subgroup of . Then for any . Thus and . By induction implies , so that implies for all ; that is, is a over .
If , then implies and hence , so that ; the condition holds for , and is an either-or subset of .

5. Fuzzy Upper Bounds in

Theorem 23. Let be a over with the condition and a over with the condition . If , then is a over with the condition .

Proof. Assume that for any . Then implies , so that . Hence satisfies the condition over .

Corollary 24. Let be a over both and with the condition . If , then is an over with the condition .

Proof. Straightforward.

Let be a nonempty set. If and are the collections of groupoids such that satisfies the condition (resp., ), then and that is, is a subsemigroup of .

Theorem 25. Let be a over with the condition , and let be a over with the condition . If , then is a over with the condition .

Proof. Assume that for any . Then . Since satisfies the condition over , we have . Since has the condition over , we obtain , proving the theorem.

6. Fuzzy--Subsets in Groupoids

Given a groupoid , the following are interesting properties in fuzzy subgroupoids :(1) for all ,(2) for all ,(3)if , then for all .

Definition 26. A    over a groupoid is called a fuzzy--subset of the groupoid if it satisfies conditions (1), (2), and (3).

Example 27. (a) Let be the set of all real numbers and let “+” be the usual addition on . Then every fuzzy--subset of is constant. In fact, for all , we have . If we let , then for all ; that is, is a constant function on .
(b) Let be a group with identity 0. Let be a fuzzy subset of with (2); then for all , whence is a constant function. Therefore is a fuzzy--subset of .

Proposition 28. Let be a -algebra. If is the characteristic function of ; that is, and otherwise, then is a fuzzy--subset of and .

Proof. Let . Then , . If , then . Since is a -algebra, we obtain . Hence . This shows that is a fuzzy--subset of and .

Proposition 29. Let be an either-or subset of a groupoid such that and are subsets of . Let be the characteristic function of and . Then is a fuzzy--subset of .

Proof. Since , we have for all . Since , , we obtain , . Assume that . Then , . Since is an either-or set over , either or . It follows that either or . Hence is a fuzzy--subset of .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the referee for valuable suggestions and help.