Fuzzy Logical Algebras and Its ApplicationsView this Special Issue
The Interaction between Fuzzy Subsets and Groupoids
We discuss properties of a class of real-valued functions on a set constructed as finite (real) linear combinations of functions denoted as , where is a groupoid (binary system) and is a fuzzy subset of and where . Many properties, for example, being a fuzzy subgroupoid of , can be restated as some properties of . Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of for example.
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. The present authors  introduced the notion of abelian fuzzy subgroupoids on a groupoid and discuss diagonal symmetric relations, convex sets, and fuzzy centers on .
In this paper, we discuss properties of a class of real valued functions on a set constructed as finite (real) linear combinations of functions denoted , where is a groupoid (binary system) and is a fuzzy subset of and where . Many properties, for example, being a fuzzy subgroupoid of , can be restated as some properties of . Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of , for example.
Given a nonempty set , we let denote 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 that notion, Kim and Neggers proved the following theorem.
Theorem 1 (see ). is a semigroup; that is, operation “” as defined in general is associative. Furthermore, the left zero semigroup is the identity for this operation.
Given a set , one may consider (i) a fuzzy subset of , that is, a mapping , and (ii) a groupoid (binary system) , where is a mapping. It is thus possible to consider a fuzzy subgroupoid of a groupoid as a composition , where and interact to satisfy the condition: for all .
One might easily conceive related notions such as “an almost fuzzy subgroupoid of a fuzzy groupoid,” which though here unspecified will be considered in what follows. Since “almost” itself is a typical “fuzzy math” notion, we will consider the intersection between a fuzzy subset and a groupoid as given by the function denoted by , where for all .
Note that intersection need not be a fuzzy subset of .
Example 2. Let be set of all real numbers and let + be the usual addition on . Define a map by . Then . If we let , then . Hence, is not a fuzzy subset of .
Example 3. Let be set of all real numbers. Define a binary operation on by . Then for all . It follows that . Hence, is a fuzzy subset of .
Example 4. Let be set of all real numbers, and let + be the usual addition on and let for all . Define . Then . It follows that for all . Assume that . Then and hence . Hence, is not a fuzzy subgroupoid of .
Given an intersection , note that the smallest value obtainable is when and . If such a pair exists, then . We call such a pair a -orthogonal pair on .
Example 5. Let be the set of all real numbers, and let be the groupoid defined by , for all . Let be the characteristic function of the rationals; that is, if is rational, then and if is irrational, then . Suppose that and are both rational. Then is irrational. In that case, we have . Thus, is an -orthogonal pair on .
Proposition 6. Given , if is a fuzzy subalgebra of , then there is no pair such that is a -orthogonal pair on .
Proof. If is a fuzzy subalgebra of , then for all . Assume that is a -orthogonal pair on . Then . It follows that , which shows that . Since is a fuzzy subalgebra of , we obtain . Hence, , a contradiction.
Given an intersection , a pair is said to be a -parallel pair on if . In Example 5, is a -parallel pair on . In fact, .
Given an element and an interaction , we define by for all . Using these notions, we will consider that a generating set for a vector space of groupoids can be expressed as finite sums , where for , and is a fuzzy subset of and where as usual for real-valued functions. If we let , then for all . Given a fuzzy subset of , we define a map by for all .
Proposition 7. Given , if is a fuzzy subset of , then for all .
Proof. Let . Then, for all , we have
Note that such an in Proposition 7 is independent of any groupoid .
Corollary 8. Given , we have
Proof. It follows immediately from Proposition 7.
Theorem 9. Given and a fuzzy subset , there exist a groupoid and a fuzzy subset such that for all .
Proof. Given and a fuzzy subset , we define a surjective map . Since , . Define a binary operation on as follows: for any , for some . It follows that . This means that .
Corollary 10. Given and a fuzzy subset , there exists a groupoid such that .
Proof. In the proof of Theorem 9, if we let , then by Proposition 7.
4. Composition of Interactions
Given a groupoid , we define a set . We consider the composition of fuzzy subsets as a function whose variables are and ; that is, . For example, ,, or .
In general, . In fact, if we define , then , while . A fuzzy subset is said to be -idempotent if .
Given interactions , we define a new interaction as follows: where and for all .
It is easy to see that if , then ( for any ,.
Example 11. Given a groupoid and , we define . It follows that
Given an interaction , we have already seen that . If , then is a fuzzy subgroupoid of and conversely. Thus, we will be interested in bounds as classification parameters of the . Hence, we usually take and for the “narrowest” fit.
Theorem 12. Given and , if the interactions have the following bounds: , , then the interaction has the bound .
Proof. Given , we let and . If , , then . Since and , we have . It follows that = ; that is, . Thus, = = = .
Note that map in Theorem 12 need not be a fuzzy subset of if .
Theorem 13. Let and . If there exist such that for all , then
Proof. If , then
for all . If , then
Assume that there exist such that for all . Let for some . Then . It follows that . By applying (14), we obtain
Similarly, since , we obtain By applying (14), we obtain Using (12), (17), and (15), we obtain the following: which shows that
Let . A groupoid is said to be -commutative if for all .
Proposition 14. If is -commutative, then is also -commutative for all .
Proof. For all , we have .
Note that Proposition 14 shows that the set of all -commutative groupoids forms a left ideal of the semigroup .
Theorem 15. Given , if are fuzzy subsets of and , then
Proof. Let . Given , we have . It follows that, for all , proving the theorem.
5. Representable Functions by Interactions
A function is said to be representable if can be represented as for some groupoid and a fuzzy subset . We denote it by .
Let be a sequence of real numbers such that, for any and , for some real numbers . Then is called a special sequence of type .
We have the following observations.(1)If and , then a special sequence of type is also a special sequence of type .(2)If is a special sequence of type and is a special sequence of type , then, for real numbers , is a special sequence of type . If and , then and . Such a special sequence of type is called a standard special sequence.(3)If and if is a special sequence of type , then is also a special sequence of type . It follows that if is a special sequence, then, for all , .(4)Notice that if , then , whence, implies that is not a special sequence.(5)Notice that , since . However, there is no such that, for any and , ; that is, since diverges. Hence, the sequence is not special for any pair , with , a pair of real numbers.
Let be a groupoid. For , determine a sequence as follows: .. We consider to be a doubling sequence for relative to .
Theorem 16. If , then the doubling sequence must be a standard special sequence.
Proof. Let for some groupoid and a fuzzy subset . If is a doubling sequence in , then Since , we obtain . Replace by by , the same inequalities are obtained. Hence, we conclude that is a standard special sequence.
Corollary 17. If or , then is not representable as .
Proof. If , then, for any expression , we obtain a value in excess of , and thus it is not possible to obtain a special sequence of type for any finite , since then also.
Example 18. Let and let for . Then is a groupoid. Assume that a mapping is defined by , , then . If we let , then ; then the doubling sequence is not a standard special sequence. By Theorem 16, the mapping cannot be representable by the groupoid and any fuzzy subset of .
Problem 19. Let be the set of all real numbers. Define a function by , the sine function, for all . The problem is to decide whether or not the mapping is representable.
If we assume that for some groupoid and a fuzzy subset , then we may deduce many properties of the representation. Note that and . In fact, . Much other information is available via the use of trigonometric properties. Thus, for example, so that . Thus, for example, implies and implies , whence for any whatsoever. Hence, if , then is a right ideal of .
In the sine function case, doubling sequences may generate sequences with positive and negative values whose average over sections must behave properly to yield at least the possibility of a representation.
Here is another example of a negative solution to the representation problem. Suppose that is a sequence of elements in and that has a sign pattern on of the following type and that when it is positive. If is representable, that is, , and if is a doubling sequence, then for , there are slots of positive value between and . Thus, summing over all these slots, one obtains an upper bound of value (the number of slots). Another associated value is the average of the functional values of that segment from to , and thus at least . This violates the condition that the doubling sequence obtained from must be a standard special sequence. Hence, is not representable.
Problem 20. Given , if it is believed/known that , develop a method of finding a groupoid and a fuzzy subset such that . So far, the best results have been of the negative kind, but interesting nevertheless.
As a closing comment, we observe that the class of representable functions is quite enormous with common properties to be examined. The doubling sequence technique developed above is an example of such a common property which can be applied to the existence problem of representations for function of the type addressed here.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the referee for careful reading and suggestions.
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