#### Abstract

We introduce the concepts of fuzzy Petri nets and marked fuzzy Petri nets along with their appropriate morphisms, which leads to two categories of such Petri nets. Some aspects of the internal structures of these categories are then explored, for example, their reflectiveness/coreflectiveness and symmetrical monoidal closed structure.

#### 1. Introduction

Petri nets are a well-known model of concurrent systems [1]. A number of authors have been led to the study of the various * categories* of Petri nets and their appropriate morphisms [2–4], in the belief that the categorical study provides a tool to compare different models of Petri nets. At the same time, * fuzzy* Petri nets have also been studied by many authors in different ways [5, 6]. In this paper, taking a cue from [3], we introduce another concept of fuzzy Petri nets (although we model our definition on the lines of [2]). This, along with their appropriate morphisms, results in a category of fuzzy Petri nets (and also of * marked* fuzzy Petri nets). The structure of these categories is then studied on the lines similar to those in [2], showing that one of the categories is * symmetric monoidal closed*.

#### 2. Category of Fuzzy Petri Nets

For categorical concepts used here, [7] may be referred. We begin by collecting some basic definitions.

A Petri net is a bipartite graph, consisting of two kinds of nodes, namely, places and transitions, where arcs are either from a place to a transition or vice versa [3]. Graphically, the places are represented by circles, transitions by rectangles, and the arcs by arrows. We define a fuzzy Petri net as follows.

*Definition 1. *A *fuzzy Petri net* (in short, fPn) is 4-tuple , where and are sets, called the set of places and set of transitions, respectively, and , are functions, called the incidence functions.

We may interpret and defined above as follows.

For (resp., ) gives, the grade with which place is related to transition (resp., transition is related to place ). Thus, and describe some kind of * fuzzy arcs* between places and transitions.

*Definition 2. *An fPn*-morphism* from an fPn to fPn is a pair of functions, and such that the following two diagrams:
(1)
“hold”, by which it is meant that for all and .

*Remark 3. *Fuzzy Petri nets and fPn-morphisms form a category, denoted as FPN (the identity morphisms and the composition for this category are obvious to guess).

*Definition 4. *The *product* of two fPn's and is the fPn , where and are given by
for all and .

Proposition 5. *Let , be the two projections and let , be the two injections. Then , are fPn-morphisms.*

*Proof. *To prove that is an fPn-morphism, we need to show that the following two diagrams hold.
(3)
The above diagrams hold because for all and . Similarly, one can prove that is also an fPn-morphism.

Proposition 6. *The product of fPn's and is the categorical product of and in FPN.*

*Proof. *Let be an fPn together with fPn-morphisms . We show that there exists a unique fPn-morphism such that , or equivalently that and . For this purpose, we choose the following and . Let be the map given by
and . We show that the diagrams
(5)
hold, that is, for all , and , that is, , if and , if , for all as well as , if and , if , for all . But as , are fPn-morphisms, the above inequalities hold, whereby the diagrams (5) hold. Thus, is an fPn-morphism. Also, the definitions of and are such that we obviously have and .

To prove the uniqueness of , let there exist another fPn-morphism such that , that is, and . We then have , and , whereby and . Thus, , proving the uniqueness of . Hence the product is a categorical product.

*Definition 7. *The *coproduct* of two fPn's and is the fPn , where , and are given by
for all and .

Similar to Propositions 5 and 6, the following two propositions can also be proved.

Proposition 8. *Let , be the two projections and , be the two injections. Then , are fPn-morphisms.*

Proposition 9. *The coproduct of fPn's and is the categorical coproduct of and in FPN.*

*Definition 10. *Given two fPn's and , we define two new fPn's and as follows (for sets and , shall denote the set of all functions from to ): (1), where are defined as(2), where are defined as

Proposition 11. *The category FPN is a symmetric monoidal closed category (with the constructions in (1) and (2) above, respectively, giving the associated tensor product and hom-object).*

*Proof. *For convenience, the notation is used to denote the fPn . We give a sketch of the proof of closedness of FPN. For this, the two functors are denoted, respectively, as and , which map any FPN-morphism , respectively, to FPN-morphisms, , and , such that for all, and , for all .

It turns out that is a right adjoint to ; the associated unit of the adjunction, , is given for each , by , where and , are such that , and , with and , for all , and for all .

To establish the universality of , we need to produce, for any given and FPN-morphism , a unique FPN-morphism , such that the following diagram commutes.

We only describe , leaving out the verification of the commutativity of the above diagram and the uniqueness of . is given by and , such that and , with , and .

#### 3. Marked Fuzzy Petri Nets

A marked Petri net is a Petri net together with a function, called * marking* defined from the set of places to the set of natural numbers [2]. Marking at a particular place gives the number of * tokens* at that particular place. In this section, we introduce a concept of marked fuzzy Petri nets and thereby a category of marked fuzzy Petri nets.

*Definition 12. *An fPn , together with a function (called a *fuzzy marking* of ), is called a *marked fuzzy Petri net* (in short, an mfPn) and is denoted as .

Here, marking at a particular place may be interpreted as the degree of confidence to which a token can reside at that place.

*Definition 13. *Given an mfPn , a transition is said to *fire at * (or is *enabled* at ), if , for all .

In an mfPn , for fixed induces a function such that for , gives the degree of confidence to which a transition can fire at marking . Thus, a transition at a marking of mfPn can fire if the degree of confidence to which it fires does not exceed the degree of confidence to which a token can reside at places.

After firing at the fuzzy marking , we get a new fuzzy marking of , given by , for all . We say that fires at to * yield * and denote this by . Also, is then said to be * directly reachable* from through the transition .

Similar to [8], the marked fuzzy Petri net models of negation, disjunction, and conjunction of fuzzy proposition, can also be given. We illustrate these by following examples.

*Example 14. *Consider the following graphical representation of an mfPn, which gives the truth value of the negation of a fuzzy proposition. For this, take an mfPn with and . Given a fuzzy proposition, the initial marking is so chosen that is the truth value of the fuzzy proposition and . Also, is so chosen that (so that the transition can fire) and we also take to be . After the firing of the transition , at marking , the marking at is given by , the truth value of the negation of the fuzzy proposition.

*Example 15. *Similar to Example 14, consider the following graphical representation of an mfPn, which gives the disjunction of truth values of two fuzzy propositions. For this, take an mfPn, with and . Given two fuzzy propositions, the initial marking is so chosen that and are the truth values of the given fuzzy propositions and . Also, and are so chosen that and (so that the transition can fire) and we also take to be . After the firing of the transition , at marking , the marking at is given by , the truth value of the disjunction of the fuzzy propositions.

(Analogous to Example 15, one can design mfPn, which gives the conjunction of the truth values of two fuzzy propositions.)

#### 4. Category of Marked Fuzzy Petri Net

In this section, a category of marked fuzzy Petri net, inspired from [2], is introduced.

*Definition 16. *For mfPn's and , a function , is said to be -ok if , for all .

*Remark 17. *MFPN shall denote the category of all mfPn's, with mfPn-morphisms being the fPn-morphisms such that is -ok.

Proposition 18. *Let and be two mfPn's and let be an mfPn-morphism. Then for is enabled at , if is enabled at .*

*Proof. *As is an mfPn-morphism, and , for all . Also, as is enabled at , we have , for all , whence for all. But , whereby, , for all. Thus, is enabled at .

Proposition 19. *Let and be two mfPn's and be an mfPn-morphism. Then for is -ok, if . *

*Proof. *From the above proposition, it is clear that . Also, as is an mfPn-morphism, , and , for all . Consequently, for all , whereby, . Thus, * *for all . Hence is -ok, for .

*Definition 20. *The *product* of two mfPn's and is the mfPn , where is the product of fPn's and and is given by
for all .

Proposition 21. *The FPN-morphisms , given in Proposition 5 are MFPN-morphisms from to .*

*Proof. *Since and , and are -ok and -ok, respectively. Hence , are MFPN-morphisms.

Using Propositions 5 and 21, the next proposition is evident.

Proposition 22. *The product of mfPn's is the categorical product in MFPN.*

*Definition 23. *The *coproduct* of two mfPn's and is the mfPn , where is the coproduct of fPn's and and is given by , for all .

Similar to Propositions 21 and 22, the following two propositions can also be proved.

Proposition 24. *The FPN-morphisms , given in Proposition 8 are MFPN-morphisms from to .*

Proposition 25. *The coproduct of mfPn's is the categorical coproduct in MFPN.*

#### 5. Relationship between FPN and MFPN

There is an obvious functor , given by and .

We omit the easy verification of the following observations.

Proposition 26. *There are full and faithful functors , which, on objects, are respectively, given by and , where 0 and 1, are respectively, the 0-valued and the 1-valued constant functions, and which leave the morphisms unchanged.*

It is easy to prove the following.

Proposition 27. *The functor (resp., ) is left adjoint (resp., right adjoint) to the functor .*

Thus, we have the following.

Proposition 28. *The category FPN is isomorphic to a full reflective subcategory, and also to a full coreflective subcategory, of MFPN.*

#### 6. Conclusion

We note that nothing has been said about the symmetric monoidal closed structure of the category MFPN of marked fuzzy Petri nets. An obvious attempt to make MFPN symmetric monoidal closed would appear to be as follows. Given mfPn's and , the fPn's and (cf. Definition 10) can be made mfPn's by taking their respective fuzzy markings to be and , defined as and , for all , for all. However, for each fixed mfPn , the resulting functors , do not turn out to be adjoint. As an attempt to repair the above situation, may be redefined as , so that the functor does, now, turn out to be right adjoint to the (modified) functor . However, the symmetry of in this modified setup is now lost (this situation is similar to the one noted in [2]). So there may be a different symmetric monoidal closed structure on MFPN which we have not been able to find presently.