On Categories of Fuzzy Petri Nets

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 𝑁=(𝑃,𝑇,𝐼⁻,𝐼⁺), where 𝑃 and 𝑇 are sets, called the set of places and set of transitions, respectively, and 𝐼⁻,𝐼⁺∶𝑃×𝑇→[0,1], are functions, called the incidence functions.

We may interpret 𝐼⁻ and 𝐼⁺ defined above as follows.

For (𝑝,𝑡)∈𝑃×𝑇,𝐼⁻(𝑝,𝑡) (resp., 𝐼⁺(𝑝,𝑡)) gives, the grade with which place 𝑝 is related to transition 𝑡 (resp., transition 𝑡 is related to place 𝑝). Thus, 𝐼⁻ and 𝐼⁺ describe some kind of fuzzy arcs between places and transitions.

Definition 2. An fPn-morphism from an fPn (𝑃₁,𝑇₁,𝐼⁻₁,𝐼⁺₁) to fPn (𝑃₂,𝑇₂,𝐼⁻₂,𝐼⁺₂) is a pair (𝑓,𝑔) of functions, 𝑓∶𝑃₂→𝑃₁ and 𝑔∶𝑇₁→𝑇₂ such that the following two diagrams: (1) “hold”, by which it is meant that for all (𝑝,𝑡)∈𝑃₂×𝑇₁,𝐼⁻₁(𝑓(𝑝),𝑡)≥𝐼⁻₂(𝑝,𝑔(𝑡)) and 𝐼⁺₁(𝑓(𝑝),𝑡)≤𝐼⁺₂(𝑝,𝑔(𝑡)).

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 𝑁₁=(𝑃₁,𝑇₁,𝐼⁻₁,𝐼⁺₁) and 𝑁₂=(𝑃₂,𝑇₂,𝐼⁻₂,𝐼⁺₂) is the fPn 𝑁₁×𝑁₂=(𝑃₁∪𝑃₂,𝑇₁×𝑇₂,𝐼⁻,𝐼⁺), where 𝑃₁=𝑃₁×{0},𝑃₂=𝑃₂×{1} and 𝐼⁻,𝐼⁺∶(𝑃₁∪𝑃₂)×(𝑇₁×𝑇₂)→[0,1] are given by 𝐼⁻𝑝,𝑛,𝑡₁,𝑡₂=𝐼⁻₁𝑝,𝑡₁𝐼if𝑛=0,₋₂𝑝,𝑡₂𝐼if𝑛=1,₊𝑝,𝑛,𝑡₁,𝑡₂=𝐼⁺₁𝑝,𝑡₁𝐼if𝑛=0,₊₂𝑝,𝑡₂if𝑛=1, (2) for all (𝑝,𝑛)∈𝑃₁∪𝑃₂ and (𝑡₁,𝑡₂)∈𝑇₁×𝑇₂.

Proposition 5. Let 𝜋_𝑖∶𝑇₁×𝑇₂→𝑇_𝑖,𝑖=1,2, be the two projections and let 𝜌_𝑖∶𝑃_𝑖→𝑃₁∪𝑃₂,𝑖=1,2, be the two injections. Then (𝜌_𝑖,𝜋_𝑖)∶𝑁₁×𝑁₂→𝑁_𝑖,𝑖=1,2, are fPn-morphisms.

Proof. To prove that (𝜌₁,𝜋₁)∶𝑁₁×𝑁₂→𝑁₁ is an fPn-morphism, we need to show that the following two diagrams hold. (3) The above diagrams hold because for all (𝑝₁,(𝑡₁,𝑡₂))∈𝑃₁×(𝑇₁×𝑇₂),𝐼⁻(𝜌₁(𝑝₁),(𝑡₁,𝑡₂))=𝐼⁻((𝑝₁,0),(𝑡₁,𝑡₂))=𝐼⁻₁(𝑝₁,𝑡₁)=𝐼⁻₁(𝑝₁,𝜋₁(𝑡₁,𝑡₂)) and 𝐼⁺(𝜌₁(𝑝₁),(𝑡₁,𝑡₂))=𝐼⁺((𝑝₁,0),(𝑡₁,𝑡₂))=𝐼⁺₁(𝑝₁,𝑡₁)=𝐼⁺₁(𝑝₁,𝜋₁(𝑡₁,𝑡₂)). Similarly, one can prove that (𝜌₂,𝜋₂)∶𝑁₁×𝑁₂→𝑁₂ is also an fPn-morphism.

Proposition 6. The product 𝑁₁×𝑁₂ of fPn's 𝑁₁ and 𝑁₂ is the categorical product of 𝑁₁ and 𝑁₂ in FPN.

Proof. Let 𝑁′=(𝑃′,𝑇′,𝐼′⁻,𝐼′⁺) be an fPn together with fPn-morphisms (𝑓_𝑖,𝑔_𝑖)∶𝑁′→𝑁_𝑖,𝑖=1,2. We show that there exists a unique fPn-morphism (𝑓,𝑔)∶𝑁′→𝑁₁×𝑁₂ such that (𝜌_𝑖,𝜋_𝑖)∘(𝑓,𝑔)=(𝑓_𝑖,𝑔_𝑖),𝑖=1,2, or equivalently that 𝑓∘𝜌_𝑖=𝑓_𝑖 and 𝜋_𝑖∘𝑔=𝑔_𝑖,𝑖=1,2. For this purpose, we choose the following 𝑓 and 𝑔. Let 𝑓∶𝑃₁∪𝑃₂→𝑃′ be the map given by 𝑓=𝑓𝑝,𝑛¹𝑝𝑓,if𝑛=0,₂𝑝,if𝑛=1, (4) and 𝑔=(𝑔₁,𝑔₂). We show that the diagrams (5) hold, that is, for all ((𝑝,𝑛),𝑡)∈(𝑃₁∪𝑃₂)×𝑇^′, 𝐼^′−(𝑓((𝑝,𝑛)),𝑡)≥𝐼⁻(𝑝,𝑔(𝑡)) and 𝐼^′+(𝑓((𝑝,𝑛)),𝑡)≤𝐼⁺(𝑝,𝑔(𝑡)), that is, 𝐼⁻₁(𝑝,𝑔₁(𝑡))≤𝐼^′−(𝑓₁(𝑝),𝑡), if 𝑛=0 and 𝐼⁻₂(𝑝,𝑔₂(𝑡))≤𝐼^′−(𝑓₂(𝑝),𝑡), if 𝑛=1, for all 𝑡∈𝑇′ as well as 𝐼⁺₁(𝑝,𝑔₁(𝑡))≥𝐼^′+(𝑓₁(𝑝),𝑡), if 𝑛=0 and 𝐼⁺₂(𝑝,𝑔₂(𝑡))≥𝐼^′+(𝑓₂(𝑝),𝑡), if 𝑛=1, for all 𝑡∈𝑇′. But as (𝑓_𝑖,𝑔_𝑖)∶𝑁′→𝑁_𝑖,𝑖=1,2, are fPn-morphisms, the above inequalities hold, whereby the diagrams (5) hold. Thus, (𝑓,𝑔)∶𝑁′→𝑁₁×𝑁₂ is an fPn-morphism. Also, the definitions of 𝑓 and 𝑔 are such that we obviously have 𝑓∘𝜌_𝑖=𝑓_𝑖 and 𝜋_𝑖∘𝑔=𝑔_𝑖,𝑖=1,2.
To prove the uniqueness of (𝑓,𝑔), let there exist another fPn-morphism (𝑓^′,𝑔^′) such that (𝜌_𝑖,𝜋_𝑖)∘(𝑓^′,𝑔^′)=(𝑓_𝑖,𝑔_𝑖),𝑖=1,2, that is, 𝑓′∘𝜌_𝑖=𝑓_𝑖 and 𝜋_𝑖∘𝑔′=𝑔_𝑖,𝑖=1,2. We then have 𝑓′∘𝜌₁=𝑓∘𝜌₁,𝑓′∘𝜌₂=𝑓∘𝜌₂,𝜋₁∘𝑔′=𝜋₁∘𝑔, and 𝜋₂∘𝑔′=𝜋₂∘𝑔, whereby 𝑓=𝑓′ and 𝑔=𝑔′. Thus, (𝑓^′,𝑔^′)=(𝑓,𝑔), proving the uniqueness of (𝑓,𝑔). Hence the product is a categorical product.

Definition 7. The coproduct of two fPn's 𝑁₁=(𝑃₁,𝑇₁,𝐼⁻₁,𝐼⁺₁) and 𝑁₂=(𝑃₂,𝑇₂,𝐼⁻₂,𝐼⁺₂) is the fPn 𝑁₁⊕𝑁₂=(𝑃₁×𝑃₂,𝑇₁∪𝑇₂,𝐼⁻,𝐼⁺), where 𝑇₁=𝑇₁×{0},𝑇₂=𝑇₂×{1}, and 𝐼⁻,𝐼⁺∶(𝑃₁×𝑃₂)×(𝑇₁∪𝑇₂)→[0,1] are given by 𝐼⁻𝑝₁,𝑝₂,()=𝐼𝑡,𝑛⁻¹𝑝₁𝐼,𝑡,if𝑛=0,₋₂𝑝₂𝐼,𝑡,if𝑛=1,₊𝑝₁,𝑝₂,()=𝐼𝑡,𝑛⁺¹𝑝₁𝐼,𝑡,if𝑛=0,₊₂𝑝₂,𝑡,if𝑛=1, (6) for all (𝑝₁,𝑝₂)∈𝑃₁×𝑃₂ and (𝑡,𝑛)∈𝑇₁∪𝑇₂.

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

Proposition 8. Let 𝜋_𝑖∶𝑃₁×𝑃₂→𝑃_𝑖,𝑖=1,2, be the two projections and 𝜌_𝑖∶𝑇_𝑖→𝑇₁∪𝑇₂,𝑖=1,2, be the two injections. Then (𝜋_𝑖,𝜌_𝑖)∶𝑁_𝑖→𝑁₁⊕𝑁₂,𝑖=1,2, are fPn-morphisms.

Proposition 9. The coproduct 𝑁₁⊕𝑁₂ of fPn's 𝑁₁ and 𝑁₂ is the categorical coproduct of 𝑁₁ and 𝑁₂ in FPN.

Definition 10. Given two fPn's 𝑁₁=(𝑃₁,𝑇₁,𝐼⁻₁,𝐼⁺₁) and 𝑁₂=(𝑃₂,𝑇₂,𝐼⁻₂,𝐼⁺₂), we define two new fPn's 𝑁₁⊗𝑁₂ and 𝑁^𝑁2₁ as follows (for sets 𝑋 and 𝑌, 𝑌^𝑋 shall denote the set of all functions from 𝑋 to 𝑌): (1)𝑁₁⊗𝑁₂=(𝑃^𝑇2₁×𝑃^𝑇1₂,𝑇₁×𝑇₂,𝐼⁻,𝐼⁺), where 𝐼⁻,𝐼⁺∶(𝑃^𝑇2₁×𝑃^𝑇1₂)×(𝑇₁×𝑇₂)→[0,1] are defined as𝐼⁻𝛼,𝛽,𝑡₁,𝑡₂=∧1,𝐼⁻₁𝛼𝑡₂,𝑡₁+𝐼⁻₂𝛽𝑡₁,𝑡₂𝐼⁺𝛼,𝛽,𝑡₁,𝑡₂=∨0,𝐼⁺₂𝛽𝑡₁,𝑡₂−𝐼⁺₁𝛼𝑡₂,𝑡₁,∀𝛼,𝛽,𝑡₁,𝑡₂∈𝑃^𝑇2₁×𝑃^𝑇1₂×𝑇₁×𝑇₂. (7)(2)𝑁^𝑁2₁=(𝑇₂×𝑃₁,𝑇^𝑇2₁×𝑃^𝑃1₂,𝐼⁻,𝐼⁺), where 𝐼⁻,𝐼⁺∶(𝑇₂×𝑃₁)×(𝑇^𝑇2₁×𝑃^𝑃1₂)→[0,1] are defined as𝐼⁻𝑡,𝑝,𝛼,𝛽=∨0,𝐼⁻₁𝑝,𝛼(𝑡)−𝐼⁻₂𝛽𝑝𝐼,𝑡₊𝑡,𝑝,𝛼,𝛽=∧1,𝐼⁺₁𝑝,𝛼(𝑡)+𝐼⁺₂𝛽𝑝∀,∈𝑇,𝑡𝑡,𝑝𝛼,𝛽₂×𝑃₁×𝑇^𝑇2₁×𝑃^𝑃1₂. (8)  

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 𝑁_𝑖 is used to denote the fPn (𝑃_𝑖,𝑇_𝑖,𝐼⁻_𝑖,𝐼⁺_𝑖). We give a sketch of the proof of closedness of FPN. For this, the two functors −⊗𝑁₀,(−)^𝑁0∶𝐹𝑃𝑁→𝐹𝑃𝑁 are denoted, respectively, as 𝐹 and 𝐺, which map any FPN-morphism (𝑓,𝑔)∶𝑁₁→𝑁₂, respectively, to FPN-morphisms, (𝑓^′,𝑔^′)∶𝐹(𝑁₁)→𝐹(𝑁₂), and (𝑓^′′g^′′)∶𝐺(𝑁₁)→𝐺(𝑁₂), such that 𝑓^′((𝜆,𝜇))=(𝑓∘𝜆,𝜇∘𝑔),𝑔^′((𝑡₂,𝑡₀))=(𝑔(𝑡₂),𝑡₀),for all((𝜆,𝜇),(𝑡₂,𝑡₀))∈(𝑃^𝑇0₂×𝑃^𝑇2₀)×(𝑇₂×𝑇₀), and 𝑓^′′((𝑡₀,𝑝₂))=(𝑡₀,𝑓(𝑝₂)),𝑔^′′((𝜌,𝜈))=(𝑔∘𝜌,𝜈∘𝑓), for all ((𝑡₀,𝑝₂),(𝜌,𝜈))∈(𝑇₀×𝑃₂)×(𝑇^𝑇0₁×𝑃^𝑃1₀).
It turns out that 𝐺 is a right adjoint to 𝐹; the associated unit of the adjunction, 𝜂∶𝐼_𝐹𝑃𝑁→𝐺𝐹, is given for each 𝑁₁∈𝐹𝑃𝑁, by 𝜂_𝑁1=(𝐹_𝑁1,𝐺_𝑁1)∶𝑁₁→𝐺𝐹(𝑁₁), where 𝐹_𝑁1∶𝑇₀×(𝑃^𝑇0₁×𝑃^𝑇1₀)→𝑃₁ and 𝐺_𝑁1∶𝑇₁→(𝑇₁×𝑇₀)^𝑇0×𝑃^{(𝑃𝑇01×𝑃𝑇10)}₀, are such that 𝐹_𝑁1((𝑡₀,(𝜆,𝜇))=𝜆(𝑡₀), and 𝐺_𝑁1(𝑡₁)=(𝛼^𝑡1,𝛽^𝑡1), with 𝛼^𝑡1(𝑡₀)=(𝑡₁,𝑡₀) and 𝛽^𝑡1((𝜆,𝜇))=𝜇(𝑡₁), for all (𝑡₀,(𝜆,𝜇))∈𝑇₀×(𝑃^𝑇0₁×𝑃^𝑇1₀), and for all 𝑡₁∈𝑇₁.
To establish the universality of 𝜂_𝑁1, we need to produce, for any given 𝑁₂∈FPN and FPN-morphism (𝑓,𝑔)∶𝑁₁→𝐺(𝑁₂), a unique FPN-morphism (𝑓^∗,𝑔^∗)∶𝐹(𝑁₁)→𝑁₂, such that the following diagram commutes.

We only describe (𝑓^∗,𝑔^∗), leaving out the verification of the commutativity of the above diagram and the uniqueness of (𝑓^∗,𝑔^∗). (𝑓^∗,𝑔^∗) is given by 𝑓^∗∶𝑃₂→𝑃^𝑇0₁×𝑃^𝑇1₀ and 𝑔^∗∶𝑇₁×𝑇₀→𝑇₂, such that 𝑓^∗(𝑝₂)=(𝑓^𝑝2₁,𝑓^𝑝2₂) and 𝑔^∗(𝑡₁,𝑡₀)=𝑔^𝑡1₁(𝑡₀), with 𝑓^𝑝2₁(𝑡₀)=𝑓(𝑡₀),𝑓^𝑝2₂(𝑡₁)=𝑔^𝑡1₂(𝑝₂), and 𝑔(𝑡₁)=(𝑔^𝑡1₁,𝑔^𝑡1₂).

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 𝑁=(𝑃,𝑇,𝐼⁻,𝐼⁺), together with a function 𝑀∶𝑃→[0,1] (called a fuzzy marking of 𝑁), is called a marked fuzzy Petri net (in short, an mfPn) and is denoted as (𝑁,𝑀).

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 (𝑁,𝑀), a transition 𝑡∈𝑇 is said to fire at 𝑀 (or 𝑡 is enabled at 𝑀), if 𝐼⁻(𝑝,𝑡)≤𝑀(𝑝), for all 𝑝∈𝑃.

In an mfPn (𝑃,𝑇,𝐼⁻,𝐼⁺,𝑀),𝐼⁻∶𝑃×𝑇→[0,1], for fixed 𝑡∈𝑇 induces a function 𝐼⁻_𝑡∶𝑃→[0,1] such that for 𝑝∈𝑃,𝐼⁻_𝑡(𝑝), gives the degree of confidence to which a transition 𝑡∈𝑇 can fire at marking 𝑀. Thus, a transition 𝑡 at a marking 𝑀 of mfPn (𝑁,𝑀) 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 𝑡 firing at the fuzzy marking 𝑀, we get a new fuzzy marking 𝑀_𝑡 of 𝑁, given by 𝑀_𝑡(𝑝)=min{1,𝑀(𝑝)−𝐼⁻(𝑝,𝑡)+𝐼⁺(𝑝,𝑡)}, for all 𝑝∈𝑃. We say that 𝑡∈𝑇 fires at 𝑀 to yield 𝑀_𝑡 and denote this by 𝑀→−^𝑡𝑀_𝑡. Also, 𝑀_𝑡 is then said to be directly reachable from 𝑀 through the transition 𝑡.

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 𝑃={𝑝₁,𝑝₂} and 𝑇={𝑡}. Given a fuzzy proposition, the initial marking 𝑀 is so chosen that 𝑀(𝑝₁) is the truth value of the fuzzy proposition and 𝑀(𝑝₂)=0. Also, 𝐼⁻(𝑝₁,𝑡) is so chosen that 𝐼⁻(𝑝₁,𝑡)≤𝑀(𝑝₁) (so that the transition 𝑡 can fire) and we also take 𝐼⁺(𝑝₂,𝑡) to be 1−𝑀(𝑝₁). After the firing of the transition 𝑡, at marking 𝑀, the marking at 𝑝₂ is given by 𝑀_𝑡(𝑝₂)=min{1,𝑀(𝑝₂)−𝐼⁻(𝑝₂,𝑡)+𝐼⁺(𝑝₂,𝑡)}=1−𝑀(𝑝₁), 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 𝑃={𝑝₁,𝑝₂,𝑝₃} and 𝑇={𝑡}. Given two fuzzy propositions, the initial marking 𝑀 is so chosen that 𝑀(𝑝₁) and 𝑀(𝑝₂) are the truth values of the given fuzzy propositions and 𝑀(𝑝₃)=0. Also, 𝐼⁻(𝑝₁,𝑡) and 𝐼⁻(𝑝₂,𝑡) are so chosen that 𝐼⁻(𝑝₁,𝑡)≤𝑀(𝑝₁) and 𝐼⁻(𝑝₂,𝑡)≤𝑀(𝑝₂) (so that the transition 𝑡 can fire) and we also take 𝐼⁺(𝑝₃,𝑡) to be 𝑀(𝑝₁)∨𝑀(𝑝₂). After the firing of the transition 𝑡, at marking 𝑀, the marking at 𝑝₃ is given by 𝑀_𝑡(𝑝₃)=min{1,𝑀(𝑝₃)−𝐼⁻(𝑝₃,𝑡)+𝐼⁺(𝑝₃,𝑡)}=𝑀(𝑝₁)∨𝑀(𝑝₂), 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 (𝑃₁,𝑇₁,𝐼⁻₁,𝐼⁺₁,𝑀₁) and (𝑃₂,𝑇₂,𝐼⁻₂,𝐼⁺₂,𝑀₂), a function 𝑓∶𝑃₂→𝑃₁, (𝑀₁,𝑀₂) is said to be 𝑓-ok if (𝑀₁∘𝑓)(𝑝)≤𝑀₂(𝑝), for all 𝑝∈𝑃₂.

Remark 17. MFPN shall denote the category of all mfPn's, with mfPn-morphisms (𝑓,𝑔)∶(𝑁₁,𝑀₁)→(𝑁₂,𝑀₂) being the fPn-morphisms (𝑓,𝑔)∶𝑁₁→𝑁₂ such that (𝑀₁,𝑀₂) is 𝑓-ok.

Proposition 18. Let (𝑁₁,𝑀₁)=(𝑃₁,𝑇₁,𝐼⁻₁,𝐼⁺₁,𝑀₁) and (𝑁₂,𝑀₂)=(𝑃₂,𝑇₂,𝐼⁻₂,𝐼⁺₂,𝑀₂) be two mfPn's and let (𝑓,𝑔)∶(𝑁₁,𝑀₁)→(𝑁₂,𝑀₂) be an mfPn-morphism. Then for 𝑡₁∈𝑇₁,𝑔(𝑡₁) is enabled at 𝑀₂, if 𝑡₁ is enabled at 𝑀₁.

Proof. As (𝑓,𝑔)∶𝑁₁→𝑁₂ is an mfPn-morphism, 𝐼⁻₁(𝑓(𝑝₂),𝑡₁)≥𝐼⁻₂(𝑝₂,𝑔(𝑡₁)) and 𝑀₂(𝑝₂)≥𝑀₁(𝑓(𝑝₂)), for all (𝑝₂,𝑡₁)∈𝑃₂×𝑇₁. Also, as 𝑡₁ is enabled at 𝑀₁, we have 𝐼⁻₁(𝑝₁,𝑡₁)≤𝑀₁(𝑝₁), for all 𝑝₁∈𝑃₁, whence 𝐼⁻₁(𝑓(𝑝₂),𝑡₁)≤𝑀₁(𝑓(𝑝₂)),for all𝑝₂∈𝑃₂. But 𝐼⁻₂(𝑝₂,𝑔(𝑡₁))≤𝐼⁻₁(𝑓(𝑝₂),𝑡₁)≤𝑀₁(𝑓(𝑝₂))≤𝑀₂(𝑝₂), whereby, 𝐼⁻₂(𝑝₂,𝑔(𝑡₁))≤𝑀₂(𝑝₂), for all𝑝₂∈𝑃₂. Thus, 𝑔(𝑡₁) is enabled at 𝑀₂.

Proposition 19. Let (𝑁₁,𝑀₁)=(𝑃₁,𝑇₁,𝐼⁻₁,𝐼⁺₁,𝑀₁) and (𝑁₂,𝑀₂)=(𝑃₂,𝑇₂,𝐼⁻₂,𝐼⁺₂,𝑀₂) be two mfPn's and (𝑓,𝑔)∶(𝑁₁,𝑀₁)→(𝑁₂,𝑀₂) be an mfPn-morphism. Then for 𝑡₁∈𝑇₁,((𝑀₁)_𝑡1,(𝑀₂)_𝑔(𝑡1)) is 𝑓-ok, if 𝑀−→₁^𝑡1(𝑀₁)_𝑡1.

Proof. From the above proposition, it is clear that 𝑀−−−−→₂^𝑔(𝑡1)(𝑀₂)_𝑔(𝑡1). Also, as (𝑓,𝑔)∶(𝑁₁,𝑀₁)→(𝑁₂,𝑀₂) is an mfPn-morphism, 𝐼⁻₁(𝑓(𝑝₂,𝑡₁))≥𝐼⁻₂(𝑝₂,𝑔(𝑡₁)),𝐼⁺₁(𝑓(𝑝₂,𝑡₁))≤𝐼⁺₂(𝑝₂,𝑔(𝑡₁)), and 𝑀₁(𝑓(𝑝₂))≤𝑀₂(𝑝₂), for all 𝑝₂∈𝑃₂. Consequently, for all (𝑝₂,𝑡₁)∈𝑃₂×𝑇₁,𝑀₁(𝑓(𝑝₂))−𝐼⁻₁(𝑓(𝑝₂),𝑡₁)+𝐼⁺₁(𝑓(𝑝₂),𝑡₁)≤𝑀₂(𝑝₂)−𝐼⁻₂(𝑝₂,𝑔(𝑡₁))+𝐼⁺₂(𝑝₂,𝑔(𝑡₁)), whereby, min{1,𝑀₁(𝑓(𝑝₂))−𝐼⁻₁(𝑓(𝑝₂),𝑡₁)+𝐼⁺₁(𝑓(𝑝₂),𝑡₁)}≤min{1,(𝑀₂(𝑝₂)−𝐼⁻₂(𝑝₂,𝑔(𝑡₁))+𝐼⁺₂(𝑝₂,𝑔(𝑡₁))}. Thus, (𝑀₁)_𝑡1(𝑓(𝑝₂))≤(𝑀₂)_𝑔(𝑡1)(𝑝₂) for all (𝑝₂,𝑡₁)∈𝑃₂×𝑇₁. Hence ((𝑀₁)_𝑡1,(𝑀₂)_𝑔(𝑡1)) is 𝑓-ok, for 𝑡₁∈𝑇₁.

Definition 20. The product of two mfPn's (𝑁₁,𝑀₁) and (𝑁₂,𝑀₂) is the mfPn (𝑁₁×𝑁₂,𝑀₁⊕𝑀₂), where 𝑁₁×𝑁₂ is the product of fPn's 𝑁₁ and 𝑁₂ and 𝑀₁⊕𝑀₂∶𝑃₁∪𝑃₂→[0,1] is given by 𝑀₁⊕𝑀₂=𝑀𝑝,𝑛¹𝑝𝑀if𝑛=0,₂𝑝if𝑛=1, (9) for all (𝑝,𝑛)∈𝑃₁∪𝑃₂.

Proposition 21. The FPN-morphisms (𝜌_𝑖,𝜋_𝑖)∶𝑁₁×𝑁₂→𝑁_𝑖, given in Proposition 5 are MFPN-morphisms from (𝑁₁×𝑁₂,𝑀₁⊕𝑀₂) to (𝑁_𝑖,𝑀_𝑖),𝑖=1,2.

Proof. Since (𝑀₁⊕𝑀₂)(𝜌₁((𝑝,𝑛)))=(𝑀₁⊕𝑀₂)((𝑝,0))=𝑀₁(𝑝) and (𝑀₁⊕𝑀₂)(𝜌₂((𝑝,𝑛)))=(𝑀₁⊕𝑀₂)((𝑝,1))=𝑀₂(𝑝), (𝑀₁⊕𝑀₂,𝑀₁) and (𝑀₁⊕𝑀₂,𝑀₂) are 𝜌₁-ok and 𝜌₂-ok, respectively. Hence (𝜌_𝑖,𝜋_𝑖),𝑖=1,2, 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 (𝑁₁,𝑀₁) and (𝑁₂,𝑀₂) is the mfPn (𝑁₁⊕𝑁₂,𝑀₁∨𝑀₂), where 𝑁₁⊕𝑁₂ is the coproduct of fPn's 𝑁₁ and 𝑁₂ and 𝑀₁∨𝑀₂∶𝑃₁×𝑃₂→[0,1] is given by (𝑀₁∨𝑀₂)(𝑝₁,𝑝₂)=𝑀₁(𝑝₁)∨𝑀₂(𝑝₂), for all (𝑝₁,𝑝₂)∈𝑃₁×𝑃₂.

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

Proposition 24. The FPN-morphisms (𝜋_𝑖,𝜌_𝑖)∶𝑁_𝑖→𝑁₁⊕𝑁₂,𝑖=1,2, given in Proposition 8 are MFPN-morphisms from (𝑁_𝑖,𝑀_𝑖) to (𝑁₁⊕𝑁₂,𝑀₁∨𝑀₂).

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

5. Relationship between FPN and MFPN

There is an obvious functor 𝑘₁∶𝑀𝐹𝑃𝑁→𝐹𝑃𝑁, given by (𝑁,𝑀)↦𝑁 and (𝑓,𝑔)↦(𝑓,𝑔).

We omit the easy verification of the following observations.

Proposition 26. There are full and faithful functors 𝑘₂,𝑘₃∶𝐅𝐏𝐍→𝐌𝐅𝐏𝐍, which, on objects, are respectively, given by 𝑁↦(𝑁,𝟎) and 𝑁↦(𝑁,𝟏), 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 𝑘₂ (resp., 𝑘₃) is left adjoint (resp., right adjoint) to the functor 𝑘₁.

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 (𝑁₀,𝑀₀) and (𝑁₁,𝑀₁), the fPn's 𝑁₁⊗𝑁₀ and 𝑁^𝑁0₁ (cf. Definition 10) can be made mfPn's by taking their respective fuzzy markings to be 𝑀₁⊗𝑀₀∶𝑃^𝑇0₁×𝑃^𝑇1₀→[0,1] and 𝑀^𝑀0₁∶𝑇₀×𝑃₁→[0,1], defined as 𝑀₁⊗𝑀₀((𝜆,𝜇))=∨{∨_𝑇0{𝑀₁(𝜆(𝑡₀))},∨_𝑇1{𝑀₀(𝜇(𝑡₁))}} and 𝑀^𝑀0₁((𝑡₀,𝑝₁))=𝑀₁(𝑝₁), for all (𝜆,𝜇)∈𝑃^𝑇0₁×𝑃^𝑇1₀, for all(𝑡₀,𝑝₁)∈𝑇₀×𝑃₁. However, for each fixed mfPn (𝑁₀,𝑀₀), the resulting functors −⊗(𝑁₀,𝑀₀), (−)^{(𝑁0,𝑀0)}∶𝑀𝐹𝑃𝑁→𝑀𝐹𝑃𝑁 do not turn out to be adjoint. As an attempt to repair the above situation, 𝑀₁⊗𝑀₀ may be redefined as 𝑀₁⊗𝑀₀((𝜆,𝜇))=∨_𝑇0{𝑀₁(𝜆(𝑡₀))}, so that the functor (−)^{(𝑁0,𝑀0)} does, now, turn out to be right adjoint to the (modified) functor −⊗(𝑁₀,𝑀₀). 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.