A General Categorical Framework of Minimal Realization Theory for a Fuzzy Multiset Language

is paper is to study the minimal realization theory for a fuzzy multiset language in the framework of category theory, which has already provided the tools and techniques for the advancement of several features of theoretical computer science. Specically, by using the well-known categorical concepts, it is shown herein that there is a minimal realization (called the Nerode realization) for each fuzzy multiset language, and all minimal realizations for a given fuzzy multiset language are isomorphic to it.


Introduction
Multiset, in contrast to classical set, allows repetition of any object, and the number of repetitions of an object is called its multiplicity (cf. Blizard [1]). In other words, a multiset is a set with multiplicities associated with its elements in the form of natural numbers. e multiset theory or theory of bags has been established by Cerf and his coauthors [2] as an extension of classical set in 1971, which is further advanced by Petrson [3], Yager [4] and Blizard [1]. e theory of bag developed by Yager [4] and Blizard [1], explored in the fuzzy sense by Chakrabarty et al. in [5,6] under the notion of fuzzy shadows and fuzzy bags. e multiset theory has been shown to be useful in, chemical programming [7], parallel processing [2], mathematics [8][9][10], parallel processing [2], data analysis [11], parallel programming [12], neural network [13], information system [14], decision making [15], decipherability of codes and information transmission [16][17][18][19][20][21], DNA computing [22], and many more for details see references in [23].
An automaton (cf. Hopcroft and Ullman [24], Eilenberg [25]), a well-known model of computation, has been started to generalized using fuzzy sets theory (c.f., Zadeh [26]), by Wee [27], Wee and Fu [28], Lee and Zadeh [29], Santos [30], and Kumbhojkar & Chaudhri [31] to introduce the concept of a fuzzy automaton and fuzzy languages, which are further studied algebraically by Maliket al. [32], and Mordeson and Malik [33] under the notion of fuzzy nite machines. Wechler [34], initiated the study of fuzzy automata and fuzzy languages taking membership values in structured sets. Following Wechler [34] initiative, fuzzy automata and fuzzy languages having structure for membership values of a fuzzy set in arbitrary set [35], bipointed sets [35,36], poset [37,38], locally nite complete lattice [39], lattices [40,41], lattice order monoid [42,43], complete residuated lattices [35,[44][45][46], have been studied in di erent directions from theoretical and application point of view and became an important tool for reducing the gap between formal languages and natural languages. e multiset theory is also in closed proximity with the theory of computation. Formal power series, Petri nets, databases, logics, formal language theory, and concurrency are some areas in computer science where multiset theory frequently appears. e multiset languages (commutative languages) are sets with their objects as multisets. Crespi-Reghizzi and Mandrioli [47] used multiset grammars as a device for a formalism for generating multiset language rather than strings, and shown equivalent to vector addition systems and Petri nets model of parallel processing. Multiset languages can be characterized in terms of multiset grammar as well as multiset automata [48,49]. e multiset language and multiset grammar have been shown helpful in membrane computing [23,50].
e Mealy multiset automata with output have been discussed in [51]. In contrast, multiset pushdown automata and multiset languages have been studied in [52] and closure properties of multiset language families have been studied in [53]. e multiset theory have been also shown to have successful application in the advancement of the theory of fuzzy automata. e recent contribution with application of multiset theory in fuzzy automata theory is due to Wang et al. [54], the authors introduced the notion of fuzzy multiset finite automata as a generalized version of multiset finite automata and studied its relationship with fuzzy multiset regular grammars. In [54], it has been pointed out that fuzzy multiset language produced by a fuzzy multiset regular grammar can be recognized by a fuzzy multiset finite automaton and vice-versa. Some operations on fuzzy multiset languages, along with closure properties of the family of fuzzy multiset finite automata languages under these operations, have also been studied in [54]. Martinek, in [55] reviewed fuzzy multiset finite automata and discussed determinism, languages, and pumping lemma, whereas in [56] the author studied some closure properties of fuzzy multiset regular languages. e deterministic automaton with a fuzzy set of final states also known as crisp deterministic fuzzy automata (cdfa) is a well-known concept studied by [39,40,57]. Such automaton has been considered useful due to fact that (i) some fuzzy automaton accepts a given fuzzy language iff some crisp deterministic fuzzy automaton accept the same (cf. [39,40]), and (ii) the Nerode automaton corresponding to a fuzzy automaton is a cdfa equivalent to same fuzzy automaton (cf. [57]). e deterministic fuzzy multiset finite automata and fuzzy multiset finite automata have been shown equivalent potential in terms of acceptance of a fuzzy multiset language by Tiwari et al. [58] and in addition minimal realization of a fuzzy multiset language have also been studied by them through the right congruence relation. e minimization problem, i.e., given a behavior, how one can design a machine with the minimal number of states which recognize the given behavior, is a fundamental issue in classical automata theory and is, no doubt, a significant problem in fuzzy automata theory. Several contributions with different approaches have been reported to handle this issue (cf. [33, 35-38, 40-42, 44-46, 59-61]). Myhill-Nerode's theory [62,63], use the congruences on free monoid to study deterministic automata and formal languages. ese congruences were found to play a critical role in minimal deterministic automaton construction recognizing a given language.
e Myhill-Nerode's theory has also been successfully addressed the minimal realization problem of fuzzy languages [33,35,42]. e minimization idea of Lei and Li [42] or Mordeson and malik [33] for deterministic fuzzy recognizers take in account the reduction of the number of states of a fuzzy automaton by merging indistinguishable states, similar to the algorithm for minimization of deterministic automata. But, in [35], it has been pointed out that "minimization used in [33,42] does not mean the usual construction of the minimal one in the set of all fuzzy automata recognizing a given fuzzy language, but just the procedure of computing and merging indistinguishable states which do not necessarily result in a minimal fuzzy automaton." Further, in [35] as well as in [33], the derivative of fuzzy languages plays a key role in the minimization procedure.
e minimization problem of fuzzy multiset automata and fuzzy multiset languages having membership values in distributive lattices have been studied by Tiwari et al. [58], using the right congruence relation, a key concept of Myhill-Nerode theory, and concept of derivative of a fuzzy multiset language. Wang and Li [64] also studied the minimization problem using the right congruence relation defined on the state-set of deterministic lattice multiset automata and presented an efficient minimization algorithm where unreachable states were removed in the process of minimization.
e recent contribution to the study of fuzzy multiset automata and fuzzy multiset languages is due to Sharma et al. [76,77], Wang and Li [64], Pal and Tiwari [78], and Tiwari et al. [79]. e authors in [76], studied algebraic aspects of fuzzy multiset automata like fuzzy multiset transformation semigroup and coverings of fuzzy multiset finite automata, while authors in [77] studied fuzzy multiset regular languages and proved pumping lemma and use it to point out mandatory condition for fuzzy multiset languages being non-constant. In [64], the method of transformation of lattice valued nondeterministic multiset automata into deterministic lattice valued multiset automata (DLMA), and then its minimization algorithm; some lattice-valued regular multiset languages accepted by some special minimal DLMA the lattice-valued regular multiset language decomposition by some simple were studied with structure for truth values of a fuzzy set as distributive lattices. In Pal and Tiwari [78], minimal realization for an L-valued multiset language is studied through Brzozowski's algorithm; the authors show that the deterministic L-valued multiset automaton obtained by use of Brzozowski's algorithm is minimal.
We have gone through the cited literature related to the study of fuzzy multiset automata and fuzzy multiset languages and observed that (i) all cited work associated with the study of fuzzy multiset automata and fuzzy multiset languages uses either [0, 1], or distributive lattices as the membership structure for fuzzy sets; (ii) the minimal realization for given fuzzy multiset languages and minimization of fuzzy multiset automata studied so for uses (i) concept of Myhill-Nerode equivalence relation defined either on input multiset [58,64], or through defining equivalence relation on state set of deterministic lattice valued multiset automata and removal of unreachable states [64], (ii) derivative of fuzzy languages [58], (iii) Brzozowski's minimization algorithm [78]; (iii) except Pal and Tiwari [78], no work has been reported for studying fuzzy multiset automata and fuzzy multiset languages via categorical concept; while, the categorical approach has been shown to be helpful in the study of several aspects of classical automata and languages, as well as fuzzy automata and fuzzy languages, as mentioned above.
Keeping the above-mentioned points in account, in the present work, instead of taking [0, 1] or distributive lattice, we have taken arbitrary sets with two distinguished elements 0 and 1 for structure of truth values of fuzzy sets as in [35] and provide a general categorical framework for minimal realization theory for a fuzzy multiset language to enrich the fuzzy multiset automata theory. e substance of the paper is as follows. In Section 2, we have presented the fundamental concepts and notions related to category theory, multiset theory, fuzzy multiset automata theory, and fuzzy multiset languages, which we need in subsequent sections. In Section 3, for a fixed input alphabet Σ 0 , we define categories CDFMA(Σ 0 ) of crisp deterministic fuzzy multiset automata and their homomorphisms having membership values in L ∈ |LSET|, and then characterize the reachability and coreachability map of M ∈ |CDFMA(Σ 0 )| as a morphism of CDFMA(Σ 0 ). In Section 4, we have provided Myhill-Nerode theory base construction for minimal realization for a given fuzzy multiset language in our sense. In Section 5, we have define a category MDYN(Σ ⊕ 0 ) of MΣ ⊕ 0 -dynamics and define a forgetful functor U: MDYN(Σ ⊕ 0 ) ⟶ LSET having both left as well as right adjoint and introduce concept of free and cofree MΣ ⊕ 0 -dynamics. A different view of run map, reachability map and coreachability map introduced in previous section facilitate us to characterized them as a morphism of category MDYN(Σ ⊕ 0 ). In Section 6, we have shown that the free and cofree MΣ ⊕ 0 -dynamics are free and cofree with respect to the forgetful functor U introduced in the previous section. Finally, in Section 7, we have defined the response map of a crisp deterministic fuzzy multiset automaton. We show that a fuzzy multiset language can be fully characterized in terms of a response map. We have provided minimal realization of response map in category MDYN(Σ ⊕ 0 ) and conclude that for a given fuzzy multiset language, its all minimal realizations are isomorphic to its Nerode realization.

Preliminaries
In this section, we recall those concepts from category theory, multiset set theory and multiset automata theory which we need in subsequent sections. We begin with following.

Category eoretic Concepts.
e categorical concepts used by us are recalled here from Adámek, Herrlich, and Strecker [65], Arbib and Manes [67], and Mac Lane [80]. Definition 1. [65,67,68,80] A category T consist of: (i) a objects class (or T-objects); (ii) for every pair X and Y of T-objects, a class T(X, Y), of morphisms (or T-morphisms) with domain X and codomain Y; f ∈ T(X, Y) is written as called the identity morphism on X; (iv) for T-morphisms h: X ⟶ Y and k: Y ⟶ Z, a Tmorphism k°h: X ⟶ Z, called composition of h and k; Such that: (a) for all T-morphisms h: X ⟶ Y, k: Y ⟶ Z, and g: Z ⟶ D, g°(k°h) � (g°k)°h,

Hold.
We shall represent by |T|, the class of object of a category T. Definition 2. [65,67,68,80] A category S is said to be a subcategory of a category T, if (i) every object of S is also an object of T; (ii) ∀ objects X and Y of S, S (X, Y) ⊆ T(X, Y); and (iii) S has the same composition morphisms as in T.
(iv) S has identity morphisms same as in that of T. Definition 3. [65,67,68,80] Let T and S be two categories and F: T ⟶ S be a map which send each X ∈ |T| to a F(X) ∈ |S| and each morphism f: Definition 4. [65,67,68,80] Let T and S be categories, G: T ⟶ S be a functor, and Y be a S-object, then G is said to have a left adjoint, if for each T-object X, ∃ a pair (X, η), where η: Y ⟶ GX is a morphism of S, such that for any object X ′ of T, and any morphism f: Y ⟶ GX ′ of S, ∃ a unique morphism ψ: X ⟶ X ′ of T, which make diagram of Figure 1 commutes.

Mathematical Problems in Engineering
De nition 5. [65,67,68,80] Let T and S be categories. Let G: T ⟶ S be any functor, and Y be an object of S. We say that a pair (X, η), where X is an object of Tand η: Y ⟶ GX is a morphism of S, is free over S with respect to G, just in case η: Y ⟶ GX has the couniversal property that given any morphism f: Y ⟶ GX ′ with X ′ any T-object, there exists a unique T-morphism ψ: X ⟶ X ′ such that diagram in Figure 1 commutes. We call η as the inclusion of generators and the unique ψ such that diagram in Figure 1 commute as T-morphic extension of f with respect to G.
De nition 6. [65,67,68,80] Let T and S be categories, and F: T ⟶ S be a functor, and Y be a S-object, then F has right adjoint, if for each T-object X, there exists a pair (X, ε), where ε: FX ⟶ Y is a S-morphism such that for any Tobject X ′ and any S-morphism f: FX ′ ⟶ Y, there exists a unique T-morphism ψ: X ′ ⟶ X such that diagram in Figure 2 commutes. De nition 7. [65,67,68,80] Let T and S be categories, and F: T ⟶ S be a functor, and Y be a S-object. We say that a pair (X, ε), where X is an object of T and ε: FX ⟶ Y is a morphism of S, is cofree over S with respect to F, just in case η: Y ⟶ FX has the universal property that given any morphism f: FX ′ ⟶ X with X ′ any T-object, there exists a unique T-morphism ψ: X ′ ⟶ X such that diagram in Figure 2 commutes.

Multiset Concepts.
Here we recall those concepts of multiset which we use in this paper. ese standard concepts of multiset can be found in literature (cf. [1,48,54] for details).

Multiset Automata.
In this section we recall some concepts related to multiset automata which we need for completeness of the paper. ese concepts can be found in [54,55,58,64,78].
M is called multiset nite automaton if Q is nite. A con guration of a MA, M is a pair (p, α), where p and α denote current state and current multiset, respectively. e transition in a multiset automaton are described with the help of con gurations. e transition from con guration (p, α) leads to con guration (q, β) if there exists a multiset c ∈ Σ ⊕ with c ⊆ α, q ∈ δ(p, c) and β α⊖c, and is denoted by (p, α) ⟶ (q, β). We shall denote by ⟶ * , the reexive and transitive closure of this operation.
De nition 10. [64,78] Next, we recall the concept of an L-valued multiset automaton de ned over distributive lattice L from [64].
De nition 11. [64] An L-valued multiset automaton (LMA) is a 5-tuple M (Q, Σ, δ, I, τ), where (i) Q and Σ are nonempty sets called the state-set and input-set, respectively; (iii) I: Q ⟶ L is a map called the L-valued set of initial states; and (iv) τ: Q ⟶ L is a map called the L-valued set of final states.
M is called L-valued multiset finite automaton if Q is finite.
A configuration of L-valued multiset automaton M is a pair (p, α), where p and α denote current state and current multiset, respectively. e transitions in an L-valued multiset automaton are described with the help of configurations. e transition from configuration (p, α) leads to ⟶ k′ * denote the reflexive and transitive closure of (2) 3 , and the L-valued transition function is given as under: If α � < a 1 > ⊕ < a 2 > ⊕ < a 2 > ⊕ < a 3 > and β � 0 Σ . en the transition steps ((q 1 , α) ⟶ * (q 2 , β)) are as follows: Definition 12. [64,78] (i) For a given set Σ, an L-valued multiset language is a map f: ∀α ∈ Σ ⊕ . Further, an L-valued multiset language f: Also, L-valued multiset language f, accepted by an LMAM, is denoted by f M in [64,78].

Bipointed Set.
Here, we recall the concept of bipointed set from [35], where fuzzy sets were taken their membership values in a structure L � (L, 0, 1), where L is an arbitrary set, and 0 and 1 are two distinguished elements of L, and this structure were used to take crisp languages into consideration. In [35], no other conditions were imposed on L. In this paper, fuzzy sets will take their membership in a bipointed set L in sense of [35]. A more generalized view of bipointed set has been given in [36], and it is pointed out there that the concept of bipointed set can be viewed as generlization of the notion of pointed set in [80].
where L is an arbitrary set and a, b are two distinguished elements of L. A homomorphism between two bipointed sets (L, a, b) and

Proposition 1. [36] e bipointed sets and their homomorphisms form a category denoted by BSET.
Let B be any arbitrary set, we identifed it as a bipointed It is worth to note that, the class of bipointed sets mentioned above and their morphism also forms a category, which we denote LSET. Obviously, LSET is a subcategory of BSET. Note that 1 � ((0, 0, 1), 0, 1).
From now onward, throughout this paper, L stand for a bipointed set. For a nonempty set U, A: U ⟶ L is called a fuzzy set over U and L U denotes all fuzzy subsets of a set U.
e kernel of map f: U ⟶ V, denoted as ker(f), and is defined by

Crisp Deterministic Fuzzy Multiset Automata and Fuzzy Multiset Language
Fuzzy automata and fuzzy languages having membership values in bipointed set have been studied in [35]. Myhill-Nerode theory have been used there for minimization of fuzzy languages. Here, we define category of crisp deterministic fuzzy multiset automata, introduce the notion of fuzzy multiset languages over bipointed sets, and construct a minimal realization for a given fuzzy multiset language in sense of [35], but approach is categorical.

Mathematical Problems in Engineering
De nition 14. A crisp deterministic fuzzy multiset autom- (i) Q and Σ are nonempty sets called the state-set and the input-set, respectively.
is a map called fuzzy set of nal states. (iv) q 0 ∈ Q is a xed state called the initial state.
A con guration of a CDFMA is same as in the case of a multiset automaton.  We represent the category described above by CDFMA, and its object-class is denoted by |CDFMA|.

Remark 2
(i) e objects class of CDFMA having xed input set of alphabets Σ 0 and their morphisms as all CDFMA-morphisms of type (id Σ ⊕ 0 , b) form a category denoted by CDFMA(Σ 0 ), its object class is denoted by |CDFMA(Σ 0 )|. De nitely, the category so obtained is a subcategory of CDFMA. Figure 6 commute. roughout, this paper Σ ⊕ 0 denotes the set of all multisets over a xed input set Σ 0 . We shall denote by 0 Σ 0 , the identity element of Σ ⊕ 0 . Now, we de ne fuzzy multiset language having membership values in bipointed sets L.
De nition 16 For a given xed nite alphabet Σ 0 , a fuzzy multiset language is a map f: A fuzzy multiset language, accepted by M ∈ |CDFMA(Σ 0 )|, in state q 0 (i.e., f q 0 ) is simply called the fuzzy multiset language accepted by M, which we denote by f M . Further, a fuzzy multiset language f: Remark 3. It can also be easily seen that  Mathematical Problems in Engineering in |CDFMA(Σ 0 )| tells that corresponding to distinct states there are di erent fuzzy multiset languages associated with it.
which accepts the fuzzy multiset language f. Now, we have the following. Proof. To show that the reachability map r M : Σ ⊕ 0 ⟶ Q of M is a CDFMA(Σ 0 )-morphism from M 1 to M, we need to show only that the diagrams in Figure 7 commute. Figure 7 holds. e Triangle (B) in Figure 7 holds because (r M°τ1 )(0) r M (0 Σ 0 ) q 0 τ 2 (0).
Proof. To show that the coreachability map σ M : Q ⟶ L Σ ⊕ 0 is a CDFMA(Σ 0 )-morphism from M to M 2 , we need to show that the diagrams in Figure 8 hold.
Mathematical Problems in Engineering 7 e coreachability map σ M 1 :  Proof. Follows from commutativity of diagrams in Figure 9 below.

Myhill-Nerode Theory Based Minimal
Realization for a Fuzzy Multiset Language e minimal realization theory for a given fuzzy language was introduced in [35] by using Myhill Nerode theory. Herein, we present the same for fuzzy multiset language using Myhill Nerode theoretic concept and the concepts of reachability and coreachability map.
Hence N f realizes f. Now, to establish that N f is minimal, we must show that the reachability map r N f : where by σ N f is one-one. Hence N f is minimal.
□ Remark 6. Hereafter, we call N f , the Nerode realization. e derivative of a language have been used as a tool for minimization by Rabin and Scott [81], while in fuzzy scenario this concept have been shown useful by Mordeson and Malik [33], Petković [82], and Ignjatović et al. [35]. In this section, we have introduce the concept of derivative of a fuzzy multiset language over a bipointed set, and provide some useful results associated with interconnectivity of   Mathematical Problems in Engineering regular-ness, the set of derivatives, and Myhill-Neroode equivalence of a fuzzy multiset language.
en η Q: Q ⟶ Q × Σ ⊕ 0 is a morphism of LSET, and diagram in Figure 14 reduces to the diagram in Figure 15.
∀q ∈ Q, α, < c > ∈ Σ ⊕ 0 . en the following diagram in Figure 16 commutes, which means ψ is a morphism of MDYN(Σ ⊕ 0 ). e fact that ψ be unique is trivial. Further, de nition of ψ makes diagram in Figure 13 commute, whereby U has a left adjoint.    Remark 8. In Proposition 11, we have de ned a map the free MΣ ⊕ 0 -dynamics on Q generators. Now, the run map of a CDFMA(Σ 0 )-object can be characterized as follows.

De nition 24.
A MDYN(Σ ⊕ 0 )-object (Q, δ M ) with initial state q 0 is called reachable if its reachability map r M is epimorphism.
Proof. We must show that corresponding to every L ∈ |LSET| , ∃ a pair (ε, M), with M an object of MDYN(Σ ⊕ 0 ), and ε: UM ⟶ L a morphism of LSET with property that for any object N (Q, δ) of MDYN(Σ ⊕ 0 ) and for any morphism g: UN ⟶ L of LSET, the diagram below in Figure 19 commutes.  Figure 19 take the form of diagram in Figure 20. Now, de ne a map ψ: Q ⟶ L Σ ⊕ 0 satisfying ψ(q) f q , so that the diagram in Figure 21 commutes, e commutative diagram of Figure 22 implies that ψ: M ⟶ N is a MDYN(Σ ⊕ 0 )-morphism. Also, the fact that ψ is unique, is trivial. Further, the de nition of ψ make Figure 21 commute, whereby U has a right adjoint. □ Remark 10. In Proposition 12 we have de ne a map Proof. In proof of Proposition 12, we set g β M , we have commutative diagrams in Figure 23 Which provides [ψ(q)](α) β M (δ M (q, α)), ∀α ∈ Σ ⊕ 0 . is with use of De nitions 18 and 20, Remark 3 and uniqueness of ψ implies ψ σ M with property that ε° σ M β M .   Figure 19.
0 is a LSET-morphism, see Proposition 11), with respect to the functor U, then we can introduce a functor V: LSET ⟶ MDYN(Σ ⊕ 0 ) for which V Q is free over Q. Such a functor V is said to be left adjoint of the functor U. In category theory, every concept has its dual concept, so dually, if every member in |LSET| has a cofree object, then there is a cofree object constructing functor called the right adjoint of U.
us we conclude that U: MDYN(Σ ⊕ 0 ) ⟶ LSET has a left adjoint if every Q ∈ |LSET| has a pair (M, η Q) free over Q, and that V has a right adjoint if corresponding to every element M ∈ |MDYN(Σ ⊕ 0 )| there is a pair ( Q, η) cofree over M . Now, we have following.  Figure 24 en the collection of maps V: LSET ⟶ MDYN(Σ ⊕ 0 ) so de ned is a functor (it is to be noted here that the object map is unique up to isomorphism, and morphism maps is also xed uniquely) called the left adjoint of U.
Proof. In order to show that V: LSET ⟶ MDYN(Σ ⊕ 0 ) is a functor. We have to show that Vid Q id V Q and V(g°f) Vg°Vf. From uniqueness property, the commutativity of diagram in Figure 22 below clearly implies that Vid Q id V Q .
Also, for any two LSET morphisms f, g and the fact that U: MDYN(Σ ⊕ 0 ) ⟶ LSET is a functor, the diagrams in Figure 25 commute.
And so uniqueness of U ensure us that V( g° f) V g°V f. Hence V is a functor.
en U is a functor, which we call right adjoint of V.
Proof. Similar to Proposition 13.
In Proposition 11, we have de ned a MΣ ⊕ 0 -dynamics α⊕c), for all q ∈ Q, α, c ∈ Σ ⊕ 0 , which we call free MΣ ⊕ 0 -dynamics on Q generators and associate a map η Q: Q ⟶ Q × Σ ⊕ 0 de ned by η Q(q) (q, 0 Σ 0 ), ∀q ∈ Q, η Q(0) 0 and η Q(1) 1. In next proposition, we have shown that the free MΣ ⊕ 0 -dynamics on Q generators is free with respect to forgetful functor U: MDYN(Σ ⊕ 0 ) ⟶ LSET, i.e., the pair are de ned as above) is free over Q with respect to forgetful functor U. □ Proposition 15. Let us consider the forgetful functor U: MDYN(Σ ⊕ 0 ) ⟶ LSET, and let Q be any member of |LSET|. en the pair (M, η Q) de ned above is free over Q with respect to U. us U has a left adjoint.
In proof of Proposition 12, we have de ned a Mathematical Problems in Engineering 13 ese equations imply that f q (α) h(ϱ (q, α)), ∀q ∈ R, α ∈ Σ ⊕ 0 . Since a unique ϕ exists, (N, ε) is indeed cofree over L with respect to forgetful functor U.
We conclude that the free and cofree MΣ ⊕ 0 -dynamics are free and cofree with respect to the forgetful functor U: MDYN(Σ ⊕ 0 ) ⟶ LSET.

A Categorical View of Minimal Realization Theory for a Fuzzy Multiset Language
In Section 3, we have de ned the category CDFMA(Σ 0 ). In the subsequent section, Section 4, we have provided Myhill-Nerode theoretic minimal realization of a fuzzy multiset language, whereas, in Section 5, we have introduced the category MDYN(Σ ⊕ 0 ) of MΣ 0 ⊕-dynamics and their morphisms. In this section, we introduce the notion of response of a member M ∈ |CDFMA(Σ 0 )|. It has been pointed out here that every fuzzy multiset language can be expressed in terms of the response of a M ∈ |CDFMA(Σ 0 )|. We have shown that the response f M of M ∈ |CDFMA(Σ 0 )| is a MDYN(Σ ⊕ 0 )-morphism. Interestingly, it has been shown here that every MDYN(Σ ⊕ 0 )-morphism has a minimal realization. is section provides another view of minimal realization theory for a fuzzy multiset language in a pure arrow-based categorical theoretic setup. Finally, it is proved here that all the minimal realization of MDYN(Σ ⊕ 0 )-morphism as a response map (or equivalently, for a fuzzy multiset language) is isomorphic to Nerode realization for a fuzzy multiset language.
, for some response f M of a M ∈ CDFMA(Σ 0 ). erefore, it is interesting to see that the realization theory of a fuzzy multiset language is totally captured in the above introduced realization theory.  Figure 27: Commutative diagrams for Proposition 15. 14 Mathematical Problems in Engineering diagram in Figure 16 commute, we have the commutative diagrams in Figure 29.
Proof. Follows from commuting diagrams in Figure 30. e commutativity of diagrams in Figure 30 follows from the fact that f M σ M°rM , where r M : , ξ: h(Q) ⟶ Q ′ be an onto map, ξ be a one-one map, and h ξ°ζ. en, ∃ a unique Let k: Q 1 ⟶ Q 2 and h: Q 2 ⟶ Q 3 be the maps with property that k: (Q 1 , δ 1 ) ⟶ (Q 2 , δ 2 ) and h°k: Proof. Let k: en the upper Square (A) of diagram in Figure 32 and diagram in Figure 33 (i.e., the outer diagram of Figure 32) commute.
To show that h: , we need to show that lower Square (B) in Figure 32 must commutes. Now, let q ′ ∈ Q 2 and α ∈ Σ ⊕ 0 . en Figure 29: Commutative diagrams for Proposition 17. Q h (Q) Hence h: is a MDYN(Σ ⊕ 0 )-morphism. Now, we need following from [80], for our next results.
as their reachability maps and σ M , σ M ′ as their coreachability maps, respectively. en, reachability of M and coreachability of M ′ implies r M is onto and σ M ′ is one-one. So Proposition 21, ensure that ∃ a unique map b: Q ⟶ Q ′ which make diagram in Figure 35 Proof. Let Q f be a set such that r f :  Figure 38.
en M f (Q f , X 0 , δ f , q f , β f ) is a CDFMA(Σ 0 )-object, where q f τ f (0). Moreover, Corollaries 2 and 3, tells that r f is reachability map of M f and σ f is coreachability map of M f and σ f , whereby σ f°r f f. us M f is a realization of f. Also, as r f is onto and σ f is one-one, so M f is minimal realization for f.
Finally, we conclude that: Proposition 25. All minimal realizations for total response map f (or equivalently fuzzy multiset language f) are isomorphic to the Nerode realization of fuzzy multiset language f f(0 Σ 0 ).

Proof. Follows from Proposition 22 and 23.
□ Remark 12. From the above proposition, it is clear that the minimal realizations obtained either through Brzozowski's algorithm [78] or derivative of fuzzy multiset language [58] are isomorphic to Nerode realization of that fuzzy multiset language.

Conclusion
In this paper, we have introduced a general categorical framework for the realization theory of fuzzy multiset language. Speci cally, we have studied the crisp deterministic fuzzy multiset automata and fuzzy multiset languages over bipointed sets and provided purely arrow theoretic categorical interpretation of several concepts associated with a crisp deterministic fuzzy multiset automaton and fuzzy multiset language. In between, we have shown that the reachability and coreachability maps of a CDFMA (Σ 0 ) can be viewed as a CDFMA (Σ 0 ), as well as MDYN(Σ ⊕ 0 )-morphisms rather than viewing them simply as a map. Furthermore, we have de ned response map of CDFMA (Σ 0 ) and associated it with fuzzy multiset language, and viewed the response map of CDFMA (Σ 0 ) as a MDYN(Σ ⊕ 0 )-morphism and showed that every MDYN(Σ ⊕ 0 )-morphism has a minimal realization. Interestingly, it is shown here that the minimal realizations for a fuzzy multiset language are isomorphic to the Nerode realization of the fuzzy multiset language. In other words, the proposed theory (i) provides a platform to connect di erent minimal realization techniques (Brzozowski's algorithm [78], derivative of fuzzy multiset language [58]), and (ii) leads us to conclude that whatsoever the algorithms or approaches used to construct the minimal realization, the resultant will be isomorphic to the Nerode realization.
It is pertinent to mention that the study carried out herein is based on the structure which generalizes the theory of ordinary fuzzy sets. As ordinary fuzzy representation provides a limited platform for approximating the meaning of words and is not able to capture linguistic uncertainty, a formal interval type-2 fuzzy model of computing with words by generalizing the existing ordinary fuzzy sets-based model to an interval type-2 fuzzy environment has been studied in [83]. Further, interval type-3 fuzzy sets [84] introduced recently have been shown superior because the upper and lower bound of the footprint-of-uncertainty are not constant but fuzzy sets. Also, the secondary membership in such fuzzy sets is an interval type-2 fuzzy set, whereas it is an ordinary fuzzy set in the case of type-2 fuzzy sets. Such advancement in the theory of fuzzy sets may be used further to enrich the proposed study of this paper.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no con icts of interest.