On Algebraic Study of Type-2 Fuzzy Finite State Automata

Journal of Fuzzy Set Valued Analysis 2017 No.2 (2017) 86-95 Available online at www.ispacs.com/jfsva Volume 2017, Issue 2, Year 2017 Article ID jfsva-00366, 10 Pages doi:10.5899/2017/jfsva-00366 Research Article

On Algebraic Study of Type-2 Fuzzy Finite State Automata Anupam K. Singh1∗ , Saumya Pandey2 , S. P. Tiwari2 (1) Amity Institute of Applied Sciences, Amity University, Noida-201310, India (2) Indian Institute of Technology (ISM), Dhnabad-826004, India c Anupam K. Singh, Saumya Pandey and S. P. Tiwari. This is an open access article distributed under the Creative Commons Copyright 2017 ⃝ Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract Theories of fuzzy sets and type-2 fuzzy sets are powerful mathematical tools for modeling various types of uncertainty. In this paper we introduce the concept of type-2 fuzzy finite state automata and discuss the algebraic study of type-2 fuzzy finite state automata, i.e., to introduce the concept of homomorphisms between two type-2 fuzzy finite state automata, to associate a type-2 fuzzy transformation semigroup with a type-2 fuzzy finite state automata. Finally, we discuss several product of type-2 fuzzy finite state automata and shown that these product is a categorical product. Keywords: Type-2 fuzzy set, type-2 fuzzy finite state automata, type-2 fuzzy transformation semigroup, categorical product.

1 Introduction The study of fuzzy automata was initiated by [10] and [12] in 1960 after the introduction of fuzzy set theory by [13]. Much later a considerably simpler notion of a fuzzy finite state machine (which is almost identical to a fuzzy automata) was introduced by [6, 8]. Somewhat different notions were introduced subsequently in [3, 4] and [9]. The concept of homomorphism, transformation semigroup, product property play an important role in the study of finite state machine [2]. Much later [7] introduced these ideas for fuzzy finite state machines and explore their algebraic properties (cf.,[6] for more detail). Type-2 fuzzy sets, firstly proposed by [14] in 1975 as an extension of type-1 fuzzy set in which the membership function falls into a fuzzy set in the interval [0, 1]. Because type-1 fuzzy sets not able to directly model such uncertainties because their membership functions are totally crisp. Type-2 fuzzy sets are capable to improve such uncertainties because their membership function are already fuzzy. Now, in this paper, we mainly introduced the concept of automata theory in type-2 fuzzy sets, which consists of a state-set, an input-set, a transition map and a primary membership function which have been fuzzified. Further by this concept of type-2 fuzzy finite state automata we discuss the concepts of homomorphism, transformation semigroup and direct product for type-2 fuzzy finite state automata. This paper is organized as follows: In section 2, we recall some basic definitions and properties of fuzzy automata, type-2 fuzzy sets, type-2 fuzzy relations. In section 3, we recall the concept of type-2 fuzzy finite state automata and with the help of this type-2 fuzzy finite state automata we discuss the concept of homomorphism. In section 4, recall the concept of type-2 fuzzy transformation semigroups by using congruence relation and lastly, we discuss the direct product and the general direct product of type-2 fuzzy finite state automata and shown that this product is a categorical product. ∗ Corresponding

author. Email address: [email protected], Tel:+91-8860260528

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Journal of Fuzzy Set Valued Analysis 2017 No.2 (2017) 86-95 http://www.ispacs.com/journals/jfsva/2017/jfsva-00366/

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2 Preliminaries In this section we recall some basic notions relevant to fuzzy automata, type-2 fuzzy sets, type-2 fuzzy relations and collect some results, which we need in the subsequent sections. Throughout this paper, X is a nonempty set. Definition 2.1. [12] A fuzzy automaton is a triple M = (Q, X, δ ), where Q is a nonempty set (the set of states of M), X is a monoid (the input monoid of M), whose identity shall be denoted as eX , and δ : Q × X × Q → [0, 1] is a map, such that ∀p, q ∈ Q, ∀x, y ∈ X, we have { 1 if q = p δ (q, eX , p) = 0 if q ̸= p and δ (q, xy, p) = ∨r∈Q {δ (q, x, r) ∧ δ (r, y, p)} Definition 2.2. [5]. A type-2 fuzzy set, denoted A˜ is characterized by a type-2 membership function µA˜ (x, u) : X ×Jx → [0, 1], ∀x ∈ X, ∀u ∈∫ Jx ⊆∫ [0, 1], i.e., A˜ = {((x, u), µA˜ (x, u)) : x ∈ X, u ∈ Jx ⊆ [0, 1]}, In which 0 ≤ µA˜ (x, u) ≤ 1. A˜ can be expressed as A˜ = x∈X u∈Jx µA˜ (x, u)/(x, u), Jx ⊆ [0, 1], ∫∫ where denote the union over all admissible x and u. ˜ A class of type-2 fuzzy sets of the universe X is denoted by F(X). ˜ is the intersection between the two-dimensional plane whose Definition 2.3. [5]. A vertical slice denoted µA˜ (x) of A, ˜ i.e., axes are u and µA˜ (x, u) and the three-dimensional type-2 membership function A, ∫ µA˜ (x) = µA˜ (x = x′ , u) = u∈J ′ fx′ (u)/u, Jx′ ⊆ [0, 1] x in which 0 ≤ fx′ (u) ≤ 1. In terms of vertical slice, a type-2 fuzzy set A˜ can also be re-expressed as: A˜ = {(x, µA˜ (x)) : x ∈ X} or as the following: A˜ =

∫ x∈X

µA˜ (x)/x =

∫

∫

x∈X [ u∈Jx f x (u)/u]/x, Jx

⊆ [0, 1],

where fx (u) = µA˜ (x, u). The vertical slice µA˜ (x) is also called the secondary membership function, and its domain is called the primary membership of x, which is denoted by Jx , where Jx ⊆ [0, 1] for any x ∈ X. The amplitude of the secondary membership function is called the secondary grade. ˜ is the union of all of the primary memberships: i.e., Definition 2.4. [5]. The footprint of uncertainty denoted DA, ˜ DA˜ = ∪x∈X Jx , which represents the uncertainty in the primary memberships of a type-2 fuzzy set A. ˜ Let DA(x) = Jx , ∀x ∈ X. Then a type-2 fuzzy set A˜ can be re-expressed as: A˜ =

∫∫

(x,u)∈DA˜ µA˜ (x, u)/(x, u).

˜ × Y ) is said to be a Definition 2.5. [11]. Let X and Y be two nonempty universes. Then type-2 fuzzy set R˜ ∈ F(X type-2 fuzzy binary relation from X to Y as: R˜ = {(((x, y), u), µR˜ ((x, y), u)) : (x, y) ∈ X ×Y, u ∈ J(x,y) ⊆ [0, 1]}, in which 0 ≤ µR˜ ((x, y), u) ≤ 1. R˜ can be expressed as follows: R˜ =

∫

∫

(x,y)∈X×Y u∈J(x,y)

µR˜ ((x, y), u)/((x, y), u), J(x,y) ⊆ [0, 1].

˜ is denoted by The footprint of uncertainty FOU(R) ˜ y) = J(x,y) , (i) DR(x, ˜ = ∪(x,y)∈X×Y DR(x, ˜ y). (ii) DR˜ = FOU(R)

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3 Type-2 fuzzy finite state automata In this section, we recall the concepts related to a type-2 fuzzy finite state automata and introduced the concept of homomorphism between two type-2 fuzzy finite state automata. e = (Q, X, δ˜ , Jx ), where Q is a nonempty finite Definition 3.1. A type-2 fuzzy finite state automaton is a four tuple M e set (the set of states of M ), X is a monoid (whose elements are the input symbols), δ˜ is a type-2 fuzzy subset of (Q × X × Q) × Jx , i.e., a map δ˜ : (Q × X × Q) × Jx → [0, 1], where Jx ⊆ [0, 1] is a primary membership of x such that ∀p, q ∈ Q, ∀x, y ∈ X, we have { 1 if q = p Dδ˜ (q, e, p) = [µDδ˜ (q, e, p), µDδ˜ (q, e, p)] = 0 if q ̸= p with µDδ˜ (q, xy, p) = ∨r∈Q {µDδ˜ (q, x, r) ∧ µDδ˜ (r, y, p)} and µDδ˜ (q, xy, p) = ∨r∈Q {µDδ˜ (q, x, r) ∧ µDδ˜ (r, y, p)}. Example 3.1. Consider the type-2 fuzzy finite state automata (Q, X, δ˜ , Jx ), where Q = {p, q, r}, X = {x, y} with Jx ⊆ [0, 1] and δ˜ is a type-2 fuzzy subset of (Q × X × Q) × Jx defined as { 1 if q = p Dδ˜ (q, e, p) = [µDδ˜ (q, e, p), µDδ˜ (q, e, p)] = with 0 if q ̸= p p q r p q   p 1.0 0.5 0.6 p 1.0 0.5 = q  0.2 1.0 0.2 , µDδ˜ = q  0.2 1.0 r 0.3 0.4 1.0 r 0.3 0.4 

µDδ˜

r  0.6 0.2  1.0

also µDδ˜ (q, x, p) = 0.2, µDδ˜ (q, x, q) = 1.0, µDδ˜ (q, x, r) = 0.2, µDδ˜ (q, x, p) = 0.6, µDδ˜ (q, x, r) = 0.8, µDδ˜ (q, x, q) = 1.0, and

µDδ˜

p q r p q    p 0.9 0.5 0.4 p 1.0 0.6 = q  0.5 0.8 0.6 , µDδ˜ = q  0.8 0.9 r 0.6 0.4 0.6 r 0.7 0.5

r  0.6 0.8  0.8

with µDδ˜ (p, y, p) = 0.9, µDδ˜ (q, y, p) = 0.5, µDδ˜ (r, y, p) = 0.6, µDδ˜ (p, y, p) = 1.0, µDδ˜ (q, y, p) = 0.8, µDδ˜ (r, y, p) = 0.7. Now µDδ˜ (q, xy, p) = ∨r∈Q {µDδ˜ (q, x, r) ∧ µDδ˜ (r, y, p)} = ∨{µDδ˜ (q, x, p) ∧ µDδ˜ (p, y, p), µDδ˜ (q, x, q) ∧ µDδ˜ (q, y, p), µDδ˜ (q, x, r) ∧ µDδ˜ (r, y, p)} = ∨{0.2 ∧ 0.9, 1.0 ∧ 0.5, 0.2 ∧ 0.6} = ∨{0.2, 0.5, 0.2} = 0.5 and µDδ˜ (q, xy, p) = ∨r∈Q {µDδ˜ (q, x , r) ∧ µDδ˜ (r, y, p)} = ∨{µDδ˜ (q, x, p) ∧ µDδ˜ (p, y, p), µDδ˜ (q, x, q) ∧ µDδ˜ (q, y, p), µDδ˜ (q, x, r) ∧ µDδ˜ (r, y, p)} = ∨{0.6 ∧ 1.0, 1.0 ∧ 0.8, 0.8 ∧ 0.7} = ∨{0.6, 0.8, 0.7} = 0.8. Thus Dδ˜ (q, xy, p) = [µDδ˜ (q, xy, p), µDδ˜ (q, xy, p)] = [0.5, 0.8]. f1 = (Q1 , X1 , δ˜1 , Jx ) and M f2 = (Q2 , X2 , δ˜2 , Jx ) be type-2 fuzzy finite state automata. A homoDefinition 3.2. Let M 1 2 f1 to M f2 is a pair of maps ( f , g), where f : Q1 → Q2 and g : X1 → X2 are functions such that morphism from M Dδ˜1 (q, x, p) ≤ Dδ˜2 ( f (q), g(x), f (p)), ∀p, q ∈ Q1 , ∀x ∈ X1 . f1 → M f2 is called an isomorphism if f and g are both one-one and onto. A homomorphism ( f , g) : M f2 = (Q2 , X2 , δ˜2 , Jx ) be type-2 fuzzy finite state automata and ( f , g) : f1 = (Q1 , X1 , δ˜1 , Jx ) and M Definition 3.3. Let M 1 2 ∗ ∗ f1 → M f2 be a homomorphism. Let g : X → X ∗ be a map such that g∗ (e) = e and g∗ (wa) = g∗ (w)g(a), ∀w ∈ X ∗ , a ∈ M 1 2 1 X1 . The class of all type-2 fuzzy finite state automata and there homomorphisms obviously forms a category (under obvious composition of maps). We shall denote it by T2FFSA



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f2 = (Q2 , X2 , δ˜2 , Jx ) be type-2 fuzzy finite state automata and ( f , g) : f1 = (Q1 , X1 , δ˜1 , Jx ) and M Lemma 3.1. Let M 1 2 ∗ f1 → M f2 be a homomorphism. Then g (xy) = g∗ (x)g∗ (y), ∀x, y ∈ X ∗ . M 1

Let x, y ∈ X1∗ . We prove the result by induction on |y| = n. If n = 0, then y = e = g∗ (x) = g∗ (x)e = g∗ (x)g∗ (e) = g∗ (x)g∗ (y), whereby, the result is true for n = 0.

Proof. and so xy = xe = x. Thus g∗ (xy) Also. let the result be true ∀z ∈ X1∗ such that |z| = n − 1, n > 0 and y = za, where a ∈ X1 . Then g∗ (xy) = g∗ (xza) = g∗ (xz)g(a) = g∗ (x)g∗ (z)g(a) = g∗ (x)g∗ (za) = g∗ (x)g∗ (y). Hence the result is true for |y| = n. f2 = (Q2 , X2 , δ˜2 , Jx ) be type-2 fuzzy finite state automata and ( f , g) : f1 = (Q1 , X1 , δ˜1 , Jx ) and M Proposition 3.1. Let M 1 2 ∗ f1 → M f2 be a homomorphism. Then Dδ˜ (q, x, p) ≤ Dδ˜ ∗ ( f (q), g ∗ (x), f (p)), i.e., M 1 2 µDδ˜ ∗ (q, x, p) ≤ µDδ˜ ∗ ( f (q), g∗ (x), f (p)) and µDδ˜ ∗ (q, x, p) ≤ µDδ˜ ∗ ( f (q), g∗ (x), f (p)), ∀p, q ∈ Q1 , ∀x ∈ X1∗ . 1

2

1

2

Proof. Let p, q ∈ Q1 and x ∈ X1∗ . we prove the result by induction on |x| = n. If n = 0, then x = e and g∗ (x) = g∗ (e) = e. Now { 1 if q = p = µDδ˜ ∗ (q, e, p) = µDδ˜ ∗ (q, e, p) = 0 if q ̸= p 1 1

µDδ˜ ∗ ( f (q), e, f (p)) = µDδ˜ ∗ ( f (q), e, f (p)). 2 2 Let the result be true for all y ∈ X ∗ such that |y| = n − 1, n > 0 and x = ya, where a ∈ X1 , y ∈ X1∗ with |y| = n − 1. Then µDδ˜ ∗ (q, x, p) = µDδ˜ ∗ (q, ya, p) = ∨r∈Q1 {µDδ˜ ∗ (q, y, r) ∧ µDδ˜ ∗ (r, a, p)} ≤ ∨r∈Q1 {µDδ˜ ∗ ( f (q), g∗ (y), f (r)) ∧ 1 1 1 1 2 µDδ˜ ∗ ( f (r), g(a), f (p))} ≤ ∨r′ ∈Q2 {µDδ˜ ∗ ( f (q), g∗ (y), r′ )∧ µDδ˜ ∗ (r′ , g∗ (a), f (p))} = µDδ˜ ∗ ( f (q), g∗ (y)g(a), f (p)) = µDδ˜ ∗ 2 2 2 2 2 ( f (q), g∗ (ya), f (p)) = µDδ˜ ∗ ( f (q), g∗ (x), f (p)). Similarly, we have µDδ˜ ∗ (q, x, p) ≤ µDδ˜ ∗ ( f (q), g∗ (x), f (p)). 2

1

2

4 Type-2 fuzzy transformation semigroup In this section, we recall some basic concepts related to a transformation semigroups both in finite state automata and in type-2 fuzzy finite state automata (cf. [2], [6]). Also, we discuss the direct and the general direct product and shown that this product is a categorical product in type-2 fuzzy finite state automata. Recall from [6] that an equivalence relation ∼ on a semigroup (X, ∗) is called a congruence relation on X if ∀a, b ∈ X, a ∼ b ⇒ δ˜ (q, a, p) = δ˜ (q, b, p), ∀p, q ∈ Q. Let (Q, X, δ˜ , Jx ) be a type-2 fuzzy finite state automaton. Define a relation ≃ on X ∗ by x ≃ y ⇔ µDδ˜ (q, x, p) = µDδ˜ (q, y, p) and µDδ˜ (q, x, p) = µDδ˜ (q, y, p), ∀p, q ∈ Q and ∀x, y ∈ X ∗ . Then we have the following. Proposition 4.1. Let (Q, X, δ˜ , Jx ) be a type-2 fuzzy finite state automaton. Then the relation ≃ is a congruence relation on X ∗ . Proof. It is obvious that the relation ≃ is an equivalence relation on X ∗ . Let x, y ∈ X ∗ such that x ≃ y and z ∈ X ∗ . Then ∀p, q ∈ Q, µDδ˜ (q, xz, p) = ∨r∈Q {µDδ˜ (q, x, r) ∧ µDδ˜ (r, z, p)} = ∨r∈Q {µDδ˜ (q, y, r) ∧ µDδ˜ (r, z, p)} = µDδ˜ (q, yz, p). Similarly, we have µDδ˜ (q, xz, p) = µDδ˜ (q, yz, p). Thus xz ≃ yz. Similarly zx ≃ zy. Hence ≃ is a congruence relation on X ∗ . e = (Q, X, δ˜ , Jx ), let [x] = {y ∈ X ∗ : x ≃ y} and E(M) e = {[x] : x ∈ X ∗ }. For a given type-2 fuzzy finite state automaton M e e Define a binary operation ∗ on E(M) by [x] ∗ [y] = [xy], ∀[x], [y] ∈ E(M). Then we have the following. e = (Q, X, δ˜ , Jx ) be a type-2 fuzzy finite state automaton. Then (E(M), e ∗) is a finite semigroup Proposition 4.2. Let M with identity. e we have [x]∗[e] = [xe] = [x] = [ex] = [e]∗[x], Proof. It is obvious that ∗ is associative and well defined. For [x] ∈ E(M), e ∗). Thus (E(M), e ∗) is a semigroup with identity. Let x ∈ X ∗ and let x = x1 x2 .....xn , where [e] is the identity of (E(M), where x1 , x2 , ....., xn ∈ X. Then for all p, q ∈ Q, µDδ˜ ∗ (q, x, p) = ∨q1 ,q2 ,....,qn−1 ∈Q {µDδ˜ (q, x1 , q1 ) ∧ µDδ˜ (q1 , x2 , q2 ) ∧ ..... ∧ µDδ˜ (qn−1 , xn , p)} and µDδ˜ ∗ (q, x, p) = ∨q1 ,q2 ,....,qn−1 ∈Q {µDδ˜ (q, x1 , q1 ) ∧ µDδ˜ (q1 , x2 , q2 ) ∧ ..... ∧ µDδ˜ (qn−1 , xn , p)}. Thus as Q is finite, then µDδ˜ ∗ (q, x, p) and e ∗) is a finite semigroup with identity. µ ˜ ∗ (q, x, p) is finite. Hence (E(M), Dδ



90

e = (Q, X, δ˜ , Jx ) be a type-2 fuzzy finite state Now, we define another type of congruence relation on X ∗ . Let M ∗ automaton. Define a relation ≃ on X by x ≃ y as µDδ˜ (q, x, p) > 0 ⇔ µDδ˜ (q, y, p) > 0 and µDδ˜ (q, x, p) > 0 ⇔ µDδ˜ (q, y, p) > 0, ∀p, q ∈ Q and ∀x, y ∈ X ∗ . Then we have the following. Proposition 4.3. Let (Q, X, δ˜ , Jx ) be a type-2 fuzzy finite state automaton. Then the relation ≃ is a congruence relation on X ∗ . Proof. It is obvious that the relation ≃ is an equivalence relation on X ∗ . Let x, y ∈ X ∗ such that x ≃ y and z ∈ X ∗ . Then ∀p, q ∈ Q, µDδ˜ ∗ (q, zx, p) = ∨r∈Q {µDδ˜ ∗ (q, z, r) ∧ µDδ˜ ∗ (r, x, p)} > 0 ⇔ ∃r ∈ Q such that µDδ˜ ∗ (q, z, r) ∧ µDδ˜ ∗ (r, x, p) > 0 ⇔ ∃r ∈ Q such that µDδ˜ ∗ (q, z, r) ∧ µDδ˜ ∗ (r, y, p) > 0 ⇔ µDδ˜ ∗ (q, zy, p) = ∨r∈Q {µDδ˜ ∗ (q, z, r) ∧ µDδ˜ ∗ (r, y, p)} > 0. Similarly, we have µDδ˜ ∗ (q, zx, p) > 0 ⇔ µDδ˜ ∗ (q, zy, p) > 0. Thus zx ≃ zy. Similarly xz ≃ yz. Hence ≃ is a congruence relation on X ∗. e = (Q, X, δ˜ , Jx ). Let x ∈ X ∗ and let [[x]] = {y ∈ X ∗ : x ≃ y} and For given a type-2 fuzzy finite state automaton M ] ] ] e = {[[x]] : x ∈ X ∗ }. Define a binary operation e e by [[x]]e e Then we have E( M) ∗ on E( M) ∗[[y]] = [[xy]], ∀[[x]], [[y]] ∈ E( M). the following. ] e = (Q, X, δ˜ , Jx ) be a type-2 fuzzy finite state automaton. Then (E( e ∗) is a finite semigroup Proposition 4.4. Let M M),e ] e onto E(M). e with identity and [x] → [[x]] is a homomorphism of E(M) ] e we have [[x]]e Proof. Associativity of the e ∗ is trivial. For [[x]] ∈ E( M), ∗[[e]] = [[xe]] = [[x]] = [[ex]] = [[e]]e ∗[[x]], whereby ] ] ] e ∗). Thus (E(M),e e ∗) is a semigroup with identity. Now, define f : E(M) e → E( e by [[e]] is the identity of (E(M),e M) ∗ e f ([x]) = [[x]], ∀[x] ∈ E(M). Let x, y ∈ X and [x] ≃ [y]. Then ∀p, q ∈ Q, µDδ˜ ∗ (q, x, p) = µDδ˜ ∗ (q, y, p) and µDδ˜ ∗ (q, x, p) = µDδ˜ ∗ (q, y, p). Thus ∀p, q ∈ Q, µDδ˜ ∗ (q, x, p) > 0 ⇔ µDδ˜ ∗ (q, y, p) > 0 and µDδ˜ ∗ (q, x, p) > 0 ⇔ µDδ˜ ∗ (q, y, p) > 0. Thus e is finite x ≃ y or [[x]] = [[y]]. Hence f is well defined. Thus obviously f is an onto homomorphism. Also, as E(M) ] e is finite. E( M) e = (Q, X, δ˜ , Jx ) be a type-2 fuzzy finite state automaton. Then the type-2 fuzzy subset xMe of Definition 4.1. Let M e Q × Q by xM (q, p) = Dδ˜ ∗ (q, x, p) = [µDδ˜ ∗ (q, x, p), µDδ˜ ∗ (q, x, p)], ∀p, q ∈ Q and ∀x ∈ X ∗ . e = (Q, X, δ˜ , Jx ) be a type-2 fuzzy finite state automaton. Let S e = {xMe : x ∈ X ∗ }. Then Proposition 4.5. Let M M e

e

e

(1) xM oyM = (xy)M , ∀x, y ∈ X ∗ e e e e (2) (SMe , o) is a finite semigroup with identity, where o is defined as (xM oyM )(q, p) = ∨r∈Q {xM (q, r) ∧ yM (r, p)}. e Proof. (1) Let p, q ∈ Q. Then (xy)M (q, p) = Dδ˜ ∗ (q, x, p) = [µDδ˜ ∗ (q, x, p), µDδ˜ ∗ (q, x, p)]. Where µDδ˜ ∗ (q, xy, p) = e ∨r∈Q {µ ˜ ∗ (q, x, r) ∧ µ ˜ ∗ (r, y, p)} and µ ˜ ∗ (q, xy, p) = ∨r∈Q {µ ˜ ∗ (q, x, r) ∧ µ ˜ ∗ (r, y, p)}. Hence (xy)M (q, p) = Dδ˜ ∗ Dδ

Dδ

Dδ

Dδ

Dδ

e e e (q, xy, p) = ∨r∈Q {Dδ˜ ∗ (q, x, r) ∧ Dδ˜ ∗ (r, y, p)}. Thus (xy)M = xM oyM . e (2) SMe is finite since Q and Dδ˜ ∗ is finite and eM is the identity elements. Thus (SMe , o) is a finite semigroup with identity.

Definition 4.2. A type-2 fuzzy transformation semigroup (T2FTS, in short) is a 4-tuple A˜ = (Q, S, λ˜ , Jv ), where ˜ S is a nonempty finite semigroup, λ˜ is a type-2 fuzzy subset of Q is a nonempty finite set (the set of states of A), ˜ (Q × S × Q) × Jv , i.e., λ : (Q × S × Q) × Jv → [0, 1], where Jv ⊆ [0, 1] is a primary membership of v such that (1) if S contains the identity e, then { Dλ˜ (q, e, p) = [µDλ˜ (q, e, p), µDλ˜ (q, e, p)] =

1 if q = p 0 if q ̸= p



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(2) µDλ˜ (q, vw, p) = ∨r∈Q {µDλ˜ (q, v, r) ∧ µDλ˜ (r, w, p)} and µDλ˜ (q, vw, p) = ∨r∈Q {µDλ˜ (q, v, r) ∧ µDλ˜ (r, w, p)}, ∀p, q ∈ Q and ∀v, w ∈ S. If in addition, ∀p, q ∈ Q and ∀v, w ∈ S, µDλ˜ (q, v, p) = µDλ˜ (q, w, p) and µDλ˜ (q, v, p) = µDλ˜ (q, w, p) ⇒ v = w holds. Then (Q, S, λ˜ , Jv ) is called faithful T2FTS. Let A˜ = (Q, S, λ˜ , Jv ) be a T2FTS which is not faithful. Define a relation ∼ on S by ∀v, w ∈ S and p, q ∈ Q, v ∼ w ⇔ µDδ˜ (q, v, p) = µDδ˜ (q, w, p) and µDδ˜ (q, v, p) = µDδ˜ (q, w, p). Then it can be easily seen that ∼ is an equivalence relation on S. Also, let v, w,t ∈ S and v ∼ w. Then µDλ˜ (q, vt, p) = ∨r∈Q {µDλ˜ (q, v, r) ∧ µDλ˜ (r,t, p)} = ∨r∈Q {µDλ˜ (q, w, r) ∧ µDλ˜ (r,t, p)} = µDλ˜ (q, wt, p). Similarly, we have µDλ˜ (q, vt, p) = µDλ˜ (q, wt, p). Thus vt ∼ wt. Similarly tv ∼ tw. Hence ∼ is a congruence relation on S. Let [v] be the equivalence class of v induced by the relation ∼ and S/ ∼= {[v] : v ∈ S}. Define λ˜ ′ : (Q × S/ ∼ ×Q) × Jx → [0, 1] by µDλ˜ ′ (q, [x], p) = µDλ˜ (q, x, p) and µDλ˜ ′ (q, [x], p) = µDλ˜ (q, x, p), ∀p, q ∈ Q and ∀[x] ∈ S/ ∼. Now { 1 if q = p µDλ˜ ′ (q, [e], p) = µDλ˜ ′ (q, [e], p) = 0 if q ̸= p Also, µDλ˜ ′ (q, [x][y], p) = µDλ˜ ′ (q, [xy], p) = µDλ˜ (q, xy, p) = ∨r∈Q {µDλ˜ (q, x, r) ∧ µDλ˜ (r, y, p)} = ∨r∈Q {µDλ˜ ′ (q, [x], r) ∧ µDλ˜ ′ (r, [y], p)}. Similarly, we have µDλ˜ ′ (q, [x][y], p) = µDλ˜ ′ (q, [xy], p) = ∨r∈Q {µDλ˜ ′ (q, [x], r) ∧ µDλ˜ ′ (r, [y], p)}, ∀[x], [y] ∈ S/ ∼. Again, let µDλ˜ ′ (q, [x], p) = µDλ˜ ′ (q, [y], p) and µDλ˜ ′ (q, [x], p) = µDλ˜ ′ (q, [y], p), ∀p, q ∈ Q. Then µDλ˜ (q, x, p) = µDλ˜ (q, y, p) and µDλ˜ (q, x, p) = µDλ˜ (q, y, p), ∀p, q ∈ Q. Thus x ∼ y, whereby [x] = [y] showing that (Q, S/ ∼, λ˜ ′ , Jx ) is a faithful T2FTS. e = (Q, X, δ˜ , Jx ) be a type-2 fuzzy finite state automaton. Then (Q, E(M), e λ˜ , J[v] ) is a faithful Proposition 4.6. Let M T2FTS, where µDλ˜ (q, [x], p) = µDδ˜ ∗ (q, x, p) and µDλ˜ (q, [x], p) = µDδ˜ ∗ (q, x, p), ∀p, q ∈ Q and ∀x ∈ X ∗ . e is a finite semigroup with identity [e] from Proposition 4.2. Obviously, Proof. E(M) {

µDλ˜ (q, [e], p) = µDλ˜ (q, [e], p) =

µDδ˜ ∗ (q, e, p) = µDδ˜ ∗ (q, e, p) = 1 if q = p µDδ˜ ∗ (q, e, p) = µDδ˜ ∗ (q, e, p) = 0 if q ̸= p

e Then µ ˜ (q, [x] ∗ [y], p) = µ ˜ (q, [xy], p) = µ ˜ ∗ (q, xy, p) = ∨r∈Q {µ ˜ ∗ (q, x, r) ∧ Let q ∈ Q and [x], [y] ∈ E(M). Dλ Dλ Dδ Dδ µDδ˜ ∗ (r, y, p)} = ∨r∈Q {µDλ˜ (q, [x], r) ∧ µDλ˜ (r, [y], p)}. Similarly, we have µDλ˜ (q, [x] ∗ [y], p) = µDλ˜ (q, [xy], p) = ∨r∈Q { µDλ˜ (q, [x], r)∧ µDλ˜ (r, [y], p)}. Also, let µDλ˜ (q, [x], p) = µDλ˜ (q, [y], p) and µDλ˜ (q, [x], p) = µDλ˜ (q, [y], p), i.e., µDδ˜ ∗ (q, x e λ˜ , J[x] ) is a , p) = µDδ˜ ∗ (q, y, p) and µDδ˜ ∗ (q, x, p) = µDδ˜ ∗ (q, y, p), ∀q ∈ Q. Thus x ∼ y or [x] = [y]. Hence (Q, E(M), faithful T2FTS. e = (Q, X, δ˜ , Jx ), we shall denote by T2FTS (M), e the T2FTS (Q, E(M), e λ˜ , J[v] ), For type-2 fuzzy finite state automaton M e and call it the T2FTS associated with M. Definition 4.3. A homomorphism from a T2FTS A˜ 1 = (Q1 , S1 , λ˜ 1 , Jv1 ) to T2FTS A˜ 2 = (Q2 , S2 , λ˜ 2 , Jv2 ) is a pair of maps (α , β ), where α : Q1 → Q2 and β : S1 → S2 are functions such that (1) β (uv) = β (u)β (v), ∀u, v ∈ S1 , (2) If S1 and S2 contain the identity e1 and e2 respectively, then β (e1 ) = e2 , and (3) µDλ˜ 1 (q, v, p) ≤ µDλ˜ 2 (α (q), β (v), α (p)). and µDλ˜ 1 (q, v, p) ≤ µDλ˜ 2 (α (q), β (v), α (p)), ∀p, q ∈ Q1 , ∀v ∈ S1 . A homomorphism (α , β ) : A˜ 1 → A˜ 2 is called an isomorphism if α and β are both one-one and onto. Let S be a semigroup with identity e and (Q, S, λ˜ , Jv ) be a faithful T2FTS. Define type-2 fuzzy finite state aue = (Q, X, δ˜ , Jx ) by taking δ˜ = λ˜ . Consider type-2 fuzzy finite state automaton(M) e = (Q, E(M), e ρ , J[v] ), tomaton M ∗ e where E(M) = S / ∼ with µDρ (q, [v], p) = µDδ˜ ∗ (q, v, p) and µDρ (q, [v], p) = µDδ˜ ∗ (q, v, p) . Now, for all p, q ∈ Q,



92

µDρ (q, [e], p) = µDρ (q, [e], p) { =

µDδ˜ ∗ (q, e, p) = µDδ˜ ∗ (q, e, p) = 1 if q = p µDδ˜ ∗ (q, e, p) = µDδ˜ ∗ (q, e, p) = 0 if q ̸= p

Hence µDρ (q, [e], p) = µDρ (q, [λ˜ ], p) and µDρ (q, [e], p) = µDρ (q, [λ˜ ], p), where λ˜ is the empty word in S∗ . Thus [e] = [λ˜ ]. e = (Q, X, δ˜ , Jx ) be a type-2 fuzzy finite state automata and S be a semigroup with identity e. Proposition 4.7. Let M e Then T2FTS (M) is isomorphic to a faithful T2FTS A˜ = (Q, S, λ˜ , Jv ). e be maps such that α (q) = q and β (v) = [v], ∀[v] ∈ S and ∀q ∈ Q. Let • be the Proof. Let α : Q → Q and β : S → E(M) binary operation of S and for a, b ∈ S, a • b ∈ S, ab ∈ S∗ . Then µDδ˜ ∗ (q, a • b, p) = µDδ˜ (q, a • b, p) = µDλ˜ (q, a • b, p) = ∨r∈Q {µDλ˜ (q, a, r) ∧ µDλ˜ (r, b, p)} = ∨r∈Q {µDδ˜ (q, a, r) ∧ µDδ˜ (r, b, p)} = µDδ˜ ∗ (q, ab, p). Similarly, we have µDδ˜ ∗ (q, a • b, p) = µDδ˜ ∗ (q, ab, p), ∀p, q ∈ Q. Thus [a • b] = [ab], showing that β (a • b) = [a • b] = [ab] = [a][b] = β (a)β (b). Also, µDρ (α (q), β (v), α (p)) = µDρ (q, [v], p) = µDδ˜ ∗ (q, v, p) = µDδ˜ (q, v, p) = µDλ˜ (q, v, p). Similarly, we have µDρ (α (q), β (v), α (p)) = µDλ˜ (q, v, p). Now it remains to show that β is one-one and onto. Let v, w ∈ S be such that β (v) = β (w). Then [v] = [w]. Thus µDδ˜ ∗ (q, v, p) = µDδ˜ ∗ (q, w, p), and µDδ˜ ∗ (q, v, p) = µDδ˜ ∗ (q, w, p) or that µDδ˜ (q, v, p) = µDδ˜ (q, w, p) ⇒ µDλ˜ (q, v, p) = µDλ˜ (q, w, p) and µDδ˜ (q, v, p) = µDδ˜ (q, w, p) ⇒ µDλ˜ (q, v, p) = µDλ˜ (q, w, p), or that v = w. As A˜ is faithful. Thus β is one-one. Also, it can be easily seen that if ci ∈ S, i ∈ [1, n], then by induction [c1 • c2 • .....• cn ] = [c1 c2 .....cn ]. Finally, let [x] ∈ E(M). If x = λ˜ then [λ˜ ] = [e] and β (e) = [λ˜ ]. Let x = a1 a2 .....an , ai ∈ S, i ∈ [1, n]. Then β (a1 • a2 • ..... • an ) = [a1 • a2 • ..... • an ] = [a1 a2 .....an ] = [x]. Thus β is onto. f1 = (Q1 , X1 , δ˜1 , Jx ) and M f2 = (Q2 , X2 , δ˜2 , Jx ) be a type-2 fuzzy finite state automata. Then the Definition 4.4. Let M 1 2 f1 f1 × M f2 = (Q1 × Q2 , X1 × X2 , δ˜1 × δ˜2 , J(x ,x ) ) is called (full) direct product of M type-2 fuzzy finite state automata M 1 2 ˜ ˜ f2 , where δ1 × δ2 : ((Q1 × Q2 ) × (X1 × X2 ) × (Q1 × Q2 )) × J(x ,x ) → [0, 1], is defined as follows: and M 1 2 µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 ), (p1 , p2 )) = µDδ˜1 (q1 , x1 , p1 ) ∧ µDδ˜2 (q2 , x2 , p2 ), and µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 ), (p1 , p2 )) = µDδ˜1 (q1 , x1 , p1 ) ∧ µDδ˜2 (q2 , x2 , p2 ), ∀(q1 , q2 ), (p1 , p2 ) ∈ (Q1 × Q2 ) and ∀(x1 , x2 ) ∈ X1 × X2 . An examination of the ‘categorical product’ in the category T2FFSA leads to a concept of ‘direct product’ of type-2 f1 = (Q1 , X1 , δ˜1 , Jx ) fuzzy finite state automata, which we illustrate here for two type-2 fuzzy finite state automata M 1 ˜ f and M2 = (Q2 , X2 , δ2 , Jx2 ) as follows. (X1 × X2 , appearing below is the ‘direct product’ of the monoids X1 and X2 . Thus it is the cartesian product of X1 and X2 , considered as a monoid, whose binary operation is defined as (x1 , x2 )(x1′ , x2′ ) = (x1 x1′ , x2 x2′ ), for (x1 , x2 ), (x1′ , x2′ ) ∈ X1 × X2 , and whose identity element is (eX1 , eX2 ), where eX1 and eX2 are the identities of X1 and X2 respectively.) f2 = (Q2 , X2 , δ˜2 , Jx ) be a type-2 fuzzy finite state automata. Then the direct f1 = (Q1 , X1 , δ˜1 , Jx ) and M Define Let M 1 2 f f f f product of M1 and M2 is M1 × M2 = (Q1 × Q2 , X1 × X2 , δ˜1 × δ˜2 , J(x1 ,x2 ) ). Then

δ˜1 × δ˜2 : ((Q1 × Q2 ) × (X1 × X2 ) × (Q1 × Q2 )) × J(x1 ,x2 ) → [0, 1] as

µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 ), (p1 , p2 )) = µDδ˜1 (q1 , x1 , p1 ) ∧ µDδ˜2 (q2 , x2 , p2 ), and µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 ), (p1 , p2 )) = µDδ˜1 (q1 , x1 , p1 ) ∧ µDδ˜2 (q2 , x2 , p2 ) ∀(q1 , q2 ), (p1 , p2 ) ∈ (Q1 × Q2 ) and ∀(x1 , x2 ) ∈ X1 × X2 . Then

µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (ex1 , ex2 ), (p1 , p2 )) = µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (ex1 , ex2 ), (p1 , p2 )) =

{

1 0

if (q1 , q2 ) = (p1 , p2 ) if (q1 , q2 ) ̸= (p1 , p2 ).

Next, for (x1′ , x2′ ) ∈ X1 × X2 , µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 )(x1′ , x2′ ), (p1 , p2 )) = µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 x1′ , x2 x2′ ), (p1 , p2 )) = µDδ˜1 (q1 , x1 x1′ , p1 ) ∧ µDδ˜2 (q2 , x2 x2′ , p2 ) = [∨{µDδ˜1 (q1 , x1 , s) ∧ µDδ˜1 (s, x1′ , p1 ) : s ∈ Q1 }] ∧ [∨{µDδ˜2 (q2 , x2 ,t)



93

∧ µDδ˜2 (t, x2′ , p2 ) : t ∈ Q2 }] = ∨{(µDδ˜1 (q1 , x1 , s) ∧ µDδ˜1 (s, x1′ , p1 )) ∧ (µDδ˜2 (q2 , x2 ,t) ∧ µDδ˜2 (t, x2′ , p2 )) : s ∈ Q1 ,t ∈ Q2 }. On the other hand, ∨{µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 ), (s,t))∧ µD(δ˜1 ×δ˜2 ) ((s,t), (x1′ , x2′ ), (p1 , p2 )) : s ∈ Q1 ,t ∈ Q2 } = ∨{(µDδ˜1 (q1 , x1 , s)∧ µDδ˜2 (q2 , x2 ,t)) ∧ (µDδ˜1 (s, x1′ , p1 ) ∧ µDδ˜1 (t, x2′ , p2 )) : s ∈ Q1 ,t ∈ Q2 } = ∨{(µDδ˜1 (q1 , x1 , s) ∧ µDδ˜1 (s, x1′ , p1 )) ∧ (µDδ˜2 (q2 , x2 ,t) ∧ µDδ˜1 (t, x2′ , p2 )) : s ∈ Q1 ,t ∈ Q2 }. Thus, µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 )(x1′ , x2′ ), (p1 , p2 )) = ∨{µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 ), (s,t)) ∧ µD(δ˜1 ×δ˜2 ) ((s,t), (x1′ , x2′ ), (p1 , p2 )) : s ∈ Q1 ,t ∈ Q2 }. Similarly, we can derive µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 )(x1′ , x2′ ) , (p1 , p2 )) = ∨{µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 ), (s,t)) ∧ µD(δ˜1 ×δ˜2 ) ((s,t), (x1′ , x2′ ), (p1 , p2 )) : s ∈ Q1 ,t ∈ Q2 } This shows that (Q1 × Q2 , X1 × X2 , δ˜1 × δ˜2 , J(x1 ,x2 ) ) is a type-2 fuzzy finite state automata, which we shall refer to f2 = (Q2 , X2 , δ˜2 , Jx ) and will f1 = (Q1 , X1 , δ˜1 , Jx ) and M as the direct product of the type-2 fuzzy finite state automata M 1 2 f1 × M f2 . denote it as M f1 and M f2 cf., e.g., D¨orfler [1]. Remark 4.1. This direct product may be interpreted as the ‘parallel composition’ of M f1 × M f2 of M f1 , M f2 ∈ T2FFSA is the categorical direct product in T2FFSA. Proposition 4.8. The direct product M f1 × M f2 to M f1 and M f2 in T2FFSA. Let M f1 = Proof. We first need to identify the two ‘projection morphisms’ from M f2 = (Q2 , X2 , δ˜2 , Jx ). Let h1 : Q1 × Q2 → Q1 , h2 : Q1 × Q2 → Q2 , k1 : X1 × X2 → X1 and k2 : X1 × (Q1 , X1 , δ˜1 , Jx1 ) and M 2 X2 → X2 be the projection maps associated with the cartesian products Q1 × Q2 and X1 × X2 . We show that (h1 , k1 ) : f1 × M f2 → M f1 and (h2 , k2 ) : M f1 × M f2 → M f2 are T2FFSA-morphisms. Let ((q1 , q2 ), (x1 , x2 ), (p1 , p2 ), u) ∈ ((Q1 × M Q2 ) × (X1 × X2 ) × (Q1 × Q2 ) × J(x1 ,x2 ) . Then µDδ˜1 (h1 (q1 , q2 ), k1 (x1 , x2 ), h1 (p1 , p2 )) = µDδ˜1 (q1 , x1 , p1 ) ≥ µDδ˜1 (q1 , x1 , p1 )∧ µDδ˜2 (q2 , x2 , p2 ) = µD(δ˜1 ×δ˜2 ) ((q1 , q2 ), (x1 , x2 ), (p1 , p2 )). Similarly, we have derived µDδ˜1 (h1 (q1 , q2 ), k1 (x1 , x2 ), h1 (p1 , p2 )) ≥ µ ˜ ˜ ((q1 , q2 ), (x1 , x2 ), (p1 , p2 )). Thus, δ˜1 (h1 (q1 , q2 ), k1 (x1 , x2 ), h1 (p1 , p2 )) ≥ (δ˜1 × δ˜2 )((q1 , q2 ), (x1 , D(δ1 ×δ2 )

f1 × M f2 → M f1 x2 ), (p1 , p2 )), ∀((q1 , q2 ), (x1 , x2 ), (p1 , p2 )) ∈ (Q1 × Q2 ) × (X1 × X2 ) × (Q1 × Q2 ), whereby (h1 , k1 ) : M f1 × M f2 → M f2 can be seen to be an T2FFSA-morphism. Next, let is an T2FFSA-morphism. Similarly, (h2 , k2 ) : M e ′ = (Q′ , X ′ , δ˜ ′ , Jx′ ) ∈ T2FFSA and two T2FFSA-morphisms ( f1 , g1 ) : M e′ → M f1 and ( f2 , g2 ) : M e′ → M f2 be given. M ′ e f f We show that there exists a unique T2FFSA-morphism ( f , g) : M → M1 × M2 such that the following diagram commutes. e′ M Q Q ( f1 , g1 ) Q ( f ,g ) ( f , g) Q 2 2 Q Q ? + s Q f1 - M f2 f f M M1 × M2 (h1 , k1 ) (h2 , k2 )

f1 × M f2 → M f1 and (h2 , k2 ) : M f1 × M f2 → M f2 are the projection maps. We choose the f and g in Here, (h1 , k1 ) : M following way. Let f : Q′ → Q1 × Q2 and g : X ′ → X1 × X2 be the maps given by f (q′ ) = ( f1 (q′ ), f2 (q′ )) and g(x′ ) = (g1 (x′ ), g2 (x′ )), ∀(q′ , x′ ) ∈ Q′ × X ′ . Let ((q′ , x′ , p′ ), u′ ) ∈ ((Q′ × X ′ × Q′ ) × Jx′ ). As both ( f1 , g1 ) and ( f2 , g2 ) are T2FFSA-morphisms, µDδ˜1 ( f1 (q′ ), g1 (x′ ), f1 (p′ )) ≥ µDδ˜ ′ (q′ , x′ , p′ ) and µDδ˜1 ( f1 (q′ ), g1 (x′ ), f1 (p′ )) ≥ µDδ˜ ′ (q′ , x′ , p′ ) also µDδ˜2 ( f2 (q′ ), g2 (x′ ), f2 (p′ )) ≥ µDδ˜ ′ (q′ , x′ , p′ ) and µDδ˜2 ( f2 (q′ ), g2 (x′ ), f2 (p′ )) ≥ µDδ˜ ′ (q′ , x′ , p′ ), whereby µDδ˜1 ( f1 (q′ ), g1 (x′ ), f1 (p′ )) ∧ µDδ˜2 ( f2 (q′ ), g2 (x′ ), f2 (p′ )) ≥ µDδ˜ ′ (q′ , x′ , p′ ) and µDδ˜1 ( f1 (q′ ), g1 (x′ ), f1 (p′ ))∧ µDδ˜2 ( f2 (q′ ), g2 (x′ ), f2 (p′ )) ≥ µDδ˜ ′ (q′ , x′ , p′ ). Thus, µD(δ˜1 ×δ˜2 ) ( f (q′ ), g(x′ ), f (p′ )) = µD(δ˜1 ×δ˜2 ) (( f1 (q′ ), f2 (q′ )), (g1 (x′ ), g2 (x′ )), ( f1 (p′ ), f2 (p′ ))) = µDδ˜1 ( f1 (q′ ), g1 (x′ ) , f1 (p′ ))∧ µDδ˜2 ( f2 (q′ ), g2 (x′ ), f2 (p′ )) ≥ µDδ˜ ′ (q′ , x′ , p′ ). Similarly, we can derive µD(δ˜1 ×δ˜2 ) ( f (q′ ), g(x′ ), f (p′ )) ≥ µDδ˜ ′ ( q′ , x′ , p′ ). Hence ( f , g) is an T2FFSA-morphism. Also, the definitions of f and g are such that we obviously have (h1 , k1 ) ◦ ( f , g) = ( f1 , g1 ) and (h2 , k2 ) ◦ ( f , g) = ( f2 , g2 ). e′ → M f1 × M f2 such that To prove the uniqueness of ( f , g), let there exist another T2FFSA-morphism ( f ′ , g′ ) : M



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(h1 , k1 ) ◦ ( f ′ , g′ ) = ( f1 , g1 ) and (h2 , k2 ) ◦ ( f ′ , g′ ) = ( f2 , g2 ), i.e., h1 ◦ f ′ = f1 , k1 ◦ g′ = g1 , h2 ◦ f ′ = f2 , and k2 ◦ g′ = g2 . We then have h1 ◦ f ′ = h1 ◦ f , k1 ◦ g′ = k1 ◦ g, h2 ◦ f ′ = h2 ◦ f , and k2 ◦ g′ = k2 ◦ g, whereby f = f ′ and g = g′ . Thus f1 × M f2 is the categorical direct prod( f ′ , g′ ) = ( f , g), proving the uniqueness of ( f , g). Hence the direct product M uct. f1 = (Q1 , X1 , δ˜1 , Jx ) and M f2 = (Q2 , X2 , δ˜2 , Jx ) be a type-2 fuzzy finite state automata. Let X be a finite set and Let M 1 2 f : X → X1 × X2 be a map. Also, let ∏1 and ∏2 be the projection mapping of X1 × X2 onto X1 and X2 respectively, i.e., ∏1 : X1 × X2 → X1 and ∏2 : X1 × X2 → X2 . Then the concept of generalized direct product of type-2 fuzzy finite state automata are given below. f1 = (Q1 , X, δ˜1 , Jx ) and M f2 = (Q2 , X, δ˜2 , Jx ) be a type-2 fuzzy finite state automata. Then the Definition 4.5. Let M f f f1 and M f2 , type-2 fuzzy finite state automata M1 ∗ M2 = (Q1 × Q2 , X, δ˜1 ∗ δ˜2 , Jx ) is called general direct product of M ˜ ˜ where δ1 ∗ δ2 : ((Q1 × Q2 ) × X × (Q1 × Q2 )) × Jx → [0, 1], is defined as follows: µD(δ˜1 ∗δ˜2 ) ((q1 , q2 ), x, (p1 , p2 )) = µD(δ˜1 ∗δ˜2 ) ((q1 , q2 ), (∏1 ( f (x)), ∏2 ( f (x))), (p1 , p2 )) = µDδ˜1 (q1 , ∏1 ( f (x)), p1 )∧ µDδ˜2 (q2 , ∏2 ( f (x)), p2 ), and µD(δ˜1 ∗δ˜2 ) ((q1 , q2 ), x, (p1 , p2 )) = µD(δ˜1 ∗δ˜2 ) ((q1 , q2 ), (∏1 ( f (x)), ∏2 ( f (x))), (p1 , p2 )) = µDδ˜1 (q1 , ∏1 ( f (x)), p1 )∧ µDδ˜2 (q2 , ∏2 ( f (x)), p2 ), ∀(q1 , q2 ), (p1 , p2 ) ∈ (Q1 × Q2 ), ∀x ∈ X. f1 × M f2 reduces to the Remark 4.2. (1). If X = X1 × X2 and f is the identity map, then the general direct product M (full) direct product. (2). As, from above Proposition 4.8. If X = X1 × X2 and f is the identity map, then the general direct product f1 × M f2 is also, the categorical general direct product. M 5 Conclusion Chiefly inspired from [6] and [7]. We have introduced and studied here the concept of type-2 fuzzy finite state automata, homomorphism between two type-2 fuzzy finite state automata and transformation semigroup associated with a type-2 fuzzy finite state automata. Finally, we have introduced the concept of product of type-2 fuzzy finite state automata. We hope that, like fuzzy finite state machine, rough finite state machine, type-2 fuzzy finite state automata which is another dimension of applications of type-2 fuzzy set theory, will attract the researchers began to work on type-2 fuzzy finite state automata and finding more successful applications of type-2 fuzzy finite state automata. References [1] W. D¨orfler, The direct product of automata and quasi-automata, Mathematical Foundation of Computer Science, Lecture Notes in Comput. Sci. Springer-Verlag, 45 (1976) 270-276. https://doi.org/10.1007/3-540-07854-1 186 [2] W. M. L. Holcombe, Algebraic automata theory, Cambridge: Cambridge University Press, (1982). https://doi.org/10.1017/CBO9780511525889 [3] Y. H. Kim, J. G. Kim, S. J. Cho, Products of T -generalized state machines and T -generalized transformation semigroups, Fuzzy Sets and Systems, 93 (1998) 87-97. https://doi.org/10.1016/S0165-0114(96)00205-9 [4] H. V. Kumbhojkar, S. R. Chaudhri, On proper fuzzification of fuzzy finite state machines, Int. J. Fuzzy Math, 4 (2008) 1019-1027. [5] J. M. Mendel, R. I. John, Type-2 Fuzzy sets made simple, IEEE Trans. Fuzzy Syst, 10 (2002) 117-127. https://doi.org/10.1109/91.995115



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[6] J. N. Mordeson, D. S. Malik, Fuzzy Automata and Languages: Theory and Applications, Chapman and Hall/CRC. London/Boca Raton, (2002). https://doi.org/10.1201/9781420035643 [7] D. S. Malik, J. N. Mordeson, M. K. Sen, Semigroups of fuzzy finite state machine, Advances in Fuzzy Theory and Technology, Bookswright Press, 2 (1994) 87-98. [8] D. S. Malik, J. N. Mordeson, M. K. Sen, Submachines of fuzzy finite state machine, J. Fuzzy Math, 2 (1994) 781-792. [9] D. Qiu, Characterizations of fuzzy finite automata, Fuzzy Sets and Systems, 141 (2004) 391-414. https://doi.org/10.1016/S0165-0114(03)00202-1 [10] E. S. Santos, Maximin automata, Inform. and Control, 12 (1968) 367-377. https://doi.org/10.1016/s0019-9958(68)90864-4 [11] H. Y. Wu, Y. Y. Wu, J. P. Luo, An interval type-2 fuzzy rough sets model for attribute reduction, IEEE Trans. Fuzzy Syst, 17 (2009) 301-315. https://doi.org/10.1109/TFUZZ.2009.2013458 [12] W. G. Wee, On generalizations of adaptive algorithm and application of the fuzzy sets concept to pattern classification, Ph. D. Thesis, Purdue University, Lafayette, IN (1967). [13] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965) 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X [14] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. Part 1, Inform. Sci, 8 (1975) 199-249. https://doi.org/10.1016/0020-0255(75)90036-5
