Lattice-valued finite state machines and lattice-valued transformation semigroups

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Fuzzy Sets and Systems 208 (2012) 1 – 21 www.elsevier.com/locate/fss

Lattice-valued finite state machines and lattice-valued transformation semigroups M. Gómez, I. Lizasoain∗,1 , C. Moreno1 Departamento de Matemáticas, Universidad Pública de Navarra, Campus de Arrosadía, 31006 Pamplona, Spain Received 15 October 2010; received in revised form 25 May 2012; accepted 30 May 2012 Available online 8 June 2012

Abstract This paper provides a generalization of known results about fuzzy finite state machines, fuzzy transformation semigroups and their relationship by broading the truth values domain from the interval [0,1] to a complete lattice endowed with a t-norm and a t-conorm. So, we deal with the concepts of L-fuzzy finite state machines and L-fuzzy transformation semigroups and we prove that the cited generalization is possible if and only if the t-norm and the t-conorm satisfy a distributive property. If we consider the complete lattice of the closed intervals inside the original lattice L, we give methods to obtain an interval lattice-valued finite state machine and an interval lattice-valued transformation semigroup from two L-fuzzy finite state machines or two L-fuzzy transformation semigroups, respectively. Conversely, we show two different ways to build a faithful L-fuzzy transformation semigroup from an interval lattice-valued state machine. In fact, both methods give the same result. © 2012 Elsevier B.V. All rights reserved. Keywords: Lattice; Lattice interval; Finite state machines; Transformation semigroups; t-Norm; t-Conorm

1. Introduction Fuzzy finite state machines (ffsm) and fuzzy transformation semigroups (fts) as well as the relationship between them have been widely studied in the literature (see [15]). The basic idea in their formulation is that, unlike the crisp case, they can switch from one state to another one to a certain truth degree between 0 and 1. So, a ffsm is a triple (Q, X, ) where Q and X are finite sets and  is a map from Q × X × Q to [0, 1]. In a similar way, a fts is a triple (Q, U, ) where Q is a set, U is a semigroup and  : Q × U × Q → [0, 1] is a map satisfying:  (q, uv, p) = {(q, u, r ) ∧ (r, v, p)|r ∈ Q} for any p, q ∈ Q and u, v ∈ U. Notice that neither initial nor final states are considered in these kinds of fuzzy automata. Our paper deals with fuzzy finite state machines and fuzzy finite transformation semigroups, as well as the relationships between them, in a more general case. Our truth structure will be a complete lattice (L , ⱕ L ) endowed with a t-norm T and a t-conorm S, denoted by L = (L , ⱕ L , T, S). Our aim is to generalize known results for the usual fuzzy case L = ([0, 1], ⱕ , ∧, ∨) to this general case. ∗ Corresponding author. Tel.: +34 948 169544; fax: +34 948 166057.

E-mail addresses: [email protected] (M. Gómez), [email protected] (I. Lizasoain), [email protected] (C. Moreno). 1 The authors are partially supported by MTM2010-19938-C03-03.

0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.05.012

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The use of a complete lattice (L , ⱕ L ) as the truth structure of finite automata is common in the literature. In [2] deterministic and non-deterministic L-fuzzy automata (with initial and final states) are compared in order to show that they can accept the same L-fuzzy languages. In that paper the only t-norm and t-conorm considered on (L , ⱕ L ) are, respectively, the infimum and the supremum. In [12] the languages accepted by fuzzy automata are studied in the case that the truth structure is a monoid with identity, (L , ⱕ L , ·, 1 L ) where (L , ⱕ L ) is a complete lattice. The only t-conorm considered is the one given by the supremum. If · is a t-norm defined on (L , ⱕ L ), then (L , ⱕ L , ·, 1 L ) agrees with a particular case of the truth structure considered in the present paper. In this sense, Theorem 5.1 can be considered a generalization of Theorem 3.1 of [12]. In this paper, for any complete lattice L = (L , ⱕ L , T, S), an L-fuzzy finite state machine (L-ffsm) is a triple (Q, X, ) where  is a map from Q × X × Q to L. In the same way, an L-fuzzy transformation semigroup (L-fts) is a triple (Q, U, ) where U is a semigroup and  : Q × U × Q → L is a map satisfying (q, uv, p) = S{T ((q, u, r ), (r, v, p))|r ∈ Q} for all p, q ∈ Q, u, v ∈ U. We prove that the L-ffsms together with the L-ffsm homomorphisms constitute a category and study conditions making the composition of S-strong L-ffsm homomorphisms S-strong too. In a similar way, we show that the L-ftss together with the L-fts homomorphisms constitute a category and study conditions making the composition of S-strong L-fts homomorphisms S-strong too. It is clear that these results generalize similar well-known results for ffsms and ftss with membership values in L = ([0, 1], ⱕ , ∧, ∨). We show that the usual construction of a faithful L-fts starting from an L-ffsm is possible if and only if the t-conorm S is distributive with respect to the t-norm T defined on (L , ⱕ L ). If S is the join, the previous property is equivalent to the residuation principle for T (see [8]). Moreover, if the distributive property holds, each S-strong L-ffsm homomorphism (bijective between the sets of states) provides an S-strong L-fts homomorphism between the respective faithful L-ftss, i.e., the construction is functorial in a sense. In addition, this construction preserves the composition of L-ffsm homomorphisms. At last, if X is a semigroup and (Q, X, ) is an L-fts, then the faithful L-fts provided by (Q, X, ) is strongly isomorphic to (Q, X, ). We wonder if an admissible relation defined in an L-ffsm provides an admissible relation in the L-fts obtained from it, like in the usual fuzzy case. The answer is affirmative if L is a chain endowed with the t-conorm given by the join. As a particular case of a complete lattice, we consider the lattice L I consisting of all the closed intervals inside the lattice L and study the L I -fuzzy finite state machines and the L I -fuzzy transformation semigroups. These automata allow us to switch from one state to another one to a certain truth degree that is not an exact lattice value, due to either uncertainty or lack of knowledge, vague information, etc. In these cases, a lattice interval can model the truth degree better than a lattice element. It is known that, if L = ([0, 1], ⱕ , ∧, ∨), the L I -fuzzy sets are mathematically equivalent to the intuitionistic fuzzy sets (see [9]). Intuitionistic ffsms and intuitionistic ftss, as well as their relationship, have been recently studied by Jun (see [10,11]), but it is clear, looking through the literature, that the interval-valued fuzzy sets are more advisable in order to model uncertainty in some applications (see [4]). In this paper we develop the theory of these kinds of fuzzy automata in a general lattice-theoretic framework, including the known theory for fuzzy automata with [0, 1] as truth values domain, the interval-valued fuzzy automata or the fuzzy automata whose truth values domain is the set of intervals contained in a lattice (interval lattice-valued fuzzy automata). In order to retrieve the information given by an L I -ffsm or an L I -fts, we build both a functor from the category of the L I -ffsms to the category of the L-ffsms and a functor from the category of the L I -ftss to the category of the L-ftss. Moreover, starting from an interval lattice-valued fuzzy finite state machine, we give two different ways of getting a faithful L-fts and show that they get the same result in the case of a residuated lattice. Conversely we give ways to build an L I -fuzzy finite state machine from two L-ffsms and an L I -fuzzy transformation semigroup from two L-ftss. In the residuated lattice case, we also show that the two natural faithful L I -ftss provided by two L-ffsms are exactly the same. The paper is organized as follows. Section 2 is devoted to introduce the theoretical concepts about t-norms and t-conorms we need further. Section 3 studies the concepts of an L-ffsm and an L-ffsm homomorphism. Section 4 does the same with L-ftss and L-fts homomorphisms. Section 5 gives the way to obtain an L-fts from an L-ffsm and

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vice-versa. Section 6 shows methods to obtain an L-ffsm and an L-fts from an interval lattice-valued finite state machine and an interval lattice-valued transformation semigroup, respectively. Finally, Section 7 analyzes the converse process to obtain interval lattice-valued objects from lattice-valued ones. We finish with some conclusions, future research and the references. 2. Preliminaries Throughout this section (L , ⱕ L ) will be a complete lattice, this is to say a partially ordered set in which every subset has an infimum (greatest lower bound) and a supremum (least upper bound). We will denote by 1 L the greatest element of the whole set L and by 1 L the least element of L. Definition 2.1. A map T : L × L → L is said to be a t-norm defined in (L , ⱕ L ) if, whenever a, b, c ∈ L, it holds (i) (ii) (iii) (iv)

T (a, b) = T (b, a), T (T (a, b), c) = T (a, T (b, c)), b ⱕ L c ⇒ T (a, b) ⱕ L T (a, c), T (a, 1 L ) = a. In particular, T (1 L , 1 L ) = 1 L .

Remark 2.2.

(i) If T : L × L → L is a t-norm in L, then

T (a, b) ⱕ L T (a, 1 L ) = a for any a, b ∈ L . (ii) In particular, for any b ∈ L, T (0 L , b) ⱕ 0 L ⇒ T (0 L , b) = 0 L . (iii) T (0 L , 0 L ) = 0 L . Definition 2.3. A map S : L × L → L is said to be a t-conorm in (L , ⱕ L ) if, whenever a, b, c ∈ L, it holds (i) (ii) (iii) (iv)

S(a, b) = S(b, a). S(S(a, b), c) = S(a, S(b, c)). b ⱕ L c ⇒ S(a, b) ⱕ L S(a, c). S(0 L , a) = a. In particular, S(0 L , 0 L ) = 0 L .

Remark 2.4.

(i) If S : L × L → L is a t-conorm in (L , ⱕ L ), then

a = S(a, 0 L ) ⱕ S(a, b) for any a, b ∈ L . (ii) In particular, for any b ∈ L, 1 L ⱕ L S(1 L , b) ⇒ S(1 L , b) = 1 L . (iii) S(1 L , 1 L ) = 1 L . (iv) Because of the associativity of a t-conorm S, we can write S(a, b, c) to denote S(S(a, b), c). If I = {1, . . . , n}, we will write S{ai |i ∈ I } to denote S(a1 , a2 , . . . , an ). If Y = {a1 }, then we understand that S{a|a ∈ Y } = a1 . The same can be said for any t-norm T . Example 2.5. In any complete lattice (L , ⱕ L ), the meet (greatest lower bound) is a t-norm, denoted by ∧, and the join (least upper bound) is a t-conorm, denoted by ∨. In this paper, we will deal only with t-norms T and t-conorms S defined in a complete lattice (L , ⱕ L ) that satisfy the following finiteness property. Definition 2.6. Let (L , ⱕ L ) be a complete lattice. We say that a t-norm T satisfies the finite property if, for each finite subset Y ⊆ L, {T (a1 , . . . , an )|a1 ∈ Y, . . . , an ∈ Y ; n ⱖ 1} is finite. The same finite property is defined for a t-conorm S.

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Remark 2.7. Notice that both the t-norm given by the meet and the t-conorm given by the join in a complete lattice (L , ⱕ L ) satisfy the finite property. Nevertheless, if L = [0, 1], the t-norm T : [0, 1] × [0, 1] → [0, 1] given by T (a, b) = a · b does not satisfy the finite property. Indeed, if Y = {1/2}, then {T (a1 , . . . , an )|a1 ∈ Y, . . . , an ∈ Y ; n ⱖ 1} = {1/2n ; n ⱖ 1}, which is clearly infinite. Throughout this paper, L = (L , ⱕ L , T, S) will be a complete lattice endowed with a t-norm T and a t-conorm S, both satisfying the finite property. Example 2.8 (Deschrijver and Kerre [6]). Consider the lattice L = {(x1 , x2 ) ∈ [0, 1]2 |x1 + x2 ⱕ 1} with the partial order given by (x1 , x2 ) ⱕ L (y1 , y2 ) ⇐⇒ x1 ⱕ y1 and y2 ⱕ x2 . It is shown in [6] that (L , ⱕ L ) is a complete lattice with 0 L = (0, 1) and 1 L = (1, 0). The map T : L × L → L given, for any (x1 , x2 ), (y1 , y2 ) ∈ [0, 1]2 , by T ((x1 , x2 ), (y1 , y2 )) = (max{0, x1 + y1 − 1}, min{1, x2 + y2 }), is a t-norm defined on (L , ⱕ L ). In addition, the map S : L × L → L given, for any (x1 , x2 ), (y1 , y2 ) ∈ [0, 1]2 , by S((x1 , x2 ), (y1 , y2 )) = (min{1, x1 + y1 }, max{0, x2 + y2 − 1}), is a t-conorm defined on (L , ⱕ L ). Both T and S satisfy the finite property. 3. L-fuzzy finite state machines The concept of a fuzzy finite state machine (ffsm) is given in [15] as a finite automaton (without initial nor final states) whose transition functions take value in the real interval [0, 1]. The basic idea is that they can switch from one state to another one to a certain truth degree between 0 and 1. We substitute the truth structure [0, 1] by any complete lattice L = (L , ⱕ L , T, S) in order to generalize known results about ffsms and ffsm homomorphisms. First we give the following definitions that generalize the concepts appearing in the book of Mordeson and Malik (see [15, Chapter 6]). Definition 3.1. Let X be a set. An L-fuzzy set in X is a new set {(x, A(x))|x ∈ X }, where A : X → L is a map named the membership function. Definition 3.2. An L-fuzzy finite state machine (L-ffsm) is a triple M = (Q, X, ), where Q and X are nonempty finite sets, called the set of states and the set of input symbols, respectively, and  : Q × X × Q → L is the membership function of an L-fuzzy set. Definition 3.3. Let M1 = (Q 1 , X 1 , 1 ) and M2 = (Q 2 , X 2 , 2 ) be L-ffsms. An L-ffsm homomorphism from M1 to M2 is a pair ( f, h) of mappings, f : Q 1 → Q 2 and h : X 1 → X 2 , such that 1 (q, x, p) ⱕ L 2 ( f (q), h(x), f ( p)) for any q, p ∈ Q 1 and x ∈ X 1 . We will write ( f, h) : M1 → M2 . The L-ffsm homomorphism ( f, h) is said to be S-strong if, in addition, for any q, p ∈ Q 1 and x ∈ X 1 , 2 ( f (q), h(x), f ( p)) = S{1 (q, x, r )| f (r ) = f ( p)}. If the t-conorm S is understood, we will say simply strong instead of S-strong. An L-ffsm isomorphism is an L-ffsm homomorphism ( f, h) such that f and h are both bijective.

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Notice that if ( f, h) : M1 → M2 is an S-strong L-ffsm homomorphism and f is one-to-one, then 1 (q, x, p) = 2 ( f (q), h(x), f ( p)) for any q, p ∈ Q 1 and x ∈ X 1 . Definition 3.4. Let M = (Q, X, ) be an L-ffsm and let ∼ be an equivalence relation on Q. Then ∼ is an admissible relation for M if it satisfies that, for any x ∈ X , p, q ∈ Q with p ∼ q and ( p, x, r ) > L 0 L for some r ∈ Q, then there exists t ∈ Q such that t ∼ r and (q, x, t) ⱖ L ( p, x, r ). The next results are generalizations of those appearing in [15] for L = ([0, 1], ⱕ , ∧, ∨). Theorem 3.5. Let L = (L , ⱕ L , T, S) be any complete lattice. The set of all the L-ffsms together with all the L-ffsm homomorphisms constitutes a category. Proof. First, notice that the pair (id Q , id X ) is an L-ffsm homomorphism for any L-ffsm M = (Q, X, ). Furthermore, if M1 , M2 and M3 are L-ffsms and we consider ( f 1 , h 1 ) : M1 → M2 and ( f 2 , h 2 ) : M2 → M3 , L-ffsm homomorphisms, then the composition ( f 2 ◦ f 1 , h 2 ◦ h 1 ) : M1 → M3 is also an L-ffsm homomorphism, denoted by ( f 2 , h 2 ) ◦ ( f 1 , h 1 ).  In order to generalize known results for L = ([0, 1], ⱕ , ∧, ∨), we study some cases in which the composition of S-strong L-ffsm homomorphisms is also S-strong. Proposition 3.6. Let ( f 1 , h 1 ) : M1 → M2 and ( f 2 , h 2 ) : M2 → M3 be S-strong L-ffsm homomorphisms between the L-ffsms M1 = (Q 1 , X 1 , 1 ), M2 = (Q 2 , X 2 , 2 ) and M3 = (Q 3 , X 3 , 3 ). (i) If f 1 is onto, then the composition ( f 2 , h 2 ) ◦ ( f 1 , h 1 ) is also an S-strong L-ffsm homomorphism. (ii) If f 1 and f 2 are one to one, then the composed L-ffsm homomorphism is also S-strong. Proof. (i) First, since ( f 1 , h 1 ) and ( f 2 , h 2 ) are both S-strong homomorphisms, we have for any q, p ∈ Q 1 and x ∈ X 1 , 3 (( f 2 ◦ f 1 )(q), (h 2 ◦ h 1 )(x), ( f 2 ◦ f 1 )( p)) = S{2 ( f 1 (q), h 1 (x), r )| f 2 (r ) = f 2 [ f 1 ( p)]}. Since f 1 is onto, every r ∈ Q 2 can be written as f 1 (s) for some s ∈ Q 1 . And then, for any r ∈ Q 2 such that f 2 (r ) = f 2 [ f 1 ( p)], 2 ( f 1 (q), h 1 (x), r ) = S{1 (q, x, s)| f 1 (s) = r }. Hence 3 (( f 2 ◦ f 1 )(q), (h 2 ◦ h 1 )(x), ( f 2 ◦ f 1 )( p)) = S{S{1 (q, x, s)| f 1 (s) = r }| f 2 (r ) = ( f 2 ◦ f 1 )( p)} = S{1 (q, x, s)|( f 2 ◦ f 1 )(s) = ( f 2 ◦ f 1 )( p)}. (ii) If f 1 and f 2 are one to one, then, for any q, p ∈ Q 1 and x ∈ X 1 , 3 (( f 2 ◦ f 1 )(q), (h 2 ◦ h 1 )(x), ( f 2 ◦ f 1 )( p)) = 2 ( f 1 (q), h 1 (x), f 1 ( p)) = 1 (q, x, p).  4. L-fuzzy transformation semigroups The concept of a fuzzy transformation semigroup (fts) is also given in [15] by allowing transition functions take values in the real interval [0, 1]. We generalize known results about ftss and fts homomorphisms from the truth structure [0, 1] to any complete lattice L = (L , ⱕ L , T, S), where T and S are, respectively, a t-norm and a t-conorm defined on (L , ⱕ L ) both satisfying the finite property. First we give the following definitions that generalize the concepts appearing in the book of Mordeson and Malik (see [15, Chapter 6]).

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Definition 4.1. An L-fuzzy transformation semigroup (or an L- f ts) is a triple G = (Q, U, ), where Q is a nonempty finite set, U is a finite semigroup and  : Q × U × Q → L is a map satisfying the following conditions: (TS1) If U is a semigroup with an identity element e then, for all p, q ∈ Q,  1 L if p = q, (q, e, p) = 0 L if p  q. (TS2) (q, uv, p) = S{T ((q, u, r ), (r, v, p))|r ∈ Q} for any p, q ∈ Q and u, v ∈ U . If in addition, for any u, v ∈ U , the following property holds (q, u, p) = (q, v, p) for any q, p ∈ Q ⇒ u = v, then the L-fts (Q, U, ) is called faithful. Definition 4.2. Let G 1 = (Q 1 , U1 , 1 ) and G 2 = (Q 2 , U2 , 2 ) be L-ftss. An L-fts homomorphism from G 1 to G 2 is a pair ( f, ) such that (i) f : Q 1 → Q 2 is a map. (ii)  : U1 → U2 is a semigroup homomorphism. (iii) If U1 and U2 are both semigroups with identity elements, e1 ∈ U1 and e2 ∈ U2 , then (e1 ) = e2 . (iv) 1 (q, u, p) ⱕ L 2 ( f (q), (u), f ( p)) for any q, p ∈ Q 1 and u ∈ U1 . The L-fts homomorphism ( f, ) is said to be S-strong if, in addition, for any q, p ∈ Q 1 and u ∈ U1 , 2 ( f (q), (u), f ( p)) = S{1 (q, u, r )| f (r ) = f ( p)}. An L-fts homomorphism ( f, ) : (Q 1 , U1 , 1 ) → (Q 2 , U2 , 2 ) is called an L-fts isomorphism if f and  are both bijective. Notice that if ( f, ) : (Q 1 , U1 , 1 ) → (Q 2 , U2 , 2 ) is an S-strong L-fts homomorphism and f is one to one, then 1 (q, u, p) = 2 ( f (q), (u), f ( p)) for any q, p ∈ Q 1 and u ∈ U1 . Theorem 4.3. The set of all the L-ftss together with all the L-fts homomorphisms constitutes a category. Proof. First, notice that the pair (id Q , idU ) is an L-fts homomorphism for any L-fts (Q, U, ). Furthermore, if G 1 , G 2 and G 3 are L-ftss and we consider the L-fts homomorphisms ( f 1 , 1 ) : G 1 → G 2 and ( f 2 , 2 ) : G 2 → G 3 , then the composition ( f 2 ◦ f 1 , 2 ◦ 1 ) : G 1 → G 3 is also an L-fts homomorphism, denoted by ( f 2 , 2 ) ◦ ( f 1 , 1 ).  Remark 4.4. In [14] Moˇckoˇr studies a category of fuzzy automata. Its objects are the L-ftss (Q, U, ) when L = ([0, 1], ⱕ , ∧, ∨) where U is a fixed semigroup. However, the morphisms between two fuzzy automata (Q 1 , U, 1 ) and (Q 2 , U, 2 ), considered in [14] are maps f × idU × h : (Q 1 , U, Q 1 ) → (Q 2 , U, Q 2 ) satisfying  2 ( f (q), u, h( p)) = {1 (r, u, s)| f (r ) = f (q), h(s) = h( p)}, which obviously differ from those considered in the previous theorem. The next result, as well as his proof, is similar to Proposition 3.6. Proposition 4.5. Let G 1 , G 2 and G 3 be L-ftss ant let ( f 1 , 1 ) : G 1 → G 2 and ( f 2 , 2 ) : G 2 → G 3 be both S-strong L-fts homomorphisms. (i) If f 1 is onto, then ( f 2 , 2 ) ◦ ( f 1 , 1 ) is also an S-strong L-fts homomorphism. (ii) If f 1 and f 2 are one to one, then the composed L-fts homomorphism is also S-strong. The construction of a faithful transformation semigroup from any transformation semigroup (see [15]) is also possible in the lattice-valued case.

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Proposition 4.6. Let (Q, U, ) be an L-fts. If we define an equivalence relation in U by means of u ∼ v ⇐⇒ (q, u, p) = (q, v, p) for every q, p ∈ Q and denote by u the equivalence class of each u ∈ U , then: (i) The equivalence relation ∼ is a congruence and hence the quotient set U/∼ is a finite semigroup. (ii) The triple (Q, U/∼ , ) is a faithful L-fts, where the map  : Q × (U/∼ ) × Q → L is given by (q, u, p) = (q, u, p) for any p, q ∈ Q, u ∈ U . Proof. (i) Assume that u, v ∈ U are such that u ∼ v. If w ∈ U , then, for any p, q ∈ Q, (q, uw, p) = S{T ((q, u, r ), (r, w, p))|r ∈ Q} = S{T ((q, v, r ), (r, w, p))|r ∈ Q} = (q, vw, p). So uw ∼ vw. Analogously, v ∼ w ⇒ uv ∼ uw for any u ∈ U . Hence U/∼ is a semigroup with the product defined by u · w = uw. (ii) First, notice that the map  is well defined. Next we see that (Q, U/∼ , ) is an L-fts. If U is a semigroup with identity e, then e¯ is the identity element of U/∼ . Since  1 L if p = q, (q, e, ¯ p) = (q, e, p) = 0 L if p  q, we conclude that (TS1) holds. If we consider any v, w ∈ U and any p, q ∈ Q, then (q, vw, p) = (q, vw, p) = S{T ((q, v, r ), (r, w, p))|r ∈ Q} = S{T ((q, v, r ), (r, w, ¯ p))|r ∈ Q} and (TS2) holds too. It is clear that the L-fts (Q, U/∼ , ) is faithful.  5. Construction of an L-fts from an L-ffsm and vice-versa This section shows that the usual mechanism to build a faithful fuzzy transformation semigroup from a fuzzy finite state machine (see [15]) works in the case that the truth structure is a complete lattice L = (L , ⱕ L , T, S) if and only if the t-norm T is distributive with respect to the t-conorm S. The last-mentioned property does not hold in general. For instance, the t-norm T given in Example 2.8 is not distributive with respect to the t-conorm S defined there. Moreover, in any complete lattice L = (L , ⱕ L , ∧, ∨), the t-norm ∧ is not always distributive with respect to the t-conorm ∨. Indeed, this property is equivalent to the residuation principle in L (see [8]). Let M = (Q, X, ) be an L-ffsm. From now on we will denote by (i) X ∗ the set of all the words of elements of X of finite length, including the empty word . Notice that X ∗ is not finite. (ii) |u| the length of any word u ∈ X ∗ . (iii) ∗ : Q × X ∗ × Q → L the map defined for any q, p ∈ Q as follows:  1 L if p = q, (a) ∗ (q, , p) = 0 L if p  q, ∗ (b)  (q, ux, p) = S{T (∗ (q, u, r ), (r, x, p))|r ∈ Q} for every u ∈ X ∗ and x ∈ X . (iv) X ∗ /∼ and ∗ : Q × (X ∗ /∼ ) × Q → L defined like in Proposition 4.6. The next result can be seen as a generalization of Theorem 3.1 of [12] from a monoid L = (L , ⱕ L , ·, ∨) as the truth structure of an automaton to L = (L , ⱕ L , T, S).

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Theorem 5.1. Let L = (L , ⱕ L , T, S) be any complete lattice with T and S satisfying the finite property. The following assertions are equivalent: (i) The triple (Q, X ∗ /∼ , ∗ ) is a faithful L-fts for any L-ffsm M = (Q, X, ). (ii) The t-norm T and the t-conorm S satisfy T (a, S(b, c)) = S(T (a, b), T (a, c)) f or any a, b, c ∈ L . (D) Remark 5.2. Notice that, for any x ∈ X and any q, p ∈ Q, ∗ (q, x, p) = S{T (∗ (q, , r ), (r, x, p))|r ∈ Q} = (q, x, p). Proof. The binary operation given by the concatenation makes X ∗ a semigroup with an identity element, , which verifies  1 L if p = q, ∗  (q, , p) = 0 L if p  q. So ∗ satisfies (T S1). Suppose first that (ii) holds. In order to show that ∗ satisfies (T S2), we must prove that ∗ (q, uv, p) = S{T [∗ (q, u, r ), ∗ (r, v, p)]|r ∈ Q} for any u, v ∈ X ∗ . We prove it by using induction on the length of v. If |v| = 0, then v =  and ∗ (q, u, p) = ∗ (q, u, p). In addition, S{T [∗ (q, u, r ), ∗ (r, , p)]|r ∈ Q} = T [∗ (q, u, p), ∗ ( p, , p)] = ∗ (q, u, p) and the equality holds. If |v| > 0, then v = wx for some w ∈ X ∗ with |w| < |v| and x ∈ X . Then ∗ (q, uv, p) = ∗ (q, uwx, p) = S{T [∗ (q, uw, s), (s, x, p)]|s ∈ Q} = S{T [S{T [∗ (q, u, r ), ∗ (r, w, s)]|r ∈ Q}, (s, x, p)]|s ∈ Q} by the induction hypothesis. Since assertion (ii) holds, then the last expression is equal to S{S[T {T [∗ (q, u, r ), ∗ (r, w, s)], (s, x, p)]|r ∈ Q}|s ∈ Q} = S{S[T {∗ (q, u, r ), T [∗ (r, w, s), (s, x, p)]]|r ∈ Q}|s ∈ Q} = S{S[T {∗ (q, u, r ), T [∗ (r, w, s), (s, x, p)]]|s ∈ Q}|r ∈ Q} = S{T {∗ (q, u, r ), S[T [∗ (r, w, s), (s, x, p)]]|s ∈ Q}|r ∈ Q} = S{T [∗ (q, u, r ), ∗ (r, wx, p)]|r ∈ Q} = S{T [∗ (q, u, r ), ∗ (r, v, p)]|r ∈ Q}. Therefore ∗ satisfies (T S1) and (T S2) and so does ∗ . Notice that X ∗ /∼ is finite because  only takes a finite number of values in L and both S and T satisfy the finite property. Hence (Q, X ∗ /∼ , ∗ ) is a faithful L-fts. Conversely, suppose that (i) holds and fix a, b, c ∈ L. In order to see that T (a, S(b, c)) = S(T (a, b), T (a, c)), we build the L-fuzzy finite state machine M = (Q, X, ) with Q = {q1 , q2 , q3 , q4 , q5 }, X = {x1 , x2 , x3 } and  : Q × X × Q → L given by (q1 , x1 , q2 ) = a, (q2 , x2 , q3 ) = 1 L = (q2 , x2 , q4 ) (q3 , x3 , q5 ) = b, (q4 , x3 , q5 ) = c and (qi , xk , q j ) = 0 L otherwise.

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Call u 1 = x1 and u 2 = x2 x3 . By using the definition of ∗ , we get ∗ (q1 , u 1 u 2 , q5 ) = S{T [∗ (q1 , x1 x2 , r ), ∗ (r, x3 , q5 )]|r ∈ Q} = S{T [∗ (q1 , x1 x2 , q3 ), ∗ (q3 , x3 , q5 )], T [∗ (q1 , x1 x2 , q4 ), ∗ (q4 , x3 , q5 )]} = S(T (a, b), T (a, c)). On the other side, since (Q, X ∗ /∼ , ∗ ), and so (Q, X ∗ , ∗ ), satisfies (TS2) by assumption (i), we have ∗ (q1 , u 1 u 2 , q5 ) = S{T [∗ (q1 , x1 , r ), ∗ (r, x2 x3 , q5 )]|r ∈ Q} = T (∗ (q1 , x1 , q2 ), ∗ (q2 , x2 x3 , q5 )) = T (a, S(b, c)). Therefore, T (a, S(b, c)) = S(T (a, b), T (a, c)) for the fixed a, b, c ∈ L. Since we can build a suitable L-ffsm for each triple a, b, c ∈ L, the assertion (ii) holds.  Remark 5.3. From now to the final of this section, L = (L , ⱕ L , T, S) will be a complete lattice with T and S satisfying the distributive property (D). Then, for any L-fuzzy finite state machine M = (Q, X, ), we denote by G L (M) = (Q, X ∗ /∼ , ∗ ) the faithful L-fuzzy transformation semigroup that is built in the previous theorem. The next results show that the map G L behaves in a sense like a functor from the category of the L-ffsms to the L-ffss, like it happened in L = ([0, 1], ⱕ , ∧, ∨). Theorem 5.4. Let M1 = (Q 1 , X 1 , 1 ) and M2 = (Q 2 , X 2 , 2 ) be L-ffsms. If for each L-ffsm S-strong homomorphism ( f, h) : M1 → M2 with f bijective, we define h ∗ : X 1∗ → X 2∗ by means of (i) h ∗ () = . (ii) h ∗ (ux) = h ∗ (u)h(x) whenever u ∈ X 1∗ and x ∈ X 1 and the map h ∗ : X 1∗ /∼ → X 2∗ /≈ by means of h ∗ (u) = h ∗ (u) for any u ∈ X 1∗ /∼ , then the pair ( f, h ∗ ) : G L (M1 ) → G L (M2 ) is an S-strong L-fts homomorphism, named G L ( f, h). Remark 5.5. Notice that h ∗ (x) = h(x) and then h ∗ (x) = h(x) for any x ∈ X 1 . Proof. Consider the faithful L-ftss G L (M1 ) = (Q 1 , X 1∗ /∼ , ∗1 ) and G L (M2 ) = (Q 2 , X 2∗ /≈ , ∗2 ). First we show that h ∗ : X 1∗ → X 2∗ is a semigroup homomorphism that maps the identity element of X 1∗ to that of ∗ X 2 . Since h ∗ () = , we only have to prove that h ∗ (uv) = h ∗ (u)h ∗ (v) for every u, v ∈ X 1∗ . Thus, we use an inductive argument on the length of v: If |v| = 0, then v =  and h ∗ (uv) = h ∗ (u) = h ∗ (u) = h ∗ (u) = h ∗ (u)h ∗ (v). Let |v| = n > 0 and write v = wx with w ∈ X 1∗ and x ∈ X 1 . Then h ∗ (uv) = h ∗ (uwx) = h ∗ (uw)h(x) = h ∗ (u)h ∗ (w)h(x) = h ∗ (u)h ∗ (v) by using the induction hypothesis in the last but one equality. By using induction on |v|, we show that, for any q, p ∈ Q 1 and v ∈ X 1∗ , ∗2 ( f (q), h ∗ (v), f ( p)) = ∗1 (q, v, p). If |v| = 0, then v =  and we have ∗2 ( f (q), h ∗ (),

f ( p)) =

which is equal to ∗1 (q, , p).

∗2 ( f (q), ,

 f ( p)) =

1 L if f ( p) = f (q), 0 L if f ( p)  f (q),

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If |v| = 1, then v ∈ X and h ∗ (v) = h(v). So ∗2 ( f (q), h ∗ (v), f ( p)) = 2 ( f (q), h(v), f ( p)) = 1 (q, v, p) = ∗1 (q, v, p). Let |v| = n > 1 and write v = wx with w ∈ X 1∗ and x ∈ X 1 . Then ∗2 ( f (q), h ∗ (v), f ( p)) = ∗2 ( f (q), h ∗ (w)h(x), f ( p))   = S{T ∗2 ( f (q), h ∗ (w), s), 2 (s, h(x), f ( p)) |s ∈ Q 2 }   = S{T ∗2 ( f (q), h ∗ (w), f (r )), 2 ( f (r ), h(x), f ( p)) |r ∈ Q 1 }   = S{T ∗1 (q, w, r ), 1 (r, x, p) |r ∈ Q 1 } = ∗1 (q, wx, p) = ∗1 (q, v, p). Finally we show that h ∗ : X 1∗ /∼ → X 2∗ /≈ is well-defined. Indeed, if u, v ∈ X 1∗ with u ∼ v, we have, for any q, p ∈ Q 1 , ∗2 ( f (q), h ∗ (u), f ( p)) = ∗1 (q, u, p) = ∗1 (q, v, p) = ∗2 ( f (q), h ∗ (v), f ( p)), so that h ∗ (u) ≈ h ∗ (v). We conclude that h ∗ is a semigroup homomorphism from X 1∗ /∼ to X 2∗ /≈ which maps  to .  Proposition 5.6. Let M1 , M2 and M3 be L-ffsms. If ( f 1 , h 1 ) : M1 → M2 and ( f 2 , h 2 ) : M2 → M3 are L-ffsm S-strong homomorphisms with f 1 and f 2 bijective, then (i) the pair ( f 2 ◦ f 1 , h 2 ◦ h 1 ) is also an S-strong L-ffsm homomorphism with f 2 ◦ f 1 bijective and (h 2 ◦ h 1 )∗ = h ∗2 ◦ h ∗1 . (ii) G L ( f 2 ◦ f 1 , h 2 ◦ h 1 ) = G L ( f 2 , h 2 ) ◦ G L ( f 1 , h 1 ). Moreover, for any L-ffsm M = (Q, X, ), it is G L (id Q , id X ) = (id Q , id(X ∗ /∼ ) ). Proof. It is straightforward.  Definition 5.7. Let G = (Q, U, ) be an L-fts and let ∼ be an equivalence relation defined on Q. Then ∼ is an admissible relation for G if it satisfies that, for any u ∈ U , p, q ∈ Q with p ∼ q and ( p, u, r ) > L 0 L for some r ∈ Q, then there exists t ∈ Q such that t ∼ r and (q, u, t) ⱖ L ( p, u, r ). Theorem 5.8 is formulated in [15] for L-ffsms and L-ftss when L = ([0, 1], ⱕ , ∧, ∨). It is not possible to generalize it to the case of any complete lattice L = (L , ⱕ L , T, S) even though T and S satisfy (D). However, if we impose that S(a, b) ∈ {a, b} for any a, b ∈ L, it can be proven. This condition is equivalent to say that S is the join defined on a totally ordered set (L , ⱕ L ). Theorem 5.8. Let L = (L , ⱕ L , T, S) be a complete lattice with T and S satisfying (D). Consider any L-ffsm M = (Q, X, ) and define an equivalence relation ≈ on Q. (i) If ≈ is an admissible relation for the faithful L-fts G L (M), then it is an admissible relation for M. (ii) Conversely, if ≈ is an admissible relation for M, (L , ⱕ L ) is a chain and the t-conorm S is the join, then ≈ is an admissible relation for the faithful L-fts G L (M) = (Q, X ∗ /∼ , ∗ ). Proof. (i) Consider any p, q ∈ Q with p ≈ q and some x ∈ X with ( p, x, r ) > L 0 L for some r ∈ Q. Since ∗ ( p, x, r ) = ∗ ( p, x, r ) = ( p, x, r ) > L 0 L , there exists some t ∈ Q with t ≈ r such that ∗ (q, x, t) ⱖ L ∗ ( p, x, r ) and so (q, x, t) ⱖ L ( p, x, r ). (ii) Assume that (L , ⱕ L ) is a chain, S is the join and ≈ is an equivalence relation defined on Q that is admissible for the L-ffsm M. Consider any p, q ∈ Q with p ≈ q and some u ∈ X ∗ with ∗ ( p, u, r ) > L 0 L for some r ∈ Q. We see inductively that there exists t ∈ Q with t ≈ r such that ∗ (q, u, t) = ∗ (q, u, t) ⱖ L ∗ ( p, u, r ) = ∗ ( p, u, r ).

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If |u| = 0, then u =  and ∗ ( p, u, r ) > L 0 L means that r = p and ∗ ( p, u, r ) = 1 L . It is enough to take t = q ∈ Q to get ∗ (q, u, t) = ∗ ( p, u, r ) with t = q ≈ p = r . If |u| = 1, then u ∈ X and ( p, u, r ) = ∗ ( p, u, r ) > L 0 L . Since ≈ is an admissible relation for M, there exists some t ∈ Q with t ≈ r such that ∗ (q, u, t) = (q, u, t) ⱖ L ( p, u, r ) = ∗ ( p, u, r ). Assume that |u| = n > 1 and write u = wx with w ∈ X ∗ and x ∈ X . Since  {T [∗ ( p, w, s), (s, x, r )]|s ∈ Q} > L 0 L , ∗ ( p, u, r ) = there must exist some s0 ∈ Q with T [∗ ( p, w, s0 ), (s0 , x, r )] > L 0 L . So we call Q 0 = {s0 ∈ Q|∗ ( p, w, s0 ) > L 0 L and (s0 , x, r ) > L 0 L }. The induction hypothesis assures that, for each s0 ∈ Q 0 , there exists some s0 ∈ Q with s0 ≈ s0 such that ∗ (q, w, s0 ) ⱖ L ∗ ( p, w, s0 ). Moreover, as s0 ≈ s0 and (s0 , x, r ) > L 0 L , there exists some t(s0 ) ∈ Q with t(s0 ) ≈ r such that (s0 , x, t(s0 )) ⱖ L (s0 , x, r ). Since we are considering the t-conorm given by the join on a totally ordered set, there exists some s1 ∈ Q 0 such that  ∗ (q, u, t(s1 )) = {∗ (q, u, t(s0 ))|s0 ∈ Q 0 }. Notice that t(s1 ) ≈ r and   ∗ (q, u, t(s1 )) = {∗ (q, u, t(s0 ))|s0 ∈ Q 0 } = { {T [∗ (q, w, s), (s, x, t(s0 ))]|s ∈ Q}|s0 ∈ Q 0 }   ⱖL {T [∗ (q, w, s0 ), (s0 , x, t(s0 ))]|s0 ∈ Q 0 } ⱖ L {T [∗ ( p, w, s0 ), (s0 , x, r )]|s0 ∈ Q 0 } = ∗ ( p, u, r ), as desired.  Example 6.7 will show that if (L , ⱕ L ) is not a chain Theorem 5.8(ii) is not true in general. The following result connecting ffsms to faithful ftss (see [15]) can also be generalized to L-ffsms and faithful L-ftss. Theorem 5.9. Let (Q, U, ) be an L-fts, where U is a semigroup with an identity element e. Then the faithful L-fts (Q, U/∼ , ) defined in Proposition 4.6 is S-strongly isomorphic to the faithful L-fts G L (M) = (Q, U ∗ /≈ , ∗ ) obtained when the triple (Q, U, ) is considered as an L-ffsm M. Proof. Consider the faithful L-fts (Q, U/∼ , ) defined as in Proposition 4.6 and denote by u¯ the equivalence class of any u ∈ U . Moreover, M = (Q, U, ) can be seen as an L-ffsm and hence we can construct the faithful L-fts (Q, U ∗ /≈ , ∗ ) as it has been done in Theorem 5.1. Denote by [w] the equivalence class of any w ∈ U ∗ . Now we take the maps id Q : Q → Q and  : U/∼ → U ∗ /≈ , that is given by (u) ¯ = [u], the equivalence class of an unilengthed word, for any u ∈ U . In order to see that  is well defined, assume that u ∼ v in U . This means that (q, u, p) = (q, v, p) for every p, q ∈ Q. Now, by Remark 5.2 ∗ (q, u, p) = (q, u, p) = (q, v, p) = ∗ (q, v, p) for every p, q ∈ Q. Therefore u ≈ v as unilengthed words of U ∗ and hence [u] = [v]. We show now that (id Q , ) is an L-fts homomorphism. Since (e) ¯ = [e] and  1 L if p = q ∈ Q, ∗ (q, e, p) = (q, e, p) = 0 L if p  q, ¯ = []. Moreover, we conclude that e ≈ , the empty word of U ∗ , and so (e) ¯ w) ¯ for all v, w ∈ U. (v¯ · w) ¯ = (v · w) = [v · w] = [v][w] = (v)( In order to see that  : U/∼ → U ∗ /≈ is one to one, let u, v ∈ U with [u] = [v]. Then, for any p, q ∈ Q, we have (q, u, p) = ∗ (q, u, p) = ∗ (q, v, p) = (q, v, p), so that u¯ = v¯ ∈ U/∼ . To show that  is onto, take any v ∈ U ∗ and now, by using induction on the length of v, we will find some s ∈ U such that (s) = [v].

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If |v| = 0, then v =  and we have already proven that (e) ¯ = [e] = []. Assume now that |v| = n > 0 and that there exists some u ∈ U with (u) ¯ = [w] for every word w with |w| < n. Write v = wx with w ∈ U ∗ and x ∈ U . Since |w| = n − 1, the induction hypothesis gives us some u ∈ U with (u) ¯ = [w], i.e., [u] = [w]. We must see that (u · x) = [v]. Indeed since (Q, U, ) is an L-fts and u · x ∈ U , for any p, q ∈ Q, we have ∗ (q, u · x, p) = (q, u · x, p) = S{T ((q, u, r ), (r, x, p))|r ∈ Q} = S{T (∗ (q, u, r ), ∗ (r, x, p))|r ∈ Q} = S{T (∗ (q, w, r ), ∗ (r, x, p))|r ∈ Q} = ∗ (q, v, p) because u, x ∈ U and [u] = [w]. Therefore, [v] = [u · x] = (u · x). Finally, we show that  : U/∼ → U ∗ /≈ satisfies, for any p, q ∈ Q and u ∈ U , that (q, u, ¯ p) = ∗ (q, (u), ¯ p), but this is clear because u ∈ U and then ∗ (q, (u), ¯ p) = ∗ (q, [u], p) = ∗ (q, u, p) = (q, u, p) = (q, u, ¯ p). Consequently, (id Q , ) is an S-strong L-fts isomorphism.  6. Generation of L-ffsms from L I -ffsms and L-fts from L I -fts Let (L , ⱕ L ) be a complete lattice. Denote by a or [a1 , a2 ] any closed interval of L, a = [a1 , a2 ] = {c ∈ L|a1 ⱕ c ⱕ a2 }. Consider the set of all closed intervals of L, L I = {[a1 , a2 ]|a1 , a2 ∈ L , a1 ⱕ L a2 }, with the partial order given by [a1 , a2 ] ⱕ L I [b1 , b2 ] ⇐⇒ a1 ⱕ L b1 and a2 ⱕ L b2 . It is known that (L I , ⱕ L I ) is a complete lattice (see [8]). In addition, if 0 L and 1 L are, respectively, the smallest and the greatest element of L, then the intervals [0 L , 0 L ] and [1 L , 1 L ] are the minimum and the maximum of L I , respectively. In this section, we consider only the t-conorm given by the join both in the lattice (L , ⱕ L ) and in the lattice (L I , ⱕ L I ). In [8] next result is proven. Theorem 6.1. Let (L , ⱕ L ) be a complete lattice and consider the lattice (L I , ⱕ L I ). If T is a t-norm defined on L I such that T(a, b∨c) = T(a, b)∨T(a, c) f or any a, b, c ∈ L I and we write T(a, b) = [T1 (a, b), T2 (a, b)], then: (i) For any a = [a1 , a2 ] and b = [b1 , b2 ], T1 (a, b) depends only on a1 , b1 ∈ L. (ii) The map T1 : L × L → L given by T1 (a1 , b1 ) = T1 ([a1 , a1 ], [b1 , b1 ]) for any a1 , b1 ∈ L is a t-norm defined on (L , ⱕ L ) satisfying T1 (a1 , b1 ∨ c1 ) = T1 (a1 , b1 ) ∨ T1 (a1 , c1 ) f or ever y a1 , b1 , c1 ∈ L . Remark 6.2. Notice that T2 ([a1 , a2 ], [b1 , b2 ]) does not depend only on a2 , b2 ∈ L in general. By using the previous theorem, the proof of the next result is immediate. Proposition 6.3. Let (L , ⱕ L ) be a complete lattice. Suppose that T is a t-norm defined in (L I , ⱕ L I ) satisfying (D) with respect to the join and let T1 be the t-norm in (L , ⱕ L ) given by Theorem 6.1. If T satisfies the finite property in (L I , ⱕ L I ), then T1 satisfies the finite property in (L , ⱕ L ).

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Remark 6.4 (see Deschrijver [8]). A t-norm T defined on a complete lattice (L , ⱕ L ) satisfies (D) with respect to the join if and only if it satisfies the residuation principle given by T (a, b) ⱕ c ⇐⇒ b ⱕ



{| ∈ L , T (a, ) ⱕ c} for any a, b, c ∈ L .

Definition 6.5. Let L = (L , ⱕ L ) be a complete lattice. A t-norm T defined in L I = (L I , ⱕ L I ) is said to be t-representable if there exist some t-norms T1 , T2 defined in L = (L , ⱕ L ) such that T(a, b) = [T1 (a1 , b1 ), T2 (a2 , b2 )] for any a = [a1 , a2 ], b = [b1 , b2 ] ∈ L I . We will write T = [T1 , T2 ] in that case. If the t-norm T is t-representable, then Proposition 6.3 can be improved. Theorem 6.6. Let (L , ⱕ L ) be a complete lattice. Suppose that T is a t-norm defined in (L I , ⱕ L I ) which is t-representable with T = [T1 , T2 ]. Then: (i) both T1 and T2 satisfy the finite property in (L , ⱕ L ) if and only if T satisfies the finite property in (L I , ⱕ L I ); (ii) both T1 and T2 satisfy property (D) with respect to the join in (L , ⱕ L ) if and only if T satisfies property (D) with respect to the join in (L I , ⱕ L I ). Proof. (i) It is immediate. (ii) For any a = [a1 , a2 ], b = [b1 , b2 ], c = [c1 , c2 ] ∈ L I , T(a∨b, c) = T([a1 ∨ b1 , a2 ∨ b2 ], [c1 , c2 ]) = [T1 (a1 ∨ b1 , c1 ), T2 (a2 ∨ b2 , c2 )]. In addition T(a, c)∨T(b, c) = [T1 (a1 , c1 ), T2 (a2 , c2 )] ∨ [T1 (b1 , c1 ), T2 (b2 , c2 )] = [T1 (a1 , c1 ) ∨ T1 (b1 , c1 ), T2 (a2 , c2 ) ∨ T2 (b2 , c2 )]. Hence, the result follows.  Notation. For any set Y , any map  : Y → L I and any y ∈ Y , we will write (y) = [1 (y), 2 (y)]. So we denote  = [1 , 2 ], where i is a map from Y to L for i ∈ {1, 2}. We are now ready to show the next example. Example 6.7. Let L = [0, 1], L = (L , ⱕ , ∧, ∨) and L I = (L I , ⱕ L I , ∧, ∨). Consider the L I -ffsm M = (Q, X, ) with Q = {q, p, r }, X = {x0 } and  : Q × X × Q → L I given by (q, x0 , q) = [0.1, 0.3], (q, x0 , p) = [0.4, 0.6], (q, x0 , r ) = [0.5, 0.5], ( p, x0 , q) = [0.2, 0.4], ( p, x0 , p) = [0.4, 0.6], ( p, x0 , r ) = [0.5, 0.5], (r, x0 , q) = [0.4, 0.6], (r, x0 , p) = [0.5, 0.5], (r, x0 , r ) = [0.4, 0.4]. The equivalence relation ≈ defined in Q by q ≈ p ≈ r is admissible for the L I -ffsm M. However, it is not admissible for the faithful L I -fts G L I (M) = (Q, X ∗ /∼ , ∗ ). Indeed, ∗ (q, x0 x0 , p) = [0.5, 0.6] > L I [0, 0], q ≈ r but ∗ (r, x0 x0 , q) = [0.4, 0.4], ∗ (r, x0 x0 , p) = [0.4, 0.6] and ∗ (r, x0 x0 , r ) = [0.5, 0.5], none of them greater of equal than ∗ (q, x0 x0 , p) = [0.5, 0.6]. Proposition 6.8. Let (L , ⱕ L ) be a complete lattice. Consider L I = (L I , ⱕ L I , T, ∨) with T satisfying property (D) with respect to the join and let L1 = (L , ⱕ , T1 , ∨) where T1 is the t-norm given by Theorem 6.1.

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Let M = (Q, X, ) and N = (P, Y, ) be L I -ffsms and let f : Q → P and h : X → Y be maps. Then: (i) If ( f, h) : M → N is an L I -ffsm homomorphism, then the map ( f, h) : (Q, X, 1 ) → (P, Y, 1 ) is an L1 -ffsm homomorphism. Moreover, if ( f, h) : (Q, X, ) → (P, Y, ) is strong, then the L1 -ffsm homomorphism ( f, ) : (Q, U, 1 ) → (P, V, 1 ) is strong. (ii) Suppose that the t-norm T is t-representable with T = [T1 , T2 ]. Then the pair ( f, h) : (Q, X, ) → (P, Y, ) is an L I -ffsm homomorphism if and only if, for i ∈ {1, 2}, the pair ( f, h) : (Q, X, i ) → (P, Y, i ) is an Li -fts homomorphism where Li = (L , ⱕ L , Ti , ∨). Moreover, the pair ( f, h) : (Q, X, ) → (P, Y, ) is a strong L I -ffsm homomorphism if and only if ( f, h) : (Q, X, i ) → (P, Y, i ) is a strong Li -ffsm homomorphism for i ∈ {1, 2}. Proof. It is straightforward.  Proposition 6.9. Let (L , ⱕ L ) be a complete lattice. Consider L I = (L I , ⱕ L I , T, ∨) with T satisfying property (D) with respect to the join and let L1 = (L , ⱕ , T1 , ∨) where T1 is the t-norm given by Theorem 6.1. Then: (i) the map D1 that associates each L I -ffsm M = (Q, X, ) with the L1 -ffsm D1 (M) = (Q, X, 1 ) and each L I -ffsm homomorphism ( f, h) with the L1 -ffsm homomorphism D1 ( f, h) = ( f, h) is a functor from the category of the L I -ffsms to the category of the L1 -ffsms; (ii) if T is t-representable with T = [T1 , T2 ], then the map D2 which maps each L I -ffsm M = (Q, X, ) to D2 (M) = (Q, X, 2 ) and each L I -ffsm homomorphism ( f, h) to D2 ( f, h) = ( f, h) is also a functor from the category of the L I -ffsms to the category of the L2 -ffsms, where L2 = (L , ⱕ , T2 , ∨); (ii) if T is t-representable with T1 = T2 , then (id Q , id X ) : D1 (M) → D2 (M) is an L1 -ffsm homomorphism that is indeed a natural equivalence from the functor D1 to the functor D2 . Proof. It is straightforward.  Proposition 6.10. In the conditions of 6.9(iii), let M = (Q, X, ) be an L I -ffsm and let ∼ be an equivalence relation defined on Q. If ∼ is admissible for M, then ∼ is admissible for both L-ffsms D1 (M) and D2 (M). Proof. In order to show that ∼ is admissible for D1 (M) = (Q, X, 1 ), take p, q ∈ Q with p ∼ q and let x ∈ X with 1 ( p, x, r ) > L 0 L for some r ∈ Q. Then, it must be ( p, x, r ) > L I 0 L I and since ∼ is admissible for the L I -ffsm M, we know that there exists t ∈ Q such that t ∼ r and (q, x, t) ⱖ L I ( p, x, r ). Hence, 1 (q, x, t) ⱖ L 1 ( p, x, r ). Analogously, we can see that the assertion holds for D2 (M).  The reciprocal does not hold in general. Example 6.11. If L = ([0, 1], ⱕ , ∧, ∨), consider the L I -ffsm M = (Q, X, ) given by Q = {q, p}, X = {x0 } and the map  : Q × X × Q → L I defined by (q, x0 , q) = [0.3, 0.5], (q, x0 , p) = [0.1, 0.4], ( p, x0 , q) = [0, 0.5] and ( p, x0 , p) = [0.3, 0.3]. The equivalence relation ∼ given by q ∼ p is admissible for the ffsms M1 = (Q, X, 1 ) and M2 = (Q, X, 2 ), but it is not admissible for M = (Q, X, ). Indeed, (q, x0 , q) = [0.3, 0.5] > L I [0, 0] and p ∼ q, but there is not any r ∈ Q with ( p, x0 , r ) ⱖ L I (q, x0 , q). Lemma 6.12. Let (L , ⱕ L ) be a complete lattice. Consider L I = (L I , ⱕ L I , T, ∨) with T satisfying property (D) with respect to the join and let L1 = (L , ⱕ , T1 , ∨) where T1 is the t-norm given by Theorem 6.1. Consider a finite set Q, a finite semigroup U and a map  : Q × U × Q → L I . Then: (i) If (Q, U, ) is an L I -fuzzy transformation semigroup, then (Q, U, 1 ) is an L-fuzzy transformation semigroup. Moreover, if (Q, U, 1 ) is faithful, then (Q, U, ) is also faithful. (ii) If T is t-representable with T = [T1 , T2 ], then (Q, U, ) is an L I -fuzzy transformation semigroup if and only if for i ∈ {1, 2}, the triple (Q, U, i ) is an Li -fuzzy transformation semigroup, where Li = (L , ⱕ L , Ti , ∨). Moreover, if either (Q, U, 1 ) or (Q, U, 2 ) is faithful, then (Q, U, ) is faithful as an L I -fts.

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Proof. (i) If U has an identity element, e, then   1 L if q = p, 1 L I if q = p ⇒ 1 (q, e, p) = (q, e, p) = 0 L I if q  p 0 L if q  p. Hence 1 satisfies (TS1). Moreover as  {T([1 (q, u, r ), 2 (q, u, r )], [1 (r, v, p), 2 (r, v, p)])}, (q, uv, p) = r ∈Q

then 1 (q, uv, p) =



{T1 (1 (q, u, r ), 1 (r, v, p))}

r ∈Q

and (TS2) holds in (Q, U, 1 ). Suppose that (Q, U, 1 ) is faithful and let (q, u, p) = (q, v, p) for every q, p ∈ Q. Then 1 (q, u, p) = 1 (q, v, p) for every q, p ∈ Q and so u = v because (Q, U, 1 ) is faithful. Thus we conclude that (Q, U, ) is a faithful L I -fts. (ii) If U has an identity element, e, then  1 L I if q = p (q, e, p) = ⇐⇒ 0 L I if q  p   1 L if q = p 1 L if q = p 1 (q, e, p) = and 2 (q, e, p) = . 0 L if q  p 0 L if q  p So, (T S1) holds in (Q, U, ) if and only if (T S1) holds in (Q, U, 1 ) and in (Q, U, 2 ). Furthermore,  (q, uv, p) = {T((q, u, r ), (r, v, p))} r ∈Q

if and only if [1 (q, uv, p), 2 (q, uv, p)] =



{T([1 (q, u, r ), 2 (q, u, r )], [1 (r, v, p), 2 (r, v, p)])}

r ∈Q

if and only if ⎧  ⎪ {T1 (1 (q, u, r ), 1 (r, v, p))}, ⎨ 1 (q, uv, p) = r ∈Q {T2 (2 (q, u, r ), 2 (r, v, p))}. ⎪ ⎩ 2 (q, uv, p) = r ∈Q

Hence, (T S2) holds in (Q, U, ) if and only if (T S2) holds in (Q, U, 1 ) and in (Q, U, 2 ). The last part follows from (i) if (Q, U, 1 ) is faithful. A similar proof can be done if (Q, U, 2 ) is faithful.  The converse of Lemma 6.12(ii) is not true in general. Example 6.13. Let L = ([0, 1], ⱕ , ∧, ∨) and consider the L I -fts G = (Q, U, ) where Q = {q}, the set U = {−1, 0, 1} is a semigroup with a commutative product given by 1 · 1 = 1; 1 · 0 = 0 = 0 · 0 = (−1) · 0; (−1) · (−1) = −1 = (−1) · 1 and  : Q × U × Q → L I is defined by (q, 0, q) = [0, 0], (q, 1, q) = [1, 1], (q, −1, q) = [0, 1]. Then G = (Q, U, ) is a faithful L I -fts but G 1 = (Q, U, 1 ) and G 2 = (Q, U, 2 ) are not faithful as L-ftss.

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Proposition 6.14. Let (L , ⱕ L ) be a complete lattice. Consider L I = (L I , ⱕ L I , T, ∨) with T satisfying property (D) with respect to the join and let L1 = (L , ⱕ , T1 , ∨) where T1 is the t-norm given by Theorem 6.1. Suppose that (Q, U, ) and (P, V, ) are L I -ftss. Let f : Q → P be a map and  : U → V a semigroup homomorphism (with (e1 ) = e2 if U and V have got identity elements, e1 and e2 , respectively). (i) If ( f, ) : (Q, U, ) → (P, V, ) is an L I -fts homomorphism then ( f, ) : (Q, U, 1 ) → (P, V, 1 ) is an L1 -fts homomorphism. Moreover, if ( f, ) : (Q, U, ) → (P, V, ) is strong, then the fts homomorphism ( f, ) : (Q, U, 1 ) → (P, V, 1 ) is strong. (ii) Suppose that the t-norm T is t-representable with T = [T1 , T2 ]. Then the pair ( f, ) : (Q, U, ) → (P, V, ) is an L I -fts homomorphism if and only if, for i ∈ {1, 2}, the pair ( f, ) : (Q, U, i ) → (P, V, i ) is an Li -fts homomorphism where Li = (L , ⱕ L , Ti , ∨). Moreover, the pair ( f, ) : (Q, U, ) → (P, V, ) is a strong L I -ftss homomorphism if and only if ( f, ) : (Q, U, i ) → (P, V, i ) is a strong Li -fts homomorphism for i ∈ {1, 2}. (iii) Let T be t-representable and ∼ an equivalence relation defined on Q. If ∼ is admissible for (Q, U, ), then ∼ is admissible for both (Q, U, 1 ) and (Q, U, 2 ). Proof. It is straightforward. The last statement is proven in a similar way to Proposition 6.10.  The converse of Proposition 6.14(iii) does not hold in general, as it is shown in the next example. Example 6.15. Let M = (Q, X, ) be the L I -ffsm defined in Example 6.11. Then G L I (M) = (Q, X ∗ /∼ , ∗ ) with X ∗ /∼ = {, x0 , x0 x0 } and ∗ (q, x0 x0 , q) = [0.3, 0.5], ∗ (q, x0 x0 , p) = [0.1, 0.4], ∗ ( p, x0 x0 , q) = [0, 0.5], ∗ ( p, x0 x0 , p) = [0.3, 0.4]. If we consider the equivalence relation ≈ given in Q by q ≈ p, it is easy to check that ≈ is admissible for both (Q, X ∗ /∼ , (∗ )1 ) and (Q, X ∗ /∼ , (∗ )2 ), L-ftss, but ≈ is not admissible for G L I (M) because it is not admissible for the ivffsm M (see Example 6.11 and Theorem 5.8(i)). Proposition 6.16. Let (L , ⱕ L ) be a complete lattice. Consider L I = (L I , ⱕ L I , T, ∨) with T satisfying property (D) with respect to the join and let L1 = (L , ⱕ , T1 , ∨) where T1 is the t-norm given by Theorem 6.1. Then (i) The map F1 which maps each L I -fts G = (Q, U, ) to F1 (G) = (Q, U, 1 ) and each L I -fts homomorphism ( f, h) to F1 ( f, h) = ( f, h) is a functor from the category of the L I -ftss to the category of the L1 -ftss. (ii) If T is t-representable, then the map F2 which maps each L I -fts G = (Q, U, ) to F2 (G) = (Q, U, 2 ) and each L I -fts homomorphism ( f, h) to F2 ( f, h) = ( f, h) is also a functor from the category of the L I -ftss to the category of the L2 -ftss, where L2 = (L , ⱕ , T2 , ∨). (iii) If T is t-representable with T1 = T2 , then (id Q , idU ) : F1 (G) → F2 (G) is an L1 -fts homomorphism that is indeed a natural equivalence from the functor F1 to the functor F2 . Proof. (i) First notice that, if G = (Q, U, ) and H = (P, V, ) are both L I -ftss, then F1 (G) and F1 (H ) are L1 -ftss by Lemma 6.12. Moreover, if ( f, h) is an L I -fts homomorphism, then F1 ( f, h) = ( f, h) is an L1 -fts homomorphism from F1 (G) to F1 (H ) by Proposition 6.14. Let G, H and R be L I -ftss. If ( f, h) : G → H and (g, l) : H → R are L I -fts homomorphism, then (g ◦ f, l ◦ h) : G → R is also an L I -fts homomorphism and F1 (g ◦ f, l ◦ h) = (g ◦ f, l ◦ h) = F1 (g, l) ◦ F1 ( f, h). Moreover, if we consider the pair (id Q , idU ) : G → G as L I -fts homomorphism, then F1 (id Q , idU ) = (id Q , idU ). (ii) It is straightforward. (iii) If T1 = T2 then both F1 (G) and F2 (G) are L1 -fts homomorphisms. The rest of the proof is immediate. 

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We show now two different ways of building a faithful L1 -fts starting from an L I -ffsm M. Theorem 6.17. Let (L , ⱕ L ) be a complete lattice. Consider L I = (L I , ⱕ L I , T, ∨) with T satisfying property (D) with respect to the join and let L1 = (L , ⱕ L , T1 , ∨) where T1 is the t-norm given by Theorem 6.1. For each L I -ffsm M = (Q, X, ), the triple G L1 (D1 (M)) = (Q, X ∗ /≈1 , (1 )∗ ) is a faithful L1 -fts. Moreover, if T is t-representable with T = [T1 , T2 ], then G L2 (D2 (M)) = (Q, X ∗ /≈2 , (2 )∗ ) is also a faithful L2 -fts, where L2 = (L , ⱕ L , T2 , ∨). Proof. D1 (M) = (Q, X, 1 ) is an L1 -ffsm by Proposition 6.9. Since T satisfies property (D) with respect to the join, then T1 satisfies property (D) with respect to the join and the triple G L1 (D1 (M)) = (Q, X ∗ /≈1 , (1 )∗ ) is a faithful L1 -fts by Theorem 5.1. The equivalence relation ≈1 in X ∗ is given by u ≈1 v ⇐⇒ (1 )∗ (q, u, p) = (1 )∗ (q, v, p) for every q, p ∈ Q. In the case that T is t-representable with T = [T1 , T2 ], we can also build the faithful L2 -fts homomorphism G L2 (D2 (M)) = (Q, X ∗ /≈2 , (2 )∗ ) by using the t-norm T2 , which also satisfies property (D).  Another L1 -fts can be obtained from an L I -ffsm M through the L I -fts G L I (M). Theorem 6.18. Let (L , ⱕ L ) be a complete lattice. Consider L I = (L I , ⱕ L I , T, ∨) with T satisfying (D) with respect to the join and let L1 = (L , ⱕ L , T1 , ∨) where T1 is the t-norm given by Theorem 6.1. If M = (Q, X, ) is an L I -ffsm, then: (i) The triple F1 (G L I (M)) = (Q, [X ∗ /∼ ]1 , [(∗ )1 ]) is a faithful L1 -fts. (ii) If T is t-representable with T = [T1 , T2 ], then the triple F2 (G L I (M)) = (Q, [X ∗ /∼ ]2 , [(∗ )2 ]) is also a faithful L2 -fts, where L2 = (L , ⱕ L , T2 , ∨). Proof. (i) G L I (M) = (Q, X ∗ /∼ , ∗ ) is a faithful L I -fts by Theorem 5.1, where the equivalence relation ∼ in X ∗ is defined by u ∼ v ⇐⇒ ∗ (q, u, p) = ∗ (q, v, p) for every q, p ∈ Q. Now Proposition 6.16 assures that the triple F1 (G L I (M)) = (Q, X ∗ /∼ , (∗ )1 ) is an L1 -fts which is not faithful in general. In addition, the equivalence relation defined in X ∗ /∼ by u ≈ v ⇐⇒ (∗ )1 (q, u, p) = (∗ )1 (q, v, p) for every q, p ∈ Q allow us to construct the faithful L1 -fts F1 (G L I (M)) = (Q, [X ∗ /∼ ]1 , [(∗ )1 ]), where the map [(∗ )1 ] is given by [(∗ )1 ](q, [u], p) = (∗ )1 (q, u, p) for every q, p ∈ Q for every u ∈ X ∗ . (ii) If T is t-representable, the proof of this statement is similar to (i).  Remark 6.19. In the conditions of Theorem 6.18, for any L I -ffsm M = (Q, X, ), two faithful L1 -ftss can be build, named respectively G L1 (D1 (M)) = (Q, X ∗ /≈1 , (1 )∗ ) and F1 (G L I (M)) = (Q, [X ∗ /∼ ]1 , [(∗ )1 ]). The following theorem shows that they are indeed the same.

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Theorem 6.20. Let (L , ⱕ L ) be a complete lattice. Consider L I = (L I , ⱕ L I , T, ∨) with T satisfying property (D) with respect to the join and let L1 = (L , ⱕ L , T1 , ∨) where T1 is the t-norm given by Theorem 6.1. Then, for any L I -ffsm M = (Q, X, ), we have (i) (1 )∗ = (∗ )1 . (ii) F1 (G L I (M)) = G L1 (D1 (M)). Proof. (i) We prove for any q, p ∈ Q and u ∈ X ∗ , by using induction on the length of u ∈ X ∗ , that (1 )∗ (q, u, p) = (∗ )1 (q, u, p). If |u| = 0, then u =  and, if q = p, ∗ (q, u, p) = 1 L I = [1 L , 1 L ] ⇒ (∗ )1 (q, u, p) = 1 L = (1 )∗ (q, u, p). Otherwise, if q  p, ∗ (q, u, p) = 0 L I = [0 L , 0 L ] ⇒ (∗ )1 (q, u, p) = 0 L = (1 )∗ (q, u, p). If |u| > 0, then u = vx for some v ∈ X ∗ and some x ∈ X with |v| < |u|. Hence, by using the induction hypothesis on the second equality, we have 

   {T1 ((∗ )1 (q, v, r ), 1 (r, x, p))|r ∈ Q}, z ∗ (q, u, p) = {T ∗ (q, v, r ), (r, x, p) |r ∈ Q} =

 {T1 ((1 )∗ (q, v, r ), 1 (r, x, p))|r ∈ Q}, z = [(1 )∗ (q, u, p), z]. = So, (∗ )1 (q, u, p) = (1 )∗ (q, u, p) for every u ∈ X ∗ and q, p ∈ Q. (ii) The equivalence relation ∼ appearing in G L I (M) = (Q, X ∗ /∼ , ∗ ) is given by u ∼ v ⇐⇒ ∗ (q, u, p) = ∗ (q, v, p) for every q, p ∈ Q (u, v ∈ X ∗ ). So the equivalence relation ≈ defined in X ∗ /∼ in the construction of F1 (G L I (M)) can be given by means of u ≈ v ⇐⇒ (∗ )1 (q, u, p) = (∗ )1 (q, v, p) for any q, p ∈ Q (u, v ∈ X ∗ /∼ ). Thus u, v ∈ X ∗ belong to the same equivalence class of [X ∗ /∼ ]1 if and only if (∗ )1 (q, u, p) = (∗ )1 (q, v, p) for any q, p ∈ Q, which is equivalent to (∗ )1 (q, u, p) = (∗ )1 (q, v, p) for any q, p ∈ Q, and by using (i) equivalent to (1 )∗ (q, u, p) = (1 )∗ (q, v, p) for any q, p ∈ Q, equivalent as well to u ≈1 v, where ≈1 is the equivalence relation defined in X ∗ in the construction of G L1 (D1 (M)).  Theorem 6.21. In the conditions of Theorem 6.20, suppose in addition that T is t-representable with T = [T1 , T2 ] and call L2 = (L , ⱕ L , T2 , ∨). Then, for each L I -ffsm M = (Q, X, ), (i) (2 )∗ = (∗ )2 . (ii) F2 (G L I (M)) = G L2 (D2 (M)). Proof. It is analogous to that of the previous result. 

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7. Generation of L I -ffsms and L I -ftss from L-ffsms and L-ftss Throughout this section (L , ⱕ L ) will be a complete lattice and the unique t-conorm considered will be the join both in (L , ⱕ L ) and in (L I , ⱕ L I ). We will consider L1 = (L , ⱕ L , T1 , ∨) and L2 = (L , ⱕ L , T2 , ∨) where T1 and T2 are t-norms defined in (L , ⱕ L ) satisfying the finite property and property (D) with respect to the join and such that, for any a, b ∈ L, T1 (a, b) ⱕ L T2 (a, b). Lemma 7.1. The map T : L I × L I → L I given by T(a, b) = [T1 (a1 , b1 ), T2 (a2 , b2 )] f or any a = [a1 , a2 ], b = [b1 , b2 ] ∈ L I is a t-norm defined in (L I , ⱕ L I ) that satisfies both the finite property and property (D) with respect to the join. Proof. T is a t-norm defined in (L I , ⱕ L I ) by [8]. Theorem 6.6 assures that T satisfies the finite property and property (D) with respect to the join.  Theorem 7.2. Suppose that M = (Q, X, ) and N = (Q, X, ) are, respectively, an L1 -ffsm and an L2 -ffsm satisfying that (q, x, p) ⱕ L (q, x, p) for any q, p ∈ Q and any x ∈ X . Then: (i) The triple (Q, X, [, ]) is an L I -ffsm. (ii) G L I (Q, X, [, ]) = (Q, X ∗ /∼ , [, ]∗ ) is a faithful L I -fts, where L I = (L I , ⱕ L I , T, ∨) with T = [T1 , T2 ] is defined as in Lemma 7.1. Proof. (i) It is straightforward. (ii) This result is a direct application of Theorem 5.1. Notice that the equivalence relation ∼ is given in X ∗ by u ∼ v ⇐⇒ [, ]∗ (q, u, p) = [, ]∗ (q, v, p) for every q, p ∈ Q.



Theorem 7.3. Suppose that the triple G = (Q, U, ) is an L1 -fts and that H = (Q, U, ) is an L2 -fts satisfying that (q, u, p) ⱕ L (q, u, p) for any q, p ∈ Q and any u ∈ U . If we consider the t-norm T = [T1 , T2 ] defined as in Lemma 7.1 and call L I = (L I , ⱕ L I , T, ∨), then: (i) The triple (Q, U, [, ]) is an L I -fts. (ii) The triple (Q, U/ ≈, [, ]) is a faithful L I -fts. Proof. (i) By Theorem 6.14(ii) and Proposition 4.6. (ii) Notice that the equivalence relation ≈ is given by u ≈ v in U if and only if [, ](q, u, p) = [, ](q, v, p) for every q, p ∈ Q.



Theorem 7.4. Suppose that M = (Q, X, ) and N = (Q, X, ) are, respectively, an L1 -ffsm and an L2 -ffsm satisfying that (q, x, p) ⱕ L (q, x, p) for any q, p ∈ Q and any x ∈ X . Then ∗ (q, u, p) ⱕ L ∗ (q, u, p) for any q, p ∈ X and u ∈ X ∗ and (i) [, ]∗ = [∗ , ∗ ]. (ii) (Q, X ∗ /∼ , [, ]∗ ) = (Q, X ∗ /≈ , [∗ , ∗ ]) as faithful L I -ftss, where L I = (L I , ⱕ L I , T, ∨) with T = [T1 , T2 ] defined as in Lemma 7.1. Proof. (i) For any q, p ∈ Q and u ∈ X ∗ , by using induction on the length of u, we prove that [, ]∗ (q, u, p) = [∗ , ∗ ](q, u, p). If |u| = 0, then u =  and, for any q, p ∈ Q with q  p we have [, ]∗ (q, u, p) = 0L I = [0 L , 0 L ] = [∗ , ∗ ](q, u, p).

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Otherwise, q = p and then [, ]∗ (q, u, p) = 1L I = [1 L , 1 L ] = [∗ , ∗ ](q, u, p). If |u| > 0, then u = vx for some v ∈ X ∗ , with |v| < |u|, and x ∈ X . Hence, by using the induction hypothesis in the second equality, [, ]∗ (q, u, p) =



{T([, ]∗ (q, v, r ), [, ](r, x, p))|r ∈ Q}  = {T([∗ , ∗ ](q, v, r ), [, ](r, x, p))|r ∈ Q}

  = {T1 (∗ (q, v, r ), (r, x, p))|r ∈ Q}, {T2 (∗ (q, v, r ), (r, x, p))|r ∈ Q} = [∗ (q, u, p), ∗ (q, u, p)].

(ii) The triple (Q, X ∗ /∼ , [, ]∗ ) is a faithful L I -ftss by Theorem 7.2. Furthermore, the triple (Q, X ∗ /≈ , [∗ , ∗ ]) is a faithful L I -ftss by Theorem 7.3. Notice now that, for any u, v ∈ X ∗ , u ∼ v ⇐⇒ [, ]∗ (q, u, p) = [, ]∗ (q, v, p) for every q, p ∈ Q ⇐⇒ [∗ , ∗ ](q, u, p) = [∗ , ∗ ](q, v, p) for every q, p ∈ Q ⇐⇒ u ≈ v. So, the result holds.  8. Conclusions and future research In this paper, the concepts of fuzzy finite state machine and fuzzy transformation semigroup have been generalized by substituting the interval [0, 1] as the truth structure of the transition functions by any complete lattice (L , ⱕ L ) endowed with a t-norm T and a t-conorm S. Several known results about these kinds of fuzzy automata (in which initial and final states are not considered) are generalized to the lattice-valued fuzzy case. In particular, it is shown that the usual mechanism to build a fuzzy transformation semigroup from a fuzzy finite state machine works in the lattice-valued case if and only if the t-conorm S is distributive with respect to the t-norm T . In the case that S is the join, this property is equivalent to the residuation principle for T . For its practical interest, we study especially the lattice consisting of all the closed intervals contained in a complete lattice L, which is also complete. We wonder about the way of recovering the information provided by an interval latticevalued finite state machine and show that two L-fuzzy finite state machines can be obtained from it in a functorial way. Regarding to the transformation semigroups, the t-norm considered in the interval lattice needs to be t-representable in order to get two L-fuzzy transformation semigroups from an interval lattice-valued one. In fact the paper proves that the two different ways of building a faithful L-fuzzy transformation semigroup starting from a lattice interval-valued state machine by using the preceding results give the same result. Conversely, an interval lattice-valued finite state machine and an interval lattice-valued transformation semigroup can be obtained starting from two L-fuzzy finite state machines and two L-fuzzy transformation semigroups, respectively. In this case, the two natural ways of obtaining a faithful interval-valued fuzzy transformation semigroup from two L-fuzzy finite state machines lead to the same result. As future work, we will consider the lattice L = [0, 1]. In this case the information provided by an interval latticevalued finite state machine or an interval lattice-valued transformation semigroup can be retrieved by using Atanassov’s K  operators. We wonder if these operators preserve the properties of a transformation semigroup. In addition the product of two lattice-valued finite state machines can be explored. In particular, the product of two interval lattice-valued finite state machines can be compared with the product of the machines provided by its bounds.

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Acknowledgments The authors would like to thank Professors H. Bustince and J. Fernández (Dpto de Automática y Computación, Universidad Pública de Navarra) for their very helpful comments and suggestions in improving this paper. References [1] K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets Syst. 31 (1989) 343–349. [2] R. Belohlavek, Determinism and fuzzy automata, Inf. Sci. 143 (2002) 205–209. [3] H. Bustince, T. Calvo, B. de Baets, J. Fodor, R. Mesiar, J. Montero, D. Paternain, A. Pradera, A class of aggregation functions encompassing two-dimensional OWA operators, Inf. Sci. 180 (2010) 1977–1989. [4] H. Bustince, E. Barrenechea, M. Pagola, J. Fernández, Interval-valued fuzzy sets constructed from matrices: application to edge detection, Fuzzy Sets Syst. 160 (2009) 1819–1840. [5] H. Bustince, E. Barrenechea, M. Pagola, Generation of interval-valued fuzzy and atanassov’s intuitionistic fuzzy connectives from fuzzy connectives and from K  -operators: laws for conjunctions and disjunctions, amplitude, Int. J. Intell. Syst. 23 (2008) 680–714. [6] G. Deschrijver, E.E. Kerre, Uninorms in L ∗ -fuzzy set theory, Fuzzy Sets Syst. 148 (2004) 243–262. [7] G. Deschrijver, E.E. Kerre, On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision, Inf. Sci. 177 (2007) 1860–1866. [8] G. Deschrijver, A representation of t-norms in interval-valued L-fuzzy set theory, Fuzzy Sets Syst. 159 (2008) 1597–1618. [9] J. Gutiérrez García, S.E. Rodabaugh, Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies, Fuzzy Sets Syst. 156 (2005) 445–484. [10] Y.B. Jun, Quotient structures of intuitionistic fuzzy finite state machines, Inf. Sci. 177 (2007) 4977–4986. [11] Y.B. Jun, Intuitionistic fuzzy transformation semigroups, Inf. Sci. 179 (2009) 4284–4291. [12] Y. Li, W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids, Fuzzy Sets Syst. 156 (2005) 68–92. [13] Y. Li, Finite automata theory with membership values in lattices, Inf. Sci. 181 (2011) 1003–1017. [14] J. Moˇckoˇr, Fuzzy and non-deterministic automata, Soft Comput. 03 (1999) 221–226. [15] J.N. Mordeson, D.S. Malik, Fuzzy Automata and Languages, Chapman and Hall, 2006.