A Remark on Myhill-Nerode Theorem for Fuzzy Languages

Abstract

Regular fuzzy languages are characterized by some algebraic approaches. In particular, an extended version of Myhill-Nerode theorem for fuzzy languages is obtained.

1. Introduction

Fuzzy sets were introduced by Zadeh in [1] and since then have appeared in many fields of sciences. They have been studied within automata theory for the first time by Wee in [2]. More on recent development of algebraic theory of fuzzy automata and formal fuzzy languages can be found in the book Mordeson and Malik [3], the texts Malik et al. [4, 5], and Petković [6].

A fuzzy language is called regular if it can be recognized by a fuzzy automaton. In the texts Mordeson and Malik [3], Petković [6], Ignjatovic et al. [7], and Shen [8], regular fuzzy languages have been characterized by the principal congruences (principal right congruences, principal left congruences) determined by fuzzy languages, which are known as Myhill-Nerode theorem for fuzzy languages. Moreover, Petković [6] also considered the varieties of fuzzy languages and Ignjatovic and Ciric [9] considered regular operations of fuzzy languages.

Recently, Wang et al. [10] generalized the usual principal congruences (resp., principal right congruences, principal left congruences) to some kinds of generalized principal congruences (resp., generalized principal right congruences, generalized principal left congruences) determined by crisp languages by using prefix-suffix-free subsets (resp., prefix-free languages, suffix-free languages) and obtained some characterizations of regular crisp languages.

In this note, we will realize the idea of the text [10] for fuzzy languages. In other word, we characterize regular fuzzy languages by some kinds of generalized principal congruences (resp., generalized principal right congruences, generalized principal left congruences) determined by fuzzy languages. In particular, we obtain an extended version of Myhill-Nerode theorem for fuzzy languages.

2. Preliminaries

Throughout the paper, 𝐴 is a finite set which is called an alphabet and 𝐴^∗ is the free monoid generated by 𝐴, that is, the set of all words with letters from 𝐴. The empty word is denoted by 1. The length of a word 𝑤 in 𝐴^∗ is the number of letters appearing in 𝑤 and is denoted by |𝑤|. The complement of a subset 𝐿 of 𝐴^∗ is the set 𝐿={𝑤∈𝐴^∗∣𝑤∉𝐿}. A subset 𝐿 of 𝐴^∗ is cofinite if 𝐿 is finite. A nonempty subset 𝑆 of 𝐴^∗ is called a suffix-free language over 𝐴 if no element in 𝑆 is a suffix of another element in 𝑆. Prefix-free languages over 𝐴 can be defined dually. On the other hand, a nonempty subset 𝐿 of 𝐴^∗ is called a prefix-closed language over 𝐴 if any prefix of an element in 𝐿 is also in 𝐿.

As an analogue of prefix-free languages and suffix-free languages over 𝐴, Wang et al. [10] introduced prefix-suffix-free subsets of 𝐴^∗×𝐴^∗. A subset Δ of the set 𝐴^∗×𝐴^∗ is called a prefix-suffix-free subset if for all words 𝑠,𝑡,𝑥,𝑦 in 𝐴^∗, the following holds: if both (𝑠,𝑡) and (𝑠𝑥,𝑦𝑡) are in Δ, then 𝑥=𝑦=1.

An equivalence 𝜌 on 𝐴^∗ is called a right congruence if 𝑥𝜌𝑦 implies that 𝑥𝑧𝜌𝑦𝑧 for any 𝑥,𝑦,𝑧∈𝐴^∗. A left congruences can be defined dually. An equivalence is a congruence if it is a right congruence and also a left congruence.

A fuzzy subset 𝛼 of a set 𝑋 is a mapping 𝛼∶𝑋→[0,1]. By ∧ and ∨ infimum and supremum in the unit segment [0,1] will be denoted, respectively. Every element 𝑦 of 𝑋 can be considered as the following fuzzy subset of 𝑋: 𝑦(𝑥)=1for𝑥=𝑦,𝑦(𝑥)=0for𝑥≠𝑦.(2.1) A fuzzy language over 𝐴 is a fuzzy subset of 𝐴^∗. A fuzzy language is regular if it is recognizable by a fuzzy automaton from the book [3]. For a fuzzy language 𝜆 over 𝐴, the relations defined on 𝐴^∗ by the following: 𝑥𝑥𝑃𝑃^𝜆(𝑟)_𝜆(𝑙)𝑦if𝜆(𝑥𝑢)=𝜆(𝑦𝑢)forevery𝑢in𝐴^∗,𝑥𝑃_𝜆𝑦if𝜆(𝑢𝑥)=𝜆(𝑢𝑦)forevery𝑢in𝐴^∗,𝑦if𝜆(𝑢𝑥𝑣)=𝜆(𝑢𝑦𝑣)forevery𝑢,𝑣∈𝐴^∗,(2.2) are called the principal right congruence (resp., principal left congruence, principal congruence) determined by 𝜆, respectively.

Now, we state the well-known Myhill-Nerode theorem for fuzzy languages which gives some algebraic characterizations for regular fuzzy languages. Recall that the index of an equivalence 𝜌 on 𝐴^∗ is the number of 𝜌-classes of 𝐴^∗.

Theorem 2.1 (see [3, 6, 8], Myhill-Nerode theorem). For a fuzzy language 𝜆 over 𝐴, the following statements are equivalent:(1)𝜆is regular.(2)𝑃_𝜆 is of finite index.(3)𝑃^𝜆(𝑟) is of finite index.(4)𝑃^𝜆(𝑙) is of finite index.

In the sequel, we recall some operations of fuzzy languages. For two fuzzy languages 𝜆₁ and 𝜆₂ over 𝐴, the union, intersection, product, and left and right quotients of 𝜆₁ and 𝜆₂ are defined, respectively, by the following: 𝜆𝜆₁∪𝜆₂(𝑤)=𝜆₁(𝑤)∨𝜆₂(𝑤),₁𝜆∩𝜆₂(𝑤)=𝜆₁(𝑤)∧𝜆₂(𝑤),₁𝜆𝜆₂(𝑤)=_{𝑥𝑦=𝑤}𝜆₁(𝑥)∧𝜆₂(𝑦)₁₋₁𝜆𝜆₂(𝑤)=_{𝑢∈𝐴∗}𝜆₂(𝑢𝑤)∧𝜆₁(𝑢)₂,𝜆¹⁻¹(𝑤)=_{𝑢∈𝐴∗}𝜆₂(𝑤𝑢)∧𝜆₁(𝑢).(2.3) Further, we also define left-right quotient of three fuzzy languages 𝜆₁, 𝜆₂ and 𝜆 over 𝐴 by the following: 𝜆¹⁻¹𝜆𝜆²⁻¹(𝑤)=𝜆¹⁻¹𝜆𝜆²⁻¹(𝑤).(2.4) Observe that (𝑠⁻¹𝜆𝑡⁻¹)(𝑤)=𝜆(𝑠𝑤𝑡) for any 𝑠,𝑡,𝑤∈𝐴^∗ with the above notations.

On regular fuzzy languages, we have the following.

Lemma 2.2 (see [6]). Finite unions, intersections, products, and left-right quotients of regular fuzzy languages over 𝐴 are regular.

3. Main Result

In this section, we shall introduce some kinds of generalized principal (resp., right, left) congruences determined by fuzzy languages by using prefix-suffix-free subsets (resp., prefix-free languages, suffix-free languages) and give an extended version of Myhill-Nerode theorem for fuzzy languages.

Now, let 𝑃 be a prefix-free language, 𝑆 be a suffix-free language over 𝐴, Δ be a prefix-suffix-free subset of 𝐴^∗×𝐴^∗, and 𝜆 be a fuzzy language over 𝐴, respectively. For a prefix-suffix-free subset Δ, denote Ω_Δ=(𝑠𝑥,𝑦𝑡)∣(𝑠,𝑡)∈Δ,𝑥,𝑦∈𝐴^∗,𝑁(Δ)=_{(𝑠,𝑡)∈Δ}𝑠𝐴^∗𝑡.(3.1) Define the following relations on 𝐴^∗: 𝑥𝑥𝑃𝑃^{(𝑙)𝑃,𝜆}_{(𝑟)𝑆,𝜆}𝑦if𝜆(𝑢𝑥)=𝜆(𝑢𝑦)forevery𝑢in𝑃𝐴^∗,𝑥𝑃_Δ,𝜆𝑦if𝜆(𝑥𝑢)=𝜆(𝑦𝑢)forevery𝑢in𝐴^∗𝑆,𝑥𝑃_{(𝑟)ℱ,𝑆,𝜆}𝑦if𝜆(𝑢𝑥𝑣)=𝜆(𝑢𝑦𝑣)forevery(𝑢,𝑣)inΩ_Δ,𝑦if𝜆th(𝑥er𝑢e)e=xi𝜆st(s𝑦𝑢so)mefofirneivteersyub𝑢sient𝐹of𝐴^∗suchthat𝑥𝑃_{(𝑙)ℱ,𝑃,𝜆}𝐹𝐴^∗𝑆,𝑦if𝜆th(e𝑢r𝑥e)e=xi𝜆st(s𝑢s𝑦o)mefofirneivteersyub𝑢seint𝐹of𝐴^∗suchthat𝑃𝐴^∗𝐹.(3.2) Then we have the following observations.

Proposition 3.1. The above 𝑃^{(𝑟)𝑆,𝜆},𝑃^{(𝑟)ℱ,𝑆,𝜆}(𝑟𝑒𝑠𝑝.,𝑃^{(𝑙)𝑃,𝜆},𝑃^{(𝑙)ℱ,𝑃,𝜆};𝑃_Δ,𝜆) are right congruences (resp., left congruences; congruence) on 𝐴^∗. Furthermore, 𝑃^𝜆(𝑟)⊆𝑃^{(𝑟)𝑆,𝜆}⊆𝑃^{(𝑟)ℱ,𝑆,𝜆},𝑃^𝜆(𝑙)⊆𝑃^{(𝑙)𝑃,𝜆}⊆𝑃^{(𝑙)ℱ,𝑃,𝜆},𝑃_𝜆⊆𝑃_Δ,𝜆.(3.3)

Proof. It is easy to check that 𝑃^{(𝑟)𝑆,𝜆} (resp., 𝑃^{(𝑙)𝑃,𝜆}) is a right (resp., left) congruence, 𝑃_Δ,𝜆 is a congruence, and 𝑃^𝜆(𝑟)⊆𝑃^{(𝑟)𝑆,𝜆}⊆𝑃^{(𝑟)ℱ,𝑆,𝜆},𝑃^𝜆(𝑙)⊆𝑃^{(𝑙)𝑃,𝜆}⊆𝑃^{(𝑙)ℱ,𝑃,𝜆},𝑃_𝜆⊆𝑃_Δ,𝜆(3.4) by their definitions. In the sequel, we show that 𝑃^{(𝑟)ℱ,𝑆,𝜆} is a right congruence and 𝑃^{(𝑙)ℱ,𝑃,𝜆} is a left congruence. Clearly, both 𝑃^{(𝑟)ℱ,𝑆,𝜆} and 𝑃^{(𝑙)ℱ,𝑃,𝜆} are equivalences. Now, let 𝑥,𝑦 be two words in 𝐴^∗ and 𝑥𝑃^{(𝑟)ℱ,𝑆,𝜆}𝑦. Then there exists a finite subset 𝐹 of 𝐴^∗ such that 𝜆(𝑥𝑢)=𝜆(𝑦𝑢) for any 𝑢 in 𝐹𝐴^∗𝑆. Now, let 𝑧 be a word in 𝐴^∗ and 𝐹^ be the union of {𝑤∈𝐴^∗∣𝑧𝑤∈𝐹} and {1}. Then 𝑧𝑢 is in 𝐹𝐴^∗𝑆 for any 𝑢 in 𝐹^𝐴^∗𝑆. This implies that 𝜆(𝑥𝑧𝑢)=𝜆(𝑦𝑧𝑢) for any 𝑢 in 𝐹^𝐴^∗𝑆 whence 𝑥𝑧𝑃^{(𝑟)ℱ,𝑆,𝜆}𝑦𝑧 since 𝐹^ is finite. Thus, 𝑃^{(𝑟)ℱ,𝑆,𝜆} is a right congruence. Dually, 𝑃^{(𝑙)ℱ,𝑃,𝜆} is a left congruence.

Remark 3.2. Note that the above inclusions are all proper in general. For example, let 𝐴={𝑎},𝑆={𝑎²} and 𝐹={1,𝑎,𝑎²,𝑎³}. Then 𝐹𝐴^∗𝑆=𝐴^∗𝑎⁵. Define a fuzzy language over 𝐴 as follows: 𝜆(𝑤)=𝛼for𝑤∈𝑎²,𝑎³,𝜆(𝑤)=𝛽for𝑤∈𝐴^∗⧵𝑎²,𝑎³,(3.5) where 𝛼,𝛽 are in [0,1] and 𝛼≠𝛽. Then we have 𝑎³,𝑎⁴∉𝑃^𝜆(𝑟),𝑎³,𝑎⁴∈𝑃^(𝑟)_S,𝜆,1,𝑎²∉𝑃^{(𝑟)𝑆,𝜆},1,𝑎²∈𝑃^{(𝑟)ℱ,𝑆,𝜆}.(3.6) Similarly, we can show that the remainder inclusions are all proper.

To obtain our main result, we need a series of lemmas. First, we recall the following alphabetic order “≤” on 𝐴^∗: For two words 𝑢 and 𝑣 in 𝐴^∗ with different lengths, 𝑢<𝑣 if |𝑢|<|𝑣|, for two words with same length, the order is the lexicographic order. Observe that the alphabetic order is a well order on 𝐴^∗. We have the following result.

Lemma 3.3. Let 𝐿 be an infinite prefix-closed language over 𝐴. Then there exists an infinite subset {1,𝑎₁,𝑎₁𝑎₂,…,𝑎₁𝑎₂,…,𝑎_𝑛,…} of 𝐿, where 𝑎_𝑖∈𝐴.

Proof. Denote Pre𝑓_𝐴(𝐿)=𝑎∈𝐴∣∃𝑦∈𝐴^∗𝑎𝑦∈𝐿.(3.7) Observe that 𝐴 is finite and 𝐿 is infinite, there exists 𝐿₁⊆𝐿 and 𝑎₁∈𝐴 such that 𝐿₁ is infinite and Pre𝑓_𝐴(𝐿₁)={𝑎₁}. Denote 𝑎¹⁻¹𝐿₁=𝑤∈𝐴^∗∣𝑎₁𝑤∈𝐿₁.(3.8) Then 𝑎¹⁻¹𝐿₁ is infinite. Hence, there also exists 𝐿₂⊆𝑎¹⁻¹𝐿₁ and 𝑎₂∈𝐴 such that 𝐿₂ is infinite and Pre𝑓_𝐴(𝐿₂)={𝑎₂}. In general, for any positive integer 𝑛, there exists 𝐿_𝑛+1⊆𝑎^𝑛−1𝐿_𝑛 and 𝑎_𝑛+1∈𝐴 such that 𝐿_𝑛+1 is infinite and Pre𝑓_𝐴(𝐿_𝑛+1)={𝑎_𝑛+1}. Let 𝐶=1,𝑎₁,𝑎₁𝑎₂,𝑎₁𝑎₂𝑎₃,…,𝑎₁𝑎₂𝑎₃⋯𝑎_𝑛,….(3.9) Clearly, 𝐶 is infinite. We claim that 𝐶⊆𝐿. Let 𝑎₁𝑎₂𝑎₃⋯𝑎_𝑛∈𝐶. Observe that 𝐿_𝑛⊆𝑎^−1𝑛−1𝐿_𝑛−1⊆𝑎_𝑛𝑎^−1𝑛−1𝑎^−1𝑛−2𝐿_𝑛−2⊆⋯⊆𝑎^−1𝑛−1𝑎^−1𝑛−2⋯𝑎¹⁻¹𝐿₁=𝑎₁𝑎₂⋯𝑎_𝑛−1⁻¹𝐿₁,=Pre𝑓_𝐴𝐿_𝑛⊆Pre𝑓_𝐴𝑎¹𝑎₂⋯𝑎_𝑛−1⁻¹𝐿₁.(3.10) Therefore, there exists 𝑦∈𝐴^∗ such that 𝑎_𝑛𝑦∈(𝑎₁𝑎₂⋯𝑎_𝑛−1)⁻¹𝐿₁. And hence, 𝑎₁𝑎₂⋯𝑎_𝑛−1𝑎_𝑛𝑦∈𝐿₁⊆𝐿. Since 𝐿 is prefix-closed, 𝑎₁𝑎₂⋯𝑎_𝑛−1𝑎_𝑛∈𝐿. This implies that 𝐶⊆𝐿.

Lemma 3.4. Let 𝜌 be a right congruence on 𝐴^∗ and {𝐿_𝑖∣𝑖∈𝐼} be the set of all 𝜌-classes of 𝐴^∗. Then, 𝐿_𝜌=𝑠_𝑖∣𝑠_𝑖𝑖𝑠𝑡ℎ𝑒𝑙𝑒𝑎𝑠𝑡𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑖𝑛𝐿_𝑖𝑤𝑖𝑡ℎ𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑡𝑜”≤”,𝑖∈𝐼(3.11) is prefix-closed.

Proof. Clearly, 1 is in 𝐿_𝜌. Let 𝑠_𝑗 be in 𝐿_𝜌 and 𝑠_𝑗=𝑎₁𝑎₂⋯𝑎_𝑡 for some positive integer 𝑡>1 and 𝑎₁,𝑎₂,…,𝑎_𝑡 in 𝐴. Then, 𝑎₁𝑎₂⋯𝑎_𝑡−1 is not in 𝐿_𝑗. Suppose that 𝑎₁𝑎₂⋯𝑎_𝑡−1 is in 𝐿_𝑘. Then, 𝑠_𝑘≤𝑎₁𝑎₂⋯𝑎_𝑡−1. This implies that 𝑠_𝑘𝑎_𝑡≤𝑎₁𝑎₂⋯𝑎_𝑡−1𝑎_𝑡=𝑠_𝑗. On the other hand, since 𝑠_𝑘𝜌𝑎₁𝑎₂⋯𝑎_𝑡−1 and 𝜌 is a right congruence, we have 𝑠_𝑘𝑎_𝑡𝜌𝑠_𝑗. Hence, 𝑠_𝑘𝑎_𝑡 is in 𝐿_𝑗 and so 𝑠_𝑘𝑎_𝑡≥𝑠_𝑗. Thus, 𝑠_𝑘𝑎_𝑡=𝑠_𝑗=𝑎₁𝑎₂𝑎₃⋯𝑎_𝑡−1𝑎_𝑡. This implies that 𝑠_𝑘=𝑎₁𝑎₂𝑎₃⋯𝑎_𝑡−1 whence 𝑎₁𝑎₂𝑎₃⋯𝑎_𝑡−1 is in 𝐿_𝜌.

Lemma 3.5. Let 𝑆 be a suffix-free language and 𝜆 be a fuzzy language over 𝐴.(1)𝑃_{{1}×𝑆,𝜆} is of finite index if and only if 𝑃^{(𝑟)𝑆,𝜆} is of finite index.(2)𝑃^{(𝑟)𝑆,𝜆} is of finite index if and only if 𝑃^{(𝑟)ℱ,𝑆,𝜆} is of finite index.

Proof. (1)  Similar to the proof of Proposition  3.11 in [10].
(2) Observe that 𝑃^{(𝑟)𝑆,𝜆}⊆𝑃^{(𝑟)ℱ,𝑆,𝜆}, the necessity holds. Conversely, if 𝑃^{(𝑟)ℱ,𝑆,𝜆} is of finite index and 𝑃^{(𝑟)𝑆,𝜆} is of infinite index, then by Lemma 3.4, 𝐿_𝑃^{(𝑟)𝑆,𝜆} is infinite and prefix-closed. By Lemma 3.3, there exists an infinite subset 𝐶=1,𝑎₁,𝑎₁𝑎₂,⋯,𝑎₁𝑎₂⋯𝑎_𝑛,…,(3.12) of 𝐿_𝑃^{(𝑟)𝑆,𝜆}, where 𝑎_𝑖∈𝐴. Since 𝑃^{(𝑟)ℱ,𝑆,𝜆} is of finite index, there exist two distinct elements 𝑥,𝑦∈𝐶 such that 𝑥𝑃^{(𝑟)ℱ,𝑆,𝜆}𝑦. Therefore, there exists a finite subset 𝐹 of 𝐴^∗ such that 𝜆(𝑥𝑢)=𝜆(𝑦𝑢)forevery𝑢in𝐹𝐴^∗𝑆. Denote 𝑇=max{|𝑓|∣𝑓∈𝐹} and take 𝑢 in 𝐴^∗ satisfying |𝑢|>𝑇. We assert that 𝑢𝑣 is in 𝐹𝐴^∗𝑆 for any 𝑣 in 𝐴^∗𝑆. In fact, if 𝑢𝑣=𝑓𝑤 for some 𝑓 in 𝐹 and 𝑤 in 𝐴^∗𝑆, then by the choice of 𝑢, 𝑓 is a prefix of 𝑢 and so 𝑣 is a suffix of 𝑤 whence 𝑤 is in 𝐴^∗𝑆. A contradiction. Therefore, for any 𝑣 in 𝐴^∗𝑆, we have 𝜆(𝑥𝑢𝑣)=𝜆(𝑦𝑢𝑣). This implies that 𝑥𝑢𝑃^{(𝑟)𝑆,𝜆}𝑦𝑢.
Without loss of generality, we let 𝑥<𝑦 with respect to the alphabetic order, 𝑦=𝑎₁𝑎₂⋯𝑎_𝑡 and 𝑢=𝑎_𝑡+1⋯𝑎_𝑡+𝑇+1. Then, by the above discussions, 𝑥𝑢𝑃^{(𝑟)𝑆,𝜆}𝑦𝑢 and 𝑦𝑢 is in 𝐶. Observe that 𝐶 is a subset of 𝐿_𝑃^{(𝑟)𝑆,𝜆}, in view of the definition of 𝐿_𝑃^{(𝑟)𝑆,𝜆}, 𝑥𝑢≥𝑦𝑢. This implies that 𝑥≥𝑦. A contradiction.

Lemma 3.6. Let 𝑆 be a finite suffix-free language over 𝐴. Then 𝐴^∗𝑆 is cofinite if and only if 𝑆 is maximal.

Proof. It follows from Lemma  3.14 in [10].

Lemma 3.7. Let Δ be a finite prefix-suffix-free subset of 𝐴^∗×𝐴^∗ and 𝜆 be a fuzzy language over 𝐴. Then the following are equivalent:(1)𝑃_Δ,𝜆 is of finite index.(2)The following fuzzy language 𝜆_Δ over 𝐴 defined by 𝜆_Δ(𝑤)=𝜆(𝑤)𝑓𝑜𝑟𝑤∈𝑁(Δ),𝜆_Δ(𝑤)=0𝑓𝑜𝑟𝑤∉𝑁(Δ)(3.13) is regular.(3)𝜆=𝜆₁∪𝜆₂, where 𝜆₁ is regular and 𝜆₂(𝑤)=0 for any 𝑤 in 𝑁(Δ).

Proof. (1) implies (2). Let 𝑥,𝑦 be in 𝐴^∗, (𝑠,𝑡) be in Δ and 𝑥𝑃_Δ,𝜆𝑦. Then for any 𝑢,𝑣 in 𝐴^∗, (𝑠𝑢,𝑣𝑡) is in Ω_Δ. Therefore, 𝑠⁻¹𝜆𝑡⁻¹(𝑢𝑥𝑣)=𝜆(𝑠𝑢𝑥𝑣𝑡)=𝜆(𝑠𝑢𝑦𝑣𝑡)=𝑠⁻¹𝜆𝑡⁻¹(𝑢𝑦𝑣),(3.14) whence 𝑥𝑃_{𝑠−1𝜆𝑡−1}𝑦. Thus, 𝑃_Δ,𝜆⊆_{(𝑠,𝑡)∈Δ}𝑃_{𝑠−1𝜆𝑡−1}.(3.15) Now, if 𝑃_Δ,𝜆 is of finite index, then 𝑠⁻¹𝜆𝑡⁻¹ is regular for any (𝑠,𝑡) in Δ. Observe that 𝜆_Δ=_{(𝑠,𝑡)∈Δ}𝑠𝑠⁻¹𝜆𝑡⁻¹𝑡,(3.16) it follows that 𝜆_Δ is regular from Lemma 2.2.
(2) implies (3). By (2), 𝜆_Δ is regular. Let 𝜆₂ be the following fuzzy language over 𝐴 defined by 𝜆₂(𝑤)=0for𝑤∈𝑁(Δ),𝜆₂(𝑤)=𝜆(𝑤)for𝑤∉𝑁(Δ).(3.17) Then 𝜆=𝜆_Δ∪𝜆₂, as required.
(3) implies (1). If 𝜆=𝜆₁∪𝜆₂ for some regular fuzzy language 𝜆₁ and a fuzzy language 𝜆₂ such that 𝜆₂(𝑤)=0 for any 𝑤 in 𝑁(Δ), then 𝑃_𝜆1 is of finite index and 𝑃_Δ,𝜆2=𝐴^∗×𝐴^∗. Observe that 𝑃_𝜆1⊆𝑃_Δ,𝜆1⊆𝑃_{Δ,𝜆1∪𝜆2}=𝑃_Δ,𝜆,(3.18)𝑃_Δ,𝜆 is of finite index.

Remark 3.8. In general, for a given finite prefix-suffix-free subset of 𝐴^∗×𝐴^∗ and a fuzzy language 𝜆 over 𝐴, 𝜆 may be nonregular even if 𝑃_Δ,𝜆 is of finite index. For example, let 𝐴={𝑎,𝑏} and Δ={(𝑎,𝑏)}. Define the following fuzzy language 𝜆 over 𝐴 as follows: 𝜆(𝑤)=0for𝑤∈𝑁(Δ),𝜆(𝑤)=1|𝑤|+1for𝑤∉𝑁(Δ).(3.19) Clearly, 𝜆_Δ(𝑤)=0 for every 𝑤 in 𝐴^∗ and so 𝜆_Δ is trivially regular. By Lemma 3.7, 𝑃_Δ,𝜆 is of finite index. However, for any pair 𝑤₁,𝑤₂ in 𝑁(Δ) with different lengths, we have 𝜆(𝑤₁)≠𝜆(𝑤₂) whence 𝑤₁ is not 𝑃_𝜆 related to 𝑤₂. Observe that 𝑁(Δ) is infinite, there are infinite 𝑃_𝜆-classes of 𝐴^∗ and so 𝑃_𝜆 is of infinite index. This implies that 𝜆 is nonregular by Theorem 2.1.

Now, we have our main theorem.

Theorem 3.9 (An extended version of Myhill-Nerode theorem). For a fuzzy language 𝜆 over 𝐴, the following statements are equivalent:(1)𝜆 is regular.(2)𝑃^{(𝑟)𝑆,𝜆} is of finite index for some finite maximal suffix-free language 𝑆 over 𝐴.(3)𝑃^{(𝑙)𝑃,𝜆} is of finite index for some finite maximal prefix-free language 𝑃 over 𝐴.(4)𝑃_Δ,𝜆 is of finite index for some finite prefix-suffix-free subset Δ of 𝐴^∗×𝐴^∗ such that 𝑁(Δ) is cofinite.(5)𝑃^{(𝑙)ℱ,𝑃,𝜆} is of finite index for some finite maximal prefix-free language 𝑃 over 𝐴.(6)𝑃^{(𝑟)ℱ,𝑆,𝜆} is of finite index for some finite maximal suffix-free language 𝑆 over 𝐴.

Proof. (1) implies (2). Observe that {1} is a maximal suffix-free language over 𝐴 and 𝑃^𝜆(𝑟)=𝑃^{(𝑟){1},𝜆}, the result follows from Theorem 2.1.
(2) implies (4). Observe that {1}×𝑆 is a prefix-suffix-free subset of 𝐴^∗×𝐴^∗ and 𝑁({1}×𝑆)=𝐴^∗𝑆, the result follows from Lemma 3.5 (1) and Lemma 3.6.
(4) implies (1). By Lemma 3.7 (3), there exists a regular fuzzy language 𝜆₁ and another fuzzy language 𝜆₂ such that 𝜆=𝜆₁∪𝜆₂ and 𝜆₂(𝑤)=0 for any 𝑤 in 𝑁(Δ). However, by (4), 𝑁(Δ) is finite, which implies that 𝜆₂ is also regular. In view of Lemma 2.2, 𝜆 is regular.
By symmetry, we can prove that the facts that (1) implies (3) and (3) implies (4). On the other hand, by Lemma 3.5 (2) and its dual, it follows that (3) is equivalent to (5) and (2) is equivalent to (6).

4. Conclusions

In this short note, we have obtained an extended version of Myhill-Nerode theorem for fuzzy languages (Theorem 3.9) which provides some algebraic characterizations of regular fuzzy languages. On the other hand, for a given prefix-suffix-free subset Δ of 𝐴^∗×𝐴^∗, by Proposition 3.1 and Remark 3.8, 𝔽ℝ_Δ(𝐴)=𝜆∣𝜆isafuzzylanguageover𝐴suchthattheindexof𝑃_Δ,𝜆isfinite(4.1)

contains the class of regular fuzzy languages over 𝐴 as a proper subclass. In fact, Lemma 3.7 gives some characterizations of members in 𝔽ℝ_Δ(𝐴) for a given finite prefix-suffix-free subset Δ of 𝐴^∗×𝐴^∗. Thus the following questions could be considered as a future work. For a general prefix-suffix-free subset Δ of 𝐴^∗×𝐴^∗, what can be said about 𝔽ℝ_Δ(𝐴)? For example, can we obtain some results parallel to Theorems  3.5 and 3.17 in [10]?