On the Syntactic Monoids Associated with a Class of Synchronized Codes

Abstract

A complete code 𝐶 over an alphabet 𝐴 is called synchronized if there exist 𝑥,𝑦∈𝐶^∗ such that 𝑥𝐴^∗∩𝐴^∗𝑦⊆𝐶^∗. In this paper we describe the syntactic monoid Syn(𝐶⁺) of 𝐶⁺ for a complete synchronized code 𝐶 over 𝐴 such that 𝐶⁺, the semigroup generated by 𝐶, is a single class of its syntactic congruence 𝑃_𝐶⁺. In particular, we prove that, for such a code 𝐶, either 𝐶=𝐴 or Syn(𝐶⁺) is isomorphic to a special submonoid of 𝒯^𝑙(𝐼)×𝒯^𝑟(Λ), where 𝒯^𝑙(𝐼) and 𝒯^𝑟(Λ) are the full transformation semigroups on the nonempty sets 𝐼 and Λ, respectively.

1. Introduction

Theory of codes is an important branch in the field of information science. Many methods, including combinatorics methods, analysis methods, and algebraic methods, are applied to study codes. As a kind of algebraic methods, it is effective to study some kinds of codes by considering syntactic monoids of the semigroups and monoids generated by these codes.

As we have known, prefix codes have fundamental importance in theory of codes. Many authors are devoted to the investigation of prefix codes by using several methods (cf. [1–4]). In particular, Petrich et al. [4] investigated maximal prefix codes by considering the syntactic monoids of the semigroups generated by them in 1996. They described the semigroup structure of the syntactic monoid Syn(𝐶⁺) of 𝐶⁺, the semigroup generated by a maximal prefix code 𝐶 for which 𝐶⁺ is a single class of the syntactic congruence 𝑃_𝐶⁺.

On the other hand, synchronized codes are also important both in theory and in applications. Many interesting results are obtained on this class of codes in the text of Berstel et al. [1]. Recently, Liu [3] investigated synchronized codes by algebraic methods and obtained an algebraic characterization of complete synchronized codes (see Lemma 5 in this paper). Furthermore, Liu [3, 5] also studied some generalizations of synchronized codes.

In this paper, by using the algebraic characterization of complete synchronized codes obtained in Liu [3] and some techniques developed in Petrich et al. [4], we give a description of the syntactic monoid Syn(𝐶⁺) of 𝐶⁺ for a complete synchronized code 𝐶 over an alphabet 𝐴 such that 𝐶⁺ is a single class of its syntactic congruence 𝑃_𝐶⁺.

2. Preliminaries

A semigroup 𝑆 is a left zero semigroup if 𝑎𝑏=𝑎 for any 𝑎,𝑏∈𝑆. Dually, we have right zero semigroups. A rectangular band is a semigroup which is isomorphic to a direct product of a left zero semigroup and a right zero semigroup. For rectangular bands, we have the following obvious result.

Lemma 1 (see [6]). Let 𝑆 be a rectangular band. Then 𝑎𝑥𝑎=𝑎 for any 𝑎,𝑥∈𝑆. As a consequence, 𝑎𝑆𝑎={𝑎} for any 𝑎∈𝑆. In particular, if 𝑆 has an identity 𝑒, then 𝑆={𝑒}.

An ideal of a semigroup 𝑆 is a nonempty subset 𝐼 of 𝑆 satisfying that the union of 𝐼𝑆 and 𝑆𝐼 is contained in 𝐼. Recall that the unique minimum ideal (with respect to set inclusion) of a semigroup 𝑆 (if exists) is called the kernel of 𝑆. For the kernel of a semigroup, we have the following.

Lemma 2 (see [6]). If the kernel of a semigroup 𝑆 only consists of idempotents, then this kernel is a rectangular band.

Let 𝑆 be a semigroup. A function 𝜆 (resp., 𝜌) on 𝑆 is a left translation (resp., right translation) of 𝑆 if 𝜆(𝑥𝑦)=(𝜆𝑥)𝑦 (resp., (𝑥𝑦)𝜌=𝑥(𝑦𝜌)) for all 𝑥,𝑦∈𝑆. A left translation 𝜆 and a right translation 𝜌 are linked if 𝑥(𝜆𝑦)=(𝑥𝜌)𝑦 in which case the pair (𝜆,𝜌) is a bitranslation of 𝑆. Denote the set of left translations and that of right translations on 𝑆 by Λ(𝑆) and 𝑃(𝑆), respectively. Clearly, Λ(𝑆) forms a monoid under the usual composition of functions: (𝜆𝜆^′)𝑥=𝜆(𝜆^′𝑥) for all 𝑥∈𝑆. Dually, 𝑃(𝑆) forms a monoid under the usual composition of functions: 𝑥(𝜌𝜌^′)=(𝑥𝜌)𝜌^′ for all 𝑥∈𝑆. The set of all bitranslations of 𝑆 forms a submonoid of the direct product Λ(𝑆)×𝑃(𝑆), which is called the translation hull of 𝑆, to be denoted by Ω(𝑆).

Let 𝑠 be an element of a semigroup 𝑆. Then the function 𝜆_𝑠 defined by 𝜆_𝑠𝑥=𝑠𝑥 for all 𝑥∈𝑆 is the inner left translation induced by 𝑠. Dually, we have inner right translation 𝜌_𝑠 induced by 𝑠. Finally, the pair 𝜋_𝑠=(𝜆_𝑠,𝜌_𝑠) is the inner bitranslation induced by 𝑠. The set Π(𝑆) of all inner bitranslations is the inner part of Ω(𝑆). From Corollary III.1.7 in Petrich [7], Π(𝑆) is an ideal of Ω(𝑆).

In the sequel, the set of all transformations on a set 𝑄 written and composed as right (resp., left) operators is denoted by 𝒯^𝑟(𝑄) (resp., 𝒯^𝑙(𝑄)). The identity mapping on a set 𝑄 is denoted by 𝜄_𝑄. If 𝑞∈𝑄, then ⟨𝑞⟩ denotes the constant function on 𝑄 whose value is 𝑞. Clearly, 𝒯^𝑙(𝑄) and 𝒯^𝑟(𝑄) are semigroups with their own compositions and 𝒯^𝑙₀(𝑄)={⟨𝑞⟩∣𝑞∈𝑄} (as left operators) and 𝒯^𝑟₀(𝑄)={⟨𝑞⟩∣𝑞∈𝑄} (as right operators) are subsemigroups of 𝒯^𝑙(𝑄) and 𝒯^𝑟(𝑄), respectively.

On the translation hull of a rectangular band, we have the following results which can be found in Section III.7 in Petrich and Reilly [6].

Lemma 3 (see [6]). Let 𝐼×Λ be a rectangular band.(1)Ω(𝐼×Λ)≅𝒯^𝑙(𝐼)×𝒯^𝑟(Λ).(2)𝐼×Λ≅Π(𝐼×Λ)≅𝒯^𝑙₀(𝐼)×𝒯^𝑟₀(Λ).(3)𝒯^𝑙₀(𝐼)×𝒯^𝑟₀(Λ) is the kernel of 𝒯^𝑙(𝐼)×𝒯^𝑟(Λ).

Let 𝑆 be a semigroup and 𝐽 an ideal of 𝑆. Then 𝑆 is called an extension of 𝐽. From Definition III.5.4 in Petrich [7], an extension 𝑆 of 𝐽 is called dense if for each congruence 𝜌 on 𝑆, the fact that the restriction of 𝜌 to 𝐽 is the equality relation on 𝐽 implies that 𝜌 is the equality relation on 𝑆.

On dense extensions of rectangular bands, the following results can be obtained as a special case from Theorem III.1.12 and Corollary III.5.5 in [7] and can also be proved easily.

Lemma 4. If 𝑆 is a dense extension of a rectangular band 𝐽, then the following semigroup homomorphism 𝜏∶𝑆⟶Ω(𝐽),𝑠⟼𝑠𝜏=𝜆^𝑠,𝜌^𝑠 (1) is injective, where for each 𝑠∈𝑆, 𝜆^𝑠∶𝐽⟶𝐽,𝑗⟼𝑠𝑗,𝜌^𝑠∶𝐽⟶𝐽,𝑗⟼𝑗𝑠. (2) Clearly in this case, 𝑆 is isomorphic to 𝑆𝜏 which contains Π(𝐽) as an ideal.
Let 𝑆 be a semigroup and let 𝑆¹ be the semigroup obtained from 𝑆 by adjoining an identity if necessary. The syntactic congruence 𝑃_𝐿 determined by a subset 𝐿 of 𝑆 is the following relation on 𝑆: {(𝑥,𝑦)∈𝑆×𝑆∣𝑢𝑥𝑣∈𝐿 if and only if 𝑢𝑦𝑣∈𝐿 for all 𝑢,𝑣∈𝑆¹}. In particular, if 𝐿={𝑥}, we call  𝑃_{𝑥}  the syntactic congruence determined by 𝑥 and denote it by 𝑃_𝑥. Moreover, 𝑥 is called disjunctive in 𝑆 if 𝑃_𝑥 is the equality relation on 𝑆. It is easy to see that the relation  𝑃_𝐿  saturates  𝐿 for every subset 𝐿 of 𝑆; that is, 𝐿 is a union of some 𝑃_𝐿-classes for every subset 𝐿 of 𝑆.

Let 𝐴 be an alphabet, let 𝐴^∗ be the free monoid generated by 𝐴, and let 1 be the identity of 𝐴^∗. For any 𝐿⊆𝐴^∗, the quotient monoid 𝐴^∗/𝑃_𝐿 is called the syntactic monoid of 𝐿, to be denoted by Syn(𝐿). A nonempty set 𝐶 of 𝐴⁺=𝐴^∗⧵{1} is called a code over 𝐴 if the fact that 𝑥₁𝑥₂⋯𝑥_𝑚=𝑦₁𝑦₂⋯𝑦_𝑛,𝑥_𝑖,𝑦_𝑗∈𝐶,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛, (3) implies that 𝑚=𝑛 and 𝑥_𝑖=𝑦_𝑖, 𝑖=1,2,…,𝑛.

A submonoid 𝑆 of a monoid 𝑀 is called stable in 𝑀 if the fact that 𝑢,𝑤𝑢,𝑢𝑤∈𝑆 implies that 𝑤∈𝑆 for all 𝑢,𝑣,𝑤∈𝑀. It is well known that the monoid 𝐶^∗ generated by a code 𝐶 over 𝐴 is stable in 𝐴^∗ (see Proposition 2.2.5 in [1]).

A code 𝐶 over 𝐴 is called complete if, for any 𝑤∈𝐴^∗, there exist 𝑢,𝑣∈𝐴^∗ such that 𝑢𝑤𝑣∈𝐶^∗, where 𝐶^∗ is the monoid generated by 𝐶. Recall that a complete code 𝐶 over 𝐴 is said to be synchronized if 𝑥𝐴^∗∩𝐴^∗𝑦⊆𝐶^∗ for some 𝑥,𝑦∈𝐶^∗ (see details in Proposition 10.1.14 of [1]). On complete synchronized codes, Liu [3] obtained the following algebraic characterizations recently.

Lemma 5 (see [3]). A complete code 𝐶 over 𝐴 is synchronized if and only if the kernel of Syn(𝐶^∗) is a rectangular band.

3. A Characterization of Complete Synchronized Codes

This section gives a characterization of complete synchronized codes by using the syntactic monoid Syn(𝐶⁺) of 𝐶⁺, the semigroup generated by a code 𝐶 over 𝐴. To this aim, we need several lemmas.

Lemma 6. Let 𝐶 be a code over 𝐴.(1)For any 𝑥,𝑦∈𝐴⁺, 𝑥𝑃_𝐶⁺𝑦 if and only if 𝑥𝑃_𝐶^∗𝑦.(2)The 𝑃_𝐶⁺-class containing 1 is {1}.(3)Syn(𝐶⁺)⧵{1𝑃_𝐶⁺} is a subsemigroup of Syn(𝐶⁺).

Proof. (1) This follows from the fact that 𝑢𝑥𝑣∈𝐶⁺ if and only if 𝑢𝑥𝑣∈𝐶^∗ for any 𝑥∈𝐴⁺ and 𝑢,𝑣∈𝐴^∗.
(2) Let 𝑤∈𝐴^∗ and 𝑤𝑃_𝐶⁺1. Since 1∉𝐶⁺ and 𝐶⁺ is a union of some 𝑃_𝐶⁺-classes, it follows that 𝑤∉𝐶⁺. On the other hand, for any 𝑐∈𝐶, we have 𝑐𝑤𝑃_𝐶⁺𝑐 and 𝑤𝑐𝑃_𝐶⁺𝑐, whence 𝑐𝑤,𝑤𝑐,𝑐∈𝐶⁺ by the fact that 𝐶⁺ is a union of some 𝑃_𝐶⁺-classes. Since 𝐶 is a code over 𝐴, 𝐶^∗ is stable. Therefore, 𝑤∈𝐶^∗. This implies that 𝑤=1. Thus, the 𝑃_𝐶⁺-class containing 1 is {1}.
(3) This follows from (2).

Lemma 7. Let 𝐶 be a code over 𝐴. If the 𝑃_𝐶^∗-class containing 1 is {1}, then Syn(𝐶^∗)≅Syn(𝐶⁺). Otherwise, Syn(𝐶^∗)≅Syn(𝐶⁺)⧵{1𝑃_𝐶⁺}.

Proof. If the 𝑃_𝐶^∗-class containing 1 is {1}, by items (1) and (2) in Lemma 6, 𝑃_𝐶⁺=𝑃_𝐶^∗ in this case, and so Syn(𝐶^∗) is isomorphic to Syn(𝐶⁺).
If the 𝑃_𝐶^∗-class containing 1 is not {1}, then there exists 𝛼∈𝐴⁺ such that 1𝑃_𝐶^∗𝛼. Moreover, Syn(𝐶⁺)⧵{1𝑃_𝐶⁺} is a subsemigroup of Syn(𝐶⁺) by item (3) of Lemma 6. In the sequel, we show that the following 𝜓∶Syn𝐶^∗⟶Syn𝐶⁺⧵1𝑃_𝐶⁺,𝑥𝑃_𝐶^∗𝜓=𝛼𝑃_𝐶⁺,if𝑥=1,𝑥𝑃_𝐶⁺,if𝑥≠1, (4) is a semigroup isomorphism. We first show that 𝜓 is well defined. In fact, let 𝑥,𝑦∈𝐴^∗ and 𝑥𝑃_𝐶^∗=𝑦𝑃_𝐶^∗. We divide the discussion into the following four cases.(i)𝑥=𝑦=1. In this case, (𝑥𝑃_𝐶^∗)𝜓=(𝑦𝑃_𝐶^∗)𝜓=𝛼𝑃_𝐶⁺.(ii)𝑥=1, 𝑦≠1. In this case, (𝑥𝑃_𝐶^∗)𝜓=𝛼𝑃_𝐶⁺ and (𝑦𝑃_𝐶^∗)𝜓=𝑦𝑃_𝐶⁺. Observe that 𝛼𝑃_𝐶^∗1=𝑥𝑃_𝐶^∗𝑦; it follows that 𝛼𝑃_𝐶^∗𝑦 whence 𝛼𝑃_𝐶⁺𝑦 from item (1) in Lemma 6. This implies that (𝑥𝑃_𝐶^∗)𝜓=(𝑦𝑃_𝐶^∗)𝜓.(iii)𝑥≠1, 𝑦=1. This is the dual of case (ii).(iv)𝑥≠1, 𝑦≠1. This follows from item (1) in Lemma 6.
By similar methods, we can show that 𝜓 is injective, and the surjectivity of 𝜓 is obvious.
On the other hand, for any 𝑥𝑃_𝐶^∗,𝑦𝑃_𝐶^∗∈Syn(𝐶^∗), we assert that 𝑥𝑃_𝐶^∗𝑦𝑃_𝐶^∗𝜓=𝑥𝑃_𝐶^∗𝜓𝑦𝑃_𝐶^∗𝜓 (5) and so 𝜓 is a semigroup morphism. In fact, we have the following cases.(a)𝑥=𝑦=1. In this case, 𝑥𝑃_𝐶^∗𝑦𝑃_𝐶^∗𝜓=1𝑃_𝐶∗𝜓=𝛼𝑃_𝐶⁺,𝑥𝑃_𝐶^∗𝜓𝑦𝑃_𝐶^∗𝜓=𝛼𝑃_𝐶⁺𝛼𝑃_𝐶⁺=𝛼²𝑃_𝐶⁺. (6) Observe that 1𝑃_𝐶^∗𝛼, 𝛼²𝑃_𝐶^∗𝛼. This implies that 𝛼²𝑃_𝐶⁺𝛼 by item (1) in Lemma 6 whence [(𝑥𝑃_𝐶^∗)(𝑦𝑃_𝐶^∗)]𝜓=(𝑥𝑃_𝐶^∗)𝜓(𝑦𝑃_𝐶^∗)𝜓.(b)𝑥=1, 𝑦≠1. In this case, 𝑥𝑃_𝐶^∗𝑦𝑃_𝐶^∗𝜓=𝑦𝑃_𝐶^∗𝜓=𝑦𝑃_𝐶⁺,𝑥𝑃_𝐶^∗𝜓𝑦𝑃_𝐶^∗𝜓=𝛼𝑃_𝐶⁺𝑦𝑃_𝐶⁺=𝛼𝑦𝑃_𝐶⁺. (7) Observe that 1𝑃_𝐶^∗𝛼, 𝑦𝑃_𝐶^∗𝛼𝑦. This implies that 𝑦𝑃_𝐶⁺𝛼𝑦 by item (1) in Lemma 6 again whence [(𝑥𝑃_𝐶^∗)(𝑦𝑃_𝐶^∗)]𝜓=(𝑥𝑃_𝐶^∗)𝜓(𝑦𝑃_𝐶^∗)𝜓.(c)𝑥≠1, 𝑦=1. This is the dual of case (b).(d)𝑥≠1, 𝑦≠1. This is obvious.

Lemma 8. If 𝑀 is a monoid with identity 1 and 𝑀⧵{1} is a subsemigroup of 𝑀, then 𝑀 has a kernel if and only if 𝑀⧵{1} has a kernel. If this is the case, the two kernels are equal.

Proof. Observe that {}={{1}{1∪{𝑀};𝑁⊆𝑀∣𝑁isanidealof𝑀𝑁⊆𝑀⧵∣𝑁isanidealof𝑀⧵}} (8) the result follows.

Combining Lemmas 5, 6, 7, and 8, we have the following result.

Theorem 9. A complete code 𝐶 over 𝐴 is a synchronized code if and only if the kernel of Syn(𝐶⁺) is a rectangular band.

4. Main Results

Let 𝐼 and Λ be two nonempty sets and ∅≠𝐾⊆𝒯^𝑙(𝐼)×𝒯^𝑟(Λ). Assume that 𝑝₁ and 𝑝₂ are the projections onto the first and second components of 𝒯^𝑙(𝐼)×𝒯^𝑟(Λ), respectively. For (𝑖,𝜆),(𝑖₀,𝜆₀)∈𝐼×Λ, we denote Ω_𝑖^(𝑖,𝜆)_0,𝜆0=(𝐹,Φ)∈𝑝₁(𝐾)×𝑝₂(𝐾)∣(𝐹𝑖,𝜆Φ)=𝑖₀,𝜆₀. (9)

Theorem 10. Let 𝐶 be a complete synchronized code over an alphabet 𝐴 such that 𝐶≠𝐴 and 𝐶⁺ is a 𝑃_𝐶⁺-class. Then Syn(𝐶⁺) is isomorphic to a submonoid 𝐾 of 𝒯^𝑙(𝐼)×𝒯^𝑟(Λ) for some nonempty sets 𝐼 and Λ with (|𝐼|,|Λ|)≠(1,1), and the following conditions hold:(1)𝒯^𝑟₀(𝐼)×𝒯^𝑟₀(Λ)⊆𝐾,(2)𝐾⧵{(𝜄_𝐼,𝜄_Λ)} is a subsemigroup of 𝐾,(3)there exists (𝑖₀,𝜆₀)∈𝐼×Λ such that the identity Ω^(𝑖,𝜆)_{(𝑖0,𝜆0)}=Ω^(𝑗,𝜇)_{(𝑖0,𝜆0)} implies that (𝑖,𝜆)=(𝑗,𝜇) for all (𝑖,𝜆),(𝑗,𝜇)∈𝐼×Λ and the identity (𝐹𝑖₀,𝜆₀Φ)=(𝑖₀,𝜆₀) implies that (𝐹,Φ)∈{(𝜄_𝐼,𝜄_Λ),(⟨𝑖₀⟩,⟨𝜆₀⟩)} for all (𝐹,Φ)∈𝐾.
The above submonoid 𝐾 will be called (𝑖₀,𝜆₀)-submonoid of 𝒯^𝑙(𝐼)×𝒯^𝑟(Λ).

Proof. Let 𝑀=Syn(𝐶⁺) and denote the set of 𝑃_𝐶⁺-classes with representatives from 𝑇 by 𝑇 for 𝑇⊆𝐴^∗. Since 𝐶⁺ is a 𝑃_𝐶⁺-class, we can also let 𝑒=𝐶⁺. Obviously, 𝑒 is an idempotent in 𝑀.
We first assert that {𝑒,1} is stable and 𝑒 is disjunctive in 𝑀. Let 𝑤∈𝐴^∗ and 𝑐∈𝐶. Then 𝑒=𝑐. If 𝑤1,1𝑤∈{𝑒,1}, then 𝑤∈{𝑒,1} obviously. Moreover, let 𝑤𝑒,𝑒𝑤∈{𝑒,1}. Since the 𝑃_𝐶⁺-class containing 1 is {1} by Lemma 6, we have 𝑒=𝑤𝑒=𝑒𝑤. This implies that 𝑐=𝑤𝑐=𝑐𝑤. Since 𝐶⁺ is a single 𝑃_𝐶⁺-class, we have 𝑤𝑐,𝑐𝑤,𝑐∈𝐶⁺. Because 𝐶^∗ is stable, it follows that 𝑤∈𝐶^∗. This implies that 𝑤∈{𝑒,1}. On the other hand, let 𝑝,𝑞∈𝐴^∗ and 𝑝𝑃_𝑒𝑞 in 𝑀. Then for all 𝑠,𝑡∈𝐴^∗, 𝑠𝑝𝑡=𝑒=𝐶⁺ if and only if 𝑠𝑞𝑡=𝑒=𝐶⁺. Since 𝐶⁺ is a single 𝑃_𝐶⁺-class, it follows that 𝑠𝑝𝑡∈𝐶⁺ if and only if 𝑠𝑞𝑡∈𝐶⁺, and so 𝑝𝑃_𝐶⁺𝑞. Thus 𝑝=𝑞.
Now, let 𝐽=𝑀𝑒𝑀. We assert that 𝐽 is the kernel of 𝑀. In fact, 𝐽 is an ideal of 𝑀 clearly. Moreover, since 𝐶 is complete, there exist 𝑢,𝑣∈𝐴^∗ such that 𝑢𝑤𝑣∈𝐶⁺ for all 𝑤∈𝐴^∗. Therefore, there exist 𝑚,𝑛∈𝑀 such that 𝑚𝑥𝑛=𝑒 for all 𝑥∈𝑀. Now, let 𝐼 be an ideal of 𝑀 and 𝑥∈𝐼. Then 𝑢𝑥𝑣=𝑒 for some 𝑢,𝑣∈𝑀 whence 𝐽=𝑀𝑒𝑀=𝑀𝑢𝑥𝑣𝑀⊆𝐼. Thus, 𝐽 is the least ideal of 𝑀 and so is the kernel of 𝑀. By Theorem 9, 𝐽 is a rectangular band.
If 𝐽={𝑒}, then we have 𝑒𝑚=𝑚𝑒=𝑒 for any 𝑚∈𝑀. This implies that 𝑐𝑤=𝑤𝑐=𝑐 for all 𝑐∈𝐶 and 𝑤∈𝐴^∗. Since 𝐶⁺ is a single 𝑃_𝐶⁺-class, it follows that 𝑤𝑐,𝑐𝑤,𝑐∈𝐶⁺. Because 𝐶^∗ is stable, we have 𝑤∈𝐶^∗. Therefore, 𝐶^∗=𝐴^∗ and hence 𝐶=𝐴. A contradiction. Thus, 𝐽≠{𝑒}. Now let 𝜌 be a congruence on 𝑀 whose restriction to 𝐽 is the identity relation on 𝐽. Assume that 𝑥∈𝑀 and 𝑥𝜌𝑒. Then 𝑥𝑒𝜌𝑒𝜌𝑒𝑥, where 𝑒,𝑥𝑒,𝑒𝑥∈𝐽 and thus 𝑒=𝑥𝑒=𝑒𝑥. Since {𝑒,1} is stable in 𝑀, it follows that 𝑥∈{𝑒,1}. If 𝑥=1, then 1𝜌𝑒 and for any 𝑦∈𝐽, we obtain 𝑦𝜌𝑦𝑒, 𝑦𝜌𝑒𝑦, and 𝑦𝑒,𝑦,𝑒𝑦∈𝐽, whence 𝑦=𝑦𝑒=𝑒𝑦. Thus, 𝐽 is a rectangular band with the identity 𝑒. And hence, 𝐽={𝑒} by Lemma 1. A contradiction. It follows that 𝑥=𝑒 and {𝑒} is a 𝜌-class in 𝑀. Now, let 𝑥,𝑦∈𝑀 and 𝑥𝜌𝑦. Then for any 𝑢,𝑣∈𝑀, we have 𝑢𝑥𝑣𝜌𝑢𝑦𝑣. Observe that {𝑒} is a 𝜌-class, it follows that 𝑢𝑥𝑣=𝑒 if and only if 𝑢𝑦𝑣=𝑒. Therefore, 𝑥𝑃_𝑒𝑦 in 𝑀. By the disjunctiveness of 𝑒 in 𝑀, we have 𝑥=𝑦. We conclude that 𝜌 is the equality relation on 𝑀. Thus, 𝑀 is a dense extension of 𝐽.
By Lemma 4, 𝑀 is isomorphic to a subsemigroup of Ω(𝐽) containing the inner part Π(𝐽) as an ideal. Since 𝐽 is a rectangular band, 𝐽 is isomorphic to the product 𝐼×Λ of a left zero semigroup 𝐼 and a right zero semigroup Λ. Observe that 𝐽≠{𝑒}, |𝐽|=|𝐼×Λ|>1, whence (|𝐼|,|Λ|)≠(1,1). By Lemma 3, Ω(𝐼×Λ)≅𝒯^𝑙(𝐼)×𝒯^𝑟(Λ), and in this isomorphism Π(𝐼×Λ)≅𝒯^𝑙₀(𝐼)×𝒯^𝑟₀(Λ). Therefore, 𝑀 is isomorphic to a subsemigroup 𝐾 of 𝒯^𝑙(𝐼)×𝒯^𝑟(Λ) and 𝐾 contains 𝒯^𝑙₀(𝐼)×𝒯^𝑟₀(Λ) as an ideal. Furthermore, since 𝑀 is a monoid, 𝐾 has an identity. Let (𝐹,Φ) be the identity of 𝐾. Then for any (⟨𝑖⟩,⟨𝜆⟩)∈𝒯^𝑙₀(𝐼)×𝒯^𝑟₀(Λ), (𝐹,Φ)(⟨𝑖⟩,⟨𝜆⟩)=(⟨𝑖⟩,⟨𝜆⟩)(𝐹,Φ)=(⟨𝑖⟩,⟨𝜆⟩), (10) whence (𝐹𝑖,𝜆Φ)=(𝑖,𝜆). This implies that (𝐹,Φ)=(𝜄_𝐼,𝜄_Λ). Thus, 𝐾 is a submonoid of 𝒯^𝑙(𝐼)×𝒯^𝑟(Λ). Since 𝑀⧵{1} is a subsemigroup of 𝑀, it follows that 𝐾⧵{(𝜄_𝐼,𝜄_Λ)} is a subsemigroup of 𝐾. Thus, Conditions (1) and (2) hold.
Since 𝐽 is the kernel of 𝑀 and 𝑒 is an idempotent in 𝐽, it follows that the image of 𝑒 in 𝐾 is the form 𝑎=(⟨𝑖₀⟩,⟨𝜆₀⟩) for some 𝑖₀∈𝐼 and 𝜆₀∈Λ. Since 𝑒 is disjunctive in 𝑀, 𝑎 is disjunctive in 𝐾. Let (𝑖,𝜆),(𝑗,𝜇)∈𝐼×Λ and Ω^(𝑖,𝜆)_{(𝑖0,𝜆0)}=Ω^(𝑗,𝜇)_{(𝑖0,𝜆0)}. Then, (𝐹𝑖,𝜆Φ)=𝑖₀,𝜆₀iff𝐹𝑗,𝜇Φ=𝑖₀,𝜆₀ (11) for any (𝐹,Φ)∈𝑝₁(𝑀)×𝑝₂(𝑀). This implies that (⟨𝐹𝑖⟩,⟨𝜆Φ⟩)=𝑎iff𝐹𝑗,𝜇Φ=𝑎, (12) for any (𝐹,Φ)∈𝑝₁(𝑀)×𝑝₂(𝑀). Thus, (𝐹,Ψ)(⟨𝑖⟩,⟨𝜆⟩)(𝐺,Φ)=𝑎iff(𝐹,Ψ)𝑗,𝜇(𝐺,Φ)=𝑎, (13) for any (𝐹,Ψ),(𝐺,Φ)∈𝐾. This shows that (⟨𝑖⟩,⟨𝜆⟩)𝑃_𝑎(⟨𝑗⟩,⟨𝜇⟩) and so (⟨𝑖⟩,⟨𝜆⟩)=(⟨𝑗⟩,⟨𝜇⟩) since 𝑎 is disjunctive in 𝐾. Thus (𝑖,𝜆)=(𝑗,𝜇).
Finally, let (𝐹,Φ)∈𝐾 and (𝐹𝑖₀,𝜆₀Φ)=(𝑖₀,𝜆₀). Then, (𝐹,Φ)𝑖₀,𝜆₀=𝐹𝑖₀,𝜆₀=𝑖₀,𝜆₀. (14) Dually, we have (⟨𝑖₀⟩,⟨𝜆₀⟩)(𝐹,Φ)=(⟨𝑖₀⟩,⟨𝜆₀⟩). Since 𝑒 is stable in 𝑀, it follows that 𝑎 is stable in 𝐾. Hence, (𝐹,Φ)∈{(⟨𝑖₀⟩,⟨𝜆₀⟩),(𝜄_𝐼,𝜄_Λ)}. Therefore, Condition (3) is also satisfied.

We end our paper by giving an example to illustrate our result.

Example 11. Let 𝐴={𝑎,𝑏} and 𝐶=𝑏^∗𝑎⁺𝑏. Then 𝐶⁺=𝐴^∗𝑎𝑏. It is routine to check that 𝐶 is a synchronized code and 𝐶⁺ is a single 𝑃_𝐿-class. Moreover, we can also check that Syn(𝐶⁺)={𝐴^∗𝑎𝑏,𝐴^∗𝑏𝑏,𝐴^∗𝑎,𝑏,1}.
On the other hand, let 𝐼={1} and Λ={1,2,3}. Denote 𝑀₁=(⟨1⟩,⟨1⟩),(⟨1⟩,⟨2⟩),(⟨1⟩,⟨3⟩),(⟨1⟩,𝛼),𝜄_𝐼,𝜄_Λ, (15) where 𝛼∶Λ⟶Λ,1⟼2,2⟼2,3⟼1. (16) Then 𝑀₁ is a (1,1)-submonoid of 𝒯^𝑙(𝐼)×𝒯^𝑟(Λ). In fact, observe that 𝛼²=⟨2⟩, 𝑀₁ is submonoid of 𝒯^𝑙(𝐼)×𝒯^𝑟(Λ), and 𝑀₁⧵{(𝜄_𝐼,𝜄_Λ)} is a subsemigroup of 𝑀₁. Let 𝜆,𝜇∈Λ. If 𝜆=1, 𝜇=2 or 𝜆=1, 𝜇=3, then there exists Φ=𝜄_Λ∈𝑝₂(𝑀₁) such that 𝜆Φ=1, 𝜇Φ=𝜇. If 𝜆=2, 𝜇=3, then there exists Φ=𝛼∈𝑝₂(𝑀₁) such that 𝜆Φ=2, 𝜇Φ=1. Furthermore, if (𝐹,Φ)∈𝑀₁ with 𝐹1=1, 1Φ=1, then 𝐹=⟨1⟩=𝜄_𝐼 and Φ=⟨1⟩ or Φ=𝜄_Λ. Thus, 𝑀₁ is a (1,1)-submonoid of 𝒯^𝑙(𝐼)×𝒯^𝑟(Λ). It is easy to see that 𝜓∶Syn𝐶⁺⟶𝑀₁,𝐴^∗𝑎𝑏⟼(⟨1⟩,⟨1⟩),𝐴^∗𝑏𝑏⟼(⟨1⟩,⟨2⟩),𝐴^∗𝑎⟼(⟨1⟩,⟨3⟩),𝑏⟼(⟨1⟩,𝛼),1⟼𝜄_𝐼,𝜄_Λ (17) is an isomorphism from Syn(𝐶⁺) onto 𝑀₁.

Acknowledgments

The author expresses his profound gratitude to Professor Liu Yun for his helpful suggestions in preparing this paper. This research work is supported by the NSF Grants of China (11226049, 11301470) and the NSF Grant of Yunnan Province of China (2012FB139).