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The Scientific World Journal
Volume 2013, Article ID 691439, 5 pages
http://dx.doi.org/10.1155/2013/691439
Research Article

On the Syntactic Monoids Associated with a Class of Synchronized Codes

School of Mathematics, Yunnan Normal University, Kunming, Yunnan 650500, China

Received 2 October 2013; Accepted 5 December 2013

Academic Editors: A. Assi and G. Bosi

Copyright © 2013 Shou-feng Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A complete code over an alphabet is called synchronized if there exist such that . In this paper we describe the syntactic monoid of for a complete synchronized code over such that , the semigroup generated by , is a single class of its syntactic congruence . In particular, we prove that, for such a code , either or is isomorphic to a special submonoid of , where and are the full transformation semigroups on the nonempty sets and , 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. [14]). 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 of , the semigroup generated by a maximal prefix code for which is a single class of the syntactic congruence .

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 of for a complete synchronized code over an alphabet such that is a single class of its syntactic congruence .

2. Preliminaries

A semigroup is a left zero semigroup if for any . 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 be a rectangular band. Then for any . As a consequence, for any . In particular, if has an identity , then .

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

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

Let be a semigroup. A function (resp., ) on is a left translation (resp., right translation) of if (resp., ) for all . A left translation and a right translation are linked if in which case the pair is a bitranslation of . Denote the set of left translations and that of right translations on by and , respectively. Clearly, forms a monoid under the usual composition of functions: for all . Dually, forms a monoid under the usual composition of functions: for all . The set of all bitranslations of forms a submonoid of the direct product , which is called the translation hull of , to be denoted by .

Let be an element of a semigroup . Then the function defined by for all is the inner left translation induced by . Dually, we have inner right translation induced by . Finally, the pair is the inner bitranslation induced by . The set of all inner bitranslations is the inner part of . From Corollary III.1.7 in Petrich [7], is an ideal of .

In the sequel, the set of all transformations on a set written and composed as right (resp., left) operators is denoted by (resp., ). The identity mapping on a set is denoted by . If , then denotes the constant function on whose value is . Clearly, and are semigroups with their own compositions and (as left operators) and (as right operators) are subsemigroups of and , 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 be a rectangular band.(1).(2).(3) is the kernel of .

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

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 is a dense extension of a rectangular band , then the following semigroup homomorphism is injective, where for each , Clearly in this case, is isomorphic to which contains as an ideal.
Let be a semigroup and let be the semigroup obtained from by adjoining an identity if necessary. The syntactic congruence determined by a subset of is the following relation on : if and only if for all . In particular, if , we call    the syntactic congruence determined by and denote it by . Moreover, is called disjunctive in if is the equality relation on . It is easy to see that the relation    saturates   for every subset of ; that is, is a union of some -classes for every subset of .

Let be an alphabet, let be the free monoid generated by , and let 1 be the identity of . For any , the quotient monoid is called the syntactic monoid of , to be denoted by . A nonempty set of is called a code over if the fact that implies that and , .

A submonoid of a monoid is called stable in if the fact that implies that for all . It is well known that the monoid generated by a code over is stable in (see Proposition 2.2.5 in [1]).

A code over is called complete if, for any , there exist such that , where is the monoid generated by . Recall that a complete code over is said to be synchronized if for some (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 over is synchronized if and only if the kernel of 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 of , the semigroup generated by a code over . To this aim, we need several lemmas.

Lemma 6. Let be a code over .(1)For any , if and only if .(2)The -class containing 1 is .(3) is a subsemigroup of .

Proof. (1) This follows from the fact that if and only if for any and .
(2) Let and . Since and is a union of some -classes, it follows that . On the other hand, for any , we have and , whence by the fact that is a union of some -classes. Since is a code over , is stable. Therefore, . This implies that . Thus, the -class containing 1 is .
(3) This follows from (2).

Lemma 7. Let be a code over . If the -class containing 1 is , then . Otherwise, .

Proof. If the -class containing 1 is , by items (1) and (2) in Lemma 6, in this case, and so is isomorphic to .
If the -class containing 1 is not , then there exists such that . Moreover, is a subsemigroup of by item (3) of Lemma 6. In the sequel, we show that the following is a semigroup isomorphism. We first show that is well defined. In fact, let and . We divide the discussion into the following four cases.(i). In this case, .(ii), . In this case, and . Observe that ; it follows that whence from item (1) in Lemma 6. This implies that .(iii), . This is the dual of case (ii).(iv), . This follows from item (1) in Lemma 6.
By similar methods, we can show that is injective, and the surjectivity of is obvious.
On the other hand, for any , we assert that and so is a semigroup morphism. In fact, we have the following cases.(a). In this case, Observe that , . This implies that by item (1) in Lemma 6 whence .(b), . In this case, Observe that , . This implies that by item (1) in Lemma 6 again whence .(c), . This is the dual of case (b).(d), . This is obvious.

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

Proof. Observe that the result follows.

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

Theorem 9. A complete code over is a synchronized code if and only if the kernel of is a rectangular band.

4. Main Results

Let and be two nonempty sets and . Assume that and are the projections onto the first and second components of , respectively. For , we denote

Theorem 10. Let be a complete synchronized code over an alphabet such that and is a -class. Then is isomorphic to a submonoid of for some nonempty sets and with , and the following conditions hold:(1),(2) is a subsemigroup of ,(3)there exists such that the identity implies that for all and the identity implies that for all .
The above submonoid will be called -submonoid of .

Proof. Let and denote the set of -classes with representatives from by for . Since is a -class, we can also let . Obviously, is an idempotent in .
We first assert that is stable and is disjunctive in . Let and . Then . If , then obviously. Moreover, let . Since the -class containing 1 is by Lemma 6, we have . This implies that . Since is a single -class, we have . Because is stable, it follows that . This implies that . On the other hand, let and in . Then for all , if and only if . Since is a single -class, it follows that if and only if , and so . Thus .
Now, let . We assert that is the kernel of . In fact, is an ideal of clearly. Moreover, since is complete, there exist such that for all . Therefore, there exist such that for all . Now, let be an ideal of and . Then for some whence . Thus, is the least ideal of and so is the kernel of . By Theorem 9, is a rectangular band.
If , then we have for any . This implies that for all and . Since is a single -class, it follows that . Because is stable, we have . Therefore, and hence . A contradiction. Thus, . Now let be a congruence on whose restriction to is the identity relation on . Assume that and . Then , where and thus . Since is stable in , it follows that . If , then and for any , we obtain , , and , whence . Thus, is a rectangular band with the identity . And hence, by Lemma 1. A contradiction. It follows that and is a -class in . Now, let and . Then for any , we have . Observe that is a -class, it follows that if and only if . Therefore, in . By the disjunctiveness of in , we have . We conclude that is the equality relation on . Thus, is a dense extension of .
By Lemma 4, is isomorphic to a subsemigroup of containing the inner part as an ideal. Since is a rectangular band, is isomorphic to the product of a left zero semigroup and a right zero semigroup . Observe that , , whence . By Lemma 3, , and in this isomorphism . Therefore, is isomorphic to a subsemigroup of and contains as an ideal. Furthermore, since is a monoid, has an identity. Let be the identity of . Then for any , whence . This implies that . Thus, is a submonoid of . Since is a subsemigroup of , it follows that is a subsemigroup of . Thus, Conditions (1) and (2) hold.
Since is the kernel of and is an idempotent in , it follows that the image of in is the form for some and . Since is disjunctive in , is disjunctive in . Let and . Then, for any . This implies that for any . Thus, for any . This shows that and so since is disjunctive in . Thus .
Finally, let and . Then, Dually, we have . Since is stable in , it follows that is stable in . Hence, . Therefore, Condition (3) is also satisfied.

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

Example 11. Let and . Then . It is routine to check that is a synchronized code and is a single -class. Moreover, we can also check that .
On the other hand, let and . Denote where Then is a -submonoid of . In fact, observe that , is submonoid of , and is a subsemigroup of . Let . If , or , , then there exists such that , . If , , then there exists such that , . Furthermore, if with , , then and or . Thus, is a -submonoid of . It is easy to see that is an isomorphism from onto .

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).

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