#### Abstract

Let be an alphabet with two elements. Considering a particular class of words (the phrases) over such an alphabet, we connect with the theory of numerical semigroups. We study the properties of the family of numerical semigroups which arise from this starting point.

#### 1. Introduction

Let be a nonempty finite set called the alphabet. Elements of are called letters or symbols. A word is a sequence of letters, which can be finite or infinite. We denote by (resp., ) the set of all finite (resp., infinite) words over . The sequence of zero letters is called the empty word and is denoted by . Any subset is called a language over . The length of a word is denoted by . If are words, we define their product or concatenation as the word . We say that a word is a factor of a word if there exist two words such that . If is a factor of with (resp., ), then is a prefix (resp., suffix) of .

We have taken these definitions from . In this book (and the references given therein), the authors study problems related to Combinatorics on Words. However, we are going to consider a different point of view. We are interested in a very particular type of words (the phrases) and, more specifically, their length.

Definition 1. Let us take . We say that is a phrase if it fulfills the following conditions: (1) is not a prefix or suffix of ,(2) is not a factor of . We denote .

If we consider that represents a gap between two words, then we have a suitable justification for the above definition.

Let be a language over such that . We will denote by . In this work we are going to deal with the structure of the set for particular choices of . In fact, let be a finite set of words such that is not a factor of . Then is the language in which each phrase is obtained as product of factors belonging to . Moreover, in order to achieve the results of this paper, we assume that .

Example 2. If we take , then , , and belong to . However, , , and do not belong to .

Let be the set of nonnegative integers. A numerical semigroup is a subset of that is closed under addition, contains the zero element, and such that is finite.

In Section 2 we will show that is a numerical semigroup. We will also see that there exist numerical semigroups that cannot be obtained by this procedure. This fact allows us to give the following definition.

Definition 3. Let be the alphabet given by the set . A numerical semigroup is the set of lengths of a language of phrases (PL-semigroup for abbreviation) if there exists such that .

The next aim of Section 2 will be to characterize PL-semigroups. Concretely, we will show that a numerical semigroup is a PL-semigroup if and only if for all .

Let be a numerical semigroup. Since is a finite set, we can consider two notable invariants of (see ). On the one hand, the Frobenius number of is the maximum of and is denoted by . On the other hand, the genus of is the cardinality of and is denoted by .

A Frobenius variety is a nonempty family of numerical semigroups that fulfills the following conditions:(1)if , then ,(2)if and , then .

Let us denote is a PL-semigroup}. In Section 3, we will show that is a Frobenius variety. This fact, together with the results of , will allow us to show orderly the elements of in a tree with root . Moreover, we will also characterize the sons of a vertex, in order to build recursively such a tree.

The multiplicity of a numerical semigroup , denoted by , is the minimum of . We will study the set in Section 4. In particular, we will show that this set is finite and has maximum and minimum with respect to the inclusion order. Furthermore, we will determine the sets and . We will also see that the elements of can be ordered in a tree with root the numerical semigroup (where the symbol means that every integer greater than belongs to ).

In Section 5, we will see that a PL-semigroup is determined perfectly by a nonempty finite set of positive integers. In addition, we will show explicitly the smallest -semigroup that contains a given nonempty finite set of positive integers.

We finish this introduction pointing out that this work admits different generalizations. Some of them are in working process and other ones have already been developed (see ).

#### 2. PL-Semigroups

If is a nonempty subset of , we denote by the submonoid of generated by ; that is, It is a well-known fact (see for instance [5, Lemma 2.1]) that is a numerical semigroup if and only if (as usual, means greatest common divisor). On the other hand, every numerical semigroup is finitely generated, and therefore there exists a finite subset of such that . In addition, if no proper subset of generates , then we say that is a minimal system of generators of . In [5, Theorem 2.7] it is proved that every numerical semigroup has a unique (finite) minimal system of generators. The elements of such a system are called minimal generators of .

Let be an alphabet and let be a finite set of words. If is the language in which each word is obtained as product of factors belonging to , then it is easy to see that is a submonoid of . In addition, if , then is a numerical semigroup. Moreover, it is a simple exercise to show that we can get any numerical semigroup in this way.

As we indicated in the introduction we are going to study the particular case in which we consider lengths of phrases. Consequently, we will focus our attention in a particular family of numerical semigroups.

Proposition 4. Let be an alphabet. If , then is a numerical semigroup.

Proof. We proceed in three steps. (i)First of all, being that and , we have that .(ii)Now, let us see that, if , then . In effect, let such that and . Then the concatenation (of and ) is an element of with .(iii)Finally, let with (i.e., ). Since , we have that . By the previous step, we know that is closed under addition and, consequently, . As , we have is a numerical semigroup. Therefore, is finite. We conclude that is a numerical semigroup.

From now on, unless another thing is stated, we take . As in the introduction, we say that a numerical semigroup is a -semigroup if there exists such that . From Proposition 4, we deduce that if is a -semigroup and , then . Consequently, there exist numerical semigroups which are not of this type. For example, is not a -semigroup because .

In the next result we give a characterization of -semigroups.

Theorem 5. Let be a numerical semigroup. The following conditions are equivalent. (1) is a -semigroup.(2)If , then .

Proof. By hypothesis, for some nonempty finite set . If , then there exist such that and . It is clear that and . Therefore, .
Let be the minimal system of generators of . Let us take the set . Our aim is to show that if , then .
Since , by applying Proposition 4, we have . Now, let . In order to prove that , we are going to use induction over . If , then the result is trivially true. Let us assume that , and let such that . If is not a factor of , then the result follows immediately. In other case, there exist such that . By hypothesis of induction, . Thereby, .

Remark 6. The previous theorem leads to the concept of numerical semigroup that admit a linear nonhomogeneous pattern. For a general study of this family of numerical semigroups see, for instance, [6, 7].

Let be a numerical semigroup with minimal system of generators given by . Following , if , then we define the order of (in ) by If no ambiguity is possible, then we write .

Remark 7. From [5, Lemma 2.3 and Theorem 2.7], we have that if is the minimal system of generators of , then every system of generators of contains . Consequently, the definition of does not depend on the considered system of generators; that is, it only depends on and .

Lemma 8. Let be a numerical semigroup with minimal system of generators given by and let . (1)If and , then .(2)If , with and , then .

Proof. Assume that , with . Then , and thus .
Since , we have . Thereby, . Now, by applying the previous item, we conclude that .

In item of the next proposition, it is shown a characterization of -semigroups in terms of minimal systems of generators. Thus, we can decide if a numerical semigroup is a -semigroup in an easier way.

Proposition 9. Let be a numerical semigroup with minimal system of generators given by . The following conditions are equivalent. (1) is a -semigroup.(2)If , then .(3)If , then .(4)If , then .

Proof. It is an immediate consequence of Theorem 5.
If , then it is clear that there exist and such that . Thus, .
We reason by induction over . If , then the result is trivially true. Now, let us assume that and that are nonnegative integers such that , , and for some . By Lemma 8, we know that . Then, by hypothesis of induction, we have that . Therefore, . Moreover, . Thereby, .
If , then it is clear that . Thus, we get that . By applying Theorem 5, we can conclude that is a -semigroup.

Example 10. Let ; that is, let be the numerical semigroup with minimal system of generators given by . It is obvious that , , , , , and are elements of . Therefore, by applying Proposition 9, we have that is a -semigroup.

#### 3. The Frobenius Variety of the PL-Semigroups

The following result is straightforward to prove and appears in .

Lemma 11. Let be numerical semigroups. (1) is a numerical semigroup.(2)If , then is a numerical semigroup.

Having in mind the definition of Frobenius variety, which was given in the introduction, we get the next result.

Proposition 12. The set is a -semigroup} is a Frobenius variety.

Proof. First of all, let us observe that and, therefore, is a nonempty set.
Let . In order to show that , we are going to use Theorem 5. So, if , then and . Therefore, . Consequently, .
Now, let such that . By applying Theorem 5 again, we are going to see that . Let . If , then . On the other hand, if , then and, thereby, . We conclude that .

A graph   is a pair , where is a nonempty set and is a subset of . The elements of are called vertices of and the elements of are called edges of . A path (of length ) connecting the vertices and of is a sequence of different edges of the form such that and .

We say that a graph is a tree if there exist a vertex (known as the root of ) such that for every other vertex of , there exists a unique path connecting and . If is an edge of the tree, then we say that is a son of .

We define the graph in the following way:(i) is the set of vertices of ,(ii) is an edge of if .

As a consequence of [3, Proposition 24, Theorem 27], we have the next result.

Theorem 13. The graph is a tree with root equal to . Moreover, the sons of a vertex are , where are the minimal generators of that are greater than and such that .

Let us observe that if is a numerical semigroup and , then is a numerical semigroup if and only if is a minimal generator of . In fact, is a numerical semigroup whenever and for all . As a consequence, if we denote by the minimal system of generators of , then (see [5, Lemma 2.3] for other proof of this result). In the following proposition we obtain an analogous for -semigroups of the first commented fact in this paragraph.

Proposition 14. Let be a -semigroup and let be a minimal generator of . Then is a -semigroup if and only if .

Proof. (Necessity). If , then . Accordingly, there exist such that . In fact, it is clear that . Therefore, by applying Theorem 5 and that is a -semigroup, we have , which is a contradiction.
(Sufficiency). Let . Since is a -semigroup, by Theorem 5 we have . As , we deduce that . Thus . By applying Theorem 5 again, we conclude that is a -semigroup.

As a consequence of the previous proposition, we have the next result.

Corollary 15. Let be a -semigroup such that , and let be a minimal generator of greater than . Then is a -semigroup if and only if .

By applying Theorem 13 together with Corollary 15, we can get the sons of a vertex of as is shown in the following example.

Example 16. It is clear that is a -semigroup with Frobenius number equal to 5. From Theorem 13 and Corollary 15, we deduce that the sons of are and .

Let us observe that we can build recursively a tree, from the root, if we know the sons of each vertex. Therefore, we can build the tree such as it is shown in Figure 1.

In order to have an easier making of the tree , we are going to study the relation between the minimal generators of a numerical semigroup and the minimal generators of , where is a minimal generator of that is greater than . First of all, let us observe that if is minimally generated by (i.e., ), then is minimally generated by . In other case we have the following result.

Proposition 17. Let be a numerical semigroup with . If and , then .

Proof. Let us take . Since and , we have that . Thus, for some . Thereby, . By applying that is a minimal system of generators, we have that . Therefore, . In particular, .
Now, let . Then and, thus, there exist such that . Since , we can assume that . On the one hand, if , then . On the other hand, if , then there exists such that . If , then it is obvious that . And if , since , we have that . In any case, we conclude that .

Corollary 18. Let be a numerical semigroup with . If and , then

Proof. From Proposition 17 we deduce that is or . In addition, if and only if .
If , then there exist such that . Since is a minimal system of generators, we get that . Thus, there exists such that . Consequently, .
Conversely, if there exists such that , then for some . Thereby, . Since is a minimal system of generators, we have that and, therefore, .

We finish this section with an illustrative example about the above corollary.

Example 19. Let be the numerical semigroup with . It is obvious that . By Proposition 17, we know that . In addition, . Thereby, applying Corollary 18, we have that . On the other hand, applying Proposition 17 again, we have that . Finally, since , we conclude that .

#### 4. PL-Semigroups with a Fixed Multiplicity

Let be a positive integer. We will denote by . It is clear that is the greatest (with respect to set inclusion) -semigroup with multiplicity . Our first aim in this section will be to show that there also exists the smallest (with respect to set inclusion) -semigroup with multiplicity .

As an immediate consequence of item in Proposition 9 we have the next result.

Lemma 20. If is a -semigroup, , and , then for all .

Proposition 21. Let . Then the numerical semigroup generated by is the smallest (with respect to set inclusion) -semigroup with multiplicity .

Proof. Let . From Lemma 20, we know that any -semigroup with multiplicity has to contain . In order to conclude the proof, we will show that is a -semigroup. For this purpose, since is a system of generators of , it will be enough to check item of Proposition 9; that is, if , then . We distinguish two cases. (1)If , then .(2)If , then + = + = + .

We will denote by and by the set of all -semigroups with multiplicity equal to . Let us recall that and .

As an application of the above comment, we have the next result.

Corollary 22. The set is finite.

Proof. If , then . Since and are numerical semigroups, we have that is finite. Consequently, is also finite.

Remark 23. The previous result can be considered a particular case of [6, Theorem 6.6].

Now we are interested in computing the Frobenius number and the genus of . For that, several concepts and results are introduced.

If is a numerical semigroup and , then the Apéry set of in (see ) is . It is clear (see for instance [5, Lemma 2.4]) that , where is, for each , the least element of that is congruent with modulo .

The next result is [5, Proposition 2.12].

Lemma 24. Let be a numerical semigroup and let . Then (1),(2).

If are integers with , we denote by the remainder of the division of by . The following result is [5, Proposition 3.5].

Lemma 25. Let and let such that, for each , is congruent with modulo . Let be the numerical semigroup generated by . The following conditions are equivalent. (1).(2) for all .

Proposition 26. If , then

Proof. It is clear that is congruent with modulo for all . Let us see now that, if , then . Indeed, = + + + + = . The proof follows from Lemma 25.

Corollary 27. If , then (1),(2), (3), (4).

Proof. Items and are trivial. On the other hand, items and are immediate consequences of Lemma 24 and Proposition 26.

Remark 28. The numerical semigroup can be rewritten as Thus is a numerical semigroup generated by an arithmetic sequence with first term and common difference (see [2, 10]).

If is a numerical semigroup, then the cardinality of the minimal system of generators of is called the embedding dimension of and is denoted by . It is well known (see [5, Proposition 2.12]) that . We say that a numerical semigroup has maximal embedding dimension if . It is clear that is the minimal system of generators of . Therefore, is a numerical semigroup with maximal embedding dimension. Now we will show that has also maximal embedding dimension.

The next result is [5, Corollary 3.6].

Lemma 29. Let be a numerical semigroup with multiplicity and assume that . Then has maximal embedding dimension if and only if for all .

Let us observe that, in the proof of Proposition 26, we have shown that for all . Therefore, by applying Lemma 29, we get the following result.

Corollary 30. If , then is a numerical semigroup with maximal embedding dimension.

As a consequence of this corollary, we have that is the minimal system of generators of .

Now, we want to show orderly the elements of . Thus, we define the graph in the following way:(i) is the set of vertices of ;(ii) is an edge of if .

The next result is analogous to Theorem 13.

Theorem 31. The graph is a tree with root equal to . Moreover, the sons of a vertex are , with = .

By applying Theorem 31 and Corollaries 15 and 18, we can get easily such as is shown in the next example.

Example 32. We are going to depict , that is, the tree of the -semigroups with multiplicity equal to . If is a tree, then the height of is the maximum of the lengths of the paths that connect each vertex with the root. Let us observe that the height of is 3. In general, the height of the tree is equal to

Let us study now the possible values of the Frobenius number and the genus for -semigroups with multiplicity .

Proposition 33. If , then (1);(2).

Proof. Let us assume that .
By Corollary 27, we know that For the opposite inclusion it is enough to observe that and that .
It is clear that with Frobenius number equal to . Thus, . For the other inclusion, let us take such that . Then and, thereby, . Since , we have . Therefore, we conclude that .

Example 34. By Proposition 33, . Since and , we have that . Therefore, by applying Proposition 33 again, we conclude that .

#### 5. The Smallest PL-Semigroup That Contains a Given Set of Positive Integers

Let us observe that, in general, the infinite intersection of elements of is not a numerical semigroup. For instance, . On the other hand, it is clear that the (finite or infinite) intersection of numerical semigroups is always a submonoid of .

Let be a submonoid of . We will say that is a -monoid if it can be expressed like the intersection of elements of .

The next lemma has an immediate proof.

Lemma 35. The intersection of -monoids is a -monoid.

In view of this result, we can give the following definition.

Definition 36. Let be a subset of . The -monoid generated by (denoted by ) is the intersection of all -monoids containing .

If , then we will say that is a -system of generators of . In addition, if no proper subset of is a -system of generators of , then we will say that is a minimal -system of generators of .

Let us recall that, by Proposition 12, we know that is a Frobenius variety. Therefore, by applying [3, Corollary 19], we have the next result.

Proposition 37. Every -monoid has a unique minimal -system of generators, which in addition is finite.

The proof of the following lemma is also immediate.

Lemma 38. If , then is the intersection of all -semigroups that contain .

Proposition 39. If is a nonempty subset of , then is a -semigroup.

Proof. We know that is a submonoid of . Therefore, in order to show that is a numerical semigroup, it will be enough to see that is a finite set.
Let . If is a -semigroup containing , then (by Theorem 5) we know that and, in this way, . From Lemma 38, we have that . Since , we get that is a numerical semigroup and, thus, is finite. Consequently, is finite.
Now, let us see that is a -semigroup. Let . If is a -semigroup containing , from Lemma 38, we deduce that and from Theorem 5, we have that . By applying again Lemma 38, we have that . Therefore, by applying Theorem 5 once more, we can assert that is a -semigroup.

Remark 40. Let us observe that, in general, Proposition 39 is not true for Frobenius varieties. In fact, let be the set of all numerical semigroups. It is clear that is a Frobenius variety. If we take , then the intersection of all elements of containing is exactly , which is not a numerical semigroup.

The next result will be key for our last purpose in this section.

Theorem 41. .

Proof. By Proposition 39, we have that For the other inclusion it is enough to observe that if , then (by Proposition 37) there exists a nonempty finite subset of such that .

Since is a Frobenius variety, by applying [3, Proposition 24], we get the next result.

Proposition 42. Let be a -monoid and let . Then is a -monoid if and only if belongs to the minimal -system of generators of .

As an immediate consequence of this proposition we have the following result.

Corollary 43. Let be a nonempty subset of . Then the minimal -system of generators of is is a -semigroup}.

Example 44. By Proposition 21, is a -semigroup. By applying Proposition 14, we easily deduce that Therefore, and is the minimal -system of generators of .

Now we want to describe when is a fixed nonempty finite set of positive integers. Let us observe that by Theorem 41, we know that every -semigroup can be obtained in this way.

Let be positive integers. We will denote by the set . Our next purpose will be to show that .

Lemma 45. Let be a numerical semigroup, let , and let . Then .

Proof. Let be the minimal system of generators of . Then, for each , there exist such that . Moreover, since , we have that . Thus, Therefore,

Theorem 46. If are positive integers, then is the smallest (with respect to set inclusion) -semigroup containing .

Proof. We divide the proof into five steps. (i)Let us see that if , then . In effect, we know that there exist nonnegative integers such that , , , and . Therefore, with . Consequently, .(ii)Let us see that is finite. Since and , we have that . By applying the first step, we get that . Using the same reasoning as we did in the proof of Proposition 39, we have the result.(iii)From the previous steps, we know that is a numerical semigroup. Let us see now that is a -semigroup. In order to do that, it is enough (by Theorem 5) to show that, if , then . Indeed, arguing as in the first step, we have that with . Therefore, .(iv)Following the proof of the second step, it is clear that .(v)Finally, let us see that is the smallest -semigroup that contains . In fact, we will show that if is -semigroup containing , then . Thus, let . Then there exist such that with . Since , by Proposition 9, we have that . By applying Lemma 45, we have that and, therefore, .
In this way, we have proved the statement.

The next result is an immediate consequence of the previous theorem.

Corollary 47. If are positive integers, then .

We finish this section with an example that illustrates its content.

Example 48. It is clear that . Therefore, .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank the referee for his/her useful comments and suggestions that helped to improve this work. Both of the authors are supported by FQM-343 (Junta de Andalucía), MTM2010-15595 (MICINN, Spain), and FEDER funds. The second author is also partially supported by Junta de Andalucía/Feder Grant no. FQM-5849.