The Numerical Semigroup of Phrases' Lengths in a Simple Alphabet

Abstract

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

We have taken these definitions from [1]. 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 𝒜={𝑎,⌣}. We say that 𝑓∈𝒜^∗ is a phrase if it fulfills the following conditions: (1)⌣ is not a prefix or suffix of 𝑓,(2)⌣⌣ is not a factor of 𝑓. We denote 𝒜^ℱ={𝑓∈𝒜^∗∣𝑓isaphrase}.

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

Let 𝒞 be a language over 𝒜 such that 𝒞⊆𝒜^ℱ. We will denote by ℓ(𝒞)={|𝑐|∣𝑐∈𝒞}. In this work we are going to deal with the structure of the set ℓ(𝒞) for particular choices of 𝒞. In fact, let {𝑤₁,…,𝑤_𝑛}⊆𝒜^∗ be a finite set of words such that ⌣ is not a factor of 𝑤_𝑖,1≤𝑖≤𝑛. Then 𝒞(𝑤₁,…,𝑤_𝑛)⊆𝒜^ℱ is the language in which each phrase 𝑓 is obtained as product of factors belonging to {𝑤₁,…,𝑤_𝑛}∪{⌣}. Moreover, in order to achieve the results of this paper, we assume that 𝜀∈𝒞(𝑤₁,…,𝑤_𝑛).

Example 2. If we take {𝑎𝑎𝑎𝑎,𝑎𝑎𝑎𝑎𝑎}, then 𝑓₁=𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎, 𝑓₂=𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎, and 𝑓₃=𝑎𝑎𝑎𝑎𝑎_⌣𝑎𝑎𝑎𝑎 belong to 𝒞(𝑎𝑎𝑎𝑎,𝑎𝑎𝑎𝑎𝑎). However, 𝑓₄=𝑎𝑎𝑎𝑎_⌣𝑎𝑎𝑎𝑎𝑎𝑎, 𝑓₅=𝑎𝑎𝑎, and 𝑓₆=𝑎𝑎_⌣𝑎 do not belong to 𝒞(𝑎𝑎𝑎𝑎,𝑎𝑎𝑎𝑎𝑎).

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

In Section 2 we will show that ℓ(𝒞(𝑤₁,…,𝑤_𝑛)) 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 𝒜 be the alphabet given by the set {𝑎,⌣}. A numerical semigroup 𝑆 is the set of lengths of a language of phrases (PL-semigroup for abbreviation) if there exists 𝒞=𝒞(𝑤₁,…,𝑤_𝑛)⊆𝒜^ℱ such that 𝑆=ℓ(𝒞).

The next aim of Section 2 will be to characterize PL-semigroups. Concretely, we will show that a numerical semigroup 𝑆 is a PL-semigroup if and only if 𝑥+𝑦+1∈𝑆 for all 𝑥,𝑦∈𝑆⧵{0}.

Let 𝑆 be a numerical semigroup. Since ℕ⧵𝑆 is a finite set, we can consider two notable invariants of 𝑆 (see [2]). On the one hand, the Frobenius number of 𝑆 is the maximum of ℕ⧵𝑆 and is denoted by F(𝑆). On the other hand, the genus of 𝑆 is the cardinality of ℕ⧵𝑆 and is denoted by g(𝑆).

A Frobenius variety is a nonempty family 𝒱 of numerical semigroups that fulfills the following conditions:(1)if 𝑆,𝑇∈𝒱, then 𝑆∩𝑇∈𝒱,(2)if 𝑆∈𝒱 and 𝑆≠ℕ, then 𝑆∪{F(𝑆)}∈𝒱.

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

The multiplicity of a numerical semigroup 𝑆, denoted by m(𝑆), is the minimum of 𝑆⧵{0}. We will study the set 𝒮_PL(𝑚)={𝑆∈𝒮_PL∣m(𝑆)=𝑚} 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 {F(𝑆)∣𝑆∈𝒮_PL(𝑚)} and {g(𝑆)∣𝑆∈𝒮_PL(𝑚)}. We will also see that the elements of 𝒮_PL(𝑚) can be ordered in a tree with root the numerical semigroup 𝑆={0,𝑚,→} (where the symbol → means that every integer greater than 𝑚 belongs to 𝑆).

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 PL-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 [4]).

2. PL-Semigroups

If 𝑋 is a nonempty subset of ℕ, we denote by ⟨𝑋⟩ the submonoid of (ℕ,+) generated by 𝑋; that is, ⟨𝑋⟩=𝜆₁𝑥₁+⋯+𝜆_𝑛𝑥_𝑛{0},𝑥∣𝑛∈ℕ⧵₁,…,𝑥_𝑛∈𝑋,𝜆₁,…,𝜆_𝑛∈ℕ. (1) It is a well-known fact (see for instance [5, Lemma 2.1]) that ⟨𝑋⟩ is a numerical semigroup if and only if gcd{𝑋}=1 (as usual, gcd means greatest common divisor). On the other hand, every numerical semigroup 𝑆 is finitely generated, and therefore there exists a finite subset 𝑋 of 𝑆 such that 𝑆=⟨𝑋⟩. In addition, if no proper subset of 𝑋 generates 𝑆, then we say that 𝑋 is a minimal system of generators of 𝑆. In [5, Theorem 2.7] it is proved that every numerical semigroup 𝑆 has a unique (finite) minimal system of generators. The elements of such a system are called minimal generators of 𝑆.

Let 𝒜 be an alphabet and let {𝑤₁,…,𝑤_𝑛}⊆𝒜^∗ be a finite set of words. If 𝒞=𝒞(𝑤₁,…,𝑤_𝑛)⊆𝒜^∗ is the language in which each word is obtained as product of factors belonging to {𝑤₁,…,𝑤_𝑛}, then it is easy to see that ℓ(𝒞) is a submonoid of (ℕ,+). In addition, if gcd{|𝑤₁|,…,|𝑤_𝑛|}=1, then ℓ(𝒞) 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 𝒜={𝑎,⌣} be an alphabet. If 𝒞=𝒞(𝑤₁,…,𝑤_𝑛)⊆𝒜^ℱ, then ℓ(𝒞) is a numerical semigroup.

Proof. We proceed in three steps. (i)First of all, being that 𝜀∈𝒞 and |𝜀|=0, we have that 0∈ℓ(𝒞).(ii)Now, let us see that, if 𝑙₁,𝑙₂∈ℓ(𝒞), then 𝑙₁+𝑙₂∈ℓ(𝒞). In effect, let 𝑓₁,𝑓₂∈𝒞 such that |𝑓₁|=𝑙₁ and |𝑓₂|=𝑙₂. Then the concatenation 𝑓₁𝑓₂ (of 𝑓₁ and 𝑓₂) is an element of 𝒞 with |𝑓₁𝑓₂|=𝑙₁+𝑙₂.(iii)Finally, let 𝑓∈𝒞 with |𝑓|≠0 (i.e., 𝑓≠𝜀). Since 𝑓_⌣𝑓∈𝒞, we have that {|𝑓|,2|𝑓|+1}⊆ℓ(𝒞). By the previous step, we know that 𝒞 is closed under addition and, consequently, ⟨|𝑓|,2|𝑓|+1⟩⊆ℓ(𝒞). As gcd{|𝑓|,2|𝑓|+1}=1, we have ⟨|𝑓|,2|𝑓|+1⟩ is a numerical semigroup. Therefore, ℕ⧵ℓ(𝒞) is finite. We conclude that ℓ(𝒞) is a numerical semigroup.

From now on, unless another thing is stated, we take 𝒜={𝑎,⌣}. As in the introduction, we say that a numerical semigroup 𝑆 is a PL-semigroup if there exists 𝒞=𝒞(𝑤₁,…,𝑤_𝑛)⊆𝒜^ℱ such that 𝑆=ℓ(𝒞). From Proposition 4, we deduce that if 𝑆 is a PL-semigroup and 𝑥∈𝑆⧵{0}, then 2𝑥+1∈𝑆. Consequently, there exist numerical semigroups which are not of this type. For example, 𝑆=⟨5,7,9⟩ is not a PL-semigroup because 2⋅5+1=11∉𝑆.

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

Theorem 5. Let 𝑆 be a numerical semigroup. The following conditions are equivalent. (1)𝑆 is a 𝑃𝐿-semigroup.(2)If 𝑥,𝑦∈𝑆⧵{0}, then 𝑥+𝑦+1∈𝑆.

Proof. (1⇒2) By hypothesis, 𝑆=ℓ(𝒞(𝑤₁,…,𝑤_𝑛)) for some nonempty finite set {𝑤₁,…,𝑤_𝑛}⊆𝒜^∗. If 𝑥,𝑦∈𝑆⧵{0}, then there exist 𝑓,𝑔∈𝒞(𝑤₁,…,𝑤_𝑛)⧵{𝜀} such that |𝑓|=𝑥 and |𝑔|=𝑦. It is clear that 𝑓_⌣𝑔∈𝒞(𝑤₁,…,𝑤_𝑛) and |𝑓_⌣𝑔|=𝑥+𝑦+1. Therefore, 𝑥+𝑦+1∈𝑆.
(2⇒1) Let {𝑛₁,…,𝑛_𝑝} be the minimal system of generators of 𝑆. Let us take the set {𝑤₁,…,𝑤_𝑝∣|𝑤_𝑖|=𝑛_𝑖,1≤𝑖≤𝑝}. Our aim is to show that if 𝒞=𝒞(𝑤₁,…,𝑤_𝑝), then 𝑆=ℓ(𝒞).
Since {𝑛₁,…,𝑛_𝑝}⊆ℓ(𝒞), by applying Proposition 4, we have 𝑆=⟨𝑛₁,…,𝑛_𝑝⟩⊆ℓ(𝒞). Now, let 𝑙∈ℓ(𝒞). In order to prove that 𝑙∈𝑆, we are going to use induction over 𝑙. If 𝑙=0, then the result is trivially true. Let us assume that 𝑙>0, and let 𝑓∈𝒞 such that |𝑓|=𝑙. If ⌣ is not a factor of 𝑓, then the result follows immediately. In other case, there exist 𝑓₁,𝑓₂∈𝒞⧵{𝜀} such that 𝑓=𝑓_1⌣𝑓₂. By hypothesis of induction, |𝑓₁|,|𝑓₂|∈𝑆. Thereby, 𝑙=|𝑓|=|𝑓₁|+|𝑓₂|+1∈𝑆.

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 𝑆 be a numerical semigroup with minimal system of generators given by {𝑛₁,…,𝑛_𝑝}. Following [8], if 𝑠∈𝑆, then we define the order of 𝑠 (in 𝑆) by ord(𝑠;𝑆)=max𝛼₁+⋯+𝛼_𝑝∣𝛼₁𝑛₁+⋯+𝛼_𝑝𝑛_𝑝=𝑠,with𝛼₁,…,𝛼_𝑝∈ℕ. (2) If no ambiguity is possible, then we write ord(𝑠).

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

Lemma 8. Let 𝑆 be a numerical semigroup with minimal system of generators given by {𝑛₁,…,𝑛_𝑝} and let 𝑠∈𝑆. (1)If 𝑖∈{1,…,𝑝} and 𝑠−𝑛_𝑖∈𝑆, then ord(𝑠−𝑛_𝑖)≤ord(𝑠)−1.(2)If 𝑠=𝛼₁𝑛₁+⋯+𝛼_𝑝𝑛_𝑝, with ord(𝑠)=𝛼₁+⋯+𝛼_𝑝 and 𝛼_𝑖≠0, then ord(𝑠−𝑛_𝑖)=ord(𝑠)−1.

Proof. (1) Assume that 𝑠−𝑛_𝑖=𝛽₁𝑛₁+⋯+𝛽_𝑝𝑛_𝑝, with 𝛽₁+⋯+𝛽_𝑝=ord(𝑠−𝑛_𝑖). Then 𝑠=𝛽₁𝑛₁+⋯+(𝛽_𝑖+1)𝑛_𝑖+⋯+𝛽_𝑝𝑛_𝑝, and thus ord(𝑠−𝑛_𝑖)+1=𝛽₁+⋯+(𝛽_𝑖+1)+⋯+𝛽_𝑝≤ord(𝑠).
(2) Since 𝑠−𝑛_𝑖=𝛼₁𝑛₁+⋯+(𝛼_𝑖−1)𝑛_𝑖+⋯+𝛼_𝑝𝑛_𝑝, we have ord(𝑠−𝑛_𝑖)≥𝛼₁+⋯+(𝛼_𝑖−1)+⋯+𝛼_𝑝=ord(𝑠)−1. Thereby, ord(𝑠)−1≤ord(𝑠−𝑛_𝑖). Now, by applying the previous item, we conclude that ord(𝑠−𝑛_𝑖)=ord(𝑠)−1.

In item (2) of the next proposition, it is shown a characterization of PL-semigroups in terms of minimal systems of generators. Thus, we can decide if a numerical semigroup is a PL-semigroup in an easier way.

Proposition 9. Let 𝑆 be a numerical semigroup with minimal system of generators given by {𝑛₁,…,𝑛_𝑝}. The following conditions are equivalent. (1)𝑆 is a 𝑃𝐿-semigroup.(2)If 𝑖,𝑗∈{1,…,𝑝}, then 𝑛_𝑖+𝑛_𝑗+1∈𝑆.(3)If 𝑠∈𝑆⧵{0,𝑛₁,…,𝑛_𝑝}, then 𝑠+1∈𝑆.(4)If 𝑠∈𝑆⧵{0}, then 𝑠+{0,…,𝑜𝑟𝑑(𝑠)−1}⊆𝑆.

Proof. (1⇒2) It is an immediate consequence of Theorem 5.
(2⇒3) If 𝑠∈𝑆⧵{0,𝑛₁,…,𝑛_𝑝}, then it is clear that there exist 𝑖,𝑗∈{1,…,𝑝} and 𝑠^′∈𝑆 such that 𝑠=𝑛_𝑖+𝑛_𝑗+𝑠^′. Thus, 𝑠+1=(𝑛_𝑖+𝑛_𝑗+1)+𝑠^′∈𝑆.
(3⇒4) We reason by induction over ord(𝑠). If ord(𝑠)=1, then the result is trivially true. Now, let us assume that ord(𝑠)≥2 and that 𝛼₁,…,𝛼_𝑝 are nonnegative integers such that 𝑠=𝛼₁𝑛₁+⋯+𝛼_𝑝𝑛_𝑝, ord(𝑠)=𝛼₁+⋯+𝛼_𝑝, and 𝑎_𝑖≠0 for some 𝑖∈{1,…,𝑝}. By Lemma 8, we know that ord(𝑠−𝑛_𝑖)=ord(𝑠)−1. Then, by hypothesis of induction, we have that 𝑠−𝑛_𝑖+{0,…,ord(𝑠)−2}⊆𝑆. Therefore, 𝑠+{0,…,ord(𝑠)−2}⊆𝑆. Moreover, (𝑠−𝑛_𝑖+ord(𝑠)−2)+𝑛_𝑖+1∈𝑆. Thereby, 𝑠+{0,…,ord(𝑠)−1}⊆𝑆.
(4⇒1) If 𝑥,𝑦∈𝑆⧵{0}, then it is clear that ord(𝑥+𝑦)≥2. Thus, we get that 𝑥+𝑦+1∈𝑆. By applying Theorem 5, we can conclude that 𝑆 is a PL-semigroup.

Example 10. Let 𝑆=⟨4,5,6⟩; that is, let 𝑆 be the numerical semigroup with minimal system of generators given by {4,5,6}. It is obvious that 4+4+1=9, 4+5+1=10, 4+6+1=11, 5+5+1=11, 5+6+1=12, and 6+6+1=13 are elements of 𝑆. Therefore, by applying Proposition 9, we have that 𝑆 is a PL-semigroup.

3. The Frobenius Variety of the PL-Semigroups

The following result is straightforward to prove and appears in [5].

Lemma 11. Let 𝑆,𝑇 be numerical semigroups. (1)𝑆∩𝑇 is a numerical semigroup.(2)If 𝑆≠ℕ, then 𝑆∪{F(𝑆)} 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 𝒮_𝑃𝐿={𝑆∣𝑆 is a 𝑃𝐿-semigroup} is a Frobenius variety.

Proof. First of all, let us observe that ℕ∈𝒮_PL and, therefore, 𝒮_PL is a nonempty set.
Let 𝑆,𝑇∈𝒮_PL. In order to show that 𝑆∩𝑇∈𝒮_PL, we are going to use Theorem 5. So, if 𝑥,𝑦∈(𝑆∩𝑇)⧵{0}, then 𝑥,𝑦∈𝑆⧵{0} and 𝑥,𝑦∈𝑇⧵{0}. Therefore, 𝑥+𝑦+1∈𝑆∩𝑇. Consequently, 𝑆∩𝑇∈𝒮_PL.
Now, let 𝑆∈𝒮_PL such that 𝑆≠ℕ. By applying Theorem 5 again, we are going to see that 𝑆∪F(𝑆)∈𝒮_PL. Let 𝑥,𝑦∈(𝑆∪F(𝑆))⧵{0}. If 𝑥,𝑦∈𝑆, then 𝑥+𝑦+1∈𝑆⊆𝑆∪F(𝑆). On the other hand, if F(𝑆)∈{𝑥,𝑦}, then 𝑥+𝑦+1>F(𝑆) and, thereby, 𝑥+𝑦+1∈𝑆⊆𝑆∪F(𝑆). We conclude that 𝑆∪F(𝑆)∈𝒮_PL.

A graph  𝐺 is a pair (𝑉,𝐸), where 𝑉 is a nonempty set and 𝐸 is a subset of {(𝑣,𝑤)∈𝑉×𝑉∣𝑣≠𝑤}. The elements of 𝑉 are called vertices of 𝐺 and the elements of 𝐸 are called edges of 𝐺. A path (of length 𝑛) connecting the vertices 𝑥 and 𝑦 of 𝐺 is a sequence of different edges of the form (𝑣₀,𝑣₁),(𝑣₁,𝑣₂),…,(𝑣_𝑛−1,𝑣_𝑛) such that 𝑣₀=𝑥 and 𝑣_𝑛=𝑦.

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

We define the graph G(𝒮_𝑃𝐿) in the following way:(i)𝒮_𝑃𝐿 is the set of vertices of G(𝒮_PL),(ii)(𝑆,𝑆^′)∈𝒮_𝑃𝐿×𝒮_𝑃𝐿 is an edge of G(𝒮_𝑃𝐿) if 𝑆^′=𝑆∪{F(𝑆)}.

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

Theorem 13. The graph G(𝒮_𝑃𝐿) is a tree with root equal to ℕ. Moreover, the sons of a vertex 𝑆∈𝒮_𝑃𝐿 are 𝑆⧵{𝑥₁},…,𝑆⧵{𝑥_𝑟}, where 𝑥₁,…,𝑥_𝑟 are the minimal generators of 𝑆 that are greater than F(𝑆) and such that 𝑆⧵{𝑥₁},…,𝑆⧵{𝑥_𝑟}∈𝒮_𝑃𝐿.

Let us observe that if 𝑆 is a numerical semigroup and 𝑥∈𝑆, then 𝑆⧵{𝑥} is a numerical semigroup if and only if 𝑥 is a minimal generator of 𝑆. In fact, 𝑆⧵{𝑥} is a numerical semigroup whenever 𝑥∈𝑆⧵{0} and 𝑥≠𝑦+𝑧 for all 𝑦,𝑧∈𝑆⧵{0}. As a consequence, if we denote by msg(𝑆) the minimal system of generators of 𝑆, then msg(𝑆)=(𝑆⧵{0})⧵((𝑆⧵{0})+(𝑆⧵{0})) (see [5, Lemma 2.3] for other proof of this result). In the following proposition we obtain an analogous for PL-semigroups of the first commented fact in this paragraph.

Proposition 14. Let 𝑆 be a 𝑃𝐿-semigroup and let 𝑥 be a minimal generator of 𝑆. Then 𝑆⧵{𝑥} is a PL-semigroup if and only if 𝑥−1∈{0}∪(ℕ⧵𝑆)∪(msg(𝑆)).

Proof. (Necessity). If 𝑥−1∉{0}∪(ℕ⧵𝑆)∪(msg(𝑆)), then 𝑥−1∈𝑆⧵({0}∪(msg(𝑆))). Accordingly, there exist 𝑦,𝑧∈𝑆⧵{0} such that 𝑥−1=𝑦+𝑧. In fact, it is clear that 𝑦,𝑧∈𝑆⧵{𝑥,0}. Therefore, by applying Theorem 5 and that 𝑆⧵{𝑥} is a PL-semigroup, we have 𝑥=𝑦+𝑧+1∈𝑆⧵{𝑥}, which is a contradiction.
(Sufficiency). Let 𝑦,𝑧∈𝑆⧵{𝑥,0}. Since 𝑆 is a PL-semigroup, by Theorem 5 we have 𝑦+𝑧+1∈𝑆. As 𝑥−1∈{0}∪(ℕ⧵𝑆)∪(msg(𝑆)), we deduce that 𝑦+𝑧+1≠𝑥. Thus 𝑦+𝑧+1∈𝑆⧵{𝑥}. By applying Theorem 5 again, we conclude that 𝑆⧵{𝑥} is a PL-semigroup.

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

Corollary 15. Let 𝑆 be a 𝑃𝐿-semigroup such that 𝑆≠ℕ, and let 𝑥 be a minimal generator of 𝑆 greater than F(𝑆). Then 𝑆⧵{𝑥} is a 𝑃𝐿-semigroup if and only if 𝑥−1∈msg(𝑆)∪{F(𝑆)}.

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

Example 16. It is clear that 𝑆=⟨4,6,7,9⟩ is a PL-semigroup with Frobenius number equal to 5. From Theorem 13 and Corollary 15, we deduce that the sons of 𝑆 are 𝑆⧵{6}=⟨4,7,9,10⟩ and 𝑆⧵{7}=⟨4,6,9,11⟩.

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 G(𝒮_PL) such as it is shown in Figure 1.

Figure 1 

In order to have an easier making of the tree G(𝒮_PL), we are going to study the relation between the minimal generators of a numerical semigroup 𝑆 and the minimal generators of 𝑆⧵{𝑥}, where 𝑥 is a minimal generator of 𝑆 that is greater than F(𝑆). First of all, let us observe that if 𝑆 is minimally generated by {𝑚,𝑚+1,…,2𝑚−1} (i.e., 𝑆={0,𝑚,→}), then 𝑆⧵{𝑚}={0,𝑚+1,→} is minimally generated by {𝑚+1,𝑚+2,…,2𝑚+1}. In other case we have the following result.

Proposition 17. Let 𝑆 be a numerical semigroup with msg(𝑆)={𝑛₁,…,𝑛_𝑝}. If m(𝑆)=𝑛₁<𝑛_𝑝 and 𝑛_𝑝>F(S), then 𝑆⧵{𝑛_𝑝}=⟨𝑛₁,…,𝑛_𝑝−1,𝑛_𝑝+𝑛₁⟩.

Proof. Let us take 𝑖∈{2,…,𝑝}. Since 𝑛_𝑝>F(𝑆) and 𝑛₁<𝑛_𝑖, we have that 𝑛_𝑝+𝑛_𝑖−𝑛₁∈𝑆. Thus, 𝑛_𝑝+𝑛_𝑖−𝑛₁=𝛼₁𝑛₁+⋯+𝛼_𝑝𝑛_𝑝 for some 𝛼₁,…,𝛼_𝑝∈ℕ. Thereby, 𝑛_𝑝+𝑛_𝑖=(𝛼₁+1)𝑛₁+⋯+𝛼_𝑝𝑛_𝑝. By applying that {𝑛₁,…,𝑛_𝑝} is a minimal system of generators, we have that 𝛼_𝑝=0. Therefore, 𝑛_𝑝+𝑛_𝑖∈⟨𝑛₁,…,𝑛_𝑝−1⟩. In particular, 2𝑛_𝑝∈⟨𝑛₁,…,𝑛_𝑝−1⟩.
Now, let 𝑠∈𝑆⧵{𝑛_𝑝}. Then 𝑠∈𝑆 and, thus, there exist 𝛽₁,…,𝛽_𝑝∈ℕ such that 𝑠=𝛽₁𝑛₁+⋯+𝛽_𝑝𝑛_𝑝. Since 2𝑛_𝑝∈⟨𝑛₁,…,𝑛_𝑝−1⟩, we can assume that 𝛽_𝑝∈{0,1}. On the one hand, if 𝛽_𝑝=0, then 𝑠∈⟨𝑛₁,…,𝑛_𝑝−1⟩. On the other hand, if 𝛽_𝑝=1, then there exists 𝑖∈{1,…,𝑝−1} such that 𝛽_𝑖≠0. If 𝑖=1, then it is obvious that 𝑠∈⟨𝑛₁,…,𝑛_𝑝−1,𝑛_𝑝+𝑛₁⟩. And if 𝑖≠1, since 𝑛_𝑝+𝑛_𝑖∈⟨𝑛₁,…,𝑛_𝑝−1⟩, we have that 𝑠∈⟨𝑛₁,…,𝑛_𝑝−1⟩. In any case, we conclude that 𝑆⧵{𝑛_𝑝}=⟨𝑛₁,…,𝑛_𝑝−1,𝑛_𝑝+𝑛₁⟩.

Corollary 18. Let 𝑆 be a numerical semigroup with msg(𝑆)={𝑛₁,…,𝑛_𝑝}. If m(𝑆)=𝑛₁<𝑛_𝑝 and 𝑛_𝑝>F(𝑆), then msg𝑆⧵𝑛_𝑝=𝑛₁,…,𝑛_𝑝−1,𝑖𝑓𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑠𝑖∈2,…,𝑝−1𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑛_𝑝+𝑛₁−𝑛_𝑖𝑛∈𝑆;₁,…,𝑛_𝑝−1,𝑛_𝑝+𝑛₁,𝑖𝑛𝑜𝑡ℎ𝑒𝑟𝑐𝑎𝑠𝑒. (3)

Proof. From Proposition 17 we deduce that msg(𝑆⧵{𝑛_𝑝}) is {𝑛₁,…,𝑛_𝑝−1} or {𝑛₁,…,𝑛_𝑝−1,𝑛_𝑝+𝑛₁}. In addition, msg(𝑆⧵{𝑛_𝑝})={𝑛₁,…,𝑛_𝑝−1} if and only if 𝑛_𝑝+𝑛₁∈⟨𝑛₁,…,𝑛_𝑝−1⟩.
If 𝑛_𝑝+𝑛₁∈⟨𝑛₁,…,𝑛_𝑝−1⟩, then there exist 𝛼₁,…,𝛼_𝑝−1∈ℕ such that 𝑛_𝑝+𝑛₁=𝛼₁𝑛₁+⋯+𝛼_𝑝−1𝑛_𝑝−1. Since {𝑛₁,…,𝑛_𝑝} is a minimal system of generators, we get that 𝛼₁=0. Thus, there exists 𝑖∈{2,…,𝑝−1} such that 𝛼_𝑖≠0. Consequently, 𝑛_𝑝+𝑛₁−𝑛_𝑖∈𝑆.
Conversely, if there exists 𝑖∈{2,…,𝑝−1} such that 𝑛_𝑝+𝑛₁−𝑛_𝑖∈𝑆, then 𝑛_𝑝+𝑛₁−𝑛_𝑖=𝛽₁𝑛₁+⋯+𝛽_𝑝𝑛_𝑝 for some 𝛽₁,…,𝛽_𝑝∈ℕ. Thereby, 𝑛_𝑝+𝑛₁=𝛽₁𝑛₁+⋯+(𝛽_𝑖+1)𝑛_𝑖+⋯+𝛽_𝑝𝑛_𝑝. Since {𝑛₁,…,𝑛_𝑝} is a minimal system of generators, we have that 𝛽_𝑝=0 and, therefore, 𝑛_𝑝+𝑛₁∈⟨𝑛₁,…,𝑛_𝑝−1⟩.

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

Example 19. Let 𝑆 be the numerical semigroup with msg(𝑆)={3,5,7}. It is obvious that F(𝑆)=4. By Proposition 17, we know that 𝑆⧵{5}=⟨3,7,8⟩. In addition, 8−7=1∉𝑆. Thereby, applying Corollary 18, we have that msg(𝑆⧵{5})={3,7,8}. On the other hand, applying Proposition 17 again, we have that 𝑆⧵{7}=⟨3,5,10⟩. Finally, since 10−5=5∈𝑆, we conclude that msg(𝑆⧵{7})={3,5}.

4. PL-Semigroups with a Fixed Multiplicity

Let 𝑚 be a positive integer. We will denote by Δ(𝑚)={0,𝑚,→}. It is clear that Δ(𝑚) is the greatest (with respect to set inclusion) PL-semigroup with multiplicity 𝑚. Our first aim in this section will be to show that there also exists the smallest (with respect to set inclusion) PL-semigroup with multiplicity 𝑚.

As an immediate consequence of item (4) in Proposition 9 we have the next result.

Lemma 20. If 𝑆 is a 𝑃𝐿-semigroup, 𝑚∈𝑆⧵{0}, and 𝑘∈ℕ⧵{0}, then 𝑘𝑚+𝑖∈𝑆 for all 𝑖∈{0,…,𝑘−1}.

Proposition 21. Let 𝑚∈ℕ⧵{0}. Then the numerical semigroup generated by {(𝑖+1)𝑚+𝑖∣𝑖∈{0,…,𝑚−1}} is the smallest (with respect to set inclusion) 𝑃𝐿-semigroup with multiplicity 𝑚.

Proof. Let 𝑆=⟨𝑚,2𝑚+1,…,𝑚²+(𝑚−1)⟩. From Lemma 20, we know that any PL-semigroup with multiplicity 𝑚 has to contain 𝑆. In order to conclude the proof, we will show that 𝑆 is a PL-semigroup. For this purpose, since {(𝑖+1)𝑚+𝑖∣𝑖∈{0,…,𝑚−1}} is a system of generators of 𝑆, it will be enough to check item (2) of Proposition 9; that is, if 𝑖,𝑗∈{0,…,𝑚−1}, then (𝑖+1)𝑚+𝑖+(𝑗+1)𝑚+𝑗+1∈𝑆. We distinguish two cases. (1)If 𝑖+𝑗+1≤𝑚−1, then (𝑖+1)𝑚+𝑖+(𝑗+1)𝑚+𝑗+1=(𝑖+𝑗+2)𝑚+(𝑖+𝑗+1)∈𝑆.(2)If 𝑖+𝑗+1≥𝑚, then (𝑖+1)𝑚+𝑖+(𝑗+1)𝑚 + 𝑗+1 = (𝑖+𝑗+2)𝑚 + (𝑖+𝑗+1) = (𝑚+1)𝑚+(𝑖+𝑗−𝑚+2)𝑚 + (𝑖+𝑗−𝑚+1)∈𝑆.

We will denote by Θ(𝑚)=⟨𝑚,2𝑚+1,…,𝑚²+(𝑚−1)⟩ and by 𝒮_PL(𝑚) the set of all PL-semigroups with multiplicity equal to 𝑚. Let us recall that Δ(𝑚)=max(𝒮_PL(𝑚)) and Θ(𝑚)=min(𝒮_PL(𝑚)).

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

Corollary 22. The set 𝒮_PL(𝑚) is finite.

Proof. If 𝑆∈𝒮_PL(𝑚), then Θ(𝑚)⊆𝑆⊆Δ(𝑚). Since Δ(𝑚) and Θ(𝑚) are numerical semigroups, we have that Δ(𝑚)⧵Θ(𝑚) is finite. Consequently, 𝒮_PL(𝑚) 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 Θ(𝑚). For that, several concepts and results are introduced.

If 𝑆 is a numerical semigroup and 𝑚∈𝑆⧵{0}, then the Apéry set of 𝑚 in 𝑆 (see [9]) is Ap(𝑆,𝑚)={𝑠∈𝑆∣𝑠−𝑚∉𝑆}. It is clear (see for instance [5, Lemma 2.4]) that Ap(𝑆,𝑚)={𝜔(0)=0,𝜔(1),…,𝜔(𝑚−1)}, where 𝜔(𝑖) is, for each 𝑖∈{0,…,𝑚−1}, the least element of 𝑆 that is congruent with 𝑖 modulo 𝑚.

The next result is [5, Proposition 2.12].

Lemma 24. Let 𝑆 be a numerical semigroup and let 𝑚∈𝑆⧵{0}. Then (1)F(𝑆)=max(Ap(𝑆,𝑚))−𝑚,(2)g(𝑆)=(1/𝑚)(∑_{𝑤∈𝐴𝑝(𝑆,𝑚)}𝑤)−((𝑚−1)/2).

If 𝑎,𝑏 are integers with 𝑏≠0, we denote by 𝑎mod𝑏 the remainder of the division of 𝑎 by 𝑏. The following result is [5, Proposition 3.5].

Lemma 25. Let 𝑚∈ℕ⧵{0} and let 𝑋={𝜔(0)=0,𝜔(1),…,𝜔(𝑚−1)}⊆ℕ such that, for each 𝑖∈{1,…,𝑚−1}, 𝜔(𝑖) is congruent with 𝑖 modulo 𝑚. Let 𝑆 be the numerical semigroup generated by 𝑋∪{𝑚}. The following conditions are equivalent. (1)Ap(𝑆,𝑚)=𝑋.(2)𝜔(𝑖)+𝜔(𝑗)≥𝜔((𝑖+𝑗)mod𝑚) for all 𝑖,𝑗∈{1,…,𝑚−1}.

Proposition 26. If 𝑚∈ℕ⧵{0}, then (Θ(𝑚))=𝜔(0)(1)()𝐴𝑝,𝑚=0,𝜔=2𝑚+1,…,𝜔𝑚−1=𝑚²+𝑚−1. (4)

Proof. It is clear that 𝜔(𝑖)=(𝑖+1)𝑚+𝑖 is congruent with 𝑖 modulo 𝑚 for all 𝑖∈{1,…,𝑚−1}. Let us see now that, if 𝑖,𝑗∈{1,…,𝑚−1}, then 𝜔(𝑖)+𝜔(𝑗)≥𝜔((𝑖+𝑗)mod𝑚). Indeed, 𝜔(𝑖)+𝜔(𝑗) = (𝑖+1)𝑚 + 𝑖+(𝑗+1)𝑚 + 𝑗>(𝑖+𝑗+1)𝑚 + (𝑖+𝑗)≥((𝑖+𝑗)mod𝑚+1)𝑚 + (𝑖+𝑗)mod𝑚 = 𝜔((𝑖+𝑗)mod𝑚). The proof follows from Lemma 25.

Corollary 27. If 𝑚∈ℕ⧵{0,1}, then (1)F(Δ(𝑚))=𝑚−1,(2)g(Δ(𝑚))=𝑚−1, (3)F(Θ(𝑚))=𝑚²−1, (4)g(Θ(𝑚))=(𝑚−1)(𝑚+2)/2.

Proof. Items (1) and (2) are trivial. On the other hand, items (3) and (4) are immediate consequences of Lemma 24 and Proposition 26.

Remark 28. The numerical semigroup Θ(𝑚) can be rewritten as Θ(𝑚)={𝑚,𝑚+(𝑚+1),𝑚+2(𝑚+1),…,𝑚+(𝑚−1)(𝑚+1)}. (5) Thus Θ(𝑚) is a numerical semigroup generated by an arithmetic sequence with first term 𝑚 and common difference 𝑚+1 (see [2, 10]).

If 𝑆 is a numerical semigroup, then the cardinality of the minimal system of generators of 𝑆 is called the embedding dimension of 𝑆 and is denoted by e(𝑆). It is well known (see [5, Proposition 2.12]) that e(𝑆)≤m(𝑆). We say that a numerical semigroup 𝑆 has maximal embedding dimension if e(𝑆)=m(𝑆). It is clear that {𝑚,𝑚+1,…,2𝑚−1} is the minimal system of generators of Δ(𝑚). Therefore, Δ(𝑚) is a numerical semigroup with maximal embedding dimension. Now we will show that Θ(𝑚) has also maximal embedding dimension.

The next result is [5, Corollary 3.6].

Lemma 29. Let 𝑆 be a numerical semigroup with multiplicity 𝑚 and assume that Ap(𝑆,𝑚)={𝜔(0)=0,𝜔(1),…,𝜔(𝑚−1)}. Then 𝑆 has maximal embedding dimension if and only if 𝜔(𝑖)+𝜔(𝑗)>𝜔((𝑖+𝑗)mod𝑚) for all 𝑖,𝑗∈{1,…,𝑚−1}.

Let us observe that, in the proof of Proposition 26, we have shown that 𝜔(𝑖)+𝜔(𝑗)>𝜔((𝑖+𝑗)mod𝑚) for all 𝑖,𝑗∈{1,…,𝑚−1}. Therefore, by applying Lemma 29, we get the following result.

Corollary 30. If 𝑚∈ℕ⧵{0}, then Θ(𝑚) is a numerical semigroup with maximal embedding dimension.

As a consequence of this corollary, we have that {𝑚,2𝑚+1,…,𝑚²+𝑚−1} is the minimal system of generators of Θ(𝑚).

Now, we want to show orderly the elements of 𝒮_PL(𝑚). Thus, we define the graph G(𝒮_PL(𝑚)) in the following way:(i)𝒮_PL(𝑚) is the set of vertices of G(𝒮_PL(𝑚));(ii)(𝑆,𝑆^′)∈𝒮_PL(𝑚)×𝒮_PL(𝑚) is an edge of G(𝒮_PL(𝑚)) if 𝑆^′=𝑆∪{F(𝑆)}.

The next result is analogous to Theorem 13.

Theorem 31. The graph G(𝒮_𝑃𝐿(𝑚)) is a tree with root equal to Δ(𝑚). Moreover, the sons of a vertex 𝑆∈𝒮_𝑃𝐿(𝑚) are 𝑆⧵{𝑥₁},…,𝑆⧵{𝑥_𝑟}, with {𝑥₁,…,𝑥_𝑟} = {𝑥∈msg(𝑆)∣𝑥≠𝑚,𝑥>F(𝑆),𝑎𝑛𝑑𝑆⧵{𝑥}∈𝒮_𝑃𝐿}.

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

Example 32. We are going to depict G(𝒮_PL(3)), that is, the tree of the PL-semigroups with multiplicity equal to 3.

If 𝑇=(𝑉,𝐸) is a tree, then the height of 𝑇 is the maximum of the lengths of the paths that connect each vertex with the root. Let us observe that the height of G(𝒮_PL(3)) is 3. In general, the height of the tree G(𝒮_PL(𝑚)) is equal to g(Θ(𝑚(Δ(𝑚=()))−g))𝑚−1)(𝑚+22−()=()𝑚𝑚−1𝑚−12. (6)

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

Proposition 33. If 𝑚∈ℕ⧵{0}, then (1){g(𝑆)∣𝑆∈𝒮_PL(𝑚)}={𝑥∈ℕ∣𝑚−1≤𝑥≤(𝑚−1)(𝑚+2)/2};(2){F(𝑆)∣𝑆∈𝒮_PL(𝑚)}=(Δ(𝑚)⧵Θ(𝑚))∪{𝑚−1}.

Proof. Let us assume that Δ(𝑚)⧵Θ(𝑚)={𝑥₁>𝑥₂>⋯>𝑥_𝜌}.
(1) By Corollary 27, we know that g(𝑆)∣𝑆∈𝒮_PL(𝑚⊆())𝑚−1,…,𝑚−1)(𝑚+22. (7) For the opposite inclusion it is enough to observe that Θ(𝑚)∪{𝑥_𝑖,→}∈𝒮_PL(𝑚) and that g(Θ(𝑚)∪{𝑥_𝑖,→})=(𝑚−1)(𝑚+2)/2−𝑖.
(2) It is clear that Θ(𝑚)∪{𝑥_𝑖+1,→}∈𝒮_PL(𝑚) with Frobenius number equal to 𝑥_𝑖. Thus, (Δ(𝑚)⧵Θ(𝑚))∪{𝑚−1}⊆{F(𝑆)∣𝑆∈𝒮_PL(𝑚)}. For the other inclusion, let us take 𝑆∈𝒮_PL(𝑚) such that 𝑆≠Δ(𝑚). Then F(𝑆)>𝑚 and, thereby, F(𝑆)∈Δ(𝑚). Since Θ(𝑚)⊆𝑆, we have F(𝑆)∉Θ(𝑆). Therefore, we conclude that F(𝑆)∈Δ(𝑚)⧵Θ(𝑆).

Example 34. By Proposition 33, {g(𝑆)∣𝑆∈𝒮_PL(3)}={2,3,4,5}. Since Δ(3)=⟨3,4,5⟩ and Θ(3)=⟨3,7,11⟩, we have that (Δ(3)⧵Θ(3))∪{2}={2,4,5,8}. Therefore, by applying Proposition 33 again, we conclude that {F(𝑆)∣𝑆∈𝒮_PL(3)}={2,4,5,8}.

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 𝒮_PL is not a numerical semigroup. For instance, ⋂_𝑛∈ℕ{0,𝑛,→}={0}. On the other hand, it is clear that the (finite or infinite) intersection of numerical semigroups is always a submonoid of (ℕ,+).

Let 𝑀 be a submonoid of (ℕ,+). We will say that 𝑀 is a 𝒮_PL-monoid if it can be expressed like the intersection of elements of 𝒮_PL.

The next lemma has an immediate proof.

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

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

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

If 𝑀=𝒮_PL(𝑋), then we will say that 𝑋 is a 𝒮_PL-system of generators of 𝑀. In addition, if no proper subset of 𝑋 is a 𝒮_PL-system of generators of 𝑀, then we will say that 𝑋 is a minimal 𝒮_PL-system of generators of 𝑀.

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

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

The proof of the following lemma is also immediate.

Lemma 38. If 𝑋⊆ℕ, then 𝒮_𝑃𝐿(𝑋) is the intersection of all 𝑃𝐿-semigroups that contain 𝑋.

Proposition 39. If 𝑋 is a nonempty subset of ℕ⧵{0}, then 𝒮_𝑃𝐿(𝑋) is a 𝑃𝐿-semigroup.

Proof. We know that 𝒮_PL(𝑋) is a submonoid of (ℕ,+). Therefore, in order to show that 𝒮_PL(𝑋) is a numerical semigroup, it will be enough to see that ℕ⧵𝒮_PL(𝑋) is a finite set.
Let 𝑥∈𝑋. If 𝑆 is a PL-semigroup containing 𝑋, then (by Theorem 5) we know that {𝑥,2𝑥+1}⊆𝑆 and, in this way, ⟨𝑥,2𝑥+1⟩⊆𝑆. From Lemma 38, we have that ⟨𝑥,2𝑥+1⟩⊆𝒮_PL(𝑋). Since gcd{𝑥,2𝑥+1}=1, we get that ⟨𝑥,2𝑥+1⟩ is a numerical semigroup and, thus, ℕ⧵⟨𝑥,2𝑥+1⟩ is finite. Consequently, ℕ⧵𝒮_PL(𝑋) is finite.
Now, let us see that 𝒮_PL(𝑋) is a PL-semigroup. Let 𝑥,𝑦∈𝒮_PL(𝑋)⧵{0}. If 𝑆 is a PL-semigroup containing 𝑋, from Lemma 38, we deduce that 𝑥,𝑦∈𝑆⧵{0} and from Theorem 5, we have that 𝑥+𝑦+1∈𝑆. By applying again Lemma 38, we have that 𝑥+𝑦+1∈𝒮_PL(𝑋). Therefore, by applying Theorem 5 once more, we can assert that 𝒮_PL(𝑋) is a PL-semigroup.

Remark 40. Let us observe that, in general, Proposition 39 is not true for Frobenius varieties. In fact, let 𝒮 be the set of all numerical semigroups. It is clear that 𝒮 is a Frobenius variety. If we take 𝑋={2}, then the intersection of all elements of 𝒮 containing {2} is exactly 𝑀=⟨2⟩, which is not a numerical semigroup.

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

Theorem 41. 𝒮_𝑃𝐿={𝒮_𝑃𝐿(𝑋)∣𝑋𝑖𝑠𝑎𝑛𝑜𝑛𝑒𝑚𝑝𝑡𝑦𝑓𝑖𝑛𝑖𝑡𝑒𝑠𝑢𝑏𝑠𝑒𝑡𝑜𝑓ℕ⧵{0}}.

Proof. By Proposition 39, we have that 𝒮_PL(𝑋)∣𝑋isanonemptyfinitesubsetofℕ⧵{0}⊆𝒮_PL. (8) For the other inclusion it is enough to observe that if 𝑆∈𝒮_PL, then (by Proposition 37) there exists a nonempty finite subset 𝑋 of ℕ⧵{0} such that 𝑆=𝒮_PL(𝑋).

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

Proposition 42. Let 𝑀 be a 𝒮_𝑃𝐿-monoid and let 𝑥∈𝑀. Then 𝑀⧵{𝑥} is a 𝒮_𝑃𝐿-monoid if and only if 𝑥 belongs to the minimal 𝒮_𝑃𝐿-system of generators of 𝑀.

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

Corollary 43. Let 𝑋 be a nonempty subset of ℕ⧵{0}. Then the minimal 𝒮_𝑃𝐿-system of generators of 𝒮_𝑃𝐿(𝑋) is {𝑥∈𝑋∣𝒮_𝑃𝐿(𝑋)⧵{𝑥} is a PL-semigroup}.

Example 44. By Proposition 21, 𝑆=⟨3,7,11⟩ is a PL-semigroup. By applying Proposition 14, we easily deduce that 𝑥∈{3,7,11}∣𝑆⧵{𝑥}isaPL-semigroup={3}. (9) Therefore, 𝑆=𝒮_PL({3}) and {3} is the minimal 𝒮_PL-system of generators of 𝑆.

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

Let 𝑛₁,…,𝑛_𝑝 be positive integers. We will denote by 𝑆(𝑛₁,…,𝑛_𝑝) the set {𝛼₁𝑛₁+⋯+𝛼_𝑝𝑛_𝑝+𝑟∣𝑟,𝛼₁,…,𝛼_𝑝∈ℕ,𝑟<𝛼₁+⋯+𝛼_𝑝}∪{0}. Our next purpose will be to show that 𝑆(𝑛₁,…,𝑛_𝑝)=𝒮_PL({𝑛₁,…,𝑛_𝑝}).

Lemma 45. Let 𝑆 be a numerical semigroup, let 𝑠₁,…,𝑠_𝑡∈𝑆⧵{0}, and let 𝛼₁,…,𝛼_𝑡∈ℕ. Then ord(𝛼₁𝑠₁+⋯+𝛼_𝑡𝑠_𝑡)≥𝛼₁+⋯+𝛼_𝑡.

Proof. Let {𝑛₁,…,𝑛_𝑝} be the minimal system of generators of 𝑆. Then, for each 𝑖∈{1,…,𝑡}, there exist 𝛽_𝑖1,…,𝛽_𝑖𝑝∈ℕ such that 𝑠_𝑖=𝛽_𝑖1𝑛₁+⋯+𝛽_𝑖𝑝𝑛_𝑝. Moreover, since 𝑠_𝑖≠0, we have that 𝛽_𝑖1+⋯+𝛽_𝑖𝑝≥1. Thus, 𝛼₁𝑠₁+⋯+𝛼_𝑡𝑠_𝑡=𝛼₁𝛽₁₁𝑛₁+⋯+𝛽_1𝑝𝑛_𝑝+⋯+𝛼_𝑡𝛽_𝑡1𝑛₁+⋯+𝛽_𝑡𝑝𝑛_𝑝=𝛼₁𝛽₁₁+⋯+𝛼_𝑡𝛽_𝑡1𝑛₁+⋯+𝛼₁𝛽_1𝑝+⋯+𝛼_𝑡𝛽_𝑡𝑝𝑛_𝑝. (10) Therefore, ord𝛼₁𝑠₁+⋯+𝛼_𝑡𝑠_𝑡≥𝛼₁𝛽₁₁+⋯+𝛼_𝑡𝛽_𝑡1+⋯+𝛼₁𝛽_1𝑝+⋯+𝛼_𝑡𝛽_𝑡𝑝=𝛼₁𝛽₁₁+⋯+𝛽_1𝑝+⋯+𝛼_𝑡𝛽_𝑡1+⋯+𝛽_𝑡𝑝≥𝛼₁+⋯+𝛼_𝑡. (11)

Theorem 46. If 𝑛₁,…,𝑛_𝑝 are positive integers, then 𝑆(𝑛₁,…,𝑛_𝑝) is the smallest (with respect to set inclusion) 𝑃𝐿-semigroup containing {𝑛₁,…,𝑛_𝑝}.

Proof. We divide the proof into five steps. (i)Let us see that if 𝑥,𝑦∈𝑆(𝑛₁,…,𝑛_𝑝)⧵{0}, then 𝑥+𝑦∈𝑆(𝑛₁,…,𝑛_𝑝). In effect, we know that there exist 𝛼₁,…,𝛼_𝑝,𝛽₁,…,𝛽_𝑝,𝑟,𝑟^′ nonnegative integers such that 𝑥=𝛼₁𝑛₁+⋯+𝛼_𝑝𝑛_𝑝+𝑟, 𝑦=𝛽₁𝑛₁+⋯+𝛽_𝑝𝑛_𝑝+𝑟^′, 𝑟<𝛼₁+⋯+𝛼_𝑝, and 𝑟^′<𝛽₁+…+𝛽_𝑝. Therefore, 𝑥+𝑦=(𝛼₁+𝛽₁)𝑛₁+⋯+(𝛼_𝑝+𝛽_𝑛)𝑛_𝑝+𝑟+𝑟^′ with 𝑟+𝑟^′<(𝛼₁+𝛽₁)+⋯+(𝛼_𝑝+𝛽_𝑛). Consequently, 𝑥+𝑦∈𝑆(𝑛₁,…,𝑛_𝑝).(ii)Let us see that ℕ⧵𝑆(𝑛₁,…,𝑛_𝑝) is finite. Since 𝑛₁=1⋅𝑛₁+0⋅𝑛₂+⋯+0⋅𝑛_𝑝+0 and 2𝑛₁+1=2⋅𝑛₁+0⋅𝑛₂+⋯+0⋅𝑛_𝑝+1, we have that 𝑛₁,2𝑛₁∈𝑆(𝑛₁,…,𝑛_𝑝). By applying the first step, we get that ⟨𝑛₁,2𝑛₁⟩⊆𝑆(𝑛₁,…,𝑛_𝑝). 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 𝑆(𝑛₁,…,𝑛_𝑝) is a numerical semigroup. Let us see now that 𝑆(𝑛₁,…,𝑛_𝑝) is a PL-semigroup. In order to do that, it is enough (by Theorem 5) to show that, if 𝑥,𝑦∈𝑆(𝑛₁,…,𝑛_𝑝)⧵{0}, then 𝑥+𝑦+1∈𝑆(𝑛₁,…,𝑛_𝑝). Indeed, arguing as in the first step, we have that 𝑥+𝑦+1=(𝛼₁+𝛽₁)𝑛₁+⋯+(𝛼_𝑝+𝛽_𝑛)𝑛_𝑝+𝑟+𝑟^′+1 with 𝑟+𝑟^′+1<(𝛼₁+𝛽₁)+⋯+(𝛼_𝑝+𝛽_𝑛). Therefore, 𝑥+𝑦+1∈𝑆(𝑛₁,…,𝑛_𝑝).(iv)Following the proof of the second step, it is clear that {𝑛₁,…,𝑛_𝑝}⊆𝑆(𝑛₁,…,𝑛_𝑝).(v)Finally, let us see that 𝑆(𝑛₁,…,𝑛_𝑝) is the smallest PL-semigroup that contains {𝑛₁,…,𝑛_𝑝}. In fact, we will show that if 𝑇 is PL-semigroup containing {𝑛₁,…,𝑛_𝑝}, then 𝑆(𝑛₁,…,𝑛_𝑝)⊆𝑇. Thus, let 𝑥∈𝑆(𝑛₁,…,𝑛_𝑝)⧵{0}. Then there exist 𝛼₁,…,𝛼_𝑝,𝑟∈ℕ such that 𝑥=𝛼₁𝑛₁+⋯+𝛼_𝑝𝑛_𝑝+𝑟 with 𝑟<𝛼₁+⋯+𝛼_𝑝. Since {𝑛₁,…,𝑛_𝑝}⊆𝑇, by Proposition 9, we have that 𝛼₁𝑛₁+⋯+𝛼_𝑝𝑛_𝑝+{0,…,ord(𝛼₁𝑛₁+⋯+𝛼_𝑝𝑛_𝑝)−1}⊆𝑇. By applying Lemma 45, we have that 𝑟<ord(𝛼₁𝑛₁+⋯+𝛼_𝑝𝑛_𝑝) and, therefore, 𝑥∈𝑇.
In this way, we have proved the statement.

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

Corollary 47. If 𝑛₁,…,𝑛_𝑝 are positive integers, then 𝒮_PL({𝑛₁,…,𝑛_𝑝})=𝑆(𝑛₁,…,𝑛_𝑝).

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

Example 48. It is clear that 𝑆(4,7)={0,4,7,8,9,11,→}=⟨4,7,9⟩. Therefore, 𝒮_PL({4,7})=⟨4,7,9⟩.

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.