## Abstract

In this study, we characterize all families of saturated numerical semigroups with multiplicity four. We also present some results about invariants of these semigroups.

Show Summary Details# On the saturated numerical semigroups

#### Open Access

## Abstract

## 1 Introduction

## 2 Main results

## Acknowledgement

## References

## About the article

## Citing Articles

*Discrete Applied Mathematics*, 2018

More options …# Open Mathematics

### formerly Central European Journal of Mathematics

More options …

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

IMPACT FACTOR 2017: 0.831

5-year IMPACT FACTOR: 0.836

CiteScore 2017: 0.68

SCImago Journal Rank (SJR) 2017: 0.450

Source Normalized Impact per Paper (SNIP) 2017: 0.829

Mathematical Citation Quotient (MCQ) 2017: 0.32

In this study, we characterize all families of saturated numerical semigroups with multiplicity four. We also present some results about invariants of these semigroups.

KeyWords: Saturated numerical semigroup; Frobenius number; Gaps

MSC 2010: 20M14

Let ℕ = {1, 2, ..., *n*, ..} and ℤ be the set of integers. A subset *S* of the set ℕ of nonnegative integers is called a numerical semigroup if it satisfies the following conditions:

- (i)
0 ∈

*S*, - (ii)
*a, b*∈*S*⇒*a*+*b*∈*S*, - (iii)
ℕ \

*S*has a finite number of elements.

Condition (iii) is equivalent to *gcd*(*S*) = 1 (Here, *gcd*(*S*) is the greatest common divisor of the element of *S*).

All numerical semigroups are finitely generated, i.e.
$$S=\u3008{a}_{1},{a}_{2},...{a}_{r}\u3009\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\{\sum _{k=1}^{t}{c}_{i}{a}_{i},...,{c}_{r}\in \mathbb{N}\}$$
where *a*_{1}*, a*_{2}*, ... ,a _{r}* ∈

We define the following invariants of numerical semigroups: $$F(S)=max\{x\phantom{\rule{thinmathspace}{0ex}}:\phantom{\rule{thinmathspace}{0ex}}x\in \mathbb{Z}\mathrm{\setminus}S\}$$ and $$n(S)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\left|\{0,1,2,...,F(S)\}\cap S\right|.$$

*F* ( *S* ) and *n* ( *S* ) are called the Frobenius number of *S* and the number determiner of *S*, respectively.

We can write
$$S=\u3008{a}_{1},{a}_{2},...,{a}_{r}\u3009=\u3008{s}_{0}=0,{s}_{1},{s}_{2},...,{s}_{n-1},{s}_{n}=F(S)+1,\to ...\u3009$$
where *s _{i}* <

The set ℕ \ *S* is the gap of *S*, and the set of gaps of *S* is denoted by *H*(*S*). *g*(*S*) = |*H*(*S*)| is called the genus of *S*. It is clear that *g*(*S*) = *F*(*S*) + 1 − *n*(*S*). An element *x* ∈ *H*(*S*) is called a fundamental gap of *S* if 2*x*, 3*x* ∈ *S*. The set of all the fundamental gaps of *S* is denoted by *FH*(*S*), i.e.
$$FH(S)=\{x\in H(S):2x,3x\in S\}.$$

An element *x* ∈ ℤ is called a Pseudo-Frobenius number of *S* if *x* ∉ *S* and *x* + *s* ∈ *S*, for *s* ∈ *S* \ {0}. We denote by *PF*(*S*) the set of all Pseudo-Frobenius numbers of *S*, i.e.
$$PF(S)=\{x\phantom{\rule{thinmathspace}{0ex}}\in \mathbb{Z}\mathrm{\setminus}S:x+s\in S,\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}\text{all}\phantom{\rule{thinmathspace}{0ex}}s\phantom{\rule{thinmathspace}{0ex}}\in \phantom{\rule{thinmathspace}{0ex}}S\mathrm{\setminus}\{0\}\}$$
(see [7]). Given a numerical semigroup *S* and *x* ∈ *S* \ {0}, we define the Apery set of *x* in *S* as *Ap*(*S, x*) = {*s* ∈ *S* : *s* − *x* ∉ *S*} (for details see [9]).

If a numerical semigroup *S* satisfies the condition *x* + *y* − *z* ∈ *S*, for every *x, y, z* ∈ *S* such that *x* ≥ *y* ≥ *z*, then *S* is called Arf. If *S* is an Arf numerical semigroup, then *S* has maximal embedding dimension.

The investigation of combinatorial properties of semigroups is very important, because they often occur in applications ([1, 3, 5]) and are related to automata theory (see [4]). A numerical semigroup *S* is saturated if *s* + *c*_{1}*s*_{1} + *c*_{2}*s*_{2} + ... + *c _{k}*

In this study, we show that all families of numerical semigroups with multiplicity four are saturated numerical semigroups; these are numerical semigroups of the form *S* = 〈4, *k, k* + 2, *k* + 3〉, for *k* = 3(*mod*4) and *k* ≥ 7 and *S* = 〈4, *k, k* + *t, k* + *t* + 2〉, for *k* = 2(*mod* 4) and *k* ≥ 6 and *t* an odd integer. We also give the formulae for *F*(*S*), *n*(*S*), *PF*(*S*), *g*(*S*), *H*(*S*) and *FH*(*S*) of these numerical semigroups.

In this section we provide some results for numerical semigroups with multiplicity four; i.e. numerical semigroups of the form *S* = 〈4, *k, k* + 2, *k* + 3〉 (for *k* ≡ 3(*mod* 4) and *k* ≥ 7) and *S* = 〈4, *k, k* + *t, k* + *t* + 2〉, (for *k* ≡ 2(*mod* 4) and *k* ≥ 6 and *t* an odd integer).

([8]).*Let S be a numerical semigroup, then the following conditions are equivalent*:

- (i)
*S is a saturated numerical semigroup*. - (ii)
*a*+*d*_{S}*(a)*∈*S for all a*∈*S*,*a*> 0*where d*_{S}*(a)*=*gcd*{*x*∈*S*:*x*≤*a*}. - (iii)
*a*+*kd*_{S}*(a)*∈*S for all a*∈*S*,*a*> 0*and k*∈ ℕ.

([10]). *If S* = 〈4,*k, k* + *1, k* + 2 〉 *, then S is a saturated numerical semigroup, for k* ≡ 1(*mod* 4) *and k* ≥ 5.

*Let S* = 〈4, *k, k* + 2, *k* + 3〉 *be numerical semigroup, where k* ≡ 3( *mod* 4) *and k* ≥ 7. *Then S is saturated*.

Let *S* = 〈4,*k, k* + 2, *k* + 3〉 be numerical semigroup, where *k* ≡ 3(*mod* 4) and *k* ≥ 7. We note that *k* = *4r* + *3*, *r* ≥ *1* and *r* ∈ ℤ. Thus, we have
$$\begin{array}{c}S=\u30084,k,k+2,k+3\u3009=\{0,4,8,\cdots ,k-7,k-3,k\to \cdots ,\}\\ =\{0,4,8,\cdots ,4r-4,4r,4r+3,\to \cdots \}.\end{array}$$

In this case,

- (a)
If

*a*< 4*r*+ 3, then*d*_{S}*(a)*=*1*. So, we find that*a*+*d*_{S}*(a)*∈*S*since*a*+*d*_{S}*(a)*=*a*+*1*≥*4r*+*4*∈*S*, for all*a*∈*S*,*a*> 0. - (b)
If

*a*≥ 4*r*+ 3, then*d*_{S}*(a)*= 4. So, we have*a*+*d*_{S}*(a)*=*a*+ 4 ∈*S*, for all*a*∈*S*,*a*> 0.

In view of Proposition 2.1, we find that *S* is saturated a numerical semigroup. □

*Let S* = 〈4,*k, k* + *t, k* + *t* + 2〉 *be numerical semigroup, where k* ≡ 2(*mod* 4), *k* ≥ 6, *and t is an odd integer. Then S is saturated*.

It is trivial that *gcd* {4,*k, k* + *t, k* + *t* + 2} = 1 since *k* is even and *t* is an odd integer. If we put *k* = *4r* + 2, *r* ≥ 1 and *r* ∈ ℤ, then we have
$$\begin{array}{c}S=\u30084,k,k+t,k+t+2\u3009=\{0,4,8,...,k-6,k-2,k,k+2,...,k+t-3,k+t-1,\to ...,\}\\ =\{0,4,8,...,4r-4,4r,4r+2,4r+4,...,4r+t-1,4r+t+1,\to ...,\}.\end{array}$$

In this case,

- (i)
If

*a*> 4*r*+*t*+ 1, then*d*_{S}*(a)*= 1. So, we obtain*a*+*d*_{S}*(a)*∈*S*from the inequality*a*+*d*_{S}*(a)*=*a*+ 1 ≥*4r*+*t*+ 2 ∈*S*, for all*a*∈*S*,*a*> 0. - (ii)
If 4

*r*≤*a*≤ 4*r*+*t*+ 1, then*d*_{S}*(a)*= 2. So, we obtain*a*+*d*_{S}*(a)*=*a*+ 2 ∈*S*from the inequality*4r*≤*a*≤ 4*r*+*t*+ 1, for all*a*∈*S*,*a*> 0. - (iii)
If

*a*< 4*r*, then*d*_{S}*(a)*= 4. So, we obtain*a*+*d*_{S}*(a)*=*a*+ 4 ∈*S*since*a*+ 4*r*< 4*r*+ 4, for all*a*∈*S*,*a*> 0.

In view of Proposition 2.1, we have that *S* is a saturated numerical semigroup. □

([6]).*Let S be a numerical semigroup minimally generated by* {*n*_{1} < *n*_{2} < ... < *n _{r}*}.

([6]).*Let S be a numerical semigroup minimally generated by* {*n*_{1} < *n*_{2} < ... < *n _{r}*}.

- (1)
*If S has maximal embedding dimension, then F*(*S*) =*n*−_{r}*n*_{1}. - (2)
*S has maximal embedding dimension if and only if*$$g(S)=\frac{{n}_{2}+{n}_{3}+...+{n}_{r}}{{n}_{1}}-\frac{{n}_{1}-1}{2}.$$

*If S* = 〈4, *k k* + 2, *k* + 3〉 *is a numerical semigroup, where k* ≡ 3(*mod* 4) *and k* ≡ 7. *Then we obtain following equalities*:

- (a)
*F*(*S*) =*k*− 1, - (b)
$g\left(S\right)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{3k\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}1}{4}$,

- (c)
*PF*(*S*) = {*k*− 4,*k*− 2,*k*− 1}, - (d)
$n\left(S\right)\phantom{\rule{thinmathspace}{0ex}}=\frac{k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}1}{4}$,

- (e)
*H*(*S*) = {1, 2, 3, 5, 6, 7, ... ,*k*− 6,*k*− 5,*5k*− 4,*k*− 2,*k*− 1}.

We have *Ap*(*S*, 4) = {0, *k*, *k* − 2, *k* − 3} since *S* has maximal embedding dimension. Thus,

- (a)
We have

*F*(*S*) = (*k*+ 3) − 4 =*k*− 1 from Corollary 2.6 (1). - (b)
We obtain $g\left(S\right)\phantom{\rule{thinmathspace}{0ex}}=\frac{k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}2\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}3}{4}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\frac{4\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}1}{2}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{3k\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}1}{4}$ from Corollary 2.6 (2).

- (c)
It is obvious that

*PF*(*S*) = {*k*− 4,*k*+*2*− 4,*k*+ 3 − 4}. So we find*PF*(*S*) = {*k*− 4,*k*− 2,*k*− 1}. - (d)
We have $n\left(S\right)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\left(k\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}1\right)\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}1\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\frac{3k\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}1}{4}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}1}{4}$ from

*g*(*S*) =*F*(*S*) + 1 −*n*(*S*). - (e)
We find that

*H*(*S*) = {1, 2, 3, 5, 6, 7, ... , 4*r*− 3, 4*r*− 2, 4*r*− 1, 4*r*+ 1, 4*r*+*2*} = {1, 2, 3, 5, 6, 7, ... ,*k*− 6,*k*− 5,*k*− 4,*k*− 2,*k*− 1} from the equality*S*= < 4,*k*,*k*+ 2,*k*+ 3 > {0, 4, 8, ... ,*k*− 7,*k*− 3,*k*, → ... ,} = {0, 4, 8, ... , 4*r*− 4, 4*r*, 4*r*− 3, → ... ,}.

*Let S*= 〈4, *k*, *k* + *t, k* + *t* + *2*〉 *be a numerical semigroup, where k* ≡ 2(*mod* 4), *k* ≡ 6*, and t is an odd integer. Then, we have following equalities*:

- (a)
*F*(*S*) =*k*+*t*− 2, - (b)
$g\left(S\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{3k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}2t\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}4}{4}$,

- (c)
*PF*(*S*) = {*k*− 4,*k*+*t*− 4,*k*+*t*− 2}, - (d)
$n\left(S\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}2t}{4}$,

- (e)
*H*(*S*) = {1, 2, 3, 5, 6, 7, ... ,*k*− 5,*k*− 4,*k*− 3,*k*− 1,*k*− 1,*k*+ 3, ... ,*k*+*t*− 2}.

We have *Ap*(*S*, 4) = {0, *k, k* + *t, k* + *t* + 2} since *S* has maximal embedding dimension. Thus,

- (a)
We have

*F*(*S*) = (*k*+*t*+ 2) − 4 =*k*+*t*+ 2 from Corollary 2.6 (1). - (b)
We obtain $g\left(S\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}t\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}t\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}2}{4}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{4\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}1}{2}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{3k\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}2t\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}4}{4}$ from Corollary 2.6 (2).

- (c)
It is obvious that

*PF*(*S*) = {*k*− 4,*k*+*t*− 4,*k*+*t*− 2 − 4}. So, we find*PF*(*S*) = {*k*− 4,*k*+*t*− 4,*k*+*t*− 2}. - (d)
We have $n(S)=(k+t-2)+1-\frac{3k+2t-4}{4}=\frac{k+2t}{4}$ from

*g*(*S*) =*F*(*S*) + 1 −*n*(*S*). - (e)
We observe that

*H*(*S*) = {1, 2, 3, 5, 6, 7, ... , 4*r*− 3, 4*r*− 2, 4*r*− 1, 4*r*− 1, 4*r*+ 3, 4*r*+ 5, ... , 4*r*+*t*} = {1, 2, 3, 5, 6, 7, ... ,*k*− 5,*k*− 4,*k*− 3,*k*− 1,*k*+ 1,*k*+ 3, ... ,*k*+*t*+ 2} since*S*= 〈4,*k, k*+*t, k*+*t*+ 2〉 = {0, 4, 8, ... ,*k*− 6,*k*− 2,*k, k*+ 2, ... ,*k*+*t*− 3,*k*+*t*−*1*, → ... ,} = {0, 4, 8, ... ,4*r*− 4, 4*r*, 4*r*+ 2, 4*r*+ 4, ... , 4*r*−*t*− 1, 4*r*+*t*+*1*, → ... ,}.

*Consider the numerical semigroup S*= 〈4, *k, k* + 2, *k* + 3〉. *If we put k* = 15 *then we have that S* = 〈4, *k, k* + 2, *k* + 3〉 = 〈4, 15, 17, 18〉 = {0, 4, 8, 12, 15, → ... ,} *is saturated. Hence, we find that*

- (a)
*F*(*S*) =*k*− 1 = 15 − 1 = 14, - (b)
$g(S)=\frac{3k-1}{4}=\frac{45-1}{4}=11$,

- (c)
*PF*(*S*) = {*k*− 4,*k*− 2,*k*− 1} = {11, 13, 14}, - (d)
$n(S)=\frac{k+1}{4}=\frac{15+1}{4}=4$,

- (e)
*H*(*S*) = {1, 2, 3, 5, 6, 7, ... ,*k*− 6,*k*− 5,*k*− 4,*k*− 2,*k*− 1} = {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14}*and Ap*(*S*, 4) = {0,*k, k*+ 2,*k*+ 3} = {0, 15, 17, 18}.

*Consider the numerical semigroup S*= 〈4, *k, k* + *t, k* + *t* + 2〉. *If we put k* = 14 *and t* = *13 then we find that S*= 〈4, *k, k* + *t, k* + *t* + 2〉 = 〈4, 14, 27, 29〉 = {0, 4, 8, 12, 14, 16, 18, 20, 22, 24, 26, → ... ,} *is saturated. Thus, we observe that*

- (a)
*F*(*S*) =*k*+*t*− 2 = 14 + 13 − 2 = 25, - (b)
$g(S)=\frac{3k+2t-4}{4}=\frac{64}{4}=16$,

- (c)
*PF*(*S*) = {*k*− 4,*k*+*t*− 4,*k*+*t*− 2} = {10, 23, 25}, - (d)
$n(S)=\frac{k+2t}{4}=\frac{40}{4}=10$,

- (e)
*H*(*S*) = {1, 2, 3, 5, 6, 7, ... ,*k*− 5,*k*− 4,*k*+ 3,*k*− 1,*k*− 1,*k*+3, ... ,*k*−*t*− 2} = {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 15, 17, 19, 21, 23, 25}*and Ap*(*S*, 4) = {0,*k, k*+*t, k*+*t*+ 2} = {0, 14, 27, 29}.

The authors thank the anonymous referee for his/her remarks which helped them to improve the presentation of the paper.

- [1]
Abawajy J., Kelarev A.V., Chowdhury M., Power graphs: a survey, Electronic J. Graph Theory and Applications, 2013, 1(2), 125-147 Google Scholar

- [2]
Barucci V, Dobbs D.E., Fontana M., Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains,Memoirs of American Mathematical Society 1997, 125(598), 1-77 Google Scholar

- [3]
Kelarev A.V., Combinatorial properties of sequences in groups and semigroups, "Combinatorics, Complexity and Logic", Discrete Mathematics and Theoretical Computer Science, 1996, 289-298 Google Scholar

- [4]
Kelarev A.V., Graph Algebras and Automata, Marcel Dekker, New York, 2003 Google Scholar

- [5]
Kelarev A.V., Ryan J., Yearwood J., Cayley graphs as classifiers for data mining:The influence of asymmetries, Discrete Mathematics, 2009, 309(17), 5360-5369 Google Scholar

- [6]
Rosales J.C., Garcia-Sánchez P.A., Numerical Semigroups, Springer, 2009 Google Scholar

- [7]
Rosales J.C., Garcia-Sánchez P.A., Garcia-Garcia J.I., Jimenez Madrid J.A., Fundamental gaps in numerical semigroups, Journal Pure and Applied Algebra, 2004, 189, 301-313 Google Scholar

- [8]
Rosales J.C., Garcia-Sánchez P.A., Garcia-Garcia J.I., Branco M.B., Saturated numerical semigroups, Houston J.of Math. 2004, 30, 321-330 Google Scholar

- [9]
Rosales J.C., Numerical Semigroups with Apery sets of unique expression, Journal of Algebra, 2000, 226, 479-487 Google Scholar

- [10]
Suer M., Ilhan S., On a family of saturated numerical semigroup with multiplicity four, Turkish Journal of Math., (in press), DOI:10.3906 Google Scholar

**Received**: 2016-08-04

**Accepted**: 2016-09-23

**Published Online**: 2016-11-03

**Published in Print**: 2016-01-01

**Citation Information: **Open Mathematics, Volume 14, Issue 1, Pages 827–831, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0074.

© Ilhan and Süer, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]

Miguel V. Carriegos, Noemí DeCastro-García, and Ángel Luis Muñoz Castañeda

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.