On the saturated numerical semigroups

Sedat Ilhan 1  and Meral Süer 2
• 1 Department of Mathematics, Faculty of Sciences, Dicle University, Diyarbakır, Turkey
• 2 Faculty of Science and Literature, Batman University, Batman, Turkey
Sedat Ilhan
and Meral Süer

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.

1 Introduction

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:

1. (i)0 ∈ S,
2. (ii)a, bSa + bS,
3. (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.
$S=〈a1,a2,...ar〉={∑k=1tciai,...,cr∈N}$
where a1, a2, ... ,arS and r ≥ 1. In this case, {a1, a2, ... , ar} is a minimal system of generators if no proper subset of {a1, a2, ... , ar} generates S. The numbers e(S) = r and m(S) = min{aS : a > 0} are called the embedding dimension and multiplicity of S respectively. In general, it holds that e(S) ≤ m(S). We say that S has maximal embedding dimension if e(S) = m(S) (see [6]).
We define the following invariants of numerical semigroups:
$F(S)=max{x:x∈Z∖S}$
and
$n(S)=|{0,1,2,...,F(S)}∩S|.$

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

We can write
$S=〈a1,a2,...,ar〉=〈s0=0,s1,s2,...,sn−1,sn=F(S)+1,→...〉$
where si < si+1 and n = n(S). The arrow means that every integer greater than F(S) + 1 belongs to S, for i = 1, 2, ... , n = n(S) (see [2]).
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 xH(S) is called a fundamental gap of S if 2x, 3xS. The set of all the fundamental gaps of S is denoted by FH(S), i.e.
$FH(S)={x∈H(S):2x,3x∈S}.$
An element x ∈ ℤ is called a Pseudo-Frobenius number of S if xS and x + sS, for sS \ {0}. We denote by PF(S) the set of all Pseudo-Frobenius numbers of S, i.e.
$PF(S)={x∈Z∖S:x+s∈S,foralls∈S∖{0}}$
(see [7]). Given a numerical semigroup S and xS \ {0}, we define the Apery set of x in S as Ap(S, x) = {sS : sxS} (for details see [9]).

If a numerical semigroup S satisfies the condition x + yzS, for every x, y, zS such that xyz, 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 + c1s1 + c2s2 + ... + ckskS, where s, siS and ci ∈ ℤ such that c1s1 + c2s2 + ... + cksk ≥ 0 and sis for i = 1, 2, ... , k. Also, all saturated numerical semigroup are Arf. However an Arf numerical semigroup need not be to be saturated. The numerical semigroup
$S=〈7,12,15,16,17,18,20〉$
is Arf, but it is not saturated since 12 + (−5).7 + 3.12 = 13 ∉ S.

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(mod4) 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.

2 Main results

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:

1. (i)S is a saturated numerical semigroup.
2. (ii)a + dS(a)S for all aS, a > 0 where dS(a) = gcd{xS : xa }.
3. (iii)a + kdS(a)S for all aS, 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, r1 and r ∈ ℤ. Thus, we have
$S=〈4,k,k+2,k+3〉={0,4,8,⋯,k−7,k−3,k→⋯,}={0,4,8,⋯,4r−4,4r,4r+3,→⋯}.$

In this case,

1. (a)If a < 4r + 3, then dS(a) = 1. So, we find that a + dS(a)S since a + dS(a) = a + 14r + 4S, for all aS, a > 0.
2. (b)If a ≥ 4r + 3, then dS(a) = 4. So, we have a + dS(a) = a + 4 ∈ S, for all aS, 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
$S=〈4,k,k+t,k+t+2〉={0,4,8,...,k−6,k−2,k,k+2,...,k+t−3,k+t−1,→...,}={0,4,8,...,4r−4,4r,4r+2,4r+4,...,4r+t−1,4r+t+1,→...,}.$

In this case,

1. (i)If a > 4r + t + 1, then dS(a) = 1. So, we obtain a + dS(a)S from the inequality a + dS(a) = a + 1 ≥ 4r + t + 2 ∈ S, for all aS, a > 0.
2. (ii)If 4ra ≤ 4r + t + 1, then dS(a) = 2. So, we obtain a + dS(a) = a + 2 ∈ S from the inequality 4ra ≤ 4r + t + 1, for all aS, a > 0.
3. (iii)If a < 4r, then dS(a) = 4. So, we obtain a + dS(a) = a + 4 ∈ S since a + 4r < 4r + 4, for all aS, 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 {n1 < n2 < ... < nr}. Then S has maximal embedding dimension if and only if Ap(S, n1) = {0,n2, n3,...,nr}.

([6]).Let S be a numerical semigroup minimally generated by {n1 < n2 < ... < nr}. Then the following conditions are true:

1. (1)If S has maximal embedding dimension, then F(S) = nrn1.
2. (2)S has maximal embedding dimension if and only if
$g(S)=n2+n3+...+nrn1−n1−12.$

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:

1. (a)F(S) = k − 1,
2. (b)$g(S)=3k−14$,
3. (c)PF(S) = {k − 4, k − 2, k − 1},
4. (d)$n(S)=k+14$,
5. (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,

1. (a)We have F(S) = (k + 3) − 4 = k − 1 from Corollary 2.6 (1).
2. (b)We obtain $g(S)=k+k+2+k+34−4−12=3k−14$ from Corollary 2.6 (2).
3. (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}.
4. (d)We have $n(S)=(k−1)+1−3k−14=k+14$ from g(S) = F(S) + 1 − n(S).
5. (e)We find that H(S) = {1, 2, 3, 5, 6, 7, ... , 4r − 3, 4r − 2, 4r − 1, 4r + 1, 4r + 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, ... , 4r − 4, 4r, 4r − 3, → ... ,}.

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

1. (a)F(S) = k + t − 2,
2. (b)$g(S)=3k+2t−44$,
3. (c)PF(S) = {k − 4, k + t − 4, k + t − 2},
4. (d)$n(S)=k+2t4$,
5. (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,

1. (a)We have F(S) = (k + t + 2) − 4 = k + t + 2 from Corollary 2.6 (1).
2. (b)We obtain $g(S)=k+k+t+k+t+24−4−12=3k+2t−44$ from Corollary 2.6 (2).
3. (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}.
4. (d)We have $n(S)=(k+t−2)+1−3k+2t−44=k+2t4$ from g(S) = F(S) + 1 − n(S).
5. (e)We observe that H(S) = {1, 2, 3, 5, 6, 7, ... , 4r − 3, 4r − 2, 4r − 1, 4r − 1, 4r + 3, 4r + 5, ... , 4r + 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 + t1, → ... ,} = {0, 4, 8, ... ,4r − 4, 4r, 4r + 2, 4r + 4, ... , 4rt − 1, 4r + 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

1. (a)F(S) = k − 1 = 15 − 1 = 14,
2. (b)$g(S)=3k−14=45−14=11$,
3. (c)PF(S) = {k − 4, k − 2, k − 1} = {11, 13, 14},
4. (d)$n(S)=k+14=15+14=4$,
5. (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

1. (a)F(S) = k + t − 2 = 14 + 13 − 2 = 25,
2. (b)$g(S)=3k+2t−44=644=16$,
3. (c)PF(S) = {k − 4, k + t − 4, k + t − 2} = {10, 23, 25},
4. (d)$n(S)=k+2t4=404=10$,
5. (e)H(S) = {1, 2, 3, 5, 6, 7, ... , k − 5, k − 4, k + 3, k − 1, k − 1, k +3, ... , kt − 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}.

Acknowledgement

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

References

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

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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

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Kelarev A.V., Graph Algebras and Automata, Marcel Dekker, New York, 2003

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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

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Rosales J.C., Garcia-Sánchez P.A., Numerical Semigroups, Springer, 2009

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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

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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

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

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

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• [1]

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

• [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

• [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

• [4]

Kelarev A.V., Graph Algebras and Automata, Marcel Dekker, New York, 2003

• [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

• [6]

Rosales J.C., Garcia-Sánchez P.A., Numerical Semigroups, Springer, 2009

• [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

• [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

• [9]

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

• [10]

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

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