Sedat Ilhan and Meral Süer

On the saturated numerical semigroups

De Gruyter | 2016

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:

  • (i)

    0 ∈ S,

  • (ii)

    a, bSa + bS,

  • (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 = a 1 , a 2 , . . . a r = { k = 1 t c i a i , . . . , c r N }
where a 1 , a 2 , ... ,arS and r ≥ 1. In this case, { a 1 , a 2 , ... , ar} is a minimal system of generators if no proper subset of { a 1 , a 2 , ... , 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 = a 1 , a 2 , . . . , a r = s 0 = 0 , s 1 , s 2 , . . . , s n 1 , s n = F ( S ) + 1 , . . .
where si < s i+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.

F H ( S ) = { x H ( S ) : 2 x , 3 x 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.

P F ( S ) = { x Z S : x + s S , for all s 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).

Proposition 2.1

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

  • (i)

    S is a saturated numerical semigroup.

  • (ii)

    a + dS(a)S for all aS, a > 0 where dS(a) = gcd{xS : xa }.

  • (iii)

    a + kdS(a)S for all aS, a > 0 and k ∈ ℕ.

Theorem 2.2

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

Theorem 2.3

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

Proof

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 , , 4 r 4 , 4 r , 4 r + 3 , } .

In this case,

  • (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.

  • (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. □

Theorem 2.4

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.

Proof

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 , . . . , 4 r 4 , 4 r , 4 r + 2 , 4 r + 4 , . . . , 4 r + t 1 , 4 r + t + 1 , . . . , } .

In this case,

  • (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.

  • (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.

  • (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. □

Proposition 2.5

([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}.

Corollary 2.6

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

  • (1)

    If S has maximal embedding dimension, then F(S) = nrn1.

  • (2)

    S has maximal embedding dimension if and only if

    g ( S ) = n 2 + n 3 + . . . + n r n 1 n 1 1 2 .

Theorem 2.7

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 ( S ) = 3 k 1 4 ,

  • (c)

    PF(S) = {k − 4, k − 2, k − 1},

  • (d)

    n ( S ) = k + 1 4 ,

  • (e)

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

Proof

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 ( S ) = k + k + 2 + k + 3 4 4 1 2 = 3 k 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 ( S ) = ( k 1 ) + 1 3 k 1 4 = k + 1 4 from g(S) = F(S) + 1 − n(S).

  • (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, → ... ,}.

Theorem 2.8

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:

  • (a)

    F(S) = k + t − 2,

  • (b)

    g ( S ) = 3 k + 2 t 4 4 ,

  • (c)

    PF(S) = {k − 4, k + t − 4, k + t − 2},

  • (d)

    n ( S ) = k + 2 t 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}.

Proof

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 ( S ) = k + k + t + k + t + 2 4 4 1 2 = 3 k + 2 t 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 3 k + 2 t 4 4 = k + 2 t 4 from g(S) = F(S) + 1 − n(S).

  • (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, → ... ,}.

Example 2.9

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 ) = 3 k 1 4 = 45 1 4 = 11 ,

  • (c)

    PF(S) = {k − 4, k − 2, k − 1} = {11, 13, 14},

  • (d)

    n ( S ) = k + 1 4 = 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}.

Example 2.10

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 ) = 3 k + 2 t 4 4 = 64 4 = 16 ,

  • (c)

    PF(S) = {k − 4, k + t − 4, k + t − 2} = {10, 23, 25},

  • (d)

    n ( S ) = k + 2 t 4 = 40 4 = 10 ,

  • (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

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

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

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

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