Symmetric numerical semigroups with arbitrary multiplicity and embedding dimension

Abstract

Abstract. We construct symmetric numerical semigroups S for every minimal number of generators µ(S) and multiplicity m(S), 2 ≤ µ(S) ≤ m(S) − 1. Furthermore we show that the set of their defining congruence is minimally generated by µ(S)(µ(S) − 1)/2 − 1 elements.

Numerical Semigroups Generated by Concatenation of Arithmetic Sequences

“…We have posed a conjecture 3.3 in section 3 on the unboundedness of minimal number of generators of the defining ideal in arbitrary embedding dimension. Our results in section 4 generalizes the results proved in [9].…”

“…Given an integer e ≥ 4, it is not completely understood whether the symmetry condition on Γ in embedding dimension e ensures that the minimal cardinality of a system of generators of σ defined above is a bounded function of e. This was answered in affirmative by Bresinsky for e = 4 in [2], and for certain cases of e = 5 in [3]. Rosales [9] constructed numerical semigroups for a given multiplicity m and embedding dimension e, which are symmetric, and showed that the cardinality of a minimal presentation of these semigroups is a bounded function of the embedding dimension e. In fact using the pertinent results obtained in [2], [5], [10], [12], one can compute the cardinality of a minimal presentation of a symmetric numerical semigroup with multiplicity m ≤ 8. This remains an open question in general, whether symmetry condition on the numerical semigroup e ≥ 5 imposes an upper bound on the cardinality of a minimal presentation of a numerical semigroup Γ.…”

Affine monomial curves

“…In [36] the case a = e + 1 has been studies in detail and it has been proved that µ(p) is bounded. In [36] it has been shown that there are symmetric semigroups of this form, generalizing Roasales' construction in [46]. Our observations in [36] give rise to the following question:…”

Numerical semigroups problem list

“…Bresinski also proves in [8] that the cardinality for a minimal presentation of a symmetric numerical semigroup with embedding dimension four can only be three or five. It is conjectured in [36] that if S is a numerical semigroup with e(S) ≥ 3, then the cardinality of a minimal presentation for S is less than or equal to e(S)(e(S)−1) 2 − 1. Barucci [2] proves with the semigroup 19,23,29,31,37 that the conjecture above is not true.…”