One half of almost symmetric numerical semigroups

Abstract

Let S,T be two numerical semigroups. We study when S is one half of T, with T almost symmetric. If we assume that the type of T, t(T), is odd, then for any S there exist infinitely many such T and we prove that 1≤t(T)≤2t(S)+1. On the other hand, if t(T) is even, there exists such T if and only if S is almost symmetric and different from N; in this case the type of S is the number of even pseudo-Frobenius numbers of T. Moreover, we construct these families of semigroups using the numerical duplication with respect to a relative ideal

One-dimensional Gorenstein local rings with decreasing Hilbert function

“…Example 3.7. Consider S := 30, 33, 37, 64, 68, 69, 71, 72, 73, 75, −→ 89, 91, 92, 95 that has Hilbert function [1,27,26,25,24,27,28,29,30, →] and set K := K(S) + 66 ⊆ S. Then the semigroup S ✶ 33 K is symmetric and has Hilbert function [1,54,55,55,54,57,58,59, 60 →]. We also note that S is not almost symmetric by Proposition 1.4.…”

“…It is possible to define the numerical duplication S ✶ b E, even if the ideal E is not contained in S; in this case we have to require that E + E + b ⊆ S, that is true if E ⊆ S, otherwise the set S ✶ b E is not a numerical semigroup. In [29,Corollary 3.10] it is proved that, even if E is not proper, S ✶ b E is symmetric if and only if E is a canonical ideal; actually every symmetric numerical semigroup can be constructed as S ✶ b K(S) for some S and some odd b ∈ S (see also [29,Proposition 3.3] and [30,Section 3]). However, if the ideal is not proper, the Hilbert function of the numerical duplication can be different from the expected one; on the other hand the next examples show that also in this case it is possible to find symmetric semigroups with decreasing Hilbert function.…”

Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup

“…Since 2+2+5 / ∈ S and 0+2+7 / ∈ S, we have K +K +5 S and K +K +7 S, while K +K +13 ⊆ S because 13 is greater than f (S). Hence the symmetric double of S with minimal genus is S ✶ 13 K = {0, 10,13,14,16,17,20,23,24,26,27,28,29,30,31,32,33,34,36 →} that has genus f (S) + b+1 2 = 18. Note that the minimum genus of a double of S is g(S) + ⌈ f (S) 2 ⌉ = 13 and, according to the proof of Theorem 2.1, it is obtained by the semigroup {0, 10,13,14,15,16,17,19,20,21,23 →}.…”

“…Consider S = {0, 3, 6 →} and set d = 3. According to the proof of the previous theorem, a "triple" of S with minimal genus is {0,8,9,10,11,13,14,16 →} that has genus g(S) + ⌈ (3−1)5 = 4 + 5 = 9.…”