The numerical duplication of a numerical semigroup

Abstract

In this paper we present and study the numerical duplication of a numerical semigroup, a construction that, starting with a numerical semigroup S and a semigroup ideal E⊆S, produces a new numerical semigroup, denoted by S⋈ b E (where b is any odd integer belonging to S), such that S=(S⋈ b E)/2. In particular, we characterize the ideals E such that S⋈ b E is almost symmetric and we determine its type

A Family of Quotients of the Rees Algebra

“…If R is local, Noetherian and I a regular ideal we find a formula for the CM type of R(I) a,b (cf. Theorem 3.2) and prove that it is Gorenstein if and only if I is a canonical ideal of R. Moreover, we show the connection of the numerical duplication of a numerical semigroup (see [7]) with R(I) 0,−b , where R is a numerical semigroup ring or an algebroid branch and b has odd valuation (see Theorems 3.4 and 3.6).…”

“…A semigroup ideal E is a subset of S such that S + E ⊆ E. We set 2 · E = {2s| s ∈ E}. According to [7], the numerical duplication of S with respect to a semigroup ideal E of S and an odd integer m ∈ S is the numerical semigroup…”

Star Operations on Numerical Semigroups

“…Among these ideals, a distinguished one is M g , which is usually called the canonical ideal (or the standard canonical ideal) of S, and is denoted by K S (see for instance [12] or [5]). Corollary 4.5 and Proposition 4.6 can be seen as a reformulation of [12, Satz 4 and Hillsatz 5].…”

One-dimensional Gorenstein local rings with decreasing Hilbert function

“…where E := v(I) is the valuation of I, see [6] for more detail. We recall that I is a canonical ideal of R if and only if v(I) is a proper canonical ideal of S; hence S ✶ b E is symmetric if and only if E is a canonical ideal, see also [14,Proposition 3.1] for a simpler proof.…”