Numerical Semigroups of small and large type

Abstract

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number F, genus g and type t. It is known that for any numerical semigroup g F +1−g ≤ t ≤ 2g − F . Numerical semigroups with t = 2g − F are called almost symmetric, we introduce a new property that characterises them. We give an explicit characterisation of numerical semigroups with t = g F +1−g . We show that for a fixed α the number of numerical semigroups with Frobenius number F and type F − α is eventually constant for large F . Also the number of numerical semigroups with genus g and type g − α is also eventually constant for large g.

Frobenius allowable gaps of Generalized Numerical Semigroups

“…We give another property that is equivalent to almost-symmetry. The special case of Proposition 4.1 for numerical semigroups was proved in [11].…”