The Type of a Good Semigroup and the Almost Symmetric Condition

Abstract

We study the type and the almost symmetric condition for good subsemigoups of $${\mathbb {N}}^2$$, a class of semigroups containing the value semigroups of curve singularities with two branches. We define the type in term of a partition of a specific set associated to the semigroup and we show that this definition generalizes the well-known notion of type of a numerical semigroup and has a good behaviour with respect to the corresponding concept for algebroid curves. Then, we study almost symmetric good semigroups, their connections with maximal embedding dimension good semigroups and their Apery set, generalizing to this context several existent known results.