A numerical semigroup $S$ is a submonoid of $(\mathbb N,+)$ with finite complement in $\mathbb N$. Integers not in $S$ are called gaps of $S$. The largest gap is known as the Frobenius number of $S$, $F(S)$. A gap $x$ is a hole, if $F(S)-x$ is also a gap.

A numerical semigroup $S$ is symmetric if it has no holes. And if $F(S)$ is even, we say that it is pseudo-symmetric, if its only hole is $F(S)/2$. Symmetric and pseudo-symmetric numerical semigroups have been widely studied due to their applications in Commutative Ring Theory.

A gap $x$ of $S$ is a pseudo-Frobenius number of $S$ if $x+S\setminus\{0\}\subseteq S$. Every pseudo-Frobenius number different from $F(S)$ is a hole. A numerical semigroup $S$ is almost symmetric if every hole is a pseudo-Frobenius number. Almost symmetric numerical semigroups where introduced by Barucci and Fröberg.

Symmetric and pseudo-symmetric numerical semigroups are maximal (with respect to set inclusion) in the set of all numerical semigroups with fixed Frobenius number.

Nary proved recently that if $S$ is almost symmetric, then it is maximal in the set of all numerical semigroups with fixed Frobenius number and type (the cardinality of the its set of pseudo-Frobenius numbers).

Every almost symmetric numerical semigroup is obtained from an irreducible numerical semigroup by removing some of its minimal generators. This will allow us to construct the set of all almost symmetric numerical semigroups with a fixed Frobenius number.