Numerical Semigroups with a Fixed Fundamental Gap - María Ángeles Moreno-Frías; José Carlos Rosales (Mathematics)
A gap a of a numerical semigroup S is fundamental if <semantics>{2a,3a}⊆S.</semantics> In this work, we will study the set <semantics>B(a)=<mfenced separators="" open="{" close="}">S∣S<mspace width="4.pt"></mspace>is<mspace width="4.pt"></mspace>a<mspace width="4.pt"></mspace>numerical<mspace width="4.pt"></mspace>semigroup<mspace width="4.pt"></mspace>and<mspace width="4.pt"></mspace>a<mspace width="4.pt"></mspace>is<mspace width="4.pt"></mspace>a<mspace width="4.pt"></mspace>fundamental<mspace width="4.pt"></mspace>gap<mspace width="4.pt"></mspace>of<mspace width="4.pt"></mspace>S</mfenced>.</semantics> In particular, we will give an algorithm to compute all the elements of <semantics>B(a)</semantics> with a given genus. The intersection of two elements of <semantics>B(a)</semantics> is again one element of <semantics>B(a).</semantics> A <semantics>B(a)</semantics>-irreducible numerical semigroup is an element of <semantics>B(a)</semantics> that cannot be expressed as an intersection of two elements of <semantics>B(a)</semantics> containing it properly. In this paper, we will study the <semantics>B(a)</semantics>-irreducible numerical semigroups. In this sense we will give an algorithm to calculate all of them. Finally, we will study the submonoids of <semantics>(N,+)</semantics> that can be expressed as an intersection (finite or infinite) of elements belonging to <semantics>B(a).</semantics>