Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups Communicated by M.-F. Roy

The oversemigroups of a numerical semigroup

“…The study of such a decomposition has been recently treated in [9], [8], [10], [11]. A natural question concerns finding a decomposition with the least number of irreducibles involved.…”

The Frobenius problem for numerical semigroups with embedding dimension equal to three

“…Let S be a numerical semigroup. We say that x ∈ Z \ S is a pseudo-Frobenius number of S (see [12]) if x + s ∈ S for all s ∈ S \ {0}. We denote by PF(S) the set of pseudo-Frobenius numbers of S. From the definition it follows that F(S) = max{PF(S)}.…”

“…12 , and then a 32 = 1 (applying item 1) of Lemma 4.6). From this equality and item 2) of Lemma 4.6, we conclude that the final end of I (a 12 , a 13 , a 23 , a 32 ) is greater than 1, which is a contradiction to the hypotheses.…”

Modular diophantine inequalities and numerical semigroups

“…(2) # H(S) = A pseudo-Frobenius number [Rosales and Branco 2002] of a numerical semigroup S is an integer x / ∈ S such that x +s ∈ S for all s ∈ S \{0}. The set of pseudoFrobenius numbers of S is denoted by Pg(S), and its cardinality, called the type of S, is denoted by t(S).…”