Numerical Semigroups with a Monotonic Apery Set

We study numerical semigroups S with the property that if m is the multiplicity of S and w(i) is the least element of S congruent with i modulo m, then 0 < w(1) < ... < w(m − 1). The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups. This paper was supported by the project BFM2000-1469. The fourth author wishes to acknowledge support from the Universidade de Evora and the CIMA-UE.