Numerical Semigroups Generated by Intervals

We study numerical semigroups generated by intervals and solve the following problems related to such semigroups: the membership problem, give an explicit formula for the Frobe-nius number, decide whether the semigroup is a complete intersection and/or symmetric, and computation of the cardi-nality of a (any) minimal presentation of this kind of numerical semigroups. A numerical semigroup is a finitely generated subsemigroup of the set of nonnegative integers N, such that the group generated by it is the set of all integers Z. In this paper we study the semigroups generated by intervals of nonnegative integers, that is to say, semigroups of the form S = a, a + 2a−1; thus we may assume that x ≤ a − 1. For a semigroup of this kind we solve the following problems: 1) Membership problem. An element n ∈ N belongs to S = a, a + 1,. .. , a + x if and only if n mod a ≤ ≤ n a x, where n a is quotient of the integer division of n by a, and n mod a denotes the remainder of this division, n − − n a a. 2) Computation of the Frobenius number of the semigroup. The Frobe-nius number of a numerical semigroup (also known as the conductor of the semigroup) is the greatest integer not belonging to the given semi-group. The Frobenius number of S = a, a+1,. .. , a+x is a−1 x a−1, where q denotes the least integer greater than or equal to q ∈ Q +. 3) Symmetry of the semigroup. A numerical semigroup T with Frobenius number C is symmetric if and only if for each z ∈ Z we have that either z ∈ S or C − z ∈ S. These kinds of semigroups are specially interesting in Ring Theory as Kunz shows in [7]. The semigroup S = a, a + 1,. .. , a + x is symmetric if and only if a ≡ 2 mod x (here a ≡ b mod c denotes the fact a − b = kc for some integer k). 4) Cardinality of a minimal presentation of S. The semigroup S = a, a+ 1,. .. , a+x is isomorphic to N x+1 /σ, where σ is the kernel congruence of the semigroup morphism