Rees quotients of numerical semigroups

Abstract

We introduce a class of finite semigroups obtained by considering Rees quotients of numerical semigroups. Several natural questions concerning this class, as well as particular subclasses obtained by considering some special ideals, are answered while others remain open. We exhibit nice presentations for these semigroups and prove that the Rees quotients by ideals of N, the positive integers under addition, constitute a set of generators for the pseudovariety of commutative and nilpotent semigroups.

Constructing the set of complete intersection numerical semigroups with a given Frobenius number

“…Observe that for genus 310 it takes 70 minutes to compute the set of all complete intersections, while it takes approximately 8 minutes and 30 seconds to determine all free numerical semigroups with this genus. For genus 55, computing the set of all numerical semigroups with this genus might take several months and a few terabytes (this was communicated to us by Manuel Delgado, see [6]…”

Numerical Semigroups, Polyhedra, and Posets III: Minimal Presentations and Face Dimension

“…The map f in the proof of Theorem 17 identifies the Kunz nilsemigroup of a numerical semigroup S as a Rees quotient, wherein the elements of a semigroup ideal I ⊂ S (in this case, I = S \ Ap(S; m)) are identified into a single (nil) element. Rees quotients of numerical semigroups were investigated in [9], although the primary focus was on ideals of the form I = S ∩ Z t for some t ∈ Z 1 , a form which the complement of Ap(S; m) doesn't fit unless m = m(S).…”