On numerical semigroups with embedding dimension three

Abstract

Let $fneq1,3$ be a positive integer‎. ‎We prove that there exists a numerical semigroup $S$ with embedding dimension three such that $f$ is the Frobenius number of $S$‎. ‎We also show that‎ ‎the same fact holds for affine semigroups in higher dimensional monoids‎.