An extension of Wilf's conjecture to affine semigroups

Let $\text{CaC}\subset {\text{Q}}^p$ be a rational cone. An affine semigroup $S\subset \CaC$isa$\CaC$-semigroupwhenever$(\CaC\setminus S)\cap \N^p$ has only a finite number of elements. In this work, we study the tree of $\text{CaC}$-semigroups, give a method to generate it and study their subsemigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to $\text{CaC}$-semigroups and give some families of $\text{CaC}$-semigroups fulfilling the extended conjecture. We also check that other conjectures on numerical semigroups seem to be also satisfied by $\text{CaC}$-semigroups.