An extension of Wilf’s conjecture to affine semigroups

Abstract

Let C ⊂ Q p be a rational cone. An affine semigroup S ⊂ C is a C-semigroup whenever (C \ S) ∩ N p has only a finite number of elements.In this work, we study the tree of C-semigroups, give a method to generate it and study their subsemigroups with minimal embedding dimension. We extend Wilf's conjecture for numerical semigroups to Csemigroups and give some families of C-semigroups fulfilling the extended conjecture. We also check that other conjectures on numerical semigroups seem to be also satisfied by C-semigroups.

A generalization of Wilf’s conjecture for generalized numerical semigroups

“…This implies that n(S) = |N (S)| = |H(S)|. Since e(S) ≥ 2d by [11,Theorem 11], we have e(S)n(S) ≥ 2d|H(S)| = d(f (1) + 1) · · · (f (d) + 1) (by Theorem 1.5).…”

“…Remark 1.11. The Generalized Wilf Conjecture can also be stated for the class of C-semigroups considered in [11]. For readability (and to aid intuition) we save discussion of this for a future paper.…”

“…In [11] another generalization of Wilf's conjecture is given. That generalization involves a larger class of affine semigroups, called C-semigroups.…”

“…That generalization involves a larger class of affine semigroups, called C-semigroups. We will only consider the work of [11] in the case of generalized numerical semigroups. We will need some additional notation from [11] (and also [9]).…”

“…A problem posed in [9] was to formulate extensions of Wilf's conjecture to the setting of generalized numerical semigroups. A first possible extension was given in [11] for a larger class of semigroups called C-semigroups. Their extension is quite natural and has the additional feature of depending on a monomial order.…”