Frobenius problem for semigroups $\mathsf{S}(d_{1},d_{2},d_{3})$

Abstract

We find the matrix representation of the set Δ(d 3 ), where d 3 = (d 1 , d 2 , d 3 ), of integers that are unrepresentable by d 1 , d 2 , d 3 and develop a diagrammatic procedure for calculating the generating function Φ(d 3 ; z) for the set Δ(d 3 ). We find the Frobenius number F (d 3 ), the genus G(d 3 ), and the Hilbert series H (d 3 ; z) of a graded subring for nonsymmetric and symmetric semigroups S(d 3 ) and enhance the lower bounds of F (d 3 ) for symmetric and nonsymmetric semigroups.

Duality relation for the Hilbert series of almost symmetric numerical semigroups

“…For this purpose we'll make use of formulas for generic 3D nonsymmetric semigroups which were established in [3], Ch. 6,…”

Gaps in nonsymmetric numerical semigroups

“…Nonsymmetric semigroups S d 3 were studied by algebraic means in [8], [9], [10] and [12]. Recall its main results following [9].…”

“…The degeneracy of the matrix (a ij ) together with (3.2) results in strong equalities relating the matrix elements a ij and the generators d k : for any permutation of indices (i, j, k), i, j, k = 1, 2, 3 the following identities hold [9], [14]:…”

“…The Hilbert series H d 3 ; z , the type t d 3 , the Frobenius number F d 3 ; z and the genus G d 3 read [9]:…”

“…We finish this section with one example for the 4-dim numerical semigroup studied in [15]. The polynomial Q (d m ; z) was obtained by means of diagrammatic calculation on the set ∆ (d m ) developed for m = 3 and extended for m ≥ 4 [9]. …”