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.

We will give a pure combinatorial proof of the Eisenbud-Goto conjecture for arbitrary monomial curves. In addition to this, we show that the conjecture holds for certain simplicial affine semigroup rings.

We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has minimal reduction. When this is case, we study Cohen–Macaulay and Gorenstein properties associated graded provide several bounds for reduction number with respect to

We show that the Eisenbud-Goto conjecture holds for seminormal simplicial affine semigroup rings. Moreover we prove an upper bound for the Castelnuovo-Mumford regularity in terms of the dimension, which is similar as in the normal case. Finally we compute explicitly the regularity of full Veronese rings.

Let C ⊂ Z be an affine semigroup, R = K[C] its semigroup ring, and *modC R the category of finitely generated “C-graded” R-modules (i.e., Z -graded modules M with M = ⊕ c∈C Mc). When R is Cohen-Macaulay and simplicial, we show that information on M ∈ *modC R such as depth, CohenMacaulayness, and (Sn) condition, can be read off from numerical invariants of the minimal irreducible resolution (i.e...

We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the mo...

We provide a generalization of pseudo-Frobenius numbers numerical semigroups to the context simplicial affine semigroups. In this way, we characterize Cohen-Macaulay type semigroup ring K[S]. define S, type(S), in terms some Apéry sets S and show that it coincides with ring, when K[S] is Cohen-Macaulay. If d-dimensional embedding dimension at most d+2, then type(S)?2. Otherwise, type(S) might b...