On Cohen-Macaulay and Gorenstein simplicial affine semigroups

1998 ◽  
Vol 41 (3) ◽  
pp. 517-537 ◽  
Author(s):  
J. C. Rosales ◽  
Pedro A. García-Sánchez
Keyword(s):  
Semigroup Ring ◽  
Affine Semigroups ◽  
Affine Semigroup ◽  
Minimal Presentations

We give arithmetic characterizations which allow us to determine algorithmically when the semigroup ring associated to a simplicial affine semigroup is Cohen-Macaulay and/or Gorenstein. These characterizations are then used to provide information about presentations of this kind of semigroup and, in particular, to obtain bounds for the cardinality of their minimal presentations. Finally, we show that these bounds are reached for semigroups with maximal codimension.

scholarly journals On Segre products of affine semigroup rings

1988 ◽  
Vol 110 ◽  
pp. 113-128 ◽  
Author(s):  
Lê Tuân Hoa
Keyword(s):  
Positive Integer ◽  
Polynomial Ring ◽  
Semigroup Ring ◽  
Semigroup Rings ◽  
Finitely Generated ◽  
Affine Semigroup

Let N denote the set of non-negative integers. An affine semigroup is a finitely generated submonoid S of the additive monoid Nm for some positive integer m. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring k[t1, …, tm] generated by the monomials .

scholarly journals Isolated factorizations and their applications in simplicial affine semigroups

2019 ◽  
Vol 19 (05) ◽  
pp. 2050082 ◽  
Author(s):  
Pedro A. García-Sánchez ◽  
Andrés Herrera-Poyatos
Keyword(s):  
Complete Intersection ◽  
Minimal Element ◽  
Numerical Semigroups ◽  
Commutative Monoid ◽  
Generalize Formula ◽  
Affine Semigroups ◽  
Minimal Presentations ◽  
Minimal Generators

We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. We also generalize [Formula: see text]-rectangular numerical semigroups to the context of simplicial affine semigroups and study their isolated factorizations. As a consequence of our results, we characterize those complete intersection simplicial affine semigroups with only one Betti minimal element in several ways. Moreover, we define Betti sorted and Betti divisible simplicial affine semigroups and characterize them in terms of gluings and their minimal presentations. Finally, we determine all the Betti divisible numerical semigroups, which turn out to be those numerical semigroups that are free for any arrangement of their minimal generators.

scholarly journals $h^*$-Vectors, Eulerian Polynomials and Stable Polytopes of Graphs

10.37236/1863 ◽  
2004 ◽  
Vol 11 (2) ◽  
Author(s):  
Christos A. Athanasiadis
Keyword(s):  
Semigroup Ring ◽  
Perfect Graphs ◽  
Lattice Polytope ◽  
Eulerian Polynomials ◽  
Affine Semigroup

Conditions are given on a lattice polytope $P$ of dimension $m$ or its associated affine semigroup ring which imply inequalities for the $h^*$-vector $(h^*_0, h^*_1,\dots,h^*_m)$ of $P$ of the form $h^*_i \ge h^*_{d-i}$ for $1 \le i \le \lfloor d / 2 \rfloor$ and $h^*_{\lfloor d / 2 \rfloor} \ge h^*_{\lfloor d / 2 \rfloor + 1} \ge \cdots \ge h^*_d$, where $h^*_i = 0$ for $d < i \le m$. Two applications to order polytopes of posets and stable polytopes of perfect graphs are included.

On the structure of simplicial affine semigroups

2000 ◽  
Vol 130 (5) ◽  
pp. 1017-1028 ◽  
Author(s):  
J. C. Rosales ◽  
P. A. García-Sánchez
Keyword(s):  
Upper Bound ◽  
Structure Theorem ◽  
Semigroup Rings ◽  
Affine Semigroups ◽  
Affine Semigroup

We give a structure theorem for simplicial affine semigroups. From this result we deduce characterizations of some properties of semigroup rings of simplicial affine semigroups. We also compute an upper bound for the cardinality of a minimal presentation of a simplicial affine semigroup.

scholarly journals FACTORIZATION INVARIANTS IN HALF-FACTORIAL AFFINE SEMIGROUPS

2013 ◽  
Vol 23 (01) ◽  
pp. 111-122 ◽  
Author(s):  
P. A. GARCÍA SÁNCHEZ ◽  
I. OJEDA ◽  
A. SÁNCHEZ-R.-NAVARRO
Keyword(s):  
Upper Bound ◽  
Affine Semigroups ◽  
Affine Semigroup ◽  
Catenary Degree

Let [Formula: see text] be the monoid generated by [Formula: see text] We introduce the homogeneous catenary degree of [Formula: see text] as the smallest N ∈ ℕ with the following property: for each [Formula: see text] and any two factorizations u, v of a, there exist factorizations u = w1,…,wt = v of a such that, for every k, d (wk,wk+1) ≤ N, where d is the usual distance between factorizations, and the length of wk, |wk|, is less than or equal to max{|u|, |v|}. We prove that the homogeneous catenary degree of [Formula: see text] improves the monotone catenary degree as upper bound for the ordinary catenary degree, and we show that it can be effectively computed. We also prove that for half-factorial monoids, the tame degree and the ω-primality coincide, and that all possible catenary degrees of the elements of an affine semigroup of this kind occur as the catenary degree of one of its Betti elements.

scholarly journals THE SHORT RESOLUTION OF A SEMIGROUP ALGEBRA

2017 ◽  
Vol 96 (3) ◽  
pp. 400-411 ◽  
Author(s):  
I. OJEDA ◽  
A. VIGNERON-TENORIO
Keyword(s):  
Semigroup Algebra ◽  
Numerical Semigroups ◽  
Frobenius Numbers ◽  
Affine Semigroups ◽  
Affine Semigroup ◽  
Lattice Ideal ◽  
Apéry Sets

This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc.131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups.

scholarly journals The computation of factorization invariants for affine semigroups

2019 ◽  
Vol 18 (01) ◽  
pp. 1950019 ◽  
Author(s):  
Pedro A. García-Sánchez ◽  
Christopher O’Neill ◽  
Gautam Webb
Keyword(s):  
Commutative Algebra ◽  
Improved Method ◽  
Dynamic Algorithm ◽  
Delta Set ◽  
Affine Semigroups ◽  
Affine Semigroup ◽  
Combinatorial Commutative Algebra ◽  
Theoretical Results ◽  
New Algorithms ◽  
Standard Techniques

We present several new algorithms for computing factorization invariant values over affine semigroups. In particular, we give (i) the first known algorithm to compute the delta set of any affine semigroup, (ii) an improved method of computing the tame degree of an affine semigroup, and (iii) a dynamic algorithm to compute catenary degrees of affine semigroup elements. Our algorithms rely on theoretical results from combinatorial commutative algebra involving Gröbner bases, Hilbert bases, and other standard techniques. Implementation in the computer algebra system GAP is discussed.

scholarly journals The F-signature of an affine semigroup ring

2005 ◽  
Vol 196 (2-3) ◽  
pp. 313-321 ◽  
Author(s):  
Anurag K. Singh
Keyword(s):  
Semigroup Ring ◽  
Affine Semigroup
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