scholarly journals Proportionally modular affine semigroups

2018 ◽  
Vol 17 (01) ◽  
pp. 1850017 ◽  
Author(s):  
J. I. García-García ◽  
M. A. Moreno-Frías ◽  
A. Vigneron-Tenorio
Keyword(s):  
Minimal System ◽  
Numerical Semigroups ◽  
Diophantine Inequalities ◽  
Generating Sets ◽  
Affine Semigroups ◽  
Affine Semigroup ◽  
Minimal System Of Generators

This work introduces a new kind of semigroup of [Formula: see text] called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We give an algorithm to compute their minimal generating sets, and we specialize when [Formula: see text]. For this case, we also provide a faster algorithm to compute their minimal system of generators, prove they are Cohen–Macaulay and Buchsbaum, and determinate their (minimal) Frobenius vectors. Besides, Gorenstein proportionally modular affine semigroups are characterized.

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.

Implosive graphs: Square-free monomials on symbolic Rees algebras

2016 ◽  
Vol 16 (08) ◽  
pp. 1750145 ◽  
Author(s):  
A. Flores-Méndez ◽  
I. Gitler ◽  
E. Reyes
Keyword(s):  
Structural Properties ◽  
Monomial Ideal ◽  
Minimal System ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Vertex Set ◽  
Minimal System Of Generators

Let [Formula: see text] be the edge monomial ideal of a graph [Formula: see text], whose vertex set is [Formula: see text]. [Formula: see text] is implosive if the symbolic Rees algebra [Formula: see text] of [Formula: see text] has a minimal system of generators [Formula: see text] where [Formula: see text] are square-free monomials. We give some structural properties of implosive graphs and we prove that they are closed under clique-sums and odd subdivisions. Furthermore, we prove that universally signable graphs are implosive. We show that odd holes, odd antiholes and some Truemper configurations (prisms, thetas and even wheels) are implosive. Moreover, we study excluded families of subgraphs for the class of implosive graphs. In particular, we characterize which Truemper configurations and extensions of odd holes and antiholes are minimal nonimplosive.

scholarly journals Schubert presentation of the cohomology ring of flag manifolds

2015 ◽  
Vol 18 (1) ◽  
pp. 489-506 ◽  
Author(s):  
Haibao Duan ◽  
Xuezhi Zhao
Keyword(s):  
Lie Group ◽  
Maximal Torus ◽  
Cohomology Ring ◽  
Minimal System ◽  
Schubert Calculus ◽  
Flag Manifolds ◽  
Generators And Relations ◽  
Integral Cohomology ◽  
Minimal System Of Generators ◽  
Connected Lie Group

Let $G$ be a compact connected Lie group with a maximal torus $T$. In the context of Schubert calculus we present the integral cohomology $H^{\ast }(G/T)$ by a minimal system of generators and relations.

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.

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.

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