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.

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 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.

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 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 Affine semigroups over an arbitrary field

1965 ◽  
Vol 7 (2) ◽  
pp. 80-92 ◽  
Author(s):  
W. Edwin Clark
Keyword(s):  
Linear Subspace ◽  
Arbitrary Field ◽  
Dimensional Vector ◽  
Multiplicative Semigroup ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Affine Semigroups ◽  
Affine Semigroup ◽  
Linear Variety

Let ℒ V denote the algebra of all linear transformations on an n-dimensional vector space V over a field Φ. A subsemigroup S of the multiplicative semigroup of ℒ V will be said to be an affine semigroup over Φ if S is a linear variety, i.e., a translate of a linear subspace of ℒ V.This concept in a somewhat different form was introduced and studied by Haskell Cohen and H. S. Collins [1]. In an appendix we give their definition and outline a method of describing possibly infinite dimensional affine semigroups in terms of algebras and supplemented algebras.

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