Narrow Results

Let be the monoid generated by We introduce the homogeneous catenary degree of as the smallest N ∈ ℕ with the following property: for each and any two factorizations u, v of a, there exist factorizations u = w₁,…,w_t = v of a such that, for every k, d(w_k,w_k+1) ≤ N, where d is the usual distance between factorizations, and the length of w_k, |w_k|, is less than or equal to max{|u|, |v|}. We prove that the homogeneous catenary degree of 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.

We study the relationship between numerical semigroups and periodic subadditive functions.

We characterize affine semigroups having one Betti element and we compute some relevant non-unique factorization invariants for these semigroups. As an example, we particularize our description to numerical semigroups.

This work introduces a new kind of semigroup of ${\mathbb{N}}^p$ 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 $p=2$. 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.

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.