100 research outputs found
Factorization invariants in half-factorial affine semigroups
Let be the monoid generated by We introduce the
homogeneous catenary degree of as the smallest with the following property: for each and any two factorizations of
, there exists factorizations of such that, for every where is the
usual distance between factorizations, and the length of is less than or equal to
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 -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.Comment: 8 pages, 1 figur
numericalsgps, a GAP package for numerical semigroups
The package numericalsgps performs computations with and for numerical and affine semigroups. This manuscript is a survey of what the package does, and at the same time intends to gather the trending topics on numerical semigroups
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
Realizable sets of catenary degrees of numerical monoids
The catenary degree is an invariant that measures the distance between
factorizations of elements within an atomic monoid. In this paper, we classify
which finite subsets of occur as the set of catenary
degrees of a numerical monoid (i.e., a co-finite, additive submonoid of
). In particular, we show that, with one exception, every
finite subset of that can possibly occur as the set of
catenary degrees of some atomic monoid is actually achieved by a numerical
monoid
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