289 research outputs found
Factorization theory: From commutative to noncommutative settings
We study the non-uniqueness of factorizations of non zero-divisors into atoms
(irreducibles) in noncommutative rings. To do so, we extend concepts from the
commutative theory of non-unique factorizations to a noncommutative setting.
Several notions of factorizations as well as distances between them are
introduced. In addition, arithmetical invariants characterizing the
non-uniqueness of factorizations such as the catenary degree, the
-invariant, and the tame degree, are extended from commutative to
noncommutative settings. We introduce the concept of a cancellative semigroup
being permutably factorial, and characterize this property by means of
corresponding catenary and tame degrees. Also, we give necessary and sufficient
conditions for there to be a weak transfer homomorphism from a cancellative
semigroup to its reduced abelianization. Applying the abstract machinery we
develop, we determine various catenary degrees for classical maximal orders in
central simple algebras over global fields by using a natural transfer
homomorphism to a monoid of zero-sum sequences over a ray class group. We also
determine catenary degrees and the permutable tame degree for the semigroup of
non zero-divisors of the ring of upper triangular matrices over a
commutative domain using a weak transfer homomorphism to a commutative
semigroup.Comment: 45 page
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
Partial monoids: associativity and confluence
A partial monoid is a set with a partial multiplication (and
total identity ) which satisfies some associativity axiom. The partial
monoid may be embedded in a free monoid and the product is
simulated by a string rewriting system on that consists in evaluating the
concatenation of two letters as a product in , when it is defined, and a
letter as the empty word . In this paper we study the profound
relations between confluence for such a system and associativity of the
multiplication. Moreover we develop a reduction strategy to ensure confluence
and which allows us to define a multiplication on normal forms associative up
to a given congruence of . Finally we show that this operation is
associative if, and only if, the rewriting system under consideration is
confluent
Relative dimension of morphisms and dimension for algebraic stacks
Motivated by applications in moduli theory, we introduce a flexible and
powerful language for expressing lower bounds on relative dimension of
morphisms of schemes, and more generally of algebraic stacks. We show that the
theory is robust and applies to a wide range of situations. Consequently, we
obtain simple tools for making dimension-based deformation arguments on moduli
spaces. Additionally, in a complementary direction we develop the basic
properties of codimension for algebraic stacks. One of our goals is to provide
a comprehensive toolkit for working transparently with dimension statements in
the context of algebraic stacks.Comment: 21 page
Factorization invariants in numerical monoids
Nonunique factorization in commutative monoids is often studied using
factorization invariants, which assign to each monoid element a quantity
determined by the factorization structure. For numerical monoids (co-finite,
additive submonoids of the natural numbers), several factorization invariants
have received much attention in the recent literature. In this survey article,
we give an overview of the length set, elasticity, delta set,
-primality, and catenary degree invariants in the setting of numerical
monoids. For each invariant, we present current major results in the literature
and identify the primary open questions that remain
A Characteristic Averaging Property of the Catenary
It is well-known that the catenary is characterized by an extremal centroidal condition: It is the shape of the curve whose centroid is the lowest among all curves having a prescribed length and specified endpoints. Here, we establish a broad characteristic averaging property of the centenary that yields two new centroidal characterizations
Extremal periodic wave profiles
As a contribution to deterministic investigations into extreme fluid surface waves, in this paper wave profiles of prescribed period that have maximal crest height will be investigated. As constraints the values of the momentum and energy integrals are used in a simplified description with the KdV model. The result is that at the boundary of the feasible region in the momentum-energy plane, the only possible profiles are the well known cnoidal wave profiles. Inside the feasible region the extremal profiles of maximal crest height are ¿cornered¿ cnoidal profiles: cnoidal profiles of larger period, cut-off and periodically continued with the prescribed period so that at the maximal crest height a corner results
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