2,167 research outputs found
Monoids of modules and arithmetic of direct-sum decompositions
Let be a (possibly noncommutative) ring and let be a class
of finitely generated (right) -modules which is closed under finite direct
sums, direct summands, and isomorphisms. Then the set
of isomorphism classes of modules is a commutative semigroup with operation
induced by the direct sum. This semigroup encodes all possible information
about direct sum decompositions of modules in . If the endomorphism
ring of each module in is semilocal, then is a Krull monoid. Although this fact was observed nearly a decade ago, the
focus of study thus far has been on ring- and module-theoretic conditions
enforcing that is Krull. If
is Krull, its arithmetic depends only on the class group of and the set of classes containing prime divisors. In this paper
we provide the first systematic treatment to study the direct-sum
decompositions of modules using methods from Factorization Theory of Krull
monoids. We do this when is the class of finitely generated
torsion-free modules over certain one- and two-dimensional commutative
Noetherian local rings.Comment: Pacific Journal of Mathematics, to appea
A Geometric Approach to the Problem of Unique Decomposition of Processes
This paper proposes a geometric solution to the problem of prime
decomposability of concurrent processes first explored by R. Milner and F.
Moller in [MM93]. Concurrent programs are given a geometric semantics using
cubical areas, for which a unique factorization theorem is proved. An effective
factorization method which is correct and complete with respect to the
geometric semantics is derived from the factorization theorem. This algorithm
is implemented in the static analyzer ALCOOL.Comment: 15 page
Toric varieties, monoid schemes and descent
We give conditions for the Mayer-Vietoris property to hold for the algebraic
K-theory of blow-up squares of toric varieties in any characteristic, using the
theory of monoid schemes. These conditions are used to relate algebraic
K-theory to topological cyclic homology in characteristic p. To achieve our
goals, we develop for monoid schemes many notions from classical algebraic
geometry, such as separated and proper maps.Comment: v2 changes: field of positive characteristic replaced by regular ring
containing such a field at appropriate places. Minor changes in expositio
Non-normal affine monoids
We give a geometric description of the set of holes in a non-normal affine
monoid . The set of holes turns out to be related to the non-trivial graded
components of the local cohomology of . From this, we see how various
properties of like local normality and Serre's conditions and
are encoded in the geometry of the holes. A combinatorial upper bound
for the depth the monoid algebra is obtained and some cases where
equality holds are identified. We apply this results to seminormal affine
monoids.Comment: 18 pages, 3 figures. Simplified proof of the main result, shortened.
An even shorter version appeared with the title "Non-normal affine monoid
algebra" in manuscripta mathematic
Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory
In this paper we characterize the congruence associated to the direct sum of
all irreducible representations of a finite semigroup over an arbitrary field,
generalizing results of Rhodes for the field of complex numbers. Applications
are given to obtain many new results, as well as easier proofs of several
results in the literature, involving: triangularizability of finite semigroups;
which semigroups have (split) basic semigroup algebras, two-sided semidirect
product decompositions of finite monoids; unambiguous products of rational
languages; products of rational languages with counter; and \v{C}ern\'y's
conjecture for an important class of automata
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