3,153 research outputs found
The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length
Theories of classification distinguish classes with some good structure
theorem from those for which none is possible. Some classes (dense linear
orders, for instance) are non-classifiable in general, but are classifiable
when we consider only countable members. This paper explores such a notion for
classes of computable structures by working out a sequence of examples.
We follow recent work by Goncharov and Knight in using the degree of the
isomorphism problem for a class to distinguish classifiable classes from
non-classifiable. In this paper, we calculate the degree of the isomorphism
problem for Abelian -groups of bounded Ulm length. The result is a sequence
of classes whose isomorphism problems are cofinal in the hyperarithmetical
hierarchy. In the process, new back-and-forth relations on such groups are
calculated.Comment: 15 page
The Classification of -Modules with Partial Decomposition Bases in
Ulm's Theorem presents invariants that classify countable abelian torsion
groups up to isomorphism. Barwise and Eklof extended this result to the
classification of arbitrary abelian torsion groups up to -equivalence. In this paper, we extend this classification to a class
of mixed -modules which includes all Warfield modules and is
closed under -equivalence. The defining property of these
modules is the existence of what we call a partial decomposition basis, a
generalization of the concept of decomposition basis. We prove a complete
classification theorem in using invariants deduced from the
classical Ulm and Warfield invariants
Degeneration and orbits of tuples and subgroups in an Abelian group
A tuple (or subgroup) in a group is said to degenerate to another if the
latter is an endomorphic image of the former. In a countable reduced abelian
group, it is shown that if tuples (or finite subgroups) degenerate to each
other, then they lie in the same automorphism orbit. The proof is based on
techniques that were developed by Kaplansky and Mackey in order to give an
elegant proof of Ulm's theorem. Similar results hold for reduced countably
generated torsion modules over principal ideal domains. It is shown that the
depth and the description of atoms of the resulting poset of orbits of tuples
depend only on the Ulm invariants of the module in question (and not on the
underlying ring). A complete description of the poset of orbits of elements in
terms of the Ulm invariants of the module is given. The relationship between
this description of orbits and a very different-looking one obtained by Dutta
and Prasad for torsion modules of bounded order is explained.Comment: 13 pages, 1 figur
Canonical stratifications along bisheaves
A theory of bisheaves has been recently introduced to measure the homological
stability of fibers of maps to manifolds. A bisheaf over a topological space is
a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks
of the sheaf to the stalks of the cosheaf. In this note we describe how, given
a bisheaf constructible (i.e., locally constant) with respect to a
triangulation of its underlying space, one can explicitly determine the
coarsest stratification of that space for which the bisheaf remains
constructible.Comment: 10 pages; this is the Final Version which appeared in the Proceedings
of the 2018 Abel Symposium on Topological Data Analysi
Almost free splitters
Let R be a subring of the rationals. We want to investigate self splitting
R-modules G that is Ext_R(G,G)=0 holds. For simplicity we will call such
modules splitters. Our investigation continues math.LO/9910159. In
math.LO/9910159, we answered an open problem by constructing a large class of
splitters. Classical splitters are free modules and torsion-free, algebraically
compact ones. In math.LO/9910159 we concentrated on splitters which are larger
then the continuum and such that countable submodules are not necessarily free.
The `opposite' case of aleph_1-free splitters of cardinality less or equal to
aleph_1 was singled out because of basically different techniques. This is the
target of the present paper. If the splitter is countable, then it must be free
over some subring of the rationals by a result of Hausen. We can show that all
aleph_1-free splitters of cardinality aleph_1 are free indeed
Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces
We define an invariant of welded virtual knots from each finite crossed
module by considering crossed module invariants of ribbon knotted surfaces
which are naturally associated with them. We elucidate that the invariants
obtained are non trivial by calculating explicit examples. We define welded
virtual graphs and consider invariants of them defined in a similar way.Comment: New results. A perfected version will appear in Compositio
Mathematic
- …