23,840 research outputs found
Homomorphisms on infinite direct products of groups, rings and monoids
We study properties of a group, abelian group, ring, or monoid which (a)
guarantee that every homomorphism from an infinite direct product
of objects of the same sort onto factors through the direct product of
finitely many ultraproducts of the (possibly after composition with the
natural map or some variant), and/or (b) guarantee that when a
map does so factor (and the index set has reasonable cardinality), the
ultrafilters involved must be principal.
A number of open questions, and topics for further investigation, are noted.Comment: 26 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be
updated more frequently than arXiv copy. Version 2 has minor revisions in
wording etc. from version
On the rational subset problem for groups
We use language theory to study the rational subset problem for groups and
monoids. We show that the decidability of this problem is preserved under graph
of groups constructions with finite edge groups. In particular, it passes
through free products amalgamated over finite subgroups and HNN extensions with
finite associated subgroups. We provide a simple proof of a result of
Grunschlag showing that the decidability of this problem is a virtual property.
We prove further that the problem is decidable for a direct product of a group
G with a monoid M if and only if membership is uniformly decidable for
G-automata subsets of M. It follows that a direct product of a free group with
any abelian group or commutative monoid has decidable rational subset
membership.Comment: 19 page
A survey on spectral multiplicities of ergodic actions
Given a transformation of a standard measure space , let \Cal
M(T) denote the set of spectral multiplicities of the Koopman operator
defined in by . It is discussed in
this survey paper which subsets of are realizable as
\Cal M(T) for various : ergodic, weakly mixing, mixing, Gaussian, Poisson,
ergodic infinite measure preserving, etc. The corresponding constructions are
considered in detail. Generalizations to actions of Abelian locally compact
second countable groups are also discussed
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