2,078 research outputs found
Green's Relations in Finite Transformation Semigroups
We consider the complexity of Green's relations when the semigroup is given
by transformations on a finite set. Green's relations can be defined by
reachability in the (right/left/two-sided) Cayley graph. The equivalence
classes then correspond to the strongly connected components. It is not
difficult to show that, in the worst case, the number of equivalence classes is
in the same order of magnitude as the number of elements. Another important
parameter is the maximal length of a chain of components. Our main contribution
is an exponential lower bound for this parameter. There is a simple
construction for an arbitrary set of generators. However, the proof for
constant alphabet is rather involved. Our results also apply to automata and
their syntactic semigroups.Comment: Full version of a paper submitted to CSR 2017 on 2016-12-1
ad-Nilpotent ideals of a Borel subalgebra II
We provide an explicit bijection between the ad-nilpotent ideals of a Borel
subalgebra of a simple Lie algebra g and the orbits of \check{Q}/(h+1)\check{Q}
under the Weyl group (\check{Q} being the coroot lattice and h the Coxeter
number of g). From this result we deduce in a uniform way a counting formula
for the ad-nilpotent ideals.Comment: AMS-TeX file, 9 pages; revised version. To appear in Journal of
Algebr
Uniformity, Universality, and Computability Theory
We prove a number of results motivated by global questions of uniformity in
computability theory, and universality of countable Borel equivalence
relations. Our main technical tool is a game for constructing functions on free
products of countable groups.
We begin by investigating the notion of uniform universality, first proposed
by Montalb\'an, Reimann and Slaman. This notion is a strengthened form of a
countable Borel equivalence relation being universal, which we conjecture is
equivalent to the usual notion. With this additional uniformity hypothesis, we
can answer many questions concerning how countable groups, probability
measures, the subset relation, and increasing unions interact with
universality. For many natural classes of countable Borel equivalence
relations, we can also classify exactly which are uniformly universal.
We also show the existence of refinements of Martin's ultrafilter on Turing
invariant Borel sets to the invariant Borel sets of equivalence relations that
are much finer than Turing equivalence. For example, we construct such an
ultrafilter for the orbit equivalence relation of the shift action of the free
group on countably many generators. These ultrafilters imply a number of
structural properties for these equivalence relations.Comment: 61 Page
The subgroup identification problem for finitely presented groups
We introduce the subgroup identification problem, and show that there is a
finitely presented group G for which it is unsolvable, and that it is uniformly
solvable in the class of finitely presented locally Hopfian groups. This is
done as an investigation into the difference between strong and weak effective
coherence for finitely presented groups.Comment: 11 pages. This is the version submitted for publicatio
Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions
We attach to each finite bipartite separated graph (E,C) a partial dynamical
system (\Omega(E,C), F, \theta), where \Omega(E,C) is a zero-dimensional
metrizable compact space, F is a finitely generated free group, and {\theta} is
a continuous partial action of F on \Omega(E,C). The full crossed product
C*-algebra O(E,C) = C(\Omega(E,C)) \rtimes_{\theta} F is shown to be a
canonical quotient of the graph C*-algebra C^*(E,C) of the separated graph
(E,C). Similarly, we prove that, for any *-field K, the algebraic crossed
product L^{ab}_K(E,C) = C_K(\Omega(E,C)) \rtimes_\theta^{alg} F is a canonical
quotient of the Leavitt path algebra L_K(E,C) of (E,C). The monoid
V(L^{ab}_K(E,C)) of isomorphism classes of finitely generated projective
modules over L^{ab}_K(E,C) is explicitly computed in terms of monoids
associated to a canonical sequence of separated graphs. Using this, we are able
to construct an action of a finitely generated free group F on a
zero-dimensional metrizable compact space Z such that the type semigroup S(Z,
F, K) is not almost unperforated, where K denotes the algebra of clopen subsets
of Z. Finally we obtain a characterization of the separated graphs (E,C) such
that the canonical partial action of F on \Omega(E,C) is topologically free.Comment: Final version to appear in Advances in Mathematic
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