458 research outputs found
Finitely presentable tree series
Tree height is known to be a non-recognizable series. In this paper, we detect two remarkable classes where this series belongs: that of polynomially presentable tree series and that of almost linearly presentable tree series. Both the above classes have nice closure properties, and seem to constitute the first levels of a tree series hierarchy which starts from the class of recognizable treeseries
Sound and complete axiomatizations of coalgebraic language equivalence
Coalgebras provide a uniform framework to study dynamical systems, including
several types of automata. In this paper, we make use of the coalgebraic view
on systems to investigate, in a uniform way, under which conditions calculi
that are sound and complete with respect to behavioral equivalence can be
extended to a coarser coalgebraic language equivalence, which arises from a
generalised powerset construction that determinises coalgebras. We show that
soundness and completeness are established by proving that expressions modulo
axioms of a calculus form the rational fixpoint of the given type functor. Our
main result is that the rational fixpoint of the functor , where is a
monad describing the branching of the systems (e.g. non-determinism, weights,
probability etc.), has as a quotient the rational fixpoint of the
"determinised" type functor , a lifting of to the category of
-algebras. We apply our framework to the concrete example of weighted
automata, for which we present a new sound and complete calculus for weighted
language equivalence. As a special case, we obtain non-deterministic automata,
where we recover Rabinovich's sound and complete calculus for language
equivalence.Comment: Corrected version of published journal articl
Infinite presentability of groups and condensation
We describe various classes of infinitely presented groups that are
condensation points in the space of marked groups. A well-known class of such
groups consists of finitely generated groups admitting an infinite minimal
presentation. We introduce here a larger class of condensation groups, called
infinitely independently presentable groups, and establish criteria which allow
one to infer that a group is infinitely independently presentable. In addition,
we construct examples of finitely generated groups with no minimal
presentation, among them infinitely presented groups with Cantor-Bendixson rank
1, and we prove that every infinitely presented metabelian group is a
condensation group.Comment: 32 pages, no figure. 1->2 Major changes (the 13-page first version,
authored by Y.C. and L.G., was entitled "On infinitely presented soluble
groups") 2->3 some changes including cuts in Section
On Cayley graphs of virtually free groups
In 1985, Dunwoody showed that finitely presentable groups are accessible.
Dunwoody's result was used to show that context-free groups, groups
quasi-isometric to trees or finitely presentable groups of asymptotic dimension
1 are virtually free. Using another theorem of Dunwoody of 1979, we study when
a group is virtually free in terms of its Cayley graph and we obtain new proofs
of the mentioned results and other previously depending on them
Distortion of wreath products in some finitely presented groups
Wreath products such as Z wr Z are not finitely-presentable yet can occur as
subgroups of finitely presented groups. Here we compute the distortion of Z wr
Z as a subgroup of Thompson's group F and as a subgroup of Baumslag's
metabelian group G.
We find that Z wr Z is undistorted in F but is at least exponentially
distorted in G.Comment: 9 pages, 5 figure
Proper Functors and Fixed Points for Finite Behaviour
The rational fixed point of a set functor is well-known to capture the
behaviour of finite coalgebras. In this paper we consider functors on algebraic
categories. For them the rational fixed point may no longer be fully abstract,
i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's
notion of a proper semiring, we introduce the notion of a proper functor. We
show that for proper functors the rational fixed point is determined as the
colimit of all coalgebras with a free finitely generated algebra as carrier and
it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor
is proper if and only if that colimit is a subcoalgebra of the final coalgebra.
These results serve as technical tools for soundness and completeness proofs
for coalgebraic regular expression calculi, e.g. for weighted automata
On the finite presentation of subdirect products and the nature of residually free groups
We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion
for the finite presentability of subdirect products of groups: if
are finitely presented and
projects to a subgroup of finite index in
each , then is finitely presentable, indeed there
is an algorithm that will construct a finite presentation for .
We use the VSP criterion to characterise the finitely presented residually
free groups. We prove that the class of such groups is recursively enumerable.
We describe an algorithm that, given a finite presentation of a residually free
group, constructs a canonical embedding into a direct product of finitely many
limit groups. We solve the (multiple) conjugacy problem and membership problem
for finitely presentable subgroups of residually free groups. We also prove
that there is an algorithm that, given a finite generating set for such a
subgroup, will construct a finite presentation.
New families of subdirect products of free groups are constructed, including
the first examples of finitely presented subgroups that are neither
nor of Stallings-Bieri typeComment: 44 pages. To appear in American Journal of Mathematics. This is a
substantial rewrite of our previous Arxiv article 0809.3704, taking into
account subsequent developments, advice of colleagues and referee's comment
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