9,636 research outputs found
Amenability and geometry of semigroups
We study the connection between amenability, Følner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Følner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Følner condition is a left quasi-isometry invariant of finitely generated semigroups, and hence that left amenability is a left quasi-isometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Følner condition in terms of the existence of weak Følner sets satisfying a local injectivity condition on the relevant translation action of the semigroup
Maximal subgroups of free idempotent generated semigroups over the full linear monoid
We show that the rank r component of the free idempotent generated semigroup
of the biordered set of the full linear monoid of n x n matrices over a
division ring Q has maximal subgroup isomorphic to the general linear group
GL_r(Q), where n and r are positive integers with r < n/3.Comment: 37 pages; Transactions of the American Mathematical Society (to
appear). arXiv admin note: text overlap with arXiv:1009.5683 by other author
A strong geometric hyperbolicity property for directed graphs and monoids
We introduce and study a strong "thin triangle"' condition for directed
graphs, which generalises the usual notion of hyperbolicity for a metric space.
We prove that finitely generated left cancellative monoids whose right Cayley
graphs satisfy this condition must be finitely presented with polynomial Dehn
functions, and hence word problems in NP. Under the additional assumption of
right cancellativity (or in some cases the weaker condition of bounded
indegree), they also admit algorithms for more fundamentally
semigroup-theoretic decision problems such as Green's relations L, R, J, D and
the corresponding pre-orders.
In contrast, we exhibit a right cancellative (but not left cancellative)
finitely generated monoid (in fact, an infinite class of them) whose Cayley
graph is a essentially a tree (hence hyperbolic in our sense and probably any
reasonable sense), but which is not even recursively presentable. This seems to
be strong evidence that no geometric notion of hyperbolicity will be strong
enough to yield much information about finitely generated monoids in absolute
generality.Comment: Exposition improved. Results unchange
On Maximal Subgroups of Free Idempotent Generated Semigroups
We prove the following results: (1) Every group is a maximal subgroup of some
free idempotent generated semigroup. (2) Every finitely presented group is a
maximal subgroup of some free idempotent generated semigroup arising from a
finite semigroup. (3) Every group is a maximal subgroup of some free regular
idempotent generated semigroup. (4) Every finite group is a maximal subgroup of
some free regular idempotent generated semigroup arising from a finite regular
semigroup.Comment: 27 page
On residual finiteness of monoids, their SchĂźtzenberger groups and associated actions
RG was supported by an EPSRC Postdoctoral Fellowship EP/E043194/1 held at the University of St Andrews, Scotland.In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M ; (RFSG) residual finiteness of SchĂźtzenberger groups of M ; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table.PostprintPeer reviewe
Vector quantization
During the past ten years Vector Quantization (VQ) has developed from a theoretical possibility promised by Shannon's source coding theorems into a powerful and competitive technique for speech and image coding and compression at medium to low bit rates. In this survey, the basic ideas behind the design of vector quantizers are sketched and some comments made on the state-of-the-art and current research efforts
A Journal for the Astronomical Computing Community?
One of the Birds of a Feather (BoF) discussion sessions at ADASS XX
considered whether a new journal is needed to serve the astronomical computing
community. In this paper we discuss the nature and requirements of that
community, outline the analysis that led us to propose this as a topic for a
BoF, and review the discussion from the BoF session itself. We also present the
results from a survey designed to assess the suitability of astronomical
computing papers of different kinds for publication in a range of existing
astronomical and scientific computing journals. The discussion in the BoF
session was somewhat inconclusive, and it seems likely that this topic will be
debated again at a future ADASS or in a similar forum.Comment: 4 pages, no figures; to appear in proceedings of ADASS X
Topological finiteness properties of monoids. Part 1: Foundations
We initiate the study of higher dimensional topological finiteness properties
of monoids. This is done by developing the theory of monoids acting on CW
complexes. For this we establish the foundations of -equivariant homotopy
theory where is a discrete monoid. For projective -CW complexes we prove
several fundamental results such as the homotopy extension and lifting
property, which we use to prove the -equivariant Whitehead theorems. We
define a left equivariant classifying space as a contractible projective -CW
complex. We prove that such a space is unique up to -homotopy equivalence
and give a canonical model for such a space via the nerve of the right Cayley
graph category of the monoid. The topological finiteness conditions
left- and left geometric dimension are then defined for monoids
in terms of existence of a left equivariant classifying space satisfying
appropriate finiteness properties. We also introduce the bilateral notion of
-equivariant classifying space, proving uniqueness and giving a canonical
model via the nerve of the two-sided Cayley graph category, and we define the
associated finiteness properties bi- and geometric dimension. We
explore the connections between all of the these topological finiteness
properties and several well-studied homological finiteness properties of
monoids which are important in the theory of string rewriting systems,
including , cohomological dimension, and Hochschild
cohomological dimension. We also develop the corresponding theory of
-equivariant collapsing schemes (that is, -equivariant discrete Morse
theory), and among other things apply it to give topological proofs of results
of Anick, Squier and Kobayashi that monoids which admit presentations by
complete rewriting systems are left-, right- and bi-.Comment: 59 pages, 1 figur
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