842 research outputs found
An infinite natural sum
As far as algebraic properties are concerned, the usual addition on the class
of ordinal numbers is not really well behaved; for example, it is not
commutative, nor left cancellative etc. In a few cases, the natural Hessemberg
sum is a better alternative, since it shares most of the usual properties of
the addition on the naturals.
A countably infinite version of the natural sum has been used in a recent
paper by V\"a\"an\"anen and Wang, with applications to infinitary logics. We
provide an order theoretical characterization of this operation. We show that
this countable natural sum differs from the more usual infinite ordinal sum
only for an initial finite "head" and agrees on the remaining infinite "tail".
We show how to evaluate the countable natural sum just by computing a finite
natural sum. Various kinds of infinite mixed sums of ordinals are discussed.Comment: v3 added a remark connected with surreal number
Circle Decompositions of Surfaces
We determine which connected surfaces can be partitioned into topological
circles. There are exactly seven such surfaces up to homeomorphism: those of
finite type, of Euler characteristic zero, and with compact boundary
components. As a byproduct, we get that any circle decomposition of a surface
is upper semicontinuous.Comment: 9 pages, final version, to appear in Topology and its Applications
(2010). A few missprints have been correcte
The Bergman-Shelah Preorder on Transformation Semigroups
This is the peer-reviewed version of the following article: Mesyan, Z., Mitchell, J. D., Morayne, M. and Péresse, Y. H. (2012), Mathematical Logic Quarterly, Vol. 58: 424–433, 'The Bergman-Shelah preorder on transformation semigroups', which has been published in final form at doi:10.1002/malq.201200002. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Copyright © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. http://www.interscience.wiley.com/Let \nat^\nat be the semigroup of all mappings on the natural numbers \nat, and let and be subsets of \nat^\nat. We write if there exists a countable subset of \nat^\nat such that is contained in the subsemigroup generated by and . We give several results about the structure of the preorder . In particular, we show that a certain statement about this preorder is equivalent to the Continuum Hypothesis. The preorder is analogous to one introduced by Bergman and Shelah on subgroups of the symmetric group on \nat. The results in this paper suggest that the preorder on subsemigroups of \nat^\nat is much more complicated than that on subgroups of the symmetric group.Peer reviewe
Homotopy classes that are trivial mod F
If F is a collection of topological spaces, then a homotopy class \alpha in
[X,Y] is called F-trivial if \alpha_* = 0: [A,X] --> [A,Y] for all A in F. In
this paper we study the collection Z_F(X,Y) of all F-trivial homotopy classes
in [X,Y] when F = S, the collection of spheres, F = M, the collection of Moore
spaces, and F = \Sigma, the collection of suspensions. Clearly Z_\Sigma (X,Y)
\subset Z_M(X,Y) \subset Z_S(X,Y), and we find examples of finite complexes X
and Y for which these inclusions are strict. We are also interested in Z_F(X) =
Z_F(X,X), which under composition has the structure of a semigroup with zero.
We show that if X is a finite dimensional complex and F = S, M or \Sigma, then
the semigroup Z_F(X) is nilpotent. More precisely, the nilpotency of Z_F(X) is
bounded above by the F-killing length of X, a new numerical invariant which
equals the number of steps it takes to make X contractible by successively
attaching cones on wedges of spaces in F, and this in turn is bounded above by
the F-cone length of X. We then calculate or estimate the nilpotency of Z_F(X)
when F = S, M or \Sigma for the following classes of spaces: (1) projective
spaces (2) certain Lie groups such as SU(n) and Sp(n). The paper concludes with
several open problems.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-19.abs.html Version 2:
reference to Christensen adde
Isometry groups among topological groups
It is shown that a topological group G is topologically isomorphic to the
isometry group of a (complete) metric space iff G coincides with its
G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete).
It is also shown that for every Polish (resp. compact Polish; locally compact
Polish) group G there is a complete (resp. proper) metric d on X inducing the
topology of X such that G is isomorphic to Iso(X,d) where X = l_2 (resp. X = Q;
X = Q\{point} where Q is the Hilbert cube). It is demonstrated that there are a
separable Banach space E and a nonzero vector e in E such that G is isomorphic
to the group of all (linear) isometries of E which leave the point e fixed.
Similar results are proved for an arbitrary complete topological group.Comment: 30 page
Making triangulations 4-connected using flips
We show that any combinatorial triangulation on n vertices can be transformed
into a 4-connected one using at most floor((3n - 9)/5) edge flips. We also give
an example of an infinite family of triangulations that requires this many
flips to be made 4-connected, showing that our bound is tight. In addition, for
n >= 19, we improve the upper bound on the number of flips required to
transform any 4-connected triangulation into the canonical triangulation (the
triangulation with two dominant vertices), matching the known lower bound of 2n
- 15. Our results imply a new upper bound on the diameter of the flip graph of
5.2n - 33.6, improving on the previous best known bound of 6n - 30.Comment: 22 pages, 8 figures. Accepted to CGTA special issue for CCCG 2011.
Conference version available at
http://2011.cccg.ca/PDFschedule/papers/paper34.pd
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