Expand all Collapse all | Results 1 - 14 of 14 |
1. CMB 2013 (vol 57 pp. 225)
Small Flag Complexes with Torsion We classify flag complexes on at most $12$ vertices with torsion in
the first homology group. The result is moderately computer-aided.
As a consequence we confirm a folklore conjecture that the smallest
poset whose order complex is homotopy equivalent to the real
projective plane (and also the smallest poset with torsion in the
first homology group) has exactly $13$ elements.
Keywords:clique complex, order complex, homology, torsion, minimal model Categories:55U10, 06A11, 55P40, 55-04, 05-04 |
2. CMB Online first
Left-orderable fundamental group and Dehn surgery on the knot $5_2$ We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is
the two-bridge knot corresponding to the rational number $3/7$, has left-orderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:left-ordering, Dehn surgery Categories:57M25, 06F15 |
3. CMB Online first
Left-orderable fundamental group and Dehn surgery on the knot $5_2$ We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is
the two-bridge knot corresponding to the rational number $3/7$, has left-orderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:left-ordering, Dehn surgery Categories:57M25, 06F15 |
4. CMB 2013 (vol 57 pp. 310)
Left-orderable Fundamental Group and Dehn Surgery on the Knot $5_2$ We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is
the two-bridge knot corresponding to the rational number $3/7$, has left-orderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:left-ordering, Dehn surgery Categories:57M25, 06F15 |
5. CMB 2012 (vol 56 pp. 850)
Left-orderability and Exceptional Dehn Surgery on Twist Knots We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil,
yields a $3$-manifold whose fundamental group is left-orderable.
This is a generalization of a result of Clay, Lidman and Watson, and
also gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.
Keywords:left-ordering, twist knot, Dehn surgery Categories:57M25, 06F15 |
6. CMB 2012 (vol 56 pp. 551)
Real Dimension Groups Dimension groups (not countable) that are also real ordered vector
spaces can be obtained as direct limits (over directed sets) of
simplicial real vector spaces (finite dimensional vector spaces with
the coordinatewise ordering), but the directed set is not as
interesting as one would like, i.e., it is not true that a
countable-dimensional real vector space that has interpolation can be
represented as such a direct limit over the a countable directed
set. It turns out this is the case when the group is additionally
simple, and it is shown that the latter have an ordered tensor product
decomposition. In the Appendix, we provide a huge class of polynomial
rings that, with a pointwise ordering, are shown to satisfy
interpolation, extending a result outlined by Fuchs.
Keywords:dimension group, simplicial vector space, direct limit, Riesz interpolation Categories:46A40, 06F20, 13J25, 19K14 |
7. CMB 2011 (vol 54 pp. 277)
Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order |
Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order Let $L$ be a finite distributive lattice. Let
$\operatorname{Sub}_0(L)$ be the lattice
$$
\{S\mid S\text{ is a sublattice of }L\}\cup\{\emptyset\}
$$
and let $\ell_*[\operatorname{Sub}_0(L)]$ be the length of the shortest maximal chain in $\operatorname{Sub}_0(L)$. It is proved that if $K$ and $L$ are non-trivial finite distributive lattices, then
$$
\ell_*[\operatorname{Sub}_0(K\times L)]=\ell_*[\operatorname{Sub}_0(K)]+\ell_*[\operatorname{Sub}_0(L)].
$$
A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved.
Keywords:(distributive) lattice, maximal sublattice, (partially) ordered set Categories:06D05, 06D50, 06A07 |
8. CMB 2010 (vol 54 pp. 193)
Measurements and $G_\delta$-Subsets of Domains
In this paper we study domains, Scott
domains, and the existence of measurements. We
use a space created by D.~K. Burke to show that
there is a Scott domain $P$ for which $\max(P)$ is
a $G_\delta$-subset of $P$ and yet no measurement
$\mu$ on $P$ has $\ker(\mu) = \max(P)$. We also
correct a mistake in the literature asserting that
$[0, \omega_1)$ is a space of this type. We show
that if $P$ is a Scott domain and $X \subseteq
\max(P)$ is a $G_\delta$-subset of $P$, then $X$
has a $G_\delta$-diagonal and is weakly
developable. We show that if $X \subseteq
\max(P)$ is a $G_\delta$-subset of $P$, where
$P$ is a domain but perhaps not a Scott domain,
then $X$ is domain-representable,
first-countable, and is the union of dense,
completely metrizable subspaces. We also
show that there is a domain $P$ such that
$\max(P)$ is the usual space of countable
ordinals and is a $G_\delta$-subset of $P$ in
the Scott topology. Finally we show that the
kernel of a measurement on a Scott domain can
consistently be a normal, separable,
non-metrizable Moore space.
Keywords:domain-representable, Scott-domain-representable, measurement, Burke's space, developable spaces, weakly developable spaces, $G_\delta$-diagonal, Äech-complete space, Moore space, $\omega_1$, weakly developable space, sharp base, AF-complete Categories:54D35, 54E30, 54E52, 54E99, 06B35, 06F99 |
9. CMB 2010 (vol 54 pp. 381)
A Short Note on the Higher Level Version of the Krull--Baer Theorem
Klep and Velu\v{s}\v{c}ek generalized the Krull--Baer theorem for
higher level preorderings to the non-commutative setting. A $n$-real valuation
$v$ on a skew field $D$ induces a group homomorphism $\overline{v}$. A section
of $\overline{v}$ is a crucial ingredient of the construction of a complete
preordering on the base field $D$ such that its projection on the residue skew
field $k_v$ equals the given level $1$ ordering on $k_v$. In the article we give
a proof of the existence of the section of $\overline{v}$, which was left as an
open problem by Klep and Velu\v{s}\v{c}ek, and thus
complete the generalization of the Krull--Baer theorem for preorderings.
Keywords:orderings of higher level, division rings, valuations Categories:14P99, 06Fxx |
10. CMB 2009 (vol 52 pp. 598)
Numerical Semigroups That Are Not Intersections of $d$-Squashed Semigroups We say that a numerical semigroup is \emph{$d$-squashed} if it can
be written in the form
$$ S=\frac 1 N \langle a_1,\dots,a_d \rangle \cap \mathbb{Z}$$
for $N,a_1,\dots,a_d$ positive integers with
$\gcd(a_1,\dots, a_d)=1$.
Rosales and Urbano have shown that a numerical semigroup is
2-squashed if and only if it is proportionally modular.
Recent works by Rosales \emph{et al.} give a concrete example of a
numerical semigroup that cannot be written as an intersection of
$2$-squashed semigroups. We will show the existence of infinitely
many numerical semigroups that cannot be written as an
intersection of $2$-squashed semigroups. We also will prove the
same result for $3$-squashed semigroups. We conjecture that there
are numerical semigroups that cannot be written as the
intersection of $d$-squashed semigroups for any fixed $d$, and we
prove some partial results towards this conjecture.
Keywords:numerical semigroup, squashed semigroup, proportionally modular semigroup Categories:20M14, 06F05, 46L80 |
11. CMB 2007 (vol 50 pp. 182)
On the Coxeter Transformations for Tamari Posets A relation between the anticyclic structure of the dendriform operad
and the Coxeter transformations in the Grothendieck groups of the
derived categories of modules over the Tamari posets is obtained.
Categories:18D50, 18E30, 06A11 |
12. CMB 2004 (vol 47 pp. 191)
Congruence Class Sizes in Finite Sectionally Complemented Lattices The congruences of a finite sectionally complemented lattice $L$ are
not necessarily \emph{uniform} (any two congruence classes of a
congruence are of the same size). To measure how far a congruence
$\Theta$ of $L$ is from being uniform, we introduce $\Spec\Theta$, the
\emph{spectrum} of $\Theta$, the family of cardinalities of the
congruence classes of $\Theta$. A typical result of this paper
characterizes the spectrum $S = (m_j \mid j < n)$ of a nontrivial
congruence $\Theta$ with the following two properties:
\begin{enumerate}[$(S_2)$]
\item[$(S_1)$] $2 \leq n$ and $n \neq 3$.
\item[$(S_2)$] $2 \leq m_j$ and $m_j \neq 3$, for all $j Keywords:congruence lattice, congruence-preserving extension Categories:06B10, 06B15 |
13. CMB 1998 (vol 41 pp. 290)
Congruence lattices of finite semimodular lattices We prove that every finite distributive lattice can be represented
as the congruence lattice of a finite (planar) {\it semimodular}
lattice.
Categories:06B10, 08A05 |
14. CMB 1997 (vol 40 pp. 39)
On projective $Z$-frames This paper deals with the projective objects in the category of all
$Z$-frames, where the latter is a common generalization of
different types of frames. The main result obtained here is that a
$Z$-frame is ${\bf E}$-projective if and only if it is stably
$Z$-continuous, for a naturally arising collection ${\bf E}$ of morphisms.
Categories:06D05, 54D10, 18D15 |