1. CMB 2017 (vol 60 pp. 830)
 Motegi, Kimihiko; Teragaito, Masakazu

Generalized Torsion Elements and Biorderability of 3manifold Groups
It is known that a biorderable group has no generalized torsion
element,
but the converse does not hold in general.
We conjecture that the converse holds for the fundamental groups
of $3$manifolds,
and verify the conjecture for nonhyperbolic, geometric $3$manifolds.
We also confirm the conjecture for some infinite families of
closed hyperbolic $3$manifolds.
In the course of the proof,
we prove that each standard generator of the Fibonacci group
$F(2, m)$ ($m \gt 2$) is a generalized torsion element.
Keywords:generalized torsion element, biordering, 3manifold group Categories:57M25, 57M05, 06F15, 20F05 

2. CMB 2017 (vol 60 pp. 372)
 Rao, M. Sambasiva

Coaxer Lattices
The notion of coaxers is introduced in a pseudocomplemented
distributive lattice. Boolean algebras are characterized in terms
of coaxer ideals and congruences. The concept of coaxer lattices
is introduced in pseudocomplemented distributive lattices and
characterized in terms of coaxer ideals and maximal ideals. Finally,
the coaxer lattices are also characterized in topological terms.
Keywords:pseudocomplemented distributive lattice, coaxer ideal, coaxer lattice, maximal ideal, congruence, kernel, antikernel Category:06D99 

3. CMB 2017 (vol 60 pp. 246)
 Bhuniya, Anjan Kumar; Hansda, Kalyan

On Radicals of Green's Relations in Ordered Semigroups
In this paper, we give a new definition of radicals of Green's
relations in an ordered semigroup and characterize left regular
(right regular), intra regular ordered semigroups by radicals
of Green's relations. Also we characterize the ordered semigroups
which are unions and complete semilattices of tsimple ordered
semigroups.
Keywords:radical of Green's relation, intra regular ordered semigroup, left regular, tsimple ordered semigroup Category:06F05 

4. CMB 2016 (vol 59 pp. 472)
 Clay, Adam; Desmarais, Colin; Naylor, Patrick

Testing Biorderability of Knot Groups
We investigate the biorderability of twobridge knot groups
and the groups of knots with 12 or fewer crossings by applying
recent theorems of Chiswell, Glass and Wilson.
Amongst all knots with 12 or fewer crossings (of which there
are 2977), previous theorems were only able to determine biorderability
of 499 of the corresponding knot groups. With our methods we
are able to deal with 191 more.
Keywords:knots, fundamental groups, orderable groups Categories:57M25, 57M27, 06F15 

5. CMB 2013 (vol 57 pp. 225)
 Adamaszek, Michał

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 computeraided.
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, 5504, 0504 

6. CMB Online first


Leftorderable 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 twobridge knot corresponding to the rational number $3/7$, has leftorderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:leftordering, Dehn surgery Categories:57M25, 06F15 

7. CMB Online first


Leftorderable 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 twobridge knot corresponding to the rational number $3/7$, has leftorderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:leftordering, Dehn surgery Categories:57M25, 06F15 

8. CMB 2013 (vol 57 pp. 310)
9. CMB 2012 (vol 56 pp. 850)
 Teragaito, Masakazu

Leftorderability and Exceptional Dehn Surgery on Twist Knots
We show that any exceptional nontrivial Dehn surgery on a twist knot, except the trefoil,
yields a $3$manifold whose fundamental group is leftorderable.
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:leftordering, twist knot, Dehn surgery Categories:57M25, 06F15 

10. CMB 2012 (vol 56 pp. 551)
 Handelman, David

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
countabledimensional 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 

11. CMB 2011 (vol 54 pp. 277)
 Farley, Jonathan David

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 nontrivial 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 

12. CMB 2010 (vol 54 pp. 193)
 Bennett, Harold; Lutzer, David

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 domainrepresentable,
firstcountable, 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,
nonmetrizable Moore space.
Keywords:domainrepresentable, Scottdomainrepresentable, measurement, Burke's space, developable spaces, weakly developable spaces, $G_\delta$diagonal, Äechcomplete space, Moore space, $\omega_1$, weakly developable space, sharp base, AFcomplete Categories:54D35, 54E30, 54E52, 54E99, 06B35, 06F99 

13. CMB 2010 (vol 54 pp. 381)
 Velušček, Dejan

A Short Note on the Higher Level Version of the KrullBaer Theorem
Klep and Velu\v{s}\v{c}ek generalized the KrullBaer theorem for
higher level preorderings to the noncommutative 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 KrullBaer theorem for preorderings.
Keywords:orderings of higher level, division rings, valuations Categories:14P99, 06Fxx 

14. CMB 2009 (vol 52 pp. 598)
 Moreno, M. A.; Nicola, J.; Pardo, E.; Thomas, H.

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
2squashed 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 

15. CMB 2007 (vol 50 pp. 182)
16. CMB 2004 (vol 47 pp. 191)
 Grätzer, G.; Schmidt, E. T.

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, congruencepreserving extension Categories:06B10, 06B15 

17. CMB 1998 (vol 41 pp. 290)
18. CMB 1997 (vol 40 pp. 39)
 Zhao, Dongsheng

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 
