1. CMB Online first
 Jensen, Gerd; Pommerenke, Christian

On the structure of the Schild group in Relativity Theory
Alfred Schild has established conditions
that Lorentz transformations map worldvectors $(ct,x,y,z)$ with
integer coordinates onto vectors of the same kind. These transformations
are called integral Lorentz transformations.
The present paper contains supplements to
our earlier work
with a new focus on group theory. To relate the results to the
familiar matrix group nomenclature we associate Lorentz transformations
with matrices in $\mathrm{SL}(2,\mathbb{C})$. We consider the
lattice of subgroups of the group originated in Schild's paper
and obtain generating sets for the full group and its subgroups.
Keywords:Lorentz transformation, integer lattice, Gaussian integers, Schild group, subgroup Categories:22E43, 20H99, 83A05 

2. CMB Online first
 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 Online first
 Haase, Christian; Hofmann, Jan

Convexnormal (pairs of) polytopes
In 2012 Gubeladze (Adv. Math. 2012)
introduced the notion of $k$convexnormal polytopes to show
that
integral polytopes all of whose edges are longer than $4d(d+1)$
have
the integer decomposition property.
In the first part of this paper we show that for lattice polytopes
there is no difference between $k$ and $(k+1)$convexnormality
(for
$k\geq 3 $) and improve the bound to $2d(d+1)$. In the second
part we
extend the definition to pairs of polytopes. Given two rational
polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement
of
the normal fan of $Q$.
If every edge $e_P$ of $P$ is at least $d$ times as long as the
corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap
\mathbb{Z}^d
= (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$.
Keywords:integer decomposition property, integrally closed, projectively normal, lattice polytopes Categories:52B20, 14M25, 90C10 

4. CMB 2016 (vol 59 pp. 564)
 Li, Boyu

Normal Extensions of Representations of Abelian Semigroups
A commuting family of subnormal operators need
not have a commuting normal extension. We study when a representation
on an abelian semigroup can be extended to a normal representation,
and show that it suffices to extend the set of generators to
commuting normals. We also extend a result due to Athavale to
representations on abelian lattice ordered semigroups.
Keywords:subnormal operator, normal extension, regular dilation, lattice ordered semigroup Categories:47B20, 47A20, 47D03 

5. CMB 2015 (vol 59 pp. 123)
 Jensen, Gerd; Pommerenke, Christian

Discrete Spacetime and Lorentz Transformations
Alfred Schild has established conditions
that Lorentz transformations map worldvectors $(ct,x,y,z)$ with
integer coordinates onto vectors of the same kind. The problem
was dealt with in the context of tensor and spinor calculus.
Due to Schild's numbertheoretic arguments, the subject is also
interesting when isolated from its physical background.
The paper of Schild is not easy to understand. Therefore we first
present a streamlined version of his proof which is based on
the use of null vectors. Then we present a purely algebraic proof
that is somewhat shorter. Both proofs rely on the properties
of Gaussian integers.
Keywords:Lorentz transformation, integer lattice, Gaussian integers Categories:22E43, 20H99, 83A05 

6. CMB 2014 (vol 58 pp. 182)
7. CMB 2014 (vol 57 pp. 277)
 Elkholy, A. M.; ElLatif, M. H. Abd

On Mutually $m$permutable Product of Smooth Groups
Let $G$ be a
finite group and $H$, $K$ two subgroups of G. A group $G$ is said to
be a mutually mpermutable product of $H$ and $K$ if $G=HK$ and
every maximal subgroup of $H$ permutes with $K$ and every maximal
subgroup of $K$ permutes with $H$. In this paper, we investigate the
structure of a finite group which is a mutually mpermutable product
of two subgroups under the assumption that its maximal subgroups are
totally smooth.
Keywords:permutable subgroups, $m$permutable, smooth groups, subgroup lattices Categories:20D10, 20D20, 20E15, 20F16 

8. CMB 2012 (vol 57 pp. 132)
 Mubeena, T.; Sankaran, P.

Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups
Given a group automorphism $\phi:\Gamma\longrightarrow \Gamma$, one has
an action of $\Gamma$ on itself by $\phi$twisted conjugacy, namely, $g.x=gx\phi(g^{1})$.
The orbits of this action are called $\phi$twisted conjugacy classes. One says
that $\Gamma$ has the $R_\infty$property if there are infinitely many $\phi$twisted conjugacy
classes for every automorphism $\phi$ of $\Gamma$. In this paper we
show that $\operatorname{SL}(n,\mathbb{Z})$ and its
congruence subgroups have the $R_\infty$property. Further we show that
any (countable) abelian extension of $\Gamma$ has the $R_\infty$property where $\Gamma$ is a torsion
free nonelementary hyperbolic group, or $\operatorname{SL}(n,\mathbb{Z}),
\operatorname{Sp}(2n,\mathbb{Z})$ or a principal congruence
subgroup of $\operatorname{SL}(n,\mathbb{Z})$ or the fundamental group of a complete Riemannian
manifold of constant negative curvature.
Keywords:twisted conjugacy classes, hyperbolic groups, lattices in Lie groups Category:20E45 

9. CMB 2011 (vol 56 pp. 659)
 Yu, ZhiXian; Mei, Ming

Asymptotics and Uniqueness of Travelling Waves for NonMonotone Delayed Systems on 2D Lattices
We establish asymptotics and uniqueness (up
to translation) of travelling waves for delayed 2D lattice equations
with nonmonotone birth functions. First, with the help of
Ikehara's Theorem, the a priori asymptotic behavior of
travelling wave is exactly derived. Then, based on the obtained
asymptotic behavior, the uniqueness of the traveling waves is
proved. These results complement earlier results in the literature.
Keywords:2D lattice systems, traveling waves, asymptotic behavior, uniqueness, nonmonotone nonlinearity Category:35K57 

10. CMB 2011 (vol 54 pp. 645)
 Flores, André Luiz; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega

An Extension of Craig's Family of Lattices
Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$th root of
unity. The lattices in Craig's family are $(p1)$dimensional and
are geometrical representations of the integral $\mathbb
Z[\zeta_p]$ideals $\langle 1\zeta_p \rangle^i$, where $i$ is a
positive integer. This lattice construction technique is a powerful
one. Indeed, in dimensions $p1$ where $149 \leq p \leq 3001$,
Craig's lattices are the densest packings known. Motivated by this,
we construct $(p1)(q1)$dimensional lattices from the integral
$\mathbb Z[\zeta _{pq}]$ideals $\langle 1\zeta_p \rangle^i \langle
1\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$
and $j$ are positive integers. In terms of spherepacking density,
the new lattices and those in Craig's family have the same
asymptotic behavior. In conclusion, Craig's family is greatly
extended while preserving its spherepacking properties.
Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields Categories:11H31, 11H55, 11H50, 11R18, 11R04 

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 2008 (vol 51 pp. 15)
13. 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 

14. CMB 2002 (vol 45 pp. 483)
 Baake, Michael

Diffraction of Weighted Lattice Subsets
A Dirac comb of point measures in Euclidean space with bounded
complex weights that is supported on a lattice $\varGamma$ inherits
certain general properties from the lattice structure. In
particular, its autocorrelation admits a factorization into a
continuous function and the uniform lattice Dirac comb, and its
diffraction measure is periodic, with the dual lattice
$\varGamma^*$ as lattice of periods. This statement remains true
in the setting of a locally compact Abelian group whose topology
has a countable base.
Keywords:diffraction, Dirac combs, lattice subsets, homometric sets Categories:52C07, 43A25, 52C23, 43A05 
