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.

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.

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.

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.

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.

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.

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.

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

We prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) {\it semimodular} lattice.

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.