Abstract We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $ . In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $ . This improves the results of G. Bezhanishvili and J. Harding in [4], where the authors prove these theorems under stronger assumptions ( $\mathfrak {a=c}$ ). Our proof is also somewhat simpler.

PyVallex is a Python-based system for presenting, searching/filtering, editing/extending and automatic processing of machine-readable lexicon data originally available in a text-based format. The system consists of several components: a parser for the specific lexicon format used in several valency lexicons, a data-validation framework, a regular expression based search engine, a map-reduce style framework for querying the lexicon data and a web-based interface integrating complex search and some basic editing capabilities. PyVallex provides most of the typical functionalities of a Dictionary Writing System (DWS), such as multiple presentation modes for the underlying lexical database, automatic evaluation of consistency tests, and a mechanism of merging updates coming from multiple sources. The editing functionality is currently limited to the client-side interface and edits of existing lexical entries, but additional script-based operations on the database are also possible. The code is published under the open source MIT license and is also available in the form of a Python module for integrating into other software.

Abstract We investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection (in ${[\omega ]^\omega }$ ). We prove the following results: (1) Many classical examples of nice tall filters contain no towers (in ZFC). (2) It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well). (3) It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers. (4) The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills). Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters ( ${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$ , ${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$ , ${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$ , and ${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$ ), and the existence of Luzin type families (of size $\ge {\omega _2}$ ), that is, if ${\cal F}$ is a filter then ${\cal X} \subseteq {[\omega ]^\omega }$ is an ${\cal F}$ -Luzin family if $\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$ is countable for every $F \in {\cal F}$ .