More Set-theory around the weak Freese-Nation property

In this paper, we introduce a very weak square principle which is even weaker than the similar principle introduced by Foreman and Magidor. A characterization of this principle is given in term of sequences of elementary submodels of H(\chi). This is used in turn to prove a characterization of kappa-Freese-Nation property under the very weak square principle and a weak variant of the Singular Cardinals Hypothesis. A typical application of this characterization shows that under 2^{\aleph_0}<\aleph_\omega and our very weak square for \aleph_\omega, the partial ordering [omega_\omega]^{<\omega} (ordered by inclusion) has the aleph_1-Freese-Nation property. On the other hand we show that, under Chang's Conjecture for \aleph_\omega the partial ordering above does not have the aleph_1-Freese-Nation property. Hence we obtain the independence of our characterization of the kappa-Freese-Nation property and also of the very weak square principle from ZFC

Topological spaces with the Freese--Nation property II

The aim of this paper is to study the class of spaces with the FNS property and \pi-\FNS property. We shown that compact spaces with the FNS property for some base consisting of cozero-sets are openly generated spaces and spaces with the \pi-\FNS property are skeletally generated spaces

Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Polyatomic Logics and Generalised Blok-Esakia Theory

This paper presents a novel concept of a Polyatomic Logic and initiates its systematic study. This approach, inspired by Inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations, through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalised Blok-Esakia theory which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of Polyatomic Logics and the general Blok-Esakia theory for a wide class of interesting translations.Comment: 48 pages, 2 figure

Set Theory

This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C*-algebras, Ramsey theory, measure theory, representation theory, group theory and Banach spaces

Proto-exact categories of matroids, Hall algebras, and K-theory

This paper examines the category \mathbf {Mat}_{\bullet } of pointed matroids and strong maps from the point of view of Hall algebras. We show that \mathbf {Mat}_{\bullet } has the structure of a finitary proto-exact category - a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We define the algebraic K-theory K_* (\mathbf {Mat}_{\bullet }) of \mathbf {Mat}_{\bullet } via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections \begin{aligned} \pi ^s_n ({\mathbb {S}}) \hookrightarrow K_n (\mathbf {Mat}_{\bullet }) \end{aligned} from the stable homotopy groups of spheres for all n. Finally, we show that the Hall algebra of \mathbf {Mat}_{\bullet } is a Hopf algebra dual to Schmitt’s matroid-minor Hopf algebra.First author draf