More game-theoretic properties of boolean algebras

AbstractThe following infinite game G was investigated in [5]: Let B be a Boolean algebra. Two players, White and Black, take turns to choose successively a sequenc

A scattering of orders

A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class $\mathcal B$ of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in $\mathcal B$. More generally, we say that a partial ordering is $\kappa$-scattered if it does not contain a copy of any $\kappa$-dense linear ordering. We prove analogues of Hausdorff's result for $\kappa$-scattered linear orderings, and for $\kappa$-scattered partial orderings satisfying the finite antichain condition. We also study the $\mathbb{Q}_\kappa$-scattered partial orderings, where $\mathbb{Q}_\kappa$ is the saturated linear ordering of cardinality $\kappa$, and a partial ordering is $\mathbb{Q}_\kappa$-scattered when it embeds no copy of $\mathbb{Q}_\kappa$. We classify the $\mathbb{Q}_\kappa$-scattered partial orderings with the finite antichain condition relative to the $\mathbb{Q}_\kappa$-scattered linear orderings. We show that in general the property of being a $\mathbb{Q}_\kappa$-scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions

Stationary logic of finitely determinate structures

AbstractIn this part we develop the theory of finitely determinate structures, that is, structures on which the dual quantifiers “stat” and “unreadable” have the same meaning. Among other genera

Games with Filters

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call \emph{Welch games}. Player II having a winning strategy in the Welch game of length $\omega$ on $\kappa$ is equivalent to weak compactness. Winning the game of length $2^\kappa$ is equivalent to $\kappa$ being measurable. We show that for games of intermediate length $\gamma$, II winning implies the existence of precipitous ideals with $\gamma$-closed, $\gamma$-dense trees. The second part shows the first is not vacuous. For each $\gamma$ between $\omega$ and $\kappa^+$, it gives a model where II wins the games of length $\gamma$, but not $\gamma^+$. The technique also gives models where for all $\omega_1< \gamma\le\kappa$ there are $\kappa$-complete, normal, $\kappa^+$-distributive ideals having dense sets that are $\gamma$-closed, but not $\gamma^+$-closed