We study the possible values of the cofinality invariant for various Borel ideals on the natural numbers. We introduce the notions of a fragmented and gradually fragmented ideal and prove a dichotomy for fragmented ideals. We show that every gradually fragmented ideal has cofinality consistently strictly smaller than the cardinal invariant and produce a model where there are uncountably many pairwise distinct cofinalities of gradually fragmented ideals.

We describe a new way to construct large subdirectly irreducibles within an equational class of algebras. We use this construction to show that there are forbidden geometries of multitraces for finite algebras in residually small equational classes. The construction is first applied to show that minimal equational classes generated by simple algebras of types 2, 3 or 4 are residually small if and only if they are congruence modular. As a second application of the construction we characterize residually small locally (...) finite abelian equational classes. (shrink)

This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.

We prove that there is a -ideal of compact sets which is strictly above in the Tukey order. Here is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Veličković asked in [Alain Louveau, Boban Veličković, Analytic ideals and cofinal types, Ann. Pure Appl. Logic 99 171–195].