We investigate ideals of the form {A ⊆ : n∈A xn is unconditionally convergent} where (xn)n∈ is a sequence in a Polish group or in a Banach space. If an ideal on can be seen in this form for some sequence in X , then we say that it is representable in X .After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a nonpathological analytic P-ideal.We focus on the family of ideals representable in c 0 . We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah's ideal, and Tsirelson ideals are not representable in c 0 , and that a tall F P-ideal is representable in c 0 iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable in 1 but not in R.Finally, we summarize some open problems of this topic. §1. Introduction. Recall that an ideal I on is summable if it is defined by a measure, i.e., there is a (mass) function m :In this case, we write I = I m . Summable ideals, together with density ideals, are the flagship examples of analytic P-ideals on . However, this class contains also ideals which are, from the combinatorial viewpoint, very far from summable and density ideals (see examples in the next section).In this article, we consider some natural generalizations of summable ideals. Consider a space X equipped in enough structure to speak about convergence of series, e.g. a Polish Abelian group or a Banach space. We say that an ideal J on is representable in X if there is a function m : → X such that I ∈ J ⇐⇒ i∈I m(i) converges unconditionally in X .

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We study cardinal invariants connected to certain classical orderings on the family of ideals on ω. We give topological and analytic characterizations of these invariants using the idealized version of Fréchet-Urysohn property and, in a special case, using sequential properties of the space of finitely-supported probability measures with the weak * topology. We investigate consistency of some inequalities between these invariants and classical ones, and other related combinatorial questions. At last, we discuss maximality properties of almost disjoint families related to certain ordering on ideals.

We investigate families of subsets of ω with almost disjoint refinements in the classical case as well as with respect to given ideals on ω. More precisely, we study the following topics and questions:1) Examples of projective ideals.2) We prove the following generalization of a result due to J. Brendle:3) The existence of perfect I-almost disjoint (I-AD) families; and the existence of a "nice" ideal I on ω with the property: Every I-AD family is countable but I is nowhere maximal.4) The existence of (I, Fin)-almost disjoint refinements of families of Ipositive sets in the case of everywhere meager (e.g. analytic or coanalytic) ideals. We show that under Martin's Axiom if I is an everywhere meager ideal and H ⊆ I + with |H| < c, then H has an (I, Fin)-ADR, that is, a family {A H : H ∈ H} such that (i) A H ⊆ H, A H ∈ I + for every H and (ii) A H 0 ∩ A H 1 is finite for every distinct H 0 , H 1 ∈ H. 5) Connections between classical properties of forcing notions and adding mixing reals (and mixing injections), that is, a (one-to-one) function f :This property is relevant concerning almost disjoint refinements because it is very easy to find an almost disjoint refinement of [ω] ω ∩ V in every extension V ⊆ W containing a mixing injection over V .2010 Mathematics Subject Classification. 03E05,03E15,03E35.

We investigate which filters on ω can 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 $F \in {\cal F}$.