arXiv subjects

Łukasz Mazurkiewicz

Publications and source records attributed to Łukasz Mazurkiewicz.

4 recordsLinked to original sources

arXiv2026

Ideal Analytic sets

The aim of this paper is to give natural examples of $\mathbfΣ_1^1$-complete and $\mathbfΠ_1^1$-complete sets. In the first part, we consider ideals on $ω$. In particular, we show that the Hindman ideal $\mathcal{H}$ is $\mathbfΠ_1^1$-complete and consider a number of ideals generated in the similar fashion. Moreover, we show that the ideal $\mathcal{D}$ is also $\mathbfΠ_1^1$-complete. In the second part, we focus on families of trees (on $ω$ and $2$) containing a specific tree type. We show the connection between two topics and explore some classical tree types (like Sacks and Miller).

math.LO↗

arXiv2025

An Ideal Zoo in the Baire Space

In this paper, we study the translations into the Baire space of several well-known $σ$-ideals and families originally defined on the Cantor space, using their combinatorial characterizations. These include the ideals of null sets, small sets, those generated by closed measure-zero sets, and the meager sets, leading to their "fake" analogues in the Baire space. We also parametrize families related to null sets by functions from $ω^ω$. Several structural properties and relations between these families are investigated, including whether they form ideals, the existence of large chains and antichains, orthogonality, the $κ$-chain condition, and the determination of certain cardinal invariants.

math.GN↗

arXiv2024

On algebraic sums, trees and ideals in the Baire space

We work in the Baire space $\mathbb{Z}^ω$ equipped with the coordinate-wise addition $+$. Consider a $σ-$ideal $\mathcal{I}$ and a family $\mathbb{T}$ of some kind of perfect trees. We are interested in results of the form: for every $A\in \mathcal{I}$ and a tree $T\in\mathbb{T}$ there exists $T'\in \mathbb{T}, T'\subseteq T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\inω$. Explored tree types include perfect trees, uniformly perfect trees, Miller trees, Laver trees and $ω-$Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard $σ$-ideal $\mathcal{M}$ of meager sets, we also analyze $\mathcal{M}_-$ and fake null sets $\mathcal{N}$. The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space.

math.GN↗

arXiv2023

Sequences with increasing subsequence

We study analytic and Borel subsets defined similarily to the old example of analytic complete set given by Luzin. Luzin's example, which is essentially a subset of the Baire space, is based on the natural partial order on naturals, i.e. division. It consists of sequences which contain increasing subsequence in given order. We consider a variety of sets defined in a similar way. Some of them occurs to be Borel subsets of the Baire space, while others are analytic complete, hence not Borel. In particular, we show that an analogon of Luzin example based on the natural linear order on rationals is analytic complete. We also characterise all countable linear orders having such property.

math.GN↗