arXiv subjects

Mateusz Lichman

Publications and source records attributed to Mateusz Lichman.

4 recordsLinked to original sources

arXiv2026

Idealistic equivalence relations and the Lusin derivative

We give a proof, that does not rely on analytic determinacy, of the existence of an idealistic analytic equivalence relation with Borel classes that is not classwise Borel isomorphic to an orbit equivalence relation. This result answers a question of Becker, and Calderoni and Motto Ros. Our proof proceeds through an analysis of Borel complexity of certain sets of countable rooted trees defined via the notion of the Lusin derivative.

math.LO↗

arXiv2023

On strong algebrability of families of non-measurable functions of two variables

Recently Tomasz Natkaniec in [On lineability of families of non-measurable functions of two variable. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 115(1):Paper No. 33, 10, 2021] studied the lineability problem for several classes of non-measurable functions in two variables. In this note we improve his results in the direction of algebrability. In particular, we show that most of the classes considered by Natkaniec contain free algebras with $2^{\mathfrak{c}}$ many generators.

math.FA↗

arXiv2023

Free group of Hamel functions

We construct a free group of continuum many generators among those autobijections of $\mathbb{R}$ which are also Hamel bases of $\mathbb{R}^2$, with identity function included. We also observe two new cases when a real function is a composition of two real functions which are Hamel bases of $\mathbb{R}^2$.

math.GR↗

arXiv2020

On functions preserving products of certain classes of semimetric spaces

In the paper we continue the research of Borsík and Doboš on functions which allow us to introduce a metric to the product of metric spaces. In this paper we extend their scope on broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving $b$-metric inequality. We provide analogues of the results of Borsík and Doboš, adjusted to the new, broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.

math.MG↗