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Waldemar Hołubowski

Publications and source records attributed to Waldemar Hołubowski.

5 recordsLinked to original sources
arXiv2025

On the methods of reduction of some types of Marczewski-Burstin measurable functions to continuous functions on products of perfect sets

In this paper, we introduce product-wise generalizations of certain Marczewski-Burstin bases, including sets with the (s)-property and completely Ramsey sets. For each of these families, we establish analogs of the classical Luzin and Eggleston theorems, showing that functions measurable with respect to these families can be reduced to continuous functions on products of perfect sets. Furthermore, we provide a method for reducilng sequences of such functions to continuity, which allows us to generalize Laver's extension of Halpern-Läuchli and Harrington theorems.

math.CO↗
arXiv2024

The unitary Cayley graph of upper triangular matrix rings

The unitary Cayley graph $C_R$ of a finite unital ring $R$ is the simple graph with vertex set $R$ in which two elements $x$ and $y$ are connected by an edge if and only if $x-y$ is a unit of $R$. We characterize the unitary Cayley graph $C_{T_n (\mathbb{F})}$ of the ring of all upper triangular matrices $T_n(\mathbb{F})$ over a finite field $\mathbb{F}$. We show that $C_{T_n (\mathbb{F})}$ is isomorphic to the semistrong product of the complete graph $K_m$ and the antipodal graph of the Hamming graph $A(H(n,p^k))$, where $m=p^{\frac{kn(n-1)}{2}}$ and $|\mathbb{F}|=p^k$. In particular, if $|\mathbb{F}|=2$, then the graph $C_{T_n (\mathbb{F})}$ has $2^{n-1}$ connected components, each component is isomorphic to the complete bipartite graph $K_{m,m}$, where $m=2^{\frac{n(n-1)}{2}}$. We also compute the diameter, triameter, and clique number of the graph $C_{T_n (\mathbb{F})}$.

math.CO↗
arXiv2021

Normal subgroups in the group of column-finite infinite matrices

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of $GL(n, K)$ ($K$ - a field, $n \geq 3$) which is not contained in the center, contains $SL(n, K)$. A. Rosenberg gave description of normal subgroups of $GL(V)$, where $V$ is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations $g$ such that $g-id_V$ has finite dimensional range the proof is not complete. We fill this gap for countably dimensional $V$ giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

math.GR↗