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David Dolžan

Publications and source records attributed to David Dolžan.

At least 19 recordsLinked to original sources
arXiv2026

Products of Nilpotent and Idempotent Matrices over Finite Local Rings

Let $R$ be a finite commutative local principal ring. We study products of nilpotent and idempotent matrices in $M_2(R)$. We show that every product of nilpotent and idempotent matrices is either a product of idempotents or a product of nilpotents. We then consider IN- and NI-matrices, that is, matrices which can be written as a product of an idempotent and a nilpotent matrix in the respective orders. We prove that the classes of IN- and NI-matrices in $M_2(R)$ coincide and give an explicit description of their common class. Finally, if $|R|=q^n$ and $R/J(R)\cong GF(q)$, we show that \[ |\operatorname{IN}(M_2(R))| = |\operatorname{NI}(M_2(R))| = q^{3n-2}(q^n+q^2-1). \]

math.RA↗
arXiv2026

The Cayley graph of a quandle

In this paper, we investigate structural properties of the Cayley graph of a quandle and describe this graph for several important classes of quandles, including conjugation, Takasaki, dihedral, and Alexander quandles. In particular, we prove that for an Alexander quandle $A_t(G)$ over a finite abelian group $G$, the connected components of the Cayley graph correspond to the cosets of the subgroup $\mathrm{im}(\mathrm{id}-t)$. We also show that the Cayley graphs of generalized Alexander quandles are regular. When the defining automorphism is inner, we give an explicit description of the forward orbits and prove that the connected components correspond to cosets of the subgroup generated by commutators with the defining element.

math.GT↗
arXiv2026

Products of idempotents in a quaternion ring

Let $R$ be a finite commutative local principal ring, and let $H(R)$ denote the corresponding quaternion ring. We show that an element of $H(R)$ is a product of idempotents if and only if it can be expressed as a product of two idempotents. Moreover, we obtain an explicit formula for the number of elements of $H(R)$ admitting such a factorization.

math.RA↗
arXiv2026

Eigenvalues of the product matrices of finite commutative rings

The product matrix of a finite commutative ring $R=\{x_1,x_2,\ldots,x_n\}$ and an element $u \in R$ is the matrix $A_u(R)=[a_{ij}]$, where $a_{ij}=1$ if $x_ix_j=u$, and $a_{ij}=0$ otherwise. This provides a natural extension of the concept of the adjacency matrix of the zero-divisor graph of a ring, which has been studied extensively. In this paper, we find the characteristic polynomial of $A_u(R)$ for a local ring $R$ of odd order and a unit $u$. By studying the structure of a finite local ring, we find the characteristic polynomial of $A_u(R)$ for a local ring $R$ and any $u \in R$ in two cases: when the Jacobson radical of $R$ has either the maximal or the minimal possible index of nilpotency.

math.RA↗
arXiv2025

Products of nilpotents in a quaternion ring of odd order

Let $R$ be a finite commutative local principal ring of cardinality $q^n$, where $q = p^r$ for an odd prime $p$ and integer $r$ with $R/J(R) \simeq GF(q)$. We determine the number of elements in the quaternion ring $H(R)$ that can be expressed as a product of at least $2n-1$ nilpotent elements, and show by example that this bound is sharp.

math.RA↗
arXiv2025

Unipotent orbits of elements in a quaternion ring of odd order

Let $n \in \NN$ and let $q=p^r$ be an odd prime power. Let $R$ be a finite commutative local principal ring of cardinality $q^{n}$ with $R/J(R) \simeq GF(q)$. We study the conjugation action of the group of all unipotent elements in the quaternion ring $H(R)$ on $H(R)$ and we classify the resulting unipotent similarity classes, using a reduction to the ring of $2$-by-$2$ matrices over $R$.

math.RA↗
arXiv2025

The multiplication probability of a finite ring

We study the probability that the product of two randomly chosen elements in a finite ring $R$ is equal to some fixed element $x \in R$. We calculate this probability for semisimple rings and some special classes of local rings, and find the bounds for this probability for an arbitrary finite ring.

math.RA↗
arXiv2024

On the sumsets of exceptional units in quaternion rings

We investigate sums of exceptional units in a quaternion ring $H(R)$ over a finite commutative ring $R$. We prove that in order to find the number of representations of an element in $H(R)$ as a sum of $k$ exceptional units for some integer $k \geq 2$, we can limit ourselves to studying the quaternion rings over local rings. For a local ring $R$ of even order, we find the number of representations of an element of $H(R)$ as a sum of $k$ exceptional units for any integer $k \geq 2$. For a local ring $R$ of odd order, we find either the number or the bounds for the number of representations of an element of $H(R)$ as a sum of $2$ exceptional units.

math.RA↗
arXiv2024

Semirings generated by idempotents

We prove that a semiring multiplicatively generated by its idempotents is commutative and Boolean, if every idempotent in the semiring has an orthogonal complement. We prove that a semiring additively generated by its idempotents is commutative, if every idempotent in the semiring has an orthogonal complement and all the nilpotents in the semirings are central. We also provide examples that the assumptions on the existence of orthogonal complements of idempotents and the centrality of nilpotents cannot be omitted.

math.RA↗
arXiv2023

The unitary Cayley graph of a semiring

We study the unitary Cayley graph of a matrix semiring. We find bounds for its diameter, clique number and independence number, and determine its girth. We also find the relationship between the diameter and the clique number of a unitary Cayley graph of a semiring $S$ and a matrix semiring over $S$.

math.RA↗
arXiv2020

Some multivariate imprecise shock model copulas

Bivariate imprecise copulas have recently attracted substantial attention. However, the multivariate case seems still to be a "blank slate". It is then natural that this idea be tested first on shock model induced copulas, a family which might be the most useful in various applications. We investigate a model in which some of the shocks are assumed imprecise and develop the corresponding set of copulas. In the Marshall's case we get a coherent set of distributions and a coherent set of copulas, where the bounds are naturally corresponding to each other. The situation with the other two groups of multivariate imprecise shock model induced copulas, i.e., the maxmin and the the reflected maxmin (RMM) copulas, is substantially more involved, but we are still able to produce their properties. These are the main results of the paper that serves as the first step into a theory that should develop in this direction. In addition, we unfold the theory of bivariate imprecise RMM copulas that has not yet been done before.

math.PR↗