arXiv subjects
Publications and source records attributed to Luka Andrenšek.
Let $Γ$ and $Θ$ be finite simple oriented graphs, and let $\mathbf{HK}_Γ$ and $\mathbf{HK}_Θ$ denote the corresponding Hecke-Kiselman monoids. We present a set of Boolean matrices that is in bijection with the set of monoid homomorphisms from $\mathbf{HK}_Γ$ to $\mathbf{HK}_Θ$. In particular, we obtain a Boolean matrix monoid isomorphic to the endomorphism monoid of $\mathbf{HK}_Θ$.
We study certain dynamical and metric aspects of Kiselman's semigroup $K_n$. The level function $\mathcal{L}$ is introduced and shown to admit a simple description in terms of right multiplication by generators. We show that every sequence of partial products in $K_n$ is eventually constant. Using $\mathcal{L}$, we further study sequences of random partial products in $K_n$ and show that, in the independent and identically distributed setting where every generator is chosen with positive probability, the hitting time of the eventual constant value is distributed as a sum of $n$ independent geometric random variables. Finally, we define a natural ultrametric on $K_n$ arising from the level function and obtain some basic results on the associated metric balls and spheres.
We investigate equations in Kiselman's semigroup $K_n$, generated by $a_1, \dots, a_n$. Let $f$ denote the zero element of $K_n$. We prove that if $y \in K_n$ lies in the subsemigroup generated by $a_2, \dots, a_n$, then $x y = f$ implies $x = f$. In contrast, the equation $x a_1 = f$ admits non-trivial solutions. We describe the solution set of this equation, show that its cardinality is $1 + |K_{n-1}|$, and study its algebraic structure. Moreover, we show that $|K_{2n+1}|$ is even, whereas $|K_{2n}|$ is odd.
We investigate zero-shot cross-lingual news sentiment detection, aiming to develop robust sentiment classifiers that can be deployed across multiple languages without target-language training data. We introduce novel evaluation datasets in several less-resourced languages, and experiment with a range of approaches including the use of machine translation; in-context learning with large language models; and various intermediate training regimes including a novel task objective, POA, that leverages paragraph-level information. Our results demonstrate significant improvements over the state of the art, with in-context learning generally giving the best performance, but with the novel POA approach giving a competitive alternative with much lower computational overhead. We also show that language similarity is not in itself sufficient for predicting the success of cross-lingual transfer, but that similarity in semantic content and structure can be equally important.