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The integration of artificial intelligence (AI) into the workplace is advancing rapidly, necessitating robust metrics to evaluate its tangible impact on the labour market. Existing measures of AI occupational exposure largely focus on AI's theoretical potential to substitute or complement human labour on the basis of technical feasibility, providing limited insight into actual adoption and offering inadequate guidance for policymakers. To address this gap, we introduce the AI Startup Exposure (AISE) index-a novel metric based on occupational descriptions from O*NET and AI applications developed by startups funded by the Y Combinator accelerator. Our findings indicate that while high-skilled professions are theoretically highly exposed according to conventional metrics, they are heterogeneously targeted by startups. Roles involving routine organizational tasks-such as data analysis and office management-display significant exposure, while occupations involving tasks that are less amenable to AI automation due to ethical or high-stakes, more than feasibility, considerations -- such as judges or surgeons -- present lower AISE scores. By focusing on venture-backed AI applications, our approach offers a nuanced perspective on how AI is reshaping the labour market. It challenges the conventional assumption that high-skilled jobs uniformly face high AI risks, highlighting instead the role of today's AI players' societal desirability-driven and market-oriented choices as critical determinants of AI exposure. Contrary to fears of widespread job displacement, our findings suggest that AI adoption will be gradual and shaped by social factors as much as by the technical feasibility of AI applications. This framework provides a dynamic, forward-looking tool for policymakers and stakeholders to monitor AI's evolving impact and navigate the changing labour landscape., Comment: 24 pages, 6 figures, + Supplementary information

The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers, and that in an arbitrary supersoluble skew brace $B$ many relevant skew brace-theoretical properties are easier to identify: for example, a centrally nilpotent ideal of $B$ is $B$-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, $B$ has finite multipermutational level if and only if $(B,+)$ is nilpotent. Given a finite presentation of the structure skew brace $G(X,r)$ associated with a finite non-degenerate solution of the Yang--Baxter Equation (YBE), there is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on., Comment: 37 pages; to be published in Forum of Mathematics, Sigma

The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behaviour of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained., Comment: 23 pages

The aim of this short note is to completely answer Questions 2.34 and 2.35 of arXiv:1806.01127. In particular, we show that a finite strong-nil skew brace $B$ of abelian type need not be right-nilpotent, but that this is the case if~$B$ is of nilpotent type and $b\ast b=0$ for all $b\in B$ (our examples show that this is the best possible result)., Comment: arXiv admin note: text overlap with arXiv:2310.07474. A typo is corrected in proof of Lemma 2.1 and a step of the proof is explained with more detail

Nipotency of skew braces is related to certain types of solutions of the Yang-Baxter equation. This paper delves into the study of centrally nilpotent skew braces. In particular, we study their torsion theory (Section 4.1) and we introduce an "index" for subbraces (Section 4.2), but we also show that the product of centrally nilpotent ideals need not be centrally nilpotent (Example B), a rather peculiar fact. To cope with these examples, we introduce a special type of nilpotent ideal, using which, we define a {\it good} Fitting ideal. Also, a Frattini ideal is defined and its relationship with the Fitting ideal is investigated. A key ingredient in our work is the characterisation of the commutator of ideals in terms of absorbing polynomials (Section 3); this solves Problem 3.4 of arXiv:2109.04389. Moreover, we provide an example (Example A) showing that the idealiser of a subbrace (as defined in arXiv:2205.01572v2) does not exist in general., Comment: 44 pages, definition of centralizer is modified

Let $\sigma=\{\sigma_j\,:\, j\in J\}$ be a partition of the set $\mathbb{P}$ of all prime numbers. A subgroup $X$ of a finite group $G$ is~\textit{$\sigma$-subnormal} in $G$ if there exists a chain of subgroups $$X=X_0\leq X_1\leq\ldots\leq X_n=G$$ such that, for each $1\leq i\leq n-1$, $X_{i-1}\trianglelefteq X_i$ or $X_i/(X_{i-1})_{X_i}$ is a $\sigma_{j_i}$-group for some $j_i\in J$. Skiba~[12] studied the main properties of $\sigma$-subnormal subgroups in finite groups and showed that the set of all $\sigma$-subnormal subgroups plays a relevant role in the structure of a finite soluble group. In [5], we laid the foundation of a general theory of $\sigma$-subnormal subgroups (and $\sigma$-series) in locally finite groups. It turns out that the main difference between the finite and the locally finite case concerns the behaviour of the join of $\sigma$-subnormal subgroups: in finite groups, $\sigma$-subnormal subgroups form a sublattice of the lattice of all subgroups [3], but this is no longer true for arbitrary locally finite groups. This is similar to what happens with subnormal subgroups, so it makes sense to study the class $\mathfrak{S}_\sigma^\infty$ (resp. $\mathfrak{S}_\sigma$) of locally finite groups in which the join of (resp. of finitely many) $\sigma$-subnormal subgroups is $\sigma$-subnormal. Our aim is to study how much one can extend a group in one of these classes before going outside the same class (see for example Theorems~3.6, 3.8, 5.5 and 5.7). Also, $\sigma$-subnormality criteria for the join of $\sigma$-subnormal subgroups are obtained: similarly to a celebrated theorem of Williams (see [15]), we give a necessary and sufficient conditions for a join of two $\sigma$-subnormal subgroups to always be $\sigma$-subnormal; consequently, we show that the join of two orthogonal $\sigma$-subnormal subgroups is $\sigma$-subnormal (extending a result of Roseblade [11])., Comment: 26pp

The aim of this paper is to show that the structure skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation is finitely presented., Comment: arXiv admin note: text overlap with arXiv:2210.08598

If $(X,r)$ is a finite non-degenerate set-theoretic solution of the Yang--Baxter equation, the additive group of the structure skew brace $G(X,r)$ is an $FC$-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an $FC$-group itself. If one additionally assumes that the derived solution of $(X,r)$ is indecomposable, then for every element $b$ of $G(X,r)$ there are finitely many elements of the form $b*c$ and $c*b$, with $c\in G(X,r)$. This naturally leads to the study of a brace-theoretic analogue of the class of $FC$-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation., Comment: 24 pages. Accepted for publication in Publicacions Matem\`atiques

The aim of this short note is to provide a proof to a statement of Sierpi\'nski concerning the number of possible sums of a series (of type $\lambda<\aleph_1$) of arbitrary ordinal numbers.