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In this article, we compare two different notions of partially defined group strutures, namely partial groups and pregroups, as introduced by Chermak and Stallings respectively. In particular we prove that the category of pregroups can be…
Word-representable graphs, which are the same as semi-transitively orientable graphs, generalize several fundamental classes of graphs. In this paper we propose a novel approach to study word-representability of graphs using a technique of…
The classical Clifford correspondence for normal subgroups is considered in the more general setting of semisimple Hopf algebras. We prove that this correspondence still holds if the extension determined by the normal Hopf subalgebra is…
For both human readers and pre-trained language models (PrLMs), lexical diversity may lead to confusion and inaccuracy when understanding the underlying semantic meanings of given sentences. By substituting complex words with simple…
We investigate the class of quasitrivial semigroups and provide various characterizations of the subclass of quasitrivial and commutative semigroups as well as the subclass of quasitrivial and order-preserving semigroups. We also determine…
We introduce the quasi-ordinarization transform of a numerical semigroup. This transform will allow to organize all the semigroups of a given genus in a forest rooted at all quasi-ordinary semigroups with the given genus. This construction…
This thesis aims to serve as an introduction to the theory of quasitilings for amenable groups. In order to showcase the power of this theory, we focus on the study of the Sofic L\"uck Approximation Conjecture, which can be proven for…
Group counterfactual explanations find a set of counterfactual instances to explain a group of input instances contrastively. However, existing methods either (i) optimize counterfactuals only for a fixed group and do not generalize to new…
String diagrams are a powerful tool for reasoning about composite structures in symmetric monoidal categories. By representing string diagrams as graphs, equational reasoning can be done automatically by double-pushout rewriting. !-graphs…
We obtain a lifting property for finite quotients of algebraic groups, and applications to the structure of these groups.
Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural…
We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the…
We study certain subgroups of the full group of Hopf algebra automorphisms of a biproduct. In the process interesting subgroups of certain permutation groups come into play.
This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on…
The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group \Gamma, with quotient group isomorphic to \Gamma/N. It is shown how to enumerate such…
We prove locality of superconformal algebras: every pluperfect superconformal algebra is spanned by coefficients of a finite family of mutually local distributions. We also introduce quasi-Poisson algebras and show that they can be used to…
Particularly in low-data regimes, an outstanding challenge in machine learning is developing principled techniques for augmenting our models with suitable priors. This is to encourage them to learn in ways that are compatible with our…
Pretrained Language Models (PLMs) such as BERT have revolutionized the landscape of Natural Language Processing (NLP). Inspired by their proliferation, tremendous efforts have been devoted to Pretrained Graph Models (PGMs). Owing to the…
We develop a theory of rewriting for structured cospans in order to extend compositional methods for modeling open networks. First, we introduce a category whose objects are structured cospans, and establish conditions under which it is…
This article is an introduction to formal languages from the point of view of combinatorial group theory. Group theoretic applications are included and language classes are defined algebraically.