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Journal articles on the topic 'Skew brace'

Author: Grafiati

Published: 9 March 2023

Last updated: 31 July 2025

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1

Wang, Ximu, Chongxia Zhang, and Liangyun Zhang. "Rota–Baxter Operators on Skew Braces." Mathematics 12, no. 11 (2024): 1671. http://dx.doi.org/10.3390/math12111671.

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In this paper, we introduce the concept of Rota–Baxter skew braces, and provide classifications of Rota–Baxter operators on various skew braces, such as (Z,+,∘) and (Z/(4),+,∘). We also present a necessary and sufficient condition for a skew brace to be a co-inverse skew brace. Additionally, we describe some constructions of Rota–Baxter quasiskew braces, and demonstrate that every Rota–Baxter skew brace can induce a quasigroup and a Rota–Baxter quasiskew brace.

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2

Rump, Wolfgang. "Set-theoretic solutions to the Yang–Baxter equation, skew-braces, and related near-rings." Journal of Algebra and Its Applications 18, no. 08 (2019): 1950145. http://dx.doi.org/10.1142/s0219498819501457.

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Skew-braces have been introduced recently by Guarnieri and Vendramin. The structure group of a non-degenerate solution to the Yang–Baxter equation is a skew-brace, and every skew-brace gives a set-theoretic solution to the Yang–Baxter equation. It is proved that skew-braces arise from near-rings with a distinguished exponential map. For a fixed skew-brace, the corresponding near-rings with exponential form a category. The terminal object is a near-ring of self-maps, while the initial object is a near-ring which gives a complete invariant of the skew-brace. The radicals of split local near-ring

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3

De Commer, K. "Actions of skew braces and set-theoretic solutions of the reflection equation." Proceedings of the Edinburgh Mathematical Society 62, no. 4 (2019): 1089–113. http://dx.doi.org/10.1017/s0013091519000129.

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AbstractA skew brace, as introduced by L. Guarnieri and L. Vendramin, is a set with two group structures interacting in a particular way. When one of the group structures is abelian, one gets back the notion of brace as introduced by W. Rump. Skew braces can be used to construct solutions of the quantum Yang–Baxter equation. In this article, we introduce a notion of action of a skew brace, and show how it leads to solutions of the closely associated reflection equation.

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4

Catino, Francesco, Ilaria Colazzo, and Paola Stefanelli. "Skew left braces with non-trivial annihilator." Journal of Algebra and Its Applications 18, no. 02 (2019): 1950033. http://dx.doi.org/10.1142/s0219498819500336.

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We describe the class of all skew left braces with non-trivial annihilator through ideal extension of a skew left brace. The ideal extension of skew left braces is a generalization to the non-abelian case of the extension of left braces provided by Bachiller in [D. Bachiller, Extensions, matched products, and simple braces, J. Pure Appl. Algebra 222 (2018) 1670–1691].

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5

Koch, Alan. "Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures." Proceedings of the American Mathematical Society, Series B 8, no. 16 (2021): 189–203. http://dx.doi.org/10.1090/bproc/87.

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Let G G be a finite nonabelian group, and let ψ : G → G \psi :G\to G be a homomorphism with abelian image. We show how ψ \psi gives rise to two Hopf-Galois structures on a Galois extension L / K L/K with Galois group (isomorphic to) G G ; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the

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6

Vendramin, Leandro. "What is... a Skew Brace?" Notices of the American Mathematical Society 71, no. 01 (2024): 1. http://dx.doi.org/10.1090/noti2855.

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7

Bachiller, David. "Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks." Journal of Knot Theory and Its Ramifications 27, no. 08 (2018): 1850055. http://dx.doi.org/10.1142/s0218216518500554.

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Given a skew left brace [Formula: see text], a method is given to construct all the non-degenerate set-theoretic solutions [Formula: see text] of the Yang–Baxter equation such that the associated permutation group [Formula: see text] is isomorphic, as a skew left brace, to [Formula: see text]. This method depends entirely on the brace structure of [Formula: see text]. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation grou

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8

Mauffrey, Océane, Kevin Yu, Malvika Choudhari, Ashley Lynn Habig, Alec Pugh, and Vinay Narotam. "Bracing for Impact: A Survey Analysis of the Impact of Socioeconomic Factors on Brace Adherence in Clubfoot." Journal of Clinical Orthopaedics 10, no. 1 (2025): 4–7. https://doi.org/10.13107/jcorth.2025.v10i01.702.

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Background: Clubfoot is a congenital deformity characterized by cavus deformity of the midfoot, adductus of the forefoot and equinus and varus of the hindfoot. The Ponseti method, a series of casting and bracing protocols has become the standard of care as a highly effective non-surgical intervention. Poor adherence with stringent brace wearing protocols has been identified as one of the leading causes of deformity recurrence with the Ponseti method. The present study seeks to uncover the socioeconomic variables which may contribute to brace adherence. Methods: This survey study included 219 p

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9

Jespers, Eric, and Arne Van Antwerpen. "Left semi-braces and solutions of the Yang–Baxter equation." Forum Mathematicum 31, no. 1 (2019): 241–63. http://dx.doi.org/10.1515/forum-2018-0059.

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Abstract Let {r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra {K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solu

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10

Nasybullov, Timur. "Connections between properties of the additive and the multiplicative groups of a two-sided skew brace." Journal of Algebra 540 (December 2019): 156–67. http://dx.doi.org/10.1016/j.jalgebra.2019.05.005.

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11

Bardakov, Valeriy G., Mikhail V. Neshchadim, and Manoj K. Yadav. "Computing skew left braces of small orders." International Journal of Algebra and Computation 30, no. 04 (2020): 839–51. http://dx.doi.org/10.1142/s0218196720500216.

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We improve Algorithm 5.1 of [Math. Comp. 86 (2017) 2519–2534] for computing all nonisomorphic skew left braces, and enumerate left braces and skew left braces of orders up to 868 with some exceptions. Using the enumerated data, we state some conjectures for further research.

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12

Jespers, E., Ł. Kubat, A. Van Antwerpen, and L. Vendramin. "Factorizations of skew braces." Mathematische Annalen 375, no. 3-4 (2019): 1649–63. http://dx.doi.org/10.1007/s00208-019-01909-1.

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13

Acri, E., R. Lutowski, and L. Vendramin. "Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces." International Journal of Algebra and Computation 30, no. 01 (2019): 91–115. http://dx.doi.org/10.1142/s0218196719500656.

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Using Bieberbach groups, we study multipermutation involutive solutions to the Yang–Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right [Formula: see text]-nil

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14

Bardakov, Valeriy G., Mikhail V. Neshchadim та Manoj K. Yadav. "On λ-homomorphic skew braces". Journal of Pure and Applied Algebra 226, № 6 (2022): 106961. http://dx.doi.org/10.1016/j.jpaa.2021.106961.

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15

Acri, E., and M. Bonatto. "Skew braces of size pq." Communications in Algebra 48, no. 5 (2020): 1872–81. http://dx.doi.org/10.1080/00927872.2019.1709480.

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16

Brzeziński, Tomasz, Stefano Mereta, and Bernard Rybołowicz. "From pre-trusses to skew braces." Publicacions Matemàtiques 66 (July 1, 2022): 683–714. http://dx.doi.org/10.5565/publmat6622206.

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17

Cedó, Ferran, Agata Smoktunowicz, and Leandro Vendramin. "Skew left braces of nilpotent type." Proceedings of the London Mathematical Society 118, no. 6 (2018): 1367–92. http://dx.doi.org/10.1112/plms.12209.

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18

Koch, Alan, and Paul J. Truman. "Opposite skew left braces and applications." Journal of Algebra 546 (March 2020): 218–35. http://dx.doi.org/10.1016/j.jalgebra.2019.10.033.

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19

Gorshkov, Ilya, and Timur Nasybullov. "Finite skew braces with solvable additive group." Journal of Algebra 574 (May 2021): 172–83. http://dx.doi.org/10.1016/j.jalgebra.2021.01.027.

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20

Guarnieri, L., and L. Vendramin. "Skew braces and the Yang–Baxter equation." Mathematics of Computation 86, no. 307 (2016): 2519–34. http://dx.doi.org/10.1090/mcom/3161.

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21

Acri, E., and M. Bonatto. "Skew Braces of Size p2 q I: Abelian Type." Algebra Colloquium 29, no. 02 (2022): 297–320. http://dx.doi.org/10.1142/s1005386722000244.

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This is the first part of a series of two articles. In this paper we enumerate and classify the left braces of size [Formula: see text], where[Formula: see text] and [Formula: see text] are distinct prime numbers, by the classification of regular subgroups of the holomorph of the abelian groups of the same order. We also provide the formulas that define the constructed braces.

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22

Catino, Francesco, Ilaria Colazzo, and Paola Stefanelli. "Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces." Forum Mathematicum 33, no. 3 (2021): 757–72. http://dx.doi.org/10.1515/forum-2020-0082.

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Abstract This paper aims to introduce a construction technique of set-theoretic solutions of the Yang–Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known that braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions.

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23

Nejabati Zenouz, Kayvan. "Skew braces and Hopf–Galois structures of Heisenberg type." Journal of Algebra 524 (April 2019): 187–225. http://dx.doi.org/10.1016/j.jalgebra.2019.01.012.

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24

Caranti, A. "Bi-skew braces and regular subgroups of the holomorph." Journal of Algebra 562 (November 2020): 647–65. http://dx.doi.org/10.1016/j.jalgebra.2020.07.006.

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25

Brzeziński, Tomasz, Ilaria Colazzo, Anastasia Doikou, and Leandro Vendramin. "Mini-Workshop: Skew Braces and the Yang–Baxter Equation." Oberwolfach Reports 20, no. 1 (2023): 537–63. http://dx.doi.org/10.4171/owr/2023/9.

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26

Konovalov, A., A. Smoktunowicz, and L. Vendramin. "Erratum to the Paper “On Skew Braces and Their Ideals”." Experimental Mathematics 31, no. 1 (2021): 346. http://dx.doi.org/10.1080/10586458.2021.1980466.

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27

Childs, Lindsay N. "Skew braces and the Galois correspondence for Hopf Galois structures." Journal of Algebra 511 (October 2018): 270–91. http://dx.doi.org/10.1016/j.jalgebra.2018.06.023.

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28

Ghobadi, Aryan. "Skew braces as remnants of co-quasitriangular Hopf algebras in SupLat." Journal of Algebra 586 (November 2021): 607–42. http://dx.doi.org/10.1016/j.jalgebra.2021.07.006.

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29

Bardakov, Valeriy G., and Vsevolod Gubarev. "Rota—Baxter groups, skew left braces, and the Yang—Baxter equation." Journal of Algebra 596 (April 2022): 328–51. http://dx.doi.org/10.1016/j.jalgebra.2021.12.036.

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30

Almoosi, Y., and N. Oukaili. "The Response of a Highly Skewed Steel I-Girder Bridge with Different Cross-Frame Connections." Engineering, Technology & Applied Science Research 11, no. 4 (2021): 7349–57. http://dx.doi.org/10.48084/etasr.4137.

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Braces in straight bridge systems improve the lateral-torsional buckling resistance of the girders by reducing the unbraced length, while in horizontally curved and skew bridges, the braces are primary structural elements for controlling deformations by engaging adjacent girders to act as a system to resist the potentially large forces and torques caused by the curved or skewed geometry of the bridge. The cross-frames are usually designed as torsional braces, which increase the overall strength and stiffness of the individual girders by creating a girder system that translates and rotates as a

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31

Smoktunowicz, Agata, and Leandro Vendramin. "On skew braces (with an appendix by N. Byott and L. Vendramin)." Journal of Combinatorial Algebra 2, no. 1 (2018): 47–86. http://dx.doi.org/10.4171/jca/2-1-3.

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32

Ballester-Bolinches, A., R. Esteban-Romero, P. Jiménez-Seral, and V. Pérez-Calabuig. "Soluble skew left braces and soluble solutions of the Yang-Baxter equation." Advances in Mathematics 455 (October 2024): 109880. http://dx.doi.org/10.1016/j.aim.2024.109880.

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33

Tsang, Cindy (Sin Yi). "Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group." Proceedings of the American Mathematical Society, Series B 9, no. 36 (2022): 377–92. http://dx.doi.org/10.1090/bproc/138.

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34

Ballester-Bolinches, Adolfo, Ramón Esteban-Romero, Maria Ferrara, Vicent Pérez-Calabuig, and Marco Trombetti. "Central nilpotency of left skew braces and solutions of the Yang–Baxter equation." Pacific Journal of Mathematics 335, no. 1 (2025): 1–32. https://doi.org/10.2140/pjm.2025.335.1.

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35

Kozlovskaya, Tatyana A. "Multi-groups." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 87 (2024): 34–43. http://dx.doi.org/10.17223/19988621/87/4.

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In the present paper we define homogeneous algebraic systems. Particular cases of these systems are semigroup (monoid, group) systems. These algebraic systems were studied by J. Loday, A. Zhuchok, T. Pirashvili, and N. Koreshkov. Quandle systems were introduced and studied by V. Bardakov, D. Fedoseev, and V. Turaev. We construct some group systems on the set of square matrices over a field k. Also, we define rack systems on the set V x G , where V is a vector space of dimension n over k and G is a subgroup of GLn(k). Finally, we find the connection between skew braces and dimonoids.

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36

Mazzotta, Marzia. "A family of set-theoretic solutions of the Yang–Baxter equation coming from skew braces." Banach Center Publications 129 (2025): 115–23. https://doi.org/10.4064/bc129-7.

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37

Mitchell, Denis, Sharlie Huffman, Robert Tremblay, et al. "Damage to bridges due to the 27 February 2010 Chile earthquake." Canadian Journal of Civil Engineering 40, no. 8 (2013): 675–92. http://dx.doi.org/10.1139/l2012-045.

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This paper provides a summary of the damage to bridges in the Mw 8.8 Chile earthquake of 27 February 2010. Lessons from the different types of structural damage observed on concrete and steel bridges are discussed. The important roles played by soil liquefaction, settlement and embankment failures are highlighted. Aspects such as shear failure of steel piles, shear failure of concrete substructure elements, failures and severe buckling of steel braces, failures of shear keys and restrainers at supports, and damage to girders due to lack of diaphragms are described. Many examples of loss of sup

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38

Caranti, A., and L. Stefanello. "From endomorphisms to bi-skew braces, regular subgroups, the Yang–Baxter equation, and Hopf–Galois structures." Journal of Algebra 587 (December 2021): 462–87. http://dx.doi.org/10.1016/j.jalgebra.2021.07.029.

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39

Jespers, E., Ł. Kubat, A. Van Antwerpen, and L. Vendramin. "Radical and weight of skew braces and their applications to structure groups of solutions of the Yang–Baxter equation." Advances in Mathematics 385 (July 2021): 107767. http://dx.doi.org/10.1016/j.aim.2021.107767.

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40

Crespo, Teresa. "Hopf Galois structures on field extensions of degree twice an odd prime square and their associated skew left braces." Journal of Algebra 565 (January 2021): 282–308. http://dx.doi.org/10.1016/j.jalgebra.2020.09.005.

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41

Campedel, E., A. Caranti, and I. Del Corso. "Hopf-Galois structures on extensions of degree p2q and skew braces of order p2q: The cyclic Sylow p-subgroup case." Journal of Algebra 556 (August 2020): 1165–210. http://dx.doi.org/10.1016/j.jalgebra.2020.04.009.

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42

T, Raju, Pavan I, Sravani K, B. Manuswini, and Chenna Reddy G. "Facial Features Monitoring for Real Time Drowsiness Detection using Support Vector Machine." International Journal for Modern Trends in Science and Technology 11, no. 03 (2025): 109–17. https://doi.org/10.5281/zenodo.15084878.

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Fatigue among drivers is a major cause of road accidents every year in India. Lack of sound sleep for six to eight hours is one of the primary reasons behind this fatigue. Drivers with sleep deprivation can imbalance the reaction time and decision making when behind the wheels and this can increase the cause of accidents. This type of accidents is more likely to result in death or severe injury as they tend to be in high speed and because of the fact that driver has fallen asleep cannot apply brake or skew to avoid or reduce the impact. Therefore, it is highly essential to create a smart syste

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43

Cañadas, Agustín Moreno, Pedro Fernando Fernández Espinosa, and Adolfo Ballester-Bolinches. "Solutions of the Yang–Baxter Equation and Automaticity Related to Kronecker Modules." Computation 11, no. 3 (2023): 43. http://dx.doi.org/10.3390/computation11030043.

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The Kronecker algebra K is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over K are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form Ax=Bx′, where A and B are m×n, F-matrices with F an algebraically closed field. On the other hand, researching the Yang–Baxter equation (YBE) is a topic of great interest i

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44

Kohl, Timothy. "Characteristic subgroup lattices and Hopf–Galois structures." International Journal of Algebra and Computation 29, no. 02 (2019): 391–405. http://dx.doi.org/10.1142/s0218196719500073.

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The Hopf–Galois structures on normal field extensions [Formula: see text] with [Formula: see text] are in one-to-one correspondence with the set of regular subgroups [Formula: see text] of [Formula: see text], the group of permutations of [Formula: see text] as a set, that are normalized by the left regular representation [Formula: see text]. Each such [Formula: see text] corresponds to a Hopf algebra [Formula: see text] that acts on [Formula: see text]. Such regular subgroups need not be isomorphic to [Formula: see text] but must have the same order. One can divide all such [Formula: see text

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45

Wancket, Lyn, Ranyia Matta, John Barnard, et al. "Role of MKP-1 in an IL-10 knockout murine inflammatory bowel disease model (47.7)." Journal of Immunology 184, no. 1_Supplement (2010): 47.7. http://dx.doi.org/10.4049/jimmunol.184.supp.47.7.

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Abstract IBD is a chronic intestinal inflammatory disease that often has extra-intestinal manifestations. We examined the role of the MAPK phosphatase Mkp-1 in a mouse model of IBD. Mkp-1+/+/Il-10+/+, Mkp-1-/-/Il-10+/+, Mkp-1+/+/Il-10-/-, and Mkp-1-/-/Il-10-/- (dKO) mice on a 129 background were housed in a specific pathogen-free environment. Colitis signs were evaluated using a clinical scoring system, histological examination, and cytokine analysis. Most dKO mice developed severe rectal prolapse, peri-ocular lesions, and high clinical scores, signs generally not seen in Il-10 KO mice. dKO co

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46

Gorzelnik, Jerzy. "Kult autentyczności i powrót do słowiańskich korzeni. Projekt rzeźbiarskiej dekoracji katedry Chrystusa Króla w Katowicach a mit cyrylo-metodiański." Nasza Przeszłość 128 (December 30, 2017): 203–17. http://dx.doi.org/10.52204/np.2017.128.203-217.

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W roku 1927 w Katowicach, stolicy ustanowionej dwa lata wcześniej diecezji, rozpoczęto budowę katedry Chrystusa Króla, zaprojektowanej przez Zygmunta Gawlika. W tym samym czasie Xawery Dunikowski we współpracy z architektem przystąpił do prac nad koncepcją rzeźbiarskiej dekoracji fasady. Ich efekty w postaci gipsowego modelu zaprezentowano w roku 1931. Centralną część głównej elewacji świątyni zajmować miały figury świętych Cyryla i Metodego, flankowane przez grupy ludu i rycerstwa śląskiego. W zamyśle tym, który nie doczekał się realizacji, szczególnie wyeksponowano zatem postaci braci sołuńs

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47

Rump, Wolfgang. "Skew-braces and 𝑞-braces". Forum Mathematicum, 3 серпня 2023. http://dx.doi.org/10.1515/forum-2022-0385.

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Abstract Skew-braces are ring-like objects arising in connection with Hopf–Galois theory and set-theoretic solutions 𝑆 to the Yang–Baxter equation. Interactions between skew-braces are often related to 𝑞-braces. For example, every 𝑞-brace 𝐴 is given by a pair of skew-braces which induces a ℤ-indexed sequence of skew-braces. The sequence collapses if 𝐴 itself is a skew-brace. The free group over the underlying set of a solution 𝑆 is a 𝑞-brace. Bi-crossed products of skew-braces are shown to be 𝑞-braces, and criteria are developed when they are skew-braces. Two classes of skew-braces are put int

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48

Ballester-Bolinches, A., R. Esteban-Romero, M. Ferrara, V. Pérez-Calabuig, and M. Trombetti. "Finite skew braces of square-free order and supersolubility." Forum of Mathematics, Sigma 12 (2024). http://dx.doi.org/10.1017/fms.2024.29.

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Abstract 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

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49

Bardakov, Valeriy G., Mikhail V. Neshchadim, and Manoj K. Yadav. "Symmetric skew braces and brace systems." Forum Mathematicum, February 28, 2023. http://dx.doi.org/10.1515/forum-2022-0134.

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Abstract For a skew left brace ( G , ⋅ , ∘ ) {(G,\cdot\,,\circ)} , the map λ : ( G , ∘ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\circ)\to\operatorname{Aut}(G,\cdot\,)} , a ↦ λ a , {a\mapsto\lambda_{a},} where λ a ⁢ ( b ) = a - 1 ⋅ ( a ∘ b ) {\lambda_{a}(b)=a^{-1}\cdot(a\circ b)} for all a , b ∈ G {a,b\in G} , is a group homomorphism. Then λ can also be viewed as a map from ( G , ⋅ ) {(G,\cdot\,)} to Aut ⁡ ( G , ⋅ ) {\operatorname{Aut}(G,\cdot\,)} , which, in general, may not be a homomorphism. A skew left brace will be called λ-anti-homomorphic (λ-homomorphic) if λ : ( G , ⋅ ) → Aut ⁡ ( G , ⋅ ) {\lambd

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Malinowska, Izabela Agata. "Skew two-sided bracoids." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, September 11, 2024. http://dx.doi.org/10.1007/s13366-024-00765-8.

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AbstractIsabel Martin-Lyons and Paul J.Truman generalised the definition of a skew brace to give a new algebraic object, which they termed a skew bracoid. Their construction involves two groups interacting in a manner analogous to the compatibility condition found in the definition of a skew brace. They formulated tools for characterizing and classifying skew bracoids, and studied substructures and quotients of skew bracoids. As an application, they proved that finite skew bracoids correspond with Hopf-Galois structures on finite separable extensions of fields, generalizing the existing connec

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