dc.contributor.advisor | Drápal, Aleš | |
dc.creator | Spišák, Martin | |
dc.date.accessioned | 2021-08-03T09:20:42Z | |
dc.date.available | 2021-08-03T09:20:42Z | |
dc.date.issued | 2021 | |
dc.identifier.uri | http://hdl.handle.net/20.500.11956/128242 | |
dc.description.abstract | Neasociativita kvázigrúp je užitočná vlastnosť pre kryptografiu. A. Drápal and I. M. Wanless vo svojej nedávnej práci študovali existenciu maximálne neasociatı́vnych kvázigrúp, no táto otázka ostáva pre niektoré rády nezopove- daná. Táto práca je úvodom do novej metódy riešenia tejto otázky. Po rekapitulácii najnovšı́ch zistenı́ a naznačenı́ využitia v kryptografii vyložı́ práca konštrukciu abstraktného simpliciálneho komplexu dimenzie 3 z neasociatı́vnych trojı́c konečnej kvázigrupy. Ukážeme, že tento komplex má formu zjednotenia uzavretých orientovateľných pseudovariet dimenzie 3. Pre rády do 6 nezávisle overı́me zistenia Ježka and Kepku o spektre asociati- vity a klasifikujeme možné rozklady komplexu neasociativity na silne súvislé komponenty analýzou ich duálnych grafov. Hlavným výsledok práce je prvý krok k riešeniu singuları́t v komplexe neasociativity. Ukážeme, že linky vrcholov v komplexe majú riešiteľné sin- gularity, čo nám umožnı́ normalizovať ich algoritmicky. Nakoniec spočı́tame rody komponent v linkoch a ilustrujeme typy linkov na prı́kladoch malých kvázigrúp. 1 | cs_CZ |
dc.description.abstract | The non-associative properties of quasigroups are useful in cryptography. A. Drápal and I. M. Wanless have recently analyzed the existence of a max- imally non-associative quasigroup of order n in their work, but there remain orders n for which the existence is not known. This thesis is an introduction to a new method of tackling the problem. After presenting the most recent results and hinting at a possible crypto- graphic application, the thesis proposes the construction of a 3-dimensional abstract simplicial complex from non-associative triples of a finite quasigroup. It shows that the complex forms of a union of closed orientable pseudomani- folds of dimension 3. For orders up to 6, we independently verify the findings of Ježek and Kepka regarding the associativity spectrum of n and classify possible decompositions of the non-associativity complexes into strongly con- nected components by analyzing their dual graphs. The main result of the thesis performs the first step towards resolving the singularities in the complex. We show that links of vertices in the complex have solvable singularities, enabling us to normalize the links of vertices algorithmically. Lastly, we illustrate the types of vertex neighborhoods on examples of small quasigroups by calculating the genera of their components. 1 | en_US |
dc.language | English | cs_CZ |
dc.language.iso | en_US | |
dc.publisher | Univerzita Karlova, Matematicko-fyzikální fakulta | cs_CZ |
dc.subject | kvazigrupa|asociativní trojice|orientovaný komplex|pseudovarieta|kombinatorický povrch | cs_CZ |
dc.subject | quasigroup|associative triple|oriented complex|pseudomanifold|combinatorial surface | en_US |
dc.title | Deriving a pseudomanifold of dimension 3 from nonassociative triples | en_US |
dc.type | bakalářská práce | cs_CZ |
dcterms.created | 2021 | |
dcterms.dateAccepted | 2021-07-08 | |
dc.description.department | Department of Algebra | en_US |
dc.description.department | Katedra algebry | cs_CZ |
dc.description.faculty | Faculty of Mathematics and Physics | en_US |
dc.description.faculty | Matematicko-fyzikální fakulta | cs_CZ |
dc.identifier.repId | 235664 | |
dc.title.translated | Odvození pseudovariety dimenze 3 z neasociativních trojic | cs_CZ |
dc.contributor.referee | Patáková, Zuzana | |
thesis.degree.name | Bc. | |
thesis.degree.level | bakalářské | cs_CZ |
thesis.degree.discipline | Obecná matematika | cs_CZ |
thesis.degree.discipline | General Mathematics | en_US |
thesis.degree.program | Mathematics | en_US |
thesis.degree.program | Matematika | cs_CZ |
uk.thesis.type | bakalářská práce | cs_CZ |
uk.taxonomy.organization-cs | Matematicko-fyzikální fakulta::Katedra algebry | cs_CZ |
uk.taxonomy.organization-en | Faculty of Mathematics and Physics::Department of Algebra | en_US |
uk.faculty-name.cs | Matematicko-fyzikální fakulta | cs_CZ |
uk.faculty-name.en | Faculty of Mathematics and Physics | en_US |
uk.faculty-abbr.cs | MFF | cs_CZ |
uk.degree-discipline.cs | Obecná matematika | cs_CZ |
uk.degree-discipline.en | General Mathematics | en_US |
uk.degree-program.cs | Matematika | cs_CZ |
uk.degree-program.en | Mathematics | en_US |
thesis.grade.cs | Výborně | cs_CZ |
thesis.grade.en | Excellent | en_US |
uk.abstract.cs | Neasociativita kvázigrúp je užitočná vlastnosť pre kryptografiu. A. Drápal and I. M. Wanless vo svojej nedávnej práci študovali existenciu maximálne neasociatı́vnych kvázigrúp, no táto otázka ostáva pre niektoré rády nezopove- daná. Táto práca je úvodom do novej metódy riešenia tejto otázky. Po rekapitulácii najnovšı́ch zistenı́ a naznačenı́ využitia v kryptografii vyložı́ práca konštrukciu abstraktného simpliciálneho komplexu dimenzie 3 z neasociatı́vnych trojı́c konečnej kvázigrupy. Ukážeme, že tento komplex má formu zjednotenia uzavretých orientovateľných pseudovariet dimenzie 3. Pre rády do 6 nezávisle overı́me zistenia Ježka and Kepku o spektre asociati- vity a klasifikujeme možné rozklady komplexu neasociativity na silne súvislé komponenty analýzou ich duálnych grafov. Hlavným výsledok práce je prvý krok k riešeniu singuları́t v komplexe neasociativity. Ukážeme, že linky vrcholov v komplexe majú riešiteľné sin- gularity, čo nám umožnı́ normalizovať ich algoritmicky. Nakoniec spočı́tame rody komponent v linkoch a ilustrujeme typy linkov na prı́kladoch malých kvázigrúp. 1 | cs_CZ |
uk.abstract.en | The non-associative properties of quasigroups are useful in cryptography. A. Drápal and I. M. Wanless have recently analyzed the existence of a max- imally non-associative quasigroup of order n in their work, but there remain orders n for which the existence is not known. This thesis is an introduction to a new method of tackling the problem. After presenting the most recent results and hinting at a possible crypto- graphic application, the thesis proposes the construction of a 3-dimensional abstract simplicial complex from non-associative triples of a finite quasigroup. It shows that the complex forms of a union of closed orientable pseudomani- folds of dimension 3. For orders up to 6, we independently verify the findings of Ježek and Kepka regarding the associativity spectrum of n and classify possible decompositions of the non-associativity complexes into strongly con- nected components by analyzing their dual graphs. The main result of the thesis performs the first step towards resolving the singularities in the complex. We show that links of vertices in the complex have solvable singularities, enabling us to normalize the links of vertices algorithmically. Lastly, we illustrate the types of vertex neighborhoods on examples of small quasigroups by calculating the genera of their components. 1 | en_US |
uk.file-availability | V | |
uk.grantor | Univerzita Karlova, Matematicko-fyzikální fakulta, Katedra algebry | cs_CZ |
thesis.grade.code | 1 | |
uk.publication-place | Praha | cs_CZ |
uk.thesis.defenceStatus | O | |