|
|
Selfdistributive quasigroups of size 2^k
Nagy, Tomáš ; Stanovský, David (advisor) ; Kepka, Tomáš (referee)
We present the theory of selfdistributive quasigroups and the construction of non-affine selfdistributive quasigroup of size 216 that was presented by Onoi in 1970 and which was the least known example of such structure of size 2k . Based on this construction, we introduce the notion of Onoi structures and Onoi mappings between them which generalizes Onoi's construction and which allows us to construct non-affine selfdistributive quasigroups of size 22k for k ≥ 3. We present and implement algorithm for finding central extensions of self- distributive quasigroups which enables us to classify non-affine selfdistributive quasigroups of size 2k and prove that those quasigroup exists exactly for k ≥ 6, k ̸= 7. We use this algorithm also in order to better understand the structure of non-affine selfdistributive quasigroups of size 26 . 1
|
|
|
Enumeration of affine quasigroups
Semanišinová, Žaneta ; Stanovský, David (advisor) ; Kepka, Tomáš (referee)
The thesis deals with an enumeration of paramedial quasigroups. In the thesis it is proved that there are precisely, up to isomorphism, 2p − 1 paramedial quasi- groups of order p, where p is odd prime number. It is also showed that there are, up to isomorphism, 11 2 p2 + 3 2 p − 4 paramedial quasigroups of order p2 for any odd prime p. The corresponding calculations include the enumeration of paramedial quasigroups affine over the group Zp and the groups Zp2 and Z2 p. The enumeration algorithm is a special case of the result obtained by Aleš Drápal. The most complex case is the enumeration over the group Z2 p, which involves an analysis of square roots and conjugacy classes in the group GL(2, p).
|
|
|
Quasigroups with few associative triples
Valent, Viliam ; Drápal, Aleš (advisor) ; Kepka, Tomáš (referee)
This bachelor thesis deals with quasigroups with a small number of associative triples. They were studied from the algebraic point of view by Drápal, Ježek and Kepka, Kotzig, and recently by Grošek and Horák. The aim of this thesis is to build on the research of Grošek and Horák, replicate and improve their findings concerning the minimum number of associative triples in small quasigroups. Another important part is an establishment of a new upper bound on the minimum number of associative triples among all quasigroups of the same order. We provide an algorithm that can produce quasigroups with the number of associative triples less or equal to the second power of their order. We also present the applications of such quasigroups in cryptography, namely in hash functions and zero-knowledge protocols. Powered by TCPDF (www.tcpdf.org)
|
|
|
An elementary proof of the existence of primitive elements
Majerčík, Miroslav ; Kepka, Tomáš (advisor) ; Bulín, Jakub (referee)
Vedúcí bakalárskej práce: prof. RNDr. Tomáš Kepka, DrSc. Abstrakt: Tento text je venovaný elementárnym dôkazom dvoch významných viet teórie čísel a to Gaussovmu kvadratickému zákonu reciprocity a vete o primitívnom prvku. Dôkazy týchto viet sú vo forme menších na seba nadväzujúcich lemmat a dôkazov. Úvod je venovaný historickému priblíženiu a metóde dôkazov viet. Prvá kapitola smeruje k dôkazu Gaussovmu kvadratickému zákonu reciprocity a druhá k dôkazu vety o primitívnom prvku a k určeniu prirodzených čísel n pre ktoré existuje primitívny prvok modulo n a pre ktoré nie. K dôkazu týchto viet bolo potrebné dokázat aj niekol'ko dalších viet, napríklad malú Fermatovu vetu alebo schému rozdielu mocnín. Klúčové slová: Kvadratický zbytok, Primitívny koreň, Rád prvku modulo n, Eulerova funkcia Title: An elementary proof of the existence of primitive elements Author: Miroslav Majerčík Department: Department of Algebra Supervisor: prof. RNDr. Tomáš Kepka, DrSc. Abstract: This text is about elementary proofs of two well known number theory statements, Gauss quadratic reciprocity law and proof of the existence of primitive elements. These proofs are in form of simpler interlinked lemmas and proofs. Introduction is about historical background and about...
|
|
|
Algebraic Substructures in Cm
Kala, Vítězslav ; Kepka, Tomáš (advisor) ; Stanovský, David (referee) ; El Bashir, Robert (referee)
Title: Algebraic Substructures in ℂ Author: Vítězslav Kala Department: Department of Algebra Supervisor: Prof. RNDr. Tomáš Kepka, DrSc., Department of Algebra Abstract: We study the structure of finitely generated semirings, parasemifields and other algebraic structures, developing and applying tools based on the geom- etry of algebraic substructures of the Euclidean space ℂ . To a parasemifield which is finitely generated as a semiring we attach a certain subsemigroup of the semigroup ℕ0 (defined using elements such that + = for some ∈ and ∈ ℕ). Algebraic and geometric properties of carry important structural information about ; we use them to show that if a parasemifield is 2-generated as a semiring, then it is additively idempotent. We also provide a ring-theoretic reformulation of this conjecture in the case of -generated semirings. We also classify all additively idempotent parasemifields which are finitely gen- erated as semirings by using the fact that they correspond to certain finitely generated unital lattice ordered groups. Busaniche, Cabrer, and Mundici [4] re- cently classified these using the combinatorial and geometric notion of a stellar sequence which is a sequences of certain simplicial complexes in [0, 1] . We use their results to prove that each such parasemifield is a finite product of...
|
|
|
Sudé triangulace a Abelovy grupy
Hrbek, Michal ; Drápal, Aleš (advisor) ; Kepka, Tomáš (referee)
Title: Even triangulations and Abelian groups Author: Michal Hrbek Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal CSc., DSc. Abstract: This thesis takes interest in spherical Eulerian triangulations and the algebraic structure defined on its vertices corresponding with the latin bitrade equivalent to the triangulation. First, we introduce needed results about the properties of the triangulations and their embeddings into Abelian groups. Then we get concerned with a particular kind of almost 6-homogenous triangulations. The text presents several examples, then the groups of the simplest sequence of triangulations are explicitly described. In order to investigate more complicated cases, we introduce a recursive formula for defining relations of the groups and we show an example of its usage with modular arithmetic. The thesis is completed by discussing computed data. Keywords: latin bitrade, eulerian triangulation, Abelian group 1
|
|
|
Kategoriální metody v teorii struktur
Opršal, Jakub ; Trnková, Věra (advisor) ; Kepka, Tomáš (referee)
Title: Categorial methods in structure theory Author: Jakub Opršal Department / Institute: Mathematical Institute, Charles University Supervisor of the master thesis: prof. RNDr. Věra Trnková, DrSc. Abstract: In the first part of the thesis we investigate functor algebras. Initial algebras have distin- guished role in the study of these structures, and it can be constructed by certain transfinite construction, which is called initial algebra construction. Sooner this year Adámek and Trnková have prooved, that the construction stops in either at most three, or in κ steps where κ is a regular cardinal. We continue with their work, and we study the relation between the size of the algebra and the length of the convergence. We prove that the length of the convergence never exceeds the cardinality of the initial algebra. Another transfinite construction has been studied in 1980 by Kelly. He has described the construction of free algebras for a pointed functor and defined a class of well-pointed functors for which the construction is especially simple (and is in fact special case of the construction of relatively terminal coalgebra which has been recently defined by Adámek and Trnková). In the last chapter we describe all well-pointed functors in categories of sets and the dual category, and we provide list of...
|
|
|
Some questions of definability
Lechner, Jiří ; Stanovský, David (advisor) ; Kepka, Tomáš (referee)
We focus on first-order definability in the quasiordered class of finite digraphs ordered by embeddability. At first we will prove definability of each digraph up to size three. We will need to add to the quasiorder structure some digraphs as constants, so we try to find the needed set of constants as small as possible with small digraph as well. Gradually we make instruments that we can use to express the inner structure of each digraphs in the language of embeddability. At the end we investigate definability in the closely related lattice of universal classes of digraphs. We show that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice.
|
|
|
Semigroups of lattice points
Scholle, Marek ; Kepka, Tomáš (advisor) ; Šaroch, Jan (referee)
The thesis deals with subsemigroups of (Nm 0 , +), a special discussion is later devoted to the cases m = 1, m = 2 and m = 3. We prove that a subsemigroup of Nm 0 is finitely generated if and only if its generated cone is finitely generated (equivalently polyhedral) and we describe basic topological properties of such cones. We give a few examples illustrating that conditions sufficient for finite generation in N2 0 can not be easily trans- ferred to higher dimensions. We define the Hilbert basis and the related notion of Carathéodory's rank. Besides their basic properties we prove that Carathédory's rank of a subsemigroup of Nm 0 , m = 1, 2, 3, is less than or equal to m. A particular attention is devoted to the subsemigroups containing non-trivial subsemigroups of "subtractive" elements.
|
|
|
Even triangulations and commutative groups
Luber, Jan ; Drápal, Aleš (advisor) ; Kepka, Tomáš (referee)
Title: Even triangulations and commutative groups Author: Jan Luber Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: This thesis takes interest in latin bitrades and triangulations construc- ted from them. Firstly, we introduce needed definitions, properties of the latin bitrades, detailed construction of the triangulation and mainly possibility of em- bedding latin bitrades into abelian groups. These groups are determined by the relations definied on vertices of the triangulation. Then we get concerned with a particular kind of 3-homogeneous latin bitrades which correspond to toroidal tri- angulation whose each vertex has degree six. For these groups we express relation matrix and complement to their torsion ranks. In case of simple triangulations we present explicit description of the groups and with modular arithmetic we get partial description even for more complex triangulations. Keywords: latin bitrade, eulerian triangulation, abelian group
|
guest ::
