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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
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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.
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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.
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Constructions of Commutative Semirings and Radical Rings
Korbelář, Miroslav ; Kepka, Tomáš (advisor) ; Němec, Petr (referee) ; Příhoda, Pavel (referee)
In this dissertation we deal with constructive methods applied to the commutative semirings and commutative radical rings. In Chapter 2 we study the class S of the commutative subdirectly irreducible radical rings. We present a few constructive methods for them and using the reflection of the category of the commutative rings into the category of the commutative radical rings we derive a lot of examples of rings in S with various properties. We prove that a ring S 2 S is noetherian if and only if it is finite. We show partial results in the classification of factors of S modulo monoliths. In Chapter 3 we introduce, using the p-prime valuation for all primes p, a set of characteristic sequences that can be assign to every subsemiring of Q+. We find and classify all maximal subsemirings of positive rational numbers and show that every proper subsemiring of Q+ is contained in at least one of them. This results was published in [16]. In Chapter 4 we construct, using the approach from the Chapter 4, a new large subclass of the class CongSimp of all proper congruence-simple subsemirings of Q+, classify all the maximal elements of CongSimp and show that every element of CongSimp is contained in at least one of them. In Chapter 5 we find an equivalent condition under which is the semiring Q+[ ] C, 2 C, contained in...
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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...
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Congruence-Simple Semirings
Al-Zoubi, Khaldoun Falah Salim ; Kepka, Tomáš (advisor) ; Ježek, Jaroslav (referee) ; Philips, Jon (referee)
Následující dizertační práce je věnována podrobnému zkoumání kongrunenčně-jednoduchých (konečných) aditivně idempotentních polookruhů. sestává ze čtyř kapitol. V první kapitole jsou zkoumány kvazitriviální polomoduly a polookruhy. Speciálně jsou charakterizovány minimální a kongruenčně-jednoduché kvazitriviální polomoduly. Jsou zobeněny výsledky z knihy. Ve druhé kapitole pokračujeme ve zkoumíní polomodulů. Hlavně s důrazem na minimální a kongruenčně-jednoduché polomoduly. Ve třetí kapitole se zkoumají skoro minimální polomoduly. Klasifikujeme konečné kongruenčně jednoduché polookruhy. Ve čtvrté kapitole dokážeme, že polookruh endomorfismů netriviálního polosvazu je vždy subdirektně nerozložitelný a popíšeme jeho monolit. Polookruh endomorfismů je kongruenčně jednoduchý právě, když příslušný polosvaz má největší i nejmenší prvek. První tři kapitoly jsou souhrnem výsledků [2], [3] a [4]. Obsah poslední kapitoly je adaptován z [17]. K práci patří z ilustrativních důvodů.
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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)
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Simple Semirings
Kala, Vítězslav ; Kepka, Tomáš (advisor) ; El Bashir, Robert (referee)
A well-known statement says that if a commutative field is finitely generated as a ring, then it is finite. This thesis studies a generalization of this statement - problem, whether every finitely generated ideal-simple commutative semiring is additively idempotent or finite. Using the characterization of idealsimple semirings we prove that this question is equivalent to the question, whether every commutative parasemifield (i.e., a semiring whose multiplicative semigroup is a group), which is finitely generated as a semiring, is additively idempotent. In the thesis we deduce various useful properties of such parasemifields and use them to solve the problem in the one-generated case. Finally, we mention a way of using obtained properties of parasemifields for the solution of the two-generated case via the study of subsemigroups of Nm0.
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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...
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