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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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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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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
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Varieties determined by short identities
Koula, Jiří ; Kepka, Tomáš (referee) ; Ježek, Jaroslav (advisor)
This paper deals with searching for free algebras over the countable infinite set of variables in varieties determined by identity of form x = t, where t is term of length 5. Notion of rewriting system is introduced and used for it. The paper is based on article [3] and extends its results.
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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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Simple Semirings
Kala, Vítězslav ; El Bashir, Robert (referee) ; Kepka, Tomáš (advisor)
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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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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