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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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