Jan Chvalina

Discrete transformation hypergroups and transformation hypergroups with phase tolerance space

Tolerance spaces and algebraic structures with compatible tolerances play an important role in contemporary algebra and its applications. In this contribution we present transformation hyperstructures, namely semihypergroups and hypergroups, acting on tolerance spaces. Some basic concepts concerning the mentioned structures are introduced and their fundamental properties are examined on suitable constructions.

On Certain Proximities and Preorderings on the Transposition Hypergroups of Linear First-Order Partial Differential Operators

Abstract The contribution aims to create hypergroups of linear first-order partial differential operators with proximities, one of which creates a tolerance semigroup on the power set of the mentioned differential operators. Constructions of investigated hypergroups are based on the so called “Ends-Lemma” applied on ordered groups of differnetial operators. Moreover, there is also obtained a regularly preordered transpositions hypergroup of considered partial differntial operators.

General actions of hyperstructures and some applications

Abstract The aim of this paper is to investigate useful generalizations of the classical concept of a quasi-automaton without outputs or a discrete dynamical system, which are also called actions of semigroups or groups on given phase sets. The paper contains also certain applications of presented concepts and examples from various areas of mathematics.

Cartesian composition and the problem of generalizing the MAC condition to quasi-multiautomata

Abstract When we assume that the input-set of an automaton without output is a semihypergroup instead of a monoid, we talk about quasi-multiautomata. Even though cartesian composition of quasi-automata is a commonly used concept, the cartesian composition of quasi-multiautomata has not been successfully constructed yet. In our paper we show that the straightforward transfer of the deffinition into the multivariate context fails. We suggest two possible solutions of this problem.

Homomorphic Transformations: Why and possible ways to How

Monografie je věnovana výzkumu problemů homomorfnich transformaci algebraických struktur. V centru pozornosti se nalezaji možnosti konstrukci homomorfismů. Tyto otazky tvoři podstatnou cast mnoha problemů v matematice a dalsich disciplinach, kupřikladu v pocitacove vědě. Kniha je tudiž urcena matematikům, pokrocilým studentům a dalsim specialistům použivajicim matematicke metody, zajimajicim se o seznameni s některými konstrukcnimi metodami v teorii algebraických struktur. Jedna kapitola monografiie poskytuje odpovědi na otazku „proc homomorfismy?“. Zbývajici kapitoly podavaji odpovědi na otazku „jak (konstruovat) homomorfismy?“. Tato otazka je nastolena nejprve pro jednoduche algebraicke struktury nazývane mono-unarni algebry, dale pro algebry v obecnosti a konecně pro relacni systemy.

Characterizations of totally ordered sets by their various endomorphisms

We characterize totally ordered sets within the class of all ordered sets containing at least three-element chains using a simple relationship between their isotone transformations and the so called 2-, 3-, 4-endomorphisms which are introduced in the paper. Another characterization of totally ordered sets within the class of ordered sets of a locally finite height with at least four-element chains in terms of the regular semigroup theory is also given.

Extensions of Cascades Created by Certain Function Systems

Přispěvek vznikl na zakladě vědeckeho zkoumani v oblasti mezioborových vztahů mezi algebrou a matematickou analýzou s cilem nalezeni hlubsich souvislosti mezi těmito obory. Na přikladu tři realných funkci jedne proměnne (jedne mocninne s lichým exponentem a dvou linearnich)jsou v přispěvku zkonstruovany tzv. kaskady. Užitim jisteho rozsiřeni těchto funkci pak jsou popsany kaskady, ktere jsou navzajem izomorfni.

On Tabor's problem concerning a certain quasi-ordering of iterative roots of functions

SummaryUsing the Isaacs-Zimmermanns theory of iterative roots of functions, we prove a theorem concerning the problemP 250 posed by J. Tabor:“Letf: E → E be a given mapping. Denote byF the set of all iterative roots off. InF we define the following relation:ϕ ≦ ψ if and only ifϕ is an iterative root ofψ. The relation is obviously reflexive and transitive. The question is: Is it also antisymmetric? If we consider iterative roots of a monotonic function the answer is ‘yes’. But in general the question is open.”Here we prove that there exists a three-element decomposition {Φi;i = 1, 2, 3} of the setEE with blocks Φi of the same cardinality 2cardE such that the functions from Ф1 do not possess any proper iterative root, the quasi-ordering ≦ is not antisymmetric onF(f) for anyf ∈ Φ2, and ≦ is an ordering onF(f) for anyf ∈ Ф3. Iff is a strictly increasing continuous self-bijection ofE, then the relation ≦ is an ordering onF(f) ifff is different from the identity mapping of the setE.