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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

Kondenzacioni poredak, kondenzaciona ekvivalencija i reverzibilnost relacijskih struktura / Condensational order, condensational equivalenceand reversibility of relational structures

Morača Nenad 09 July 2018 (has links)
<p>Ako je<em> L </em>relacijski jezik, kondenzacioni pretporedak na skupu<em> Int</em><sub>L</sub> <em>(X)</em> svih <em>L-</em>interpretacija nad domenom <em>X,</em> dat je sa: &rho;≼<sub>c</sub> <em>&sigma;</em> ako postoji bijektivni homomorfizam (kondenzacija)<em> f:〈X,&rho;</em>〉&rarr;<em>〈X,&sigma;〉.</em> Odgovarajući antisimetrični količnik <em>〈Int<sub> L</sub></em> (X)/~<sub>c</sub>,&le;<sub>c</sub>〉 ~naziva se kondenzacioni poredak. Za proizvoljnu<em> L-</em>interpretaciju &rho;, klasa [&rho;]~<sub>c</sub>&nbsp; je konveksno zatvorenje klase [&rho;]_&cong; u Booleovoj mreži 〈<em>IntL (X</em>),&sube;〉. Za <em>L</em>-interpretaciju &rho; reći ćemo da je jako reverzibilna (redom, reverzibilna, slabo reverzibilna) akko je klasa [&rho;]_&cong;&nbsp; (ili, ekvivalentno, klasa [&rho;]~<sub>c </sub>)) singlton (redom, antilanac, konveksan skup) u Booleovoj mreži 〈<em>IntL (X)</em>,&sube;〉. U cilju ispitivanja poseta 〈<em>Int(<sub>Lb</sub></em><sub> </sub>) (X)/~c,&le;c〉, za &rho;&isin;<em>Irrefl<sub>X</sub></em> uveden je skup D<sub>&rho;</sub>:={[&rho;&cup;&Delta;<sub>A</sub> ](~<sub>c</sub> ):<em>A&sube;X</em>} i pokazano je kako je poduređenje 〈D<sub>&rho;</sub>,&le;<sub>c</sub> 〉 izomorfno određenom količniku partitivnog skupa<em> P(X)</em>. Fenomen reverzibilnosti relacijskih struktura igra istaknutu ulogu u istraživanju tog poduređenja.</p><p>U slučaju prebrojivog jezika <span id="cke_bm_1038S" style="display: none;">&nbsp;</span><em>L</em><span id="cke_bm_1038E" style="display: none;">&nbsp;</span> i prebrojivog domena <em>X</em>, pokazano je da su ~<sub>c</sub> i [&rho;]~<sub>c </sub>analitički skupovi u poljskim prostorima, redom, <em>Int<sub>L </sub>(&omega;)&times;Int<sub>L </sub>(&omega;) i Int<sub>L</sub> (&omega;)</em>, i pomoću toga, pokazano ja da su, u slučaju prebrojivog jezika i domena, klase [&rho;]&cong;&nbsp; i [&rho;]~<sub>c</sub> iste veličine, i da je to neki kardinal iz {1,&omega;,c}. Dalje je istražena hijerarhija između kondenzacione ekvivalencije, elementarne ekvivalencije, ekvimorfizma (bi-utopivosti) i drugih sličnosti <em>L-</em>struktura određenih nekim sličnostima njihovih monoida samoutapanja.</p><p>Naposletku, temeljno je istražen fenomen reverzibilnosti <em>L</em>-struktura. Data je karakterizacija jako reverzibilnih<em> L</em>-intepretacija kao onih čije su komponentne relacije definabilne formulama praznog jezika<em> L</em><sub>&empty;</sub>, bez kvantifikatora i parametara. Pokazano je kako su slabo reverzibilne interpretacije upravo one koje imaju svojstvo Cantor-Schrӧder-Bernstein (kraće, svojstvo CSB) za kondenzacije.</p><p>Poseban naglasak stavljen je na detektovanje relevantnih klasa reverzibilnih struktura. Pri tome, prvo su proučene strukture koje su ekstremni elementi L<sub>&infin;&omega;</sub>-definabilnih klasa interpretacija, pri određenim sintaktičkim ograničenjima, a zatim su istražene nepovezane<em> L</em><sub>b</sub>-strukture, gde je dato nekoliko karakterizacija njihove reverzibilnosti.</p> / <p>If <em>L</em> is a relational language, the condensational preorder on the set <em>Int<sub>L</sub> (X)</em> of all <em>L-</em>interpretations over the domain<em> X</em>, is given with: &rho;≼_c &sigma; iff there exists a bijective homomorphism (condensation) <em>f:〈X,&rho;〉&rarr;〈X,&sigma;〉. </em>The corresponding antisymmetric quotient 〈<em>Int<sub>L</sub> (X)/</em>~<sub><em>c</em></sub>,&le;_<sub>c</sub>〉 will be called the condensational order. For any <em>L</em>-interpretation &rho;, the class<em> [&rho;]~<sub>c</sub> )</em> is the convex closure of the class [<em>&rho;</em>]&cong; in the Boolean lattice 〈<em>IntL (X</em>),&sube;〉. An <em>L</em>-interpretation &rho; is said to be strongly reversible&nbsp; (respectively, reversible, weakly reversible) iff the class <em>[&rho;]</em>&cong;&nbsp; (or, equivalently, the class<em> [&rho;]~c )</em>) is a singleton (respectively, an antichain, a convex set) in the poset 〈 <em>IntL</em> <em>(X)</em>,&sube;〉. In order to investigate the poset 〈<em>Int<sub>(Lb</sub> ) (X)/~c,&le;_c</em>〉, for &rho;&isin;<em> IrreflX</em> the following set is defined <em>D<sub>&rho;</sub></em>:={[&rho;&cup;&Delta;<sub>A</sub> ]_~c :A&sube;X}. It is shown that the suborder 〈<em>D<sub>&rho;</sub>,</em>&le;<sub>c</sub> 〉 is isomorphic to a certain quotient of the power set <em>P(X)</em>. The phenomenon of reversibility plays prominent role in the investigation of that suborder.<br />In the case of a countable language<em> L</em> and a countable domain&nbsp; <em>X</em>, it is shown that ~c&nbsp; and [<em>&rho;]_<sub>~c&nbsp; </sub></em>are analytic sets in the Polish spaces, respectively,<em> IntL (&omega;)&times; IntL (&omega;)</em> and <em>Int<sub>L</sub> (&omega;)</em>, and, using those results, in the case of a countable language and domain it is shown that the classes <em>[&rho;]_</em>&cong;&nbsp; and <em>[&rho;]~<sub>c&nbsp; </sub></em>are of the same size, and that it is a cardinals from <sub>{1,&omega;,c}. N</sub>ext, the hierarchy between condensational equivalence, elementary equivalence, equimorphism (bi- embedability) and other similarities of <em>L</em>-structures, determined by some similarities of their self-embedding monoids, is investigated.<br />In the last part, the phenomenon of reversibility of<em> L</em>-structures is investigated. Strongly reversible <em>L</em>-intepretations are characterized as those whose component relations are definable by the formulae of the empty language<em> L<sub>&empty;</sub>, </em>without quantifiers and parameters. It is shown that weakly reversible interpretations are exactly those having the property Cantor-Schrӧder-Bernstein (shorter, the property CSB) for condensations.<br />Particular emphasis is put on detecting relevant classes of reversible structures. First, the structures that are extreme elements of<em> L</em><sub>&infin;&omega;</sub>-definable classes of interpretations, under certain syntactical restrictions, are investigated. Following that, disconnected Lb-structures are investigated, where several equivalents of their reversibility are proven.</p>

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