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

Ako je L relacijski jezik, kondenzacioni pretporedak na skupu IntL (X) svih L-interpretacija nad domenom X, dat je sa: ρ≼c σ ako postoji bijektivni homomorfizam (kondenzacija) f:〈X,ρ〉→〈X,σ〉. Odgovarajući antisimetrični količnik 〈Int L (X)/~c,≤c〉 ~naziva se kondenzacioni poredak. Za proizvoljnu L-interpretaciju ρ, klasa [ρ]~c  je konveksno zatvorenje klase [ρ]_&cong; u Booleovoj mreži 〈IntL (X),&sube;〉. Za L-interpretaciju ρ reći ćemo da je jako reverzibilna (redom, reverzibilna, slabo reverzibilna) akko je klasa [ρ]_&cong;  (ili, ekvivalentno, klasa [ρ]~c )) singlton (redom, antilanac, konveksan skup) u Booleovoj mreži 〈IntL (X),&sube;〉. U cilju ispitivanja poseta 〈Int(Lb ) (X)/~c,≤c〉, za ρ∈IrreflX uveden je skup Dρ:={[ρ∪ΔA ](~c ):A&sube;X} i pokazano je kako je poduređenje 〈Dρ,≤c 〉 izomorfno određenom količniku partitivnog skupa P(X). Fenomen reverzibilnosti relacijskih struktura igra istaknutu ulogu u istraživanju tog poduređenja.U slučaju prebrojivog jezika  L  i prebrojivog domena X, pokazano je da su ~c i [ρ]~c analitički skupovi u poljskim prostorima, redom, IntL (ω)×IntL (ω) i IntL (ω), i pomoću toga, pokazano ja da su, u slučaju prebrojivog jezika i domena, klase [ρ]&cong;  i [ρ]~c iste veličine, i da je to neki kardinal iz {1,ω,c}. Dalje je istražena hijerarhija između kondenzacione ekvivalencije, elementarne ekvivalencije, ekvimorfizma (bi-utopivosti) i drugih sličnosti L-struktura određenih nekim sličnostima njihovih monoida samoutapanja.Naposletku, temeljno je istražen fenomen reverzibilnosti L-struktura. Data je karakterizacija jako reverzibilnih L-intepretacija kao onih čije su komponentne relacije definabilne formulama praznog jezika L∅, bez kvantifikatora i parametara. Pokazano je kako su slabo reverzibilne interpretacije upravo one koje imaju svojstvo Cantor-Schrӧder-Bernstein (kraće, svojstvo CSB) za kondenzacije.Poseban naglasak stavljen je na detektovanje relevantnih klasa reverzibilnih struktura. Pri tome, prvo su proučene strukture koje su ekstremni elementi L∞ω-definabilnih klasa interpretacija, pri određenim sintaktičkim ograničenjima, a zatim su istražene nepovezane Lb-strukture, gde je dato nekoliko karakterizacija njihove reverzibilnosti. / If L is a relational language, the condensational preorder on the set IntL (X) of all L-interpretations over the domain X, is given with: ρ≼_c σ iff there exists a bijective homomorphism (condensation) f:〈X,ρ〉→〈X,σ〉. The corresponding antisymmetric quotient 〈IntL (X)/~c,≤_c〉 will be called the condensational order. For any L-interpretation ρ, the class [ρ]~c ) is the convex closure of the class [ρ]&cong; in the Boolean lattice 〈IntL (X),&sube;〉. An L-interpretation ρ is said to be strongly reversible  (respectively, reversible, weakly reversible) iff the class [ρ]&cong;  (or, equivalently, the class [ρ]~c )) is a singleton (respectively, an antichain, a convex set) in the poset 〈 IntL (X),&sube;〉. In order to investigate the poset 〈Int(Lb ) (X)/~c,≤_c〉, for ρ∈ IrreflX the following set is defined Dρ:={[ρ∪ΔA ]_~c :A&sube;X}. It is shown that the suborder 〈Dρ,≤c 〉 is isomorphic to a certain quotient of the power set P(X). The phenomenon of reversibility plays prominent role in the investigation of that suborder. In the case of a countable language L and a countable domain  X, it is shown that ~c  and [ρ]_~c  are analytic sets in the Polish spaces, respectively, IntL (ω)× IntL (ω) and IntL (ω), and, using those results, in the case of a countable language and domain it is shown that the classes [ρ]_&cong;  and [ρ]~c  are of the same size, and that it is a cardinals from {1,ω,c}. Next, the hierarchy between condensational equivalence, elementary equivalence, equimorphism (bi- embedability) and other similarities of L-structures, determined by some similarities of their self-embedding monoids, is investigated. In the last part, the phenomenon of reversibility of L-structures is investigated. Strongly reversible L-intepretations are characterized as those whose component relations are definable by the formulae of the empty language L∅, 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. Particular emphasis is put on detecting relevant classes of reversible structures. First, the structures that are extreme elements of L∞ω-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.

Identifer	oai:union.ndltd.org:uns.ac.rs/oai:CRISUNS:(BISIS)107294
Date	09 July 2018
Creators	Morača Nenad
Contributors	Kurilić Miloš, Madaras-Silađi Rozalija, Todorčević Stevo, Tanović Predrag, Šobot Boris
Publisher	Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, University of Novi Sad, Faculty of Sciences at Novi Sad
Source Sets	University of Novi Sad
Language	Serbian
Detected Language	English
Type	PhD thesis

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

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