Partial closure operators and applications in ordered set theory

Abstract

<p>In this thesis we generalize the well-known connections between closure operators, closure systems and complete lattices. We introduce a special kind of a partial closure operator, named sharp partial closure operator, and show that each sharp partial closure operator uniquely corresponds to a partial closure&nbsp; system. We further introduce a special kind of a partial clo-sure system, called principal partial closure system, and then prove the representation theorem for ordered sets with respect to the introduced partial closure operators and partial closure systems.<br />Further, motivated by a well-known connection between matroids and geometric lattices, given that the notion of matroids can be naturally generalized to partial matroids (by dening them with respect to a partial closure operator instead of with respect to a closure operator), we dene geometric poset, and show that there is a same kind of connection between partial matroids and geometric posets as there is between matroids and geometric lattices. Furthermore, we then dene semimod-ular poset, and show that it is indeed a generalization of semi-modular lattices, and that there is a same kind of connection between semimodular and geometric posets as there is between<br />semimodular and geometric lattices.</p><p>Finally, we note that the dened notions can be applied to im-plicational systems, that have many applications in real world,particularly in big data analysis.</p>

<p>U ovoj tezi uop&scaron;tavamo dobro poznate veze između operatora zatvaranja, sistema zatvaranja i potpunih mreža. Uvodimo posebnu vrstu parcijalnog operatora zatvaranja, koji nazivamo o&scaron;tar parcijalni operator zatvaranja, i pokazujemo da svaki o&scaron;tar parcijalni operator zatvaranja jedinstveno korespondira parcijalnom sistemu zatvaranja. Dalje uvodimo posebnu vrstu parcijalnog sistema zatvaranja, nazvan glavni parcijalni sistem zatvaranja, a zatim dokazujemo teoremu reprezentacije za posete u odnosu na uvedene parcijalne operatore zatvaranja i parcijalne sisteme zatvaranja. Dalje, s obzirom na dobro poznatu vezu između matroida i geometrijskih mreža, a budući da se pojam matroida može na prirodan nacin uop&scaron;titi na parcijalne&nbsp; matroide (defini&scaron;ući ih preko parcijalnih operatora zatvaranja umesto preko operatora&nbsp; zatvaranja), defini&scaron;emo geometrijske uređene skupove i pokazujemo da su povezani sa parcijalnim matroidima na isti način kao &scaron;to su povezani i matroidi i&nbsp; geometrijske mreže. Osim toga, defini&scaron;emo polumodularne uređene skupove i pokazujemo da su oni zaista uop&scaron;tenje polumodularnih mreža i da ista veza postoji&nbsp; između polumodularnih i geometrijskih poseta kao &scaron;to imamo između polumodularnih i geometrijskih mreža. Konačno, konstatujemo da definisani pojmovi&nbsp; mogu biti primenjeni na implikacione sisteme, koji imaju veliku primenu u realnom svetu, posebno u analizi velikih podataka.</p>