Philosophiae Doctor - PhD / Categorical closure operators were introduced by Dikranjan and Giuli in [DG87] and then developed by these authors and Tholen in [DGT89]. These operators have played an important role in the development of Categorical Topology by introducing topological concepts, such as connectedness, separatedness and compactness, in an arbitrary category and they provide a uni ed approach to various mathematical notions. Motivated by the theory of these operators, the categorical notion of interior operators was introduced by Vorster in [Vor00]. While there is a notational symmetry between categorical closure and interior operators, a detailed analysis shows that the two operators are not categorically dual to each other, that is: it is not true in general that whatever one does with respect to closure operators may be done relative to interior operators. Indeed, the continuity condition of categorical closure operators can be expressed in terms of images or equivalently, preimages, in the same way as the usual topological closure describes continuity in terms of images or preimages along continuous maps. However, unlike the case of categorical closure operators, the continuity condition of categorical interior operators can not be described in terms of images. Consequently, the general theory of categorical interior operators is not equivalent to the one of closure operators. Moreover, the categorical dual closure operator introduced in [DT15] does not lead to interior operators. As a consequence, the study of categorical interior operators in their own right is interesting.

Formal studies of models of concurrency are usually focused on two major models: Interleaving abstraction (Bergstra, 2001; Milner, 1990) and partially ordered causality (Diekert and Rozenberg, 1995; Jensen, 1997; Reisig, 1998). Although very mature, these models retain a known limitation: Neither of them can model the “not later than” relationship effectively, which causes problems with specifying priorities, error recovery, time testing, inhibitor nets, etc. See for reference: Best and Koutny (1992); Janicki (2008); Janicki and Koutny (1995); Juhas et al. (2006); Kleijn and Koutny (2004). A solution, proposed independently (in this order) in (Lamport, 1986; Gaifman and Pratt, 1987) and (Janicki and Koutny, 1991), suggests to model concurrent behaviours by an ordered structure, i.e. a triple (X, R1, R2), where X is the set of event occurrences, and R1 and R2 are two binary relations on X. The relation R1 is interpreted as “causality”, i.e. an abstraction of the “earlier than” relationship, and R2 is interpreted as “weak causality”, an abstraction of the “not later than” relationship. For ordered structures’ model, the following two kinds of relational structures are of special importance: stratified order structures (SO-structures) and interval order structures (IO-structures). The SO-structures can fully model concurrent behaviours when system executions (operational semantics) are described in terms of stratified orders, while the IO-structures can fully model concurrent behaviours when system executions are described in terms of interval orders (Janicki, 2008; Janicki and Koutny, 1997). It was argued in (Janicki and Koutny, 1993), and also implicitly in a 1914 Wiener’s paper Wiener (1914), that any execution that can be observed by a single observer must be an interval order. Thus, IO-structures provide a very definitive model of concurrency. However, the theory of IO-structures remains far less developed than its simpler counterpart - the theory of SO-structures. One of the most important concepts lying at the core of partial orders and algebraic structures theory is the concept of transitive closure of relations. The equivalent of transitive closure for SO-structures, called <>-closure, has been proposed in (Janicki and Koutny, 1995) and consequently used in (Janicki and Koutny, 1995; Juhas et al., 2006; Kleijn and Koutny, 2004) and others. However, a similar concept for IO-structures has not been proposed. In this thesis we define that concept. We introduce the transitive closure for IO-structures, called the []-closure. We prove that it has same properties as the standard transitive closure for partial orders and []-closure for SO-structures (published in Janicki and Zubkova (2009); Janicki et al. (2009)), and provide some comparison of different versions of transitive closure used in various relational structures. Some properties of another recently introduced *-closure (Janicki et al., 2013) are also discussed. / Thesis / Master of Science (MSc)

Philosophiae Doctor - PhD / In a category C with a proper (E; M)-factorization system for morphisms, we further investigate categorical topogenous structures and demonstrate their prominent role played in providing a uni ed approach to the theory of closure, interior and neighbourhood operators. We then introduce and study an abstract notion of C asz ar's syntopogenous structure which provides a convenient setting to investigate a quasi-uniformity on a category. We demonstrate that a quasi-uniformity is a family of categorical closure operators. In particular, it is shown that every idempotent closure operator is a base for a quasi-uniformity. This leads us to prove that for any idempotent closure operator c (interior i) on C there is at least a transitive quasi-uniformity U on C compatible with c (i). Various notions of completeness of objects and precompactness with respect to the quasi-uniformity de ned in a natural way are studied. The great relationship between quasi-uniformities and closure operators in a category inspires the investigation of categorical quasi-uniform structures induced by functors. We introduce the continuity of a C-morphism with respect to two syntopogenous structures (in particular with respect to two quasi-uniformities) and utilize it to investigate the quasiuniformities induced by pointed and copointed endofunctors. Amongst other things, it is shown that every quasi-uniformity on a re ective subcategory of C can be lifted to a coarsest quasi-uniformity on C for which every re ection morphism is continuous. The notion of continuity of functors between categories endowed with xed quasi-uniform structures is also introduced and used to describe the quasi-uniform structures induced by an M- bration and a functor having a right adjoint.

The focus of this research is the extension of topological operators with the addition of a inclusion measure. This extension is carried out in both crisp and fuzzy topological spaces. The mathematical properties of the new operators are discussed and compared with traditional operators. Ignoring small errors due to imperfections and noise in digital images is the main motivation in introducing the proposed operators. To show the effectiveness of the new operators, we demonstrate their utility in image database classification and shape classification. Each image (shape) category is modeled with a topological space and the interior of the query image is obtained with respect to different topologies. This novel way of looking at the image categories and classifying a query image shows some promising results. Moreover, the proposed interior and closure operators with inclusion degree is utilized in mathematical morphology area. The morphological operators with inclusion degree outperform traditional morphology in noise removal and edge detection in a noisy environment

The focus of this research is the extension of topological operators with the addition of a inclusion measure. This extension is carried out in both crisp and fuzzy topological spaces. The mathematical properties of the new operators are discussed and compared with traditional operators. Ignoring small errors due to imperfections and noise in digital images is the main motivation in introducing the proposed operators. To show the effectiveness of the new operators, we demonstrate their utility in image database classification and shape classification. Each image (shape) category is modeled with a topological space and the interior of the query image is obtained with respect to different topologies. This novel way of looking at the image categories and classifying a query image shows some promising results. Moreover, the proposed interior and closure operators with inclusion degree is utilized in mathematical morphology area. The morphological operators with inclusion degree outperform traditional morphology in noise removal and edge detection in a noisy environment

<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>